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Art Traynor
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© Art Traynor 2011 Logic Nomenclature Definitions Logic The field of inquiry ( discipline/science ) which evaluates arguments Argument A Set of Statements composed of a Conclusion and evidentiary Premises from which it is claimed to logically follow Statement A declarative sentence ( a sentence which declares a Fact ) that is either True or False but not both Truth Value The attribute by which a statement can be evaluated to be either True or False Sometimes “proposition” “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence Hurley Section 1.1, ( Pgs. 1 – 2 ) A system of inquiry addressed to gauging the validity of Arguments Wiki: “ Logic ” “ Inquiry ” because Logic is a process directed toward discovery “ Expression ” and “ Statement ” are essentially synonymous , with “ Expression ” preferred in Symbolic Logic contexts The terms “ Fact ” , “ Information ” , or sometimes “ Meaning ” are synonymous in this context The Expression or Statement encodes a “ Fact ” or “ Information ”
© Art Traynor 2011 Logic Nomenclature Definitions Hurley Section 1.1, ( Pg. 5 ) As in Inferential Nexus ( InfNex ) Inference A process of Reason supplying the Structure by which the Validity of an Argument or Conditional Statement can be gauged That component of a Statement encoding Information about which a Truth Value can be ascertained Proposition “ Expression ” and “ Statement ” are essentially synonymous , with “ Expression ” preferred in Symbolic Logic contexts The terms “ Fact ” , “ Information ” , or sometimes “ Meaning ” are synonymous in this context The Expression or Statement encodes a “ Fact ” or “ Information ”
© Art Traynor 2011 Logic Argument Definition Argument A prefatory composition of Propositions ( i.e. Declarative Statements ) , or Premises , the Truth Value of which can be reckoned , and by which the application of Inference will permit a Conclusion to be gauged Wiki: “ Argument ” A group of Statements the Conclusion of which is claimed to follow from its Premises “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence Hurley Section 1.1, ( Pgs. 2 – 3 ) An Argument expresses an Inference in a special way, in terms of one or more premises that present Evidence and a Conclusion that is claimed to follow from that Evidence. Hurley Section 1.2, ( Pg. 15 ) An Inferential Nexus ( InfNex )
© Art Traynor 2011 Logic Argument Statements Argument A group of Statements the Conclusion of which is claimed to follow from its Premises “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence  Premises One or more Statements asserted as Evidence  Conclusion A singular Statement from which the Premises are asserted to follow Hurley Section 1.1, ( Pgs. 2 – 3 ) Follow: to implicate with Necessity Argument conclusion statement statement statement premises statement
© Art Traynor 2011 Logic Statement Premise Argument A group of Statements the Conclusion of which is claimed to follow from its Premises “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence  Premises One or more Statements asserted as Evidence Hurley Section 1.1, ( Pgs. 2 – 3 ) Argument statement statement statement premises May be inferred from… Seeing that… For the reason that… Inasmuch as… As indicated by… Premise Indicators Given that… Since… Owing to… Because… In that… For… As…
© Art Traynor 2011 Logic Statement Conclusion Argument A group of Statements the Conclusion of which is claimed to follow from its Premises “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence  Premises One or more Statements asserted as Evidence  Conclusion A singular Statement from which the Premises are asserted to follow Hurley Section 1.1, ( Pgs. 2 – 3 ) Argument conclusion statement statement statement premises statement We may conclude… It follows that… It must be that… We may infer… Consequently… Conclusion Indicators Accordingly… Therefore… Implies that… Wherefore… As a result… Whence… Hence… Thus… So…
© Art Traynor 2011 Logic Reason Definition Reason A process of Inquiry may be discovered and Evaluated Wiki: “ Reason ” populating an Ontological System ( OntSys ) or Set “ Inquiry ” because Reason is a process directed toward discovery Akin to an Algebra by which Relations between Class Equivalent Elements “ Evaluate ” to reduce an indeterminant to determinacy  Etymology λóγος or “ logos ” , is the root of the English word “ Logic ” which also had a connotation of “ speech ” or “ explanation ” or an “ account ” ( as of money )  Translated into Latin as “ ratio ” , from which its synonymous correspondence with “ Rationality ” emerged  There is thus a pleasing sense in which Rationality or Reason connotes the abstract evaluation of Ratios or Proportions nn
© Art Traynor 2011 Logic Reason Definition Reason A process of Inquiry may be discovered and Evaluated Wiki: “ Reason ” populating an Ontological System ( OntSys ) or Set “ Inquiry ” because Reason is a process directed toward discovery Akin to an Algebra by which Relations between Class Equivalent Elements “ Evaluate ” to reduce an indeterminant to determinacy  Domain of Discourse Whereas Logic lies within Reason , Reason unconstrained by Logic supplies extra-logical processes of Inquiry Skipping Steps Working Backward Diagraming Exemplifying ( Exampling or Modeling ) Axiom Transformation ( changing the Laws of Composition – LOC’s ) Processes that lie outside of the OntSys and Logic
© Art Traynor 2011 Logic Reason Logic Logic Wiki: “ Reason ” which evaluates arguments Hurley Section 1.1, ( Pgs. 1 – 2 ) A system of inquiry addressed to gauging the validity of Arguments Wiki: “ Logic ” “ Inquiry ” because Reason is a process directed toward discovery  The field of Inquiry ( discipline/science )  That species of Reason which is applied within an Ontological System ( OntSys ) Inference Hurley Section 1.1, ( Pg. 5 ) A process of Reason supplying the Structure by which the Validity of an Argument or Conditional Statement can be gauged Wiki: “ Logic ” Observation Hurley Section 1.1, ( Pg. 5 ) A process of Reason applied to a Phenomenological System ( PhenSys ) by which the state of that system can be characteristically quantified Wiki: “ Logic ” Good PhenSys make Good Neighbors A PhenSys is thus a subspecies of an OntSys
© Art Traynor 2011 Logic Reason Logic Logic Wiki: “ Reason ” which evaluates arguments Hurley Section 1.1, ( Pgs. 1 – 2 ) A system of inquiry addressed to gauging the validity of Arguments Wiki: “ Logic ” “ Inquiry ” because Reason is a process directed toward discovery  The field of Inquiry ( discipline/science )  That species of Reason which is applied within an Ontological System ( OntSys )
© Art Traynor 2011 Logic Reason Statement Proposition Hurley Section 1.1, ( Pg. 5 ) “ Expression ” and “ Statement ” are essentially synonymous , with “ Expression ” preferred in Symbolic Logic contexts A single Statement or expression can exhibit more than one Proposition… a Compound Proposition  A Declarative Expression  That component of a Statement asserting a Fact the Truth Value of which can be ascertained encoding a Fact about which a Truth Value can be ascertained Identical Information can be expressed in a multiplicity of Statements A single Statement may express a multiplicity of Propositions The terms “ Fact ” , “ Information ” , or sometimes “ Meaning ” are synonymous in this context The Expression or Statement encodes a “ Fact ” or “ Information ” ExpressionStatement ↔ Proposition Information
© Art Traynor 2011 Logic 384 – 322 B.C. History Milestones Aristotle Syllogistic Logic 279 – 206 B.C. Crysippus Propositional Logic Co-Founder of Stoic School of Philosophy 129 – 199 A.D. Galen Compound Categorical Syllogistic Logic 480 – 524 Boethius Commentator on Aristotle & Crysippus 1079 – 1142 Peter Abelard Theory of Universals Mental Constructs v. Platonic Forms Formal Validity v. Content Validity 1285 – 1349 William of Occam Modal Logic Comprising Possibility, Necessity, Belief, Doubt Metalanguage & Linguistics 1646 – 1716 Gottfried William Leibniz Symbolic Logic 1781 – 1848 Bernard Bolzano Expanded Leibniz’s Symbolization of Logic 1806 – 1871 1815 – 1864 George Boole Symbolic Logic Augustus DeMorgan Symbolic Logic 1835 – 1882 William Stanley Jevons Symbolic Logic 1834 – 1923 John Venn Symbolic Logic Hurley Section 1.1, ( Pgs. 5 – 7 ) 1806 – 1871 1806 – 1873 John Stuart Mill Inductive Logic Gottlob Frege Symbolic Logic 1861 – 1947 Alfred North Whitehead Betrand Russell Mathematical Logic 1872 – 1970
© Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation 3 Character of the Link between Premise & Conclusion Evaluate ASC Satisfaction: Argument or Non-Argument Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ? Inferential Morphism: G ⋀ P ↔ P ⋀ G Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises Evaluation Standards  
© Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Argument A group of Statements the Conclusion of which is claimed to follow from its Premises  Premises  Conclusion Hurley Section 1.1, ( Pgs. 2 – 3 ) Is understood to assert a Claim Explicit : often preceded by an Indicator word Implicit : can be rendered explicit by restatement including an Indicator  Tantamount to the grammatical test for the object of a (transitive) Verb Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises
© Art Traynor 2011 Logic Non-Arguments Taxonomy Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 11 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Warning Advice Belief / Opinion Description Report Exposition Illustration Conditional Explanation
© Art Traynor 2011 Logic Non-Arguments Warning Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 11 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Warning A proto-command or precautionary statement tantamount to a claim lacking evidence to which a progression of consequence is minimally implicit Wiki: “ Precautionary Statement ” Warning… Notice… Danger… Caution… Hazard… Warning Indicia
© Art Traynor 2011 Logic Non-Arguments Advice Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 11 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Advice A proto-conditional tantamount to a claim lacking evidence to which a progression of consequence is explicitly asserted Advise… Invite… Suggest… Prefer… Recommend… Advice Indicia Consider…
© Art Traynor 2011 Logic Non-Arguments Belief or Opinion Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Belief / Opinion Assertion of a Conclusion in the absence of Evidentiary Premises about which a Truth Value might be ascertained Believe… Offer… Trust… Aver… Vouchsafe… Belief / Opinion Indicia Suspect…
© Art Traynor 2011 Logic Non-Arguments Description Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Description A Set of Statements in which the Evidentiary Premises are merely evocative ( as of imagery ) and which fail to assert a Conclusion for which a Truth Value might be ascertained Imagine… Picture… Descriptive Indicia One of the four Rhetorical Modes or Modes of Discourse
© Art Traynor 2011 Logic Non-Arguments Report Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Report A Set of Statements with otherwise well-formed Premises , tantamount to a congeries of information about which a Truth Value can indeed be ascertained , but which however fails to include a Conclusion Concerning… Rumored… Reportorial Indicia Authorities… Participants… Beware of Reports about Arguments !
© Art Traynor 2011 Logic Non-Arguments Exposition Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Exposition A Set of Statements typically commencing with a well formed opening , satisfying the evidentiary requirement of a proper Premise , but which then merely expands, elaborates, or offers supplemental evidence to an implied , or previously established Conclusion Moreover… Furthermore… Expository Indicia Nevertheless… In addition to which… One of the four Rhetorical Modes or Modes of Discourse A Logical Conjunction , akin to “ And ”
© Art Traynor 2011 Logic Non-Arguments Illustration Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Illustration A Set of Statements in which the Evidentiary Premises are a congeries of existential instantiations exemplifying the Conclusion which however fail to satisfy a Proof of the Proposition Illustrating… Defining… Illustrative Indicia Showing… For example… For all… There exists…
© Art Traynor 2011 Logic Conditional Statement A two-part Statement wherein an Antecedent Proposition, prefaced typically by “ if ” inferentially posits a Consequent Proposition , prefaced typically by “ then ” If antecedent then consequent consequent if antecedent Hurley Section 1.2, ( Pg. 14 ) In an Argument a True Conclusion is Inferred to follow from True Premises, yet in a Conditional neither the Antecedent nor the Consequent are necessarily True Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) can be characterized into distinct classifications Conditional Not equivalent to an Argumentnn Does not necessarily posit a Causal Relationship between propositionso Asserts no pretense of Evidence or Truth Value of propositionso Posits an Inference between Propositionso Non-Arguments Conditional Structure Criteria ( CSC ) 1. Antecedent proto-Premises And an Inferentially connected A Set of Statements composed of 2. Consequent Of no particular Truth Value The Inferential content of a Conditional may be restated to satisfy ASCo
© Art Traynor 2011 Logic Conditional Statement Hurley Section 1.2, ( Pg. 14 ) Akin to an Existential Instantiation Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) Conditional Not equivalent to an Argumentnn Non-Arguments Conditional Structure Criteria ( CSC ) 1. Antecedent proto-Premises And an Inferentially connected A Set of Statements composed of 2. Consequent Of no particular Truth Value Sufficiencynn A Condition “ A ” is a Sufficiency if some Condition “ B ” may arise in its presence , but not exclusively so Akin to an Independent Variable Necessitynn Conjunctive Necessity & Sufficiencynn A Condition “ A ” is a Necessity if some Condition “ B ” may not arise in its absence Sufficient But Not Necessaryo Sufficient And Necessaryo
© Art Traynor 2011 Logic Conditional Statement Hurley Section 1.2, ( Pg. 16 ) Akin to an Existential Instantiation Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) Conditional Not equivalent to an Argumentnn Non-Arguments Conditional Structure Criteria ( CSC ) 1. Antecedent proto-Premises And an Inferentially connected A Set of Statements composed of 2. Consequent Of no particular Truth Value Sufficiencynn Akin to an Independent Variable Necessitynn Conjunctive Sufficiency & Necessitynn Sufficient But Not Necessaryo Sufficient And Necessaryo Condition “ B ” may arise in the presence of “ A ” , but not exclusively so Condition “ B ” may not arise in the absence of “ A ” On closer inspection the Conjunction is superfluous…Necessity certainly entails Sufficiency ( redundant ) whereas Sufficiency is something far less restrictive than Necessity “ A ” is not the only thing that might occasion “ B ” No “ A ” … No “ B ”
© Art Traynor 2011 Logic Explanation “why” something is the case… Non-Arguments Hurley Section 1.2, ( Pg. 17 ) Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) can be characterized into distinct classifications A group of Statements, comprised of at least two salients purporting to describe or otherwise account for an event or phenomena Explanation The Explandum , describing the event or phenomenann The Explanans , providing the causal or precipitating conditionsnn Not asserting to offer proof, as in argument Argument Explanation conclusion explanandum more obvious less obvious less obvious more obvious proof elucidation An Argument however may also serve as an Explanation answering “ why ” something is the case as well as offering proof that it is the case… Explanation + Proof = Argument
© Art Traynor 2011 Logic Explanation “why” something is the case… Non-Arguments Hurley Section 1.2, ( Pg. 17 ) Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) can be characterized into distinct classifications A species of proto-Argument Explanation Like Conditionals , an Explanation proceeds by Inferencenn Where the Explanans exhibit proof the Explanation will satisfy ASCnn i.e. : Explanation + Proof = Argument Problematic Indicatorsnn Becauseo May ambiguously preface either a Premise or an Explanans Thuso May ambiguously preface either an Conclusion or an Illustration
© Art Traynor 2011 Logic Explanation “why” something is the case… Non-Arguments Hurley Section 1.2, ( Pg. 17 ) Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) can be characterized into distinct classifications A species of proto-Argument Explanation Like Conditionals , an Explanation proceeds by Inferencenn Where the Explanans exhibit proof the Explanation will satisfy ASCnn i.e. : Explanation + Proof = Argument Problematic Indicatorsnn Sinceo Promise to Preface your Premise… In its Temporal sense is merely a species of narrative ( i.e. Report ? ) In its Logical sense it can preface a Premise
© Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ?  Deduction  Induction Evaluation Standards
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True That a Conclusion does not follow with Necessity from the Premises does not conclusively indicate that an Argument is Inductive  It could be a poorly executed Deductive Argument Necessarily… Certainly… Deductive Indicia Absolutely… Definitely… Problematic Indicatorsnn Musto Can ambiguously indicate either Probability or Necessity
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Necessarily… Certainly… Deductive Indicia Absolutely… Definitely…  Morphism The Aristotelian convention holds that Deduction proceeds canonically from Generalized class Premises to a Conclusion directed toward Particular class elements General → Particular General → General Particular → Particular Deduction conclusion statement statement statement premises statement General Particular This need not necessarily be the case as it is valid for the argument to assume the following alternate Morphisms Canonic Particular → General This is easy to recall as “ Deduct ” suggests a subtractive morphism
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Necessarily… Certainly… Deductive Indicia Absolutely… Definitely…  Form Certain Formal constructions indicate Class inclusion in Deduction Mathematics Definitions Categorical Syllogism Hypothetical Syllogism Disjunctive Syllogism Deduction conclusion statement statement statement premises statement General Particular
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Necessarily… Certainly… Deductive Indicia Absolutely… Definitely…  Form Certain Formal constructions indicate Class inclusion in Deduction 1 Mathematics Exclusive of subject matter lodged in Probability Theorynn Beware of the inaptly named “ Mathematical Induction ” it is in point of fact Deductive nn
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Defined… Means… Definitional Indicia Connotes… Tantamount…  Form Certain Formal constructions indicate Class inclusion in Deduction 2 Definitions Identifies the Term by non-tautological correspondencesnn Isolates the Term as a proper subset from its constituent correspondencesnn Terminologynn Examplesnn Maximal parsimonyo Maximal syntactic generalityo Trivialo Superficialo
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True All… No… C-Syll Indicia Some…  Form Certain Formal constructions indicate Class inclusion in Deduction 3 Categorical Syllogism All professors are academicians Syllogism Structure Criteria ( SSC ) 1. Exactly two Premises And An Argument consisting of exactly 2. A singular Conclusion Example: No academicians are headhunters Therefore , no professors are headhunters conclusion premises Essentially a notion of Transitivity
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True  Form Certain Formal constructions indicate Class inclusion in Deduction 4 Hypothetical Syllogism If Tony is convicted of a felony, then he will go to jail Syllogism Structure Criteria ( SSC ) 1. Exactly two Premises And An Argument consisting of exactly 2. A singular Conclusion Example: If he goes to jail, then his wife will divorce him Therefore , if Tony is convicted of a felony , then his wife will divorce him conclusion premises Essentially a notion of Transitivity A Syllogism wherein the Premises include a Conditional
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Either… Or… Dj-Syll Indicia  Form Certain Formal constructions indicate Class inclusion in Deduction 5 Disjunctive Syllogism Either the battery is charged or the car won’t start Syllogism Structure Criteria ( SSC ) 1. Exactly two Premises And An Argument consisting of exactly 2. A singular Conclusion Example: The battery is not charged Therefore , the car won’t start conclusion premises Essentially a notion of Transitivity A Syllogism wherein the Premises include a Disjunction
© Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ?  Deduction  Induction
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.3, ( Pg. 25 & 27 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Induction conclusion statement statement statement premises statement Particular General  Induction has been conventionally ( i.e. historically ) regarded to proceed from Particular Premises to a more generalized Conclusion The Conclusion Content conspires to “ go beyond ” the Premises Content 
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.3, ( Pg. 25 & 27 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Induction conclusion statement statement statement premises statement Particular General  Morphism The Aristotelian convention holds that Induction proceeds canon- ically from Particularized class Premises to a Conclusion directed toward the entirety of the Generalized class elements or Set This need not necessarily be the case as it is valid for the argument to assume the following alternate Morphisms Particular → General General → General Particular → Particular Canonic General → Particular
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 26 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction Prediction Analogy Inductive Generalization Authority Signs Casual Inference Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 1 Prediction Premises assert Temporal Claims about the state of an Ontological system nn Conclusion avers a speculative Claim about the resulting state of the Ontological system at some point ensuing nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 1 Prediction There is an implicit Claim that the state of an Ontological system is parameterized by time nn The Claim however is not deterministic and thus its Truth Value is necessarily associated with a Probability distribution nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 2 Analogy Premises assert a Claim of Similarity ( i.e. Congruence ? ) between one Ontological system ( OntSys ) and another nn The Analogy is implicitly postulated to obtain from a Claim about a better known Ontsys exhibiting a similar Claim about a less familiar OntSys nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 2 Analogy Premises assert a Claim of Similarity ( i.e. Congruence ? ) between one Ontological system ( OntSys ) and another nn Conclusion avers a speculative Inference about the consequent claim of the lesser known OntSys by reference to the consequent claim of the better-known OnsSys nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 3 Inductive Generalization Premises assert Claims about a Quantitative or Qualitative state of some attribute characteristic of a sample or Subset of an OntSys nn The inductive proposition suggests that the pre- valence or state of this characteristic attribute in the sample or Subset will exhibit in similar fashion in some larger population or Superset of the Ontsys toward which the Claim is directed nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 3 Inductive Generalization Premises assert Claims about a Quantitative or Qualitative state of some attribute characteristic of a sample or Subset of an OntSys nn Conclusion avers a speculative Inference that some larger universe or Superset populating the sample or Subset will exhibit the characteristic attribute in similar fashion in that larger Set nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 4 Authority Premises assert Claims arising from the expertise of a learned or informed figure or found in a canonical text addressing the OntSys nn Conclusion avers a speculative Inference that the validity of the resulting claim is thus free from mistake , error , or prevarication nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 5 Signs Premises assert a Claim arising from the declarative or admonitory exhortation of a symbolic representation nn A “sign” is not usually produced by the phenomena it signifies nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 5 Signs Premises assert a Claim arising from the declarative or admonitory exhortation of a symbolic representation nn Conclusion avers a speculative Inference urging as an advisory or precaution that certain action be taken or avoided nn Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 6 Causality Premises assert a Claim proceeding from knowledge of a Cause to knowledge of its Effects or vice-versa nn Sometimes conflated with Arguments based on Signs , as an Effect can often be interpreted as a “sign” of a Cause… nn Induction conclusion statement statement statement premises statement Particular General An Effect , unlike a Sign , is evolved from its correlative cause ( i.e. a Sign does not produce an effect ) nn
© Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 6 Causality Premises assert a Claim proceeding from knowledge of a Cause to knowledge of its Effects or vice-versa nn Induction conclusion statement statement statement premises statement Particular General Conclusion avers a speculative Inference positing a nexus between Cause and Effect that is less than wholly deterministic nn
© Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation 3 Character of the Link between Premise & Conclusion Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ? Inferential Morphism: G ⋀ P ↔ P ⋀ G  Morphism General → General Particular → Particular General → Particular Particular → General
© Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation 3 Character of the Link between Premise & Conclusion Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ? Inferential Morphism: G ⋀ P ↔ P ⋀ G  Morphism Particular → General General → General Particular → Particular General → Particular
© Art Traynor 2011 Logic Statement Arguments Hurley Section 1.3, ( Pg. 25 & 27 )Quantification Addresses the inherent Cardinality of An Argument is Valid if its Conclusion follows with Probability from its Premises A Statement that asserts a Claim about one or more members of a Class or Elements of a Set  Particular Statement A Statement that asserts a Claim about every or all members of a Class or Elements of a Set  General Statement
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 34 )Validity Every Deductive Argument is either Valid or Invalid  An Argument in which the Conclusion follows necessarily from the Premises If the Premises are assumed True , it is impossible that the Conclusion be false   An Invalid Deductive Argument is thus one in which the Conclusion does not follow with necessity from the premises If the Premises are assumed True , it is possible that the Conclusion be false   There is thus no middle ground between Validity and Invalidity A Binary State  There are no Arguments that are “ almost ” Valid or Invalid
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 34 )Validity Every Deductive Argument is either Valid or Invalid  There is thus no middle ground between Validity and Invalidity A Binary State  There are no Arguments that are “ almost ” Valid or Invalid  There is only an indirect relation between Validity and Truth If the Premises are assumed True , the Conclusion is True on the basis of that assumption alone !   For an Argument to be Valid it is not necessary that either the Premises or the Conclusion be True ! It is sufficient for Validity to obtain merely IF the Premises are assumed True , the Conclusion is rendered True on the basis of that assumption alone ! 
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 34 )Validity Every Deductive Argument is either Valid or Invalid  There is thus no middle ground between Validity and Invalidity A Binary State  There are no Arguments that are “ almost ” Valid or Invalid  There is only an indirect relation between Validity and Truth  For an Argument to be Valid it is not necessary that either the Premises or the Conclusion be True !  Just as a False Premise and a False Conclusion does not preclude Validity , so too a True Premise and True Conclusion does not vouchsafe Validity ! The probative test for Validity is that the Conclusion follow with Necessity from the Premises  If it is possible that the Conclusion may be False , a Deductive Argument fails Validity 
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 34 )Validity Every Deductive Argument is either Valid or Invalid  There is thus no middle ground between Validity and Invalidity A Binary State  There are no Arguments that are “ almost ” Valid or Invalid  There is only an indirect relation between Validity and Truth  For an Argument to be Valid it is not necessary that either the Premises or the Conclusion be True !  Just as a False Premise and a False Conclusion does not preclude Validity , so too a True Premise and True Conclusion does not vouchsafe Validity !  The Truth Value of the Premises and Conclusion thus tell us nothing , cannot tell us anything , about the Validity of a Deductive Argument Any Deductive Argument exhibiting True Premises and a False Conclusion is thus necessarily Invalid 
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Every Deductive Argument is either Valid or Invalid Premises Conclusion T T T F Validity F T F F ? Invalid ? ? Soundness A Deductive Argument that is Valid and exhibits True Premises is regarded as a Sound Argument  In the absence of the satisfaction of these two conjunctive conditions the Deductive Argument is rendered Unsound
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Every Deductive Argument is either Valid or Invalid Soundness A Deductive Argument that is Valid and exhibits True Premises is regarded as a Sound Argument  In the absence of the satisfaction of these two conjunctive conditions a Deductive Argument is rendered Unsound All the Premises must be True Satisfying ASC , a Sound Argument will always feature a True Conclusion  Deductive Argument Structure Criteria ( DASC ) 1. If the Premises are assumed True Then An Argument ( Satisfying ASC ) for which 2. The Conclusion follows Necessarily A Sound Argument is a species of Argument featuring additional structure… i.e.: True Premises Sound Argument = Valid Argument + True Premises The Deductive analogue of a Cogent Inductive Argument
© Art Traynor 2011 Logic Induction Arguments Hurley Section 1.4, ( Pg. 36 )Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak  Strong Inductive Argument The Conclusion follows with probability presuming True premises The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum Induction ( Strong ) conclusion True True premises True ?Probable This barrel contains 100 apples Example: Eighty apples randomly chosen were confirmed ripe Therefore , probably all 100 apples are ripe Particular General
© Art Traynor 2011 Logic Induction Arguments Hurley Section 1.4, ( Pg. 36 )Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak  Weak Inductive Argument The Conclusion does not follow with probability presuming True premises The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum Induction ( Weak ) conclusion True True premises True ?Not Probable This barrel contains 100 apples Example: Three apples randomly chosen were confirmed ripe Therefore , probably all 100 apples are ripe Particular General
© Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak  Inductive Arguments can be strengthened or weakened by modifications to their Premises  Strength or Weakness are only indirectly related to Truth and Falsity Induction ( Less Weak ) statement Induction ( Weak ) True True True ?Not Probable Induction ( Less Strong ) statement Induction ( Strong ) True True True ?Probable Probable True ? True ? True True True True Not Probable
© Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak Example: This barrel contains 100 apples Eighty apples randomly chosen were confirmed ripe Therefore , probably all 100 apples are ripe This barrel contains 100 apples Eighty apples randomly chosen were confirmed ripe Therefore , probably all 100 apples are ripe Induction ( Less Strong ) statement Induction ( Strong ) True True True ?Probable Probable True ? True True One unripe apple found earlier had been removed  Inductive Arguments can be strengthened or weakened by modifications to their Premises
© Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak Example: Of the many U.S. Presidents Therefore , probably the next American President will be a Federalist Induction ( Strong ) False False False ?Probable  Strength or Weakness are only indirectly related to Truth and Falsity The vast majority have been Federalists conclusion premises A Strong IA ( SIA ) cannot render a Conclusion probably True If these False Premises however were considered True , a True Conclusion may probably follow
© Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak Example: During the past fifty years Therefore , industrial productivity will probably increase in the years ahead  Strength or Weakness are only indirectly related to Truth and Falsity Inflation has consistently reduced the value of the American dollar conclusion premises Induction ( Weak ) True True True ?Not Probable The Truth of the Premises and probable Conclusion cannot render a weak IA ( WIA ) Strong
© Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak Induction ( Weak ) True True False ?Not Probable Any SIA featuring True Premises but with a probably False Conclusion is Ipso Facto rendered into a WIA  Strong Inductive Argument ( SIA ) The Conclusion follows with probability presuming True premises  Weak Inductive Argument ( WIA ) The Conclusion does not follow with probability presuming True premises  Inductive Arguments can be strengthened or weakened by modifications to their Premises  Strength or Weakness are only indirectly related to Truth and Falsity
© Art Traynor 2011 Logic Induction Arguments Hurley Section 1.4, ( Pg. 36 )Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak  The Conclusion follows with probability presuming True premises  The Conclusion does not follow with probability presuming True premises The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum  Inductive Arguments can be strengthened or weakened by modifications to their Premises  Strength or Weakness are only indirectly related to Truth and Falsity Strong Inductive Argument ( SIA ) Weak Inductive Argument ( WIA )
© Art Traynor 2011 Logic Induction Arguments Hurley Section 1.4, ( Pg. 36 )Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum  Strength or Weakness are only indirectly related to Truth and Falsity Induction conclusion statement statement statement premises statement Particular General
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 38 ) Premises Conclusion T Probably T T Validity F F ? Weak ? ? Probably F Probably T Probably F Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum  Strength or Weakness are only indirectly related to Truth and Falsity
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 38 ) Cogency A Inductive Argument that is Strong and exhibits True Premises is regarded as a Cogent Argument  In the absence of the satisfaction of these two conjunctive conditions an Inductive Argument is rendered Uncogent All the Premises must be True Satisfying ASC , a Cogent Argument will always feature a Conclusion that is True with Probablity  Inductive Argument Structure Criteria ( IASC ) 1. If the Premises are assumed True Then An Argument ( Satisfying ASC ) for which 2. The Conclusion Probably follows A Cogent IA ( CIA ) is a species of Argument featuring additional structure… i.e.: True Premises Cogent Argument = Strong Argument + True Premises Induction An Argument is Valid if its Conclusion follows with Probability from its Premises The Inductive analogue of a Sound Deductive Argument
© Art Traynor 2011 Logic Classification Arguments Hurley Section 1.4, ( Pg. 38 ) Statements Deductive Arguments FalseTrue Truth Value InvalidValid Validity Unsound Sound Unsound Inductive Arguments WeakStrong Strength Uncogent Cogent Uncogent
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The validity of an Argument arises from the Form or Structure of the Argument The Argument is valid because of the way in which the terms of the Premises are arranged We can substitute any terms we choose in the form of the argument and obtain a Valid Argumentnn Depending on what terms we substitute for “ A ” , “ B ” , and “ C ” the Conclusion will sometimes evaluate to True or False truth values nn Example: All Adlers are Bobkins Therefore , all Adlers are Crockers All Bobkins are Crockers conclusion premises All A are B All B are C All A are CParticular Essentially Transitive! General Valid Deduction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The validity of an Argument arises from the Form or Structure of the Argument The Argument is valid because of the way in which the terms of the Premises are arranged We can substitute any terms we choose in the form of the argument and obtain a Valid Argumentnn Nevertheless if we assume the Premises True we obtain a True Conclusionnn Example: All Adlers are Bobkins Therefore , all Adlers are Crockers All Bobkins are Crockers conclusion premises All A are B All B are C All A are CParticular Validity is not affected by Truth ! General Valid Deduction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The validity of an Argument arises from the Form or Structure of the Argument The Argument is valid because of the way in which the terms of the Premises are arranged We can substitute any terms we choose in the form of the argument and obtain a Valid Argumentnn A False Conclusion however can never arise from True Premises within an otherwise Valid form nn Example: All Adlers are Bobkins Therefore , all Adlers are Crockers All Bobkins are Crockers conclusion premises All A are B All B are C All A are CParticular Validity is not affected by Truth ! General Valid Deduction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Premises Conclusion T T T F Validity F T F F V I V Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Validitynn The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Deductive Argument Structure Criteria ( DASC ) 1. If the Premises are assumed True Then An Argument ( Satisfying ASC ) for which 2. The Conclusion follows Necessarily A Sound Argument is a species of Argument featuring additional structure… i.e.: True Premises V
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The validity of an Argument arises from the Form or Structure of the Argument The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Example: All Adlers are Bobkins Therefore , all Adlers are Crockers All Crockers are Bobkins conclusion premises All A are B All C are B All A are CParticular The Form fails Transitivity… The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Validity is not Commutative… General Invalid Deduction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Example: conclusion premises All A are B All C are B All A are CParticular The Form fails Transitivity… The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Validity is not Commutative… This is problematic however because it is mathematically valid ! All Adlers are Bobkins Therefore , all Adlers are Crockers All Crockers are Bobkins General Invalid Deduction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Example: All Cats are Animals Therefore , all Cats are Dogs All Dogs are Animals conclusion premises All C are A All D are A All C are D General Particular The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o This substitution however yields a manifestly False Conclusion The Class Equivalency here is valid by Union but does not obtain as Intersection Invalid Deduction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Example: All Cats are Animals Therefore , all Cats are Earthlings All Animals are Earthlings conclusion premises All C are A All A are E All C are EParticular The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Class Equivalency is somehow implicated in the Argument Form General Valid Deduction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Premises Conclusion T T T F Validity F T F F I I I I Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o The possibility of Invalidity violates the DASC Deductive Argument Structure Criteria Validate or Vitiate Invalid Deduction All A are B All C are B All A are CParticular General
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Premises Conclusion T T T F Validity F T F F Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Deductive Argument Structure Criteria ( DASC ) 1. If the Premises are assumed True Then An Argument ( Satisfying ASC ) for which 2. The Conclusion follows Necessarily A Sound Argument is a species of Argument featuring additional structure… i.e.: True Premises I I I I
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Premises Conclusion T T T F Validity F T F F Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o I I I I An Argument is Invalid if and only if its form allows for a substitution instance having True Premises and a False Conclusion
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn If an Argument is Invalid, it will exhibit an Invalid Form ( e.g. Formally Invalid )o However if the Form is Invalid (e.g. Formally Invalid ) , there is at least one conjunctive Substitution instance featuring True Premises and False Conclusion o An Argument is Invalid if and only if its form allows for a substitution instance having True Premises and a False Conclusion Premises Conclusion T T T F Validity F T F F I I I I
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn If an Argument is Invalid, it will exhibit an Invalid Form ( e.g. Formally Invalid )o However if the Form is Invalid (e.g. Formally Invalid ) , there is at least one conjunctive Substitution instance featuring True Premises and False Conclusion o An Argument is Invalid if and only if its form allows for a substitution instance having True Premises and a False Conclusion Otherwise every substitution featuring True Premises would exhibit a True Conclusion , as required by the Valid Deductive Argument ( VDA ) form o Conversely if a substitution were to yield an Argument featuring True Premises and exhibiting a False Conclusion , that substitution violates VDA Structure Criteria ( VDASC ) and is thus evaluates Invalid 
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Otherwise every substitution featuring True Premises would exhibit a True Conclusion , as required by the Valid Deductive Argument ( VDA ) form o Conversely if a substitution were to yield an Argument featuring True Premises and exhibiting a False Conclusion , that substitution violates VDA Structure Criteria ( VDASC ) and is thus evaluates Invalid  This constructive Invalidation of an otherwise Valid Deductive form admits an alternate, direct definition of Invalidity ( i.e. a means to Prove Invalidity )  Counter Example Method ( CEM )o The Argument is thus Proven Invalid ! A Deductive form of Argumentation is Isolated – Evaluated by Inspection A Substitution is constructed featuring True Premises and a False Conclusion
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 45 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o The Argument is thus Proven Invalid ! A Deductive form of Argumentation is Isolated – Evaluated by Inspection A Substitution is constructed featuring True Premises and a False Conclusion Example: Since some employees are not social climbers We may conclude that some VP’s are not Social Climbers And all Vice Presidents are Employees conclusion premises → Some E are not S Therefore Some V are not S All V are E
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o The Argument is thus Proven Invalid ! A Deductive form of Argumentation is Isolated – Evaluated by Inspection A Substitution is constructed featuring True Premises and a False Conclusion Example: conclusion premises → Some E are not S Therefore Some V are not S All V are E All V are E Therefore Some V are not S Some E are not S BUT…Some E not S are Not V i.e. We lose some “ E ” along the way here… i.e. There are some “ E ” that are “ Not S ” and “ Not V ”
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 45 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises → Some Animals are not Mammals Therefore Some Dogs are not Mammals All Dogs are Animals E = Animals S = Mammals V = Dogs Particular General Invalid Categorical Syllogism
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 45 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises Some Animals are not Mammals Therefore Some Dogs are not Mammals All Dogs are Animals Particular General Invalid Deductive Argument 1. True Premises 2. False Conclusion Invalid Categorical Syllogism
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises → If the Government imposes import restrictions Therefore since the Government will not impose import Restrictions ( then ) the price of Automobiles will rise If G then P Not G Therefore, not P Particular General Invalid Hypothetical Syllogism It follows that the price of Automobiles will not rise This is an Invalid Deductive Argument ( IDA ) because the price of Automobiles may rise notwithstanding Government inaction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises → If G then P Therefore , not P Not G If G then P Not G Therefore, not P Particular General Invalid Hypothetical Syllogism This is an Invalid Deductive Argument ( IDA ) because the price of Automobiles may rise notwithstanding Government inaction
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises→ If Abraham committed suicide, Therefore , Abraham Lincoln is not Dead Then Abraham Lincoln is dead G = Abraham Lincoln committed suicide P = Abraham Lincoln is dead Particular General Invalid Hypothetical Syllogism This is an Invalid Deductive Argument ( IDA ) because the price of Automobiles may rise notwithstanding Government inaction Abraham Lincoln did not commit suicide
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o conclusion premises If the Government imposes import restrictions Therefore since the Government will not impose import Restrictions ( then ) the price of Automobiles will rise Particular General Invalid Hypothetical Syllogism It follows that the price of Automobiles will not rise The Form of the Argument is revealed by substitution ( i.e. Counter Example ) to be invalid – one exhibiting True Premises and a False Conclusion If Abraham committed suicide, Therefore , Abraham Lincoln is not dead Then Abraham Lincoln is dead Abraham Lincoln did not commit suicide ↔
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o conclusion premises If the Government imposes import restrictions Therefore since the Government will not impose import Restrictions ( then ) the price of Automobiles will rise Particular General Invalid Hypothetical Syllogism It follows that the price of Automobiles will not rise Applying CEM to a Conditional, it is best to construct a Substitution expressing a Necessary hypothetical nexus: e.g. as between suicide and death If Abraham committed suicide , Therefore , Abraham Lincoln is not dead Then Abraham Lincoln is dead Abraham Lincoln did not commit suicide ↔
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Only useful for Proving Arguments Invalid The only arrangement of Truth & Falsity that Proves anything in Deductive Argumentation is True Premises with a False Conclusion  Premises Conclusion T T T F Validity F T F F V I V V
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Only applicable to Deductive Argumentation Only useful for Proving Arguments Invalid A VDA featuring True Premises and a True Conclusion Proves nothing The only arrangement of Truth & Falsity that Proves anything in Deductive Argumentation is True Premises with a False Conclusion  The relative Strength or Weakness of Inductive Argumentation is only tenuously dependent on the Form of the Argument  No method that relates exclusively to the form of an Inductive Argument can be used to prove the Argument Weak 
© Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Validitynn For a Deductive Argument, Subject Matter is irrelevant to a determination of Validity Deductive Argument Form InvalidValid Subject MatterIrrelevant Germaine
© Art Traynor 2011 Logic Extended Arguments Arguments Hurley Section 1.6, ( Pg. 49 )Extended Arguments ( ExArg ) A composite Argument, satisfying ASC comprised of : one or more Standards of Evaluation , ( i.e. Deduction and/or Induction ) and possibly featuring numerous Non-Arguments ( NonArg ) The process by which extraneous subject matter is eliminated from the ExArg , and Premises and a Conclusion are isolated / identified Warning Advice Non-Arguments Belief / Opinion Description Report Exposition Illustration Conditional Explanation Hurley Section 1.2, ( Pg. 25 ) Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises Hurley Section 1.2, ( Pgs. 11 – 18 )  Logical Analysis A Conclusory Statement must be associated in a well-ordered fashion to its Premises by positing its Inferential Nexus ( InfNex ) 
© Art Traynor 2011 Logic Extended Arguments Arguments Hurley Section 1.6, ( Pg. 49 )Extended Arguments ( ExArg ) A composite Argument, satisfying ASC comprised of : one or more Standards of Evaluation , ( i.e. Deduction and/or Induction ) and possibly featuring numerous Non-Arguments ( NonArg ) Warning Advice Non-Arguments Belief / Opinion Description Report Exposition Illustration Conditional Explanation Hurley Section 1.2, ( Pg. 25 ) Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises Hurley Section 1.2, ( Pgs. 11 – 18 )  Logical Analysis Inferential Nexus ( InfNex ) Vertical Inferential Nexus ( VIN )nn Horizontal Inferential Nexus ( HIN )nn The process by which extraneous subject matter is eliminated from the ExArg , and Premises and a Conclusion are isolated / identified
© Art Traynor 2011 Logic Extended Arguments Arguments Hurley Section 1.6, ( Pg. 49 )Extended Arguments ( ExArg ) A composite Argument, satisfying ASC comprised of : one or more Standards of Evaluation , ( i.e. Deduction and/or Induction ) and possibly featuring numerous Non-Arguments ( NonArg ) Warning Advice Non-Arguments Belief / Opinion Description Report Exposition Illustration Conditional Explanation Hurley Section 1.2, ( Pgs. 11 – 18 )  Logical Analysis Inferential Nexus ( InfNex ) Vertical Inferential Nexus ( VIN )nn The process by which extraneous subject matter is eliminated from the ExArg , and Premises and a Conclusion are isolated / identified conclusion premises The contamination of underground aquifers represents a pollution problem of catastrophic proportions Half the nation’s drinking water , which comes from these aquifers , is being poisoned by chemical wastes dumped into the soil for generations ↔ 1 2 [ if ] Half the nation’s drinking water , comes from aquifers , being poisoned by chemical wastes dumped into the soil for generations 2 [ Then ] contamination of underground aquifers represents a pollution problem of catastrophic proportions 1
© Art Traynor 2011 Logic Extended Arguments Arguments Hurley Section 1.6, ( Pg. 49 )Extended Arguments ( ExArg ) A composite Argument, satisfying ASC comprised of : one or more Standards of Evaluation , ( i.e. Deduction and/or Induction ) and possibly featuring numerous Non-Arguments ( NonArg ) Warning Advice Non-Arguments Belief / Opinion Description Report Exposition Illustration Conditional Explanation Hurley Section 1.2, ( Pgs. 11 – 18 )  Logical Analysis Inferential Nexus ( InfNex ) Vertical Inferential Nexus ( VIN )nn The process by which extraneous subject matter is eliminated from the ExArg , and Premises and a Conclusion are isolated / identified conclusion premises ↔ [ if ] Half the nation’s drinking water , comes from aquifers , being poisoned by chemical wastes dumped into the soil for generations 2 [ Then ] contamination of underground aquifers represents a pollution problem of catastrophic proportions 1 2 1 Inferential Nexus ( InfNex )

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  • 1. © Art Traynor 2011 Logic Nomenclature Definitions Logic The field of inquiry ( discipline/science ) which evaluates arguments Argument A Set of Statements composed of a Conclusion and evidentiary Premises from which it is claimed to logically follow Statement A declarative sentence ( a sentence which declares a Fact ) that is either True or False but not both Truth Value The attribute by which a statement can be evaluated to be either True or False Sometimes “proposition” “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence Hurley Section 1.1, ( Pgs. 1 – 2 ) A system of inquiry addressed to gauging the validity of Arguments Wiki: “ Logic ” “ Inquiry ” because Logic is a process directed toward discovery “ Expression ” and “ Statement ” are essentially synonymous , with “ Expression ” preferred in Symbolic Logic contexts The terms “ Fact ” , “ Information ” , or sometimes “ Meaning ” are synonymous in this context The Expression or Statement encodes a “ Fact ” or “ Information ”
  • 2. © Art Traynor 2011 Logic Nomenclature Definitions Hurley Section 1.1, ( Pg. 5 ) As in Inferential Nexus ( InfNex ) Inference A process of Reason supplying the Structure by which the Validity of an Argument or Conditional Statement can be gauged That component of a Statement encoding Information about which a Truth Value can be ascertained Proposition “ Expression ” and “ Statement ” are essentially synonymous , with “ Expression ” preferred in Symbolic Logic contexts The terms “ Fact ” , “ Information ” , or sometimes “ Meaning ” are synonymous in this context The Expression or Statement encodes a “ Fact ” or “ Information ”
  • 3. © Art Traynor 2011 Logic Argument Definition Argument A prefatory composition of Propositions ( i.e. Declarative Statements ) , or Premises , the Truth Value of which can be reckoned , and by which the application of Inference will permit a Conclusion to be gauged Wiki: “ Argument ” A group of Statements the Conclusion of which is claimed to follow from its Premises “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence Hurley Section 1.1, ( Pgs. 2 – 3 ) An Argument expresses an Inference in a special way, in terms of one or more premises that present Evidence and a Conclusion that is claimed to follow from that Evidence. Hurley Section 1.2, ( Pg. 15 ) An Inferential Nexus ( InfNex )
  • 4. © Art Traynor 2011 Logic Argument Statements Argument A group of Statements the Conclusion of which is claimed to follow from its Premises “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence  Premises One or more Statements asserted as Evidence  Conclusion A singular Statement from which the Premises are asserted to follow Hurley Section 1.1, ( Pgs. 2 – 3 ) Follow: to implicate with Necessity Argument conclusion statement statement statement premises statement
  • 5. © Art Traynor 2011 Logic Statement Premise Argument A group of Statements the Conclusion of which is claimed to follow from its Premises “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence  Premises One or more Statements asserted as Evidence Hurley Section 1.1, ( Pgs. 2 – 3 ) Argument statement statement statement premises May be inferred from… Seeing that… For the reason that… Inasmuch as… As indicated by… Premise Indicators Given that… Since… Owing to… Because… In that… For… As…
  • 6. © Art Traynor 2011 Logic Statement Conclusion Argument A group of Statements the Conclusion of which is claimed to follow from its Premises “ that ” something is the case…. Purports to demonstrate a conclusion on the basis of evidence  Premises One or more Statements asserted as Evidence  Conclusion A singular Statement from which the Premises are asserted to follow Hurley Section 1.1, ( Pgs. 2 – 3 ) Argument conclusion statement statement statement premises statement We may conclude… It follows that… It must be that… We may infer… Consequently… Conclusion Indicators Accordingly… Therefore… Implies that… Wherefore… As a result… Whence… Hence… Thus… So…
  • 7. © Art Traynor 2011 Logic Reason Definition Reason A process of Inquiry may be discovered and Evaluated Wiki: “ Reason ” populating an Ontological System ( OntSys ) or Set “ Inquiry ” because Reason is a process directed toward discovery Akin to an Algebra by which Relations between Class Equivalent Elements “ Evaluate ” to reduce an indeterminant to determinacy  Etymology λóγος or “ logos ” , is the root of the English word “ Logic ” which also had a connotation of “ speech ” or “ explanation ” or an “ account ” ( as of money )  Translated into Latin as “ ratio ” , from which its synonymous correspondence with “ Rationality ” emerged  There is thus a pleasing sense in which Rationality or Reason connotes the abstract evaluation of Ratios or Proportions nn
  • 8. © Art Traynor 2011 Logic Reason Definition Reason A process of Inquiry may be discovered and Evaluated Wiki: “ Reason ” populating an Ontological System ( OntSys ) or Set “ Inquiry ” because Reason is a process directed toward discovery Akin to an Algebra by which Relations between Class Equivalent Elements “ Evaluate ” to reduce an indeterminant to determinacy  Domain of Discourse Whereas Logic lies within Reason , Reason unconstrained by Logic supplies extra-logical processes of Inquiry Skipping Steps Working Backward Diagraming Exemplifying ( Exampling or Modeling ) Axiom Transformation ( changing the Laws of Composition – LOC’s ) Processes that lie outside of the OntSys and Logic
  • 9. © Art Traynor 2011 Logic Reason Logic Logic Wiki: “ Reason ” which evaluates arguments Hurley Section 1.1, ( Pgs. 1 – 2 ) A system of inquiry addressed to gauging the validity of Arguments Wiki: “ Logic ” “ Inquiry ” because Reason is a process directed toward discovery  The field of Inquiry ( discipline/science )  That species of Reason which is applied within an Ontological System ( OntSys ) Inference Hurley Section 1.1, ( Pg. 5 ) A process of Reason supplying the Structure by which the Validity of an Argument or Conditional Statement can be gauged Wiki: “ Logic ” Observation Hurley Section 1.1, ( Pg. 5 ) A process of Reason applied to a Phenomenological System ( PhenSys ) by which the state of that system can be characteristically quantified Wiki: “ Logic ” Good PhenSys make Good Neighbors A PhenSys is thus a subspecies of an OntSys
  • 10. © Art Traynor 2011 Logic Reason Logic Logic Wiki: “ Reason ” which evaluates arguments Hurley Section 1.1, ( Pgs. 1 – 2 ) A system of inquiry addressed to gauging the validity of Arguments Wiki: “ Logic ” “ Inquiry ” because Reason is a process directed toward discovery  The field of Inquiry ( discipline/science )  That species of Reason which is applied within an Ontological System ( OntSys )
  • 11. © Art Traynor 2011 Logic Reason Statement Proposition Hurley Section 1.1, ( Pg. 5 ) “ Expression ” and “ Statement ” are essentially synonymous , with “ Expression ” preferred in Symbolic Logic contexts A single Statement or expression can exhibit more than one Proposition… a Compound Proposition  A Declarative Expression  That component of a Statement asserting a Fact the Truth Value of which can be ascertained encoding a Fact about which a Truth Value can be ascertained Identical Information can be expressed in a multiplicity of Statements A single Statement may express a multiplicity of Propositions The terms “ Fact ” , “ Information ” , or sometimes “ Meaning ” are synonymous in this context The Expression or Statement encodes a “ Fact ” or “ Information ” ExpressionStatement ↔ Proposition Information
  • 12. © Art Traynor 2011 Logic 384 – 322 B.C. History Milestones Aristotle Syllogistic Logic 279 – 206 B.C. Crysippus Propositional Logic Co-Founder of Stoic School of Philosophy 129 – 199 A.D. Galen Compound Categorical Syllogistic Logic 480 – 524 Boethius Commentator on Aristotle & Crysippus 1079 – 1142 Peter Abelard Theory of Universals Mental Constructs v. Platonic Forms Formal Validity v. Content Validity 1285 – 1349 William of Occam Modal Logic Comprising Possibility, Necessity, Belief, Doubt Metalanguage & Linguistics 1646 – 1716 Gottfried William Leibniz Symbolic Logic 1781 – 1848 Bernard Bolzano Expanded Leibniz’s Symbolization of Logic 1806 – 1871 1815 – 1864 George Boole Symbolic Logic Augustus DeMorgan Symbolic Logic 1835 – 1882 William Stanley Jevons Symbolic Logic 1834 – 1923 John Venn Symbolic Logic Hurley Section 1.1, ( Pgs. 5 – 7 ) 1806 – 1871 1806 – 1873 John Stuart Mill Inductive Logic Gottlob Frege Symbolic Logic 1861 – 1947 Alfred North Whitehead Betrand Russell Mathematical Logic 1872 – 1970
  • 13. © Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation 3 Character of the Link between Premise & Conclusion Evaluate ASC Satisfaction: Argument or Non-Argument Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ? Inferential Morphism: G ⋀ P ↔ P ⋀ G Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises Evaluation Standards  
  • 14. © Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Argument A group of Statements the Conclusion of which is claimed to follow from its Premises  Premises  Conclusion Hurley Section 1.1, ( Pgs. 2 – 3 ) Is understood to assert a Claim Explicit : often preceded by an Indicator word Implicit : can be rendered explicit by restatement including an Indicator  Tantamount to the grammatical test for the object of a (transitive) Verb Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises
  • 15. © Art Traynor 2011 Logic Non-Arguments Taxonomy Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 11 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Warning Advice Belief / Opinion Description Report Exposition Illustration Conditional Explanation
  • 16. © Art Traynor 2011 Logic Non-Arguments Warning Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 11 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Warning A proto-command or precautionary statement tantamount to a claim lacking evidence to which a progression of consequence is minimally implicit Wiki: “ Precautionary Statement ” Warning… Notice… Danger… Caution… Hazard… Warning Indicia
  • 17. © Art Traynor 2011 Logic Non-Arguments Advice Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 11 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Advice A proto-conditional tantamount to a claim lacking evidence to which a progression of consequence is explicitly asserted Advise… Invite… Suggest… Prefer… Recommend… Advice Indicia Consider…
  • 18. © Art Traynor 2011 Logic Non-Arguments Belief or Opinion Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Belief / Opinion Assertion of a Conclusion in the absence of Evidentiary Premises about which a Truth Value might be ascertained Believe… Offer… Trust… Aver… Vouchsafe… Belief / Opinion Indicia Suspect…
  • 19. © Art Traynor 2011 Logic Non-Arguments Description Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Description A Set of Statements in which the Evidentiary Premises are merely evocative ( as of imagery ) and which fail to assert a Conclusion for which a Truth Value might be ascertained Imagine… Picture… Descriptive Indicia One of the four Rhetorical Modes or Modes of Discourse
  • 20. © Art Traynor 2011 Logic Non-Arguments Report Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Report A Set of Statements with otherwise well-formed Premises , tantamount to a congeries of information about which a Truth Value can indeed be ascertained , but which however fails to include a Conclusion Concerning… Rumored… Reportorial Indicia Authorities… Participants… Beware of Reports about Arguments !
  • 21. © Art Traynor 2011 Logic Non-Arguments Exposition Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Exposition A Set of Statements typically commencing with a well formed opening , satisfying the evidentiary requirement of a proper Premise , but which then merely expands, elaborates, or offers supplemental evidence to an implied , or previously established Conclusion Moreover… Furthermore… Expository Indicia Nevertheless… In addition to which… One of the four Rhetorical Modes or Modes of Discourse A Logical Conjunction , akin to “ And ”
  • 22. © Art Traynor 2011 Logic Non-Arguments Illustration Argument Recognition An Argument can be difficult to recognize Hurley Section 1.2, ( Pg. 12 )  The presence of Indicators does not necessarily render a Set of Statements into an Argument Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises  Non-Arguments ( NonArg ) can be characterized into distinct classifications Akin to Equivalence Classes Illustration A Set of Statements in which the Evidentiary Premises are a congeries of existential instantiations exemplifying the Conclusion which however fail to satisfy a Proof of the Proposition Illustrating… Defining… Illustrative Indicia Showing… For example… For all… There exists…
  • 23. © Art Traynor 2011 Logic Conditional Statement A two-part Statement wherein an Antecedent Proposition, prefaced typically by “ if ” inferentially posits a Consequent Proposition , prefaced typically by “ then ” If antecedent then consequent consequent if antecedent Hurley Section 1.2, ( Pg. 14 ) In an Argument a True Conclusion is Inferred to follow from True Premises, yet in a Conditional neither the Antecedent nor the Consequent are necessarily True Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) can be characterized into distinct classifications Conditional Not equivalent to an Argumentnn Does not necessarily posit a Causal Relationship between propositionso Asserts no pretense of Evidence or Truth Value of propositionso Posits an Inference between Propositionso Non-Arguments Conditional Structure Criteria ( CSC ) 1. Antecedent proto-Premises And an Inferentially connected A Set of Statements composed of 2. Consequent Of no particular Truth Value The Inferential content of a Conditional may be restated to satisfy ASCo
  • 24. © Art Traynor 2011 Logic Conditional Statement Hurley Section 1.2, ( Pg. 14 ) Akin to an Existential Instantiation Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) Conditional Not equivalent to an Argumentnn Non-Arguments Conditional Structure Criteria ( CSC ) 1. Antecedent proto-Premises And an Inferentially connected A Set of Statements composed of 2. Consequent Of no particular Truth Value Sufficiencynn A Condition “ A ” is a Sufficiency if some Condition “ B ” may arise in its presence , but not exclusively so Akin to an Independent Variable Necessitynn Conjunctive Necessity & Sufficiencynn A Condition “ A ” is a Necessity if some Condition “ B ” may not arise in its absence Sufficient But Not Necessaryo Sufficient And Necessaryo
  • 25. © Art Traynor 2011 Logic Conditional Statement Hurley Section 1.2, ( Pg. 16 ) Akin to an Existential Instantiation Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) Conditional Not equivalent to an Argumentnn Non-Arguments Conditional Structure Criteria ( CSC ) 1. Antecedent proto-Premises And an Inferentially connected A Set of Statements composed of 2. Consequent Of no particular Truth Value Sufficiencynn Akin to an Independent Variable Necessitynn Conjunctive Sufficiency & Necessitynn Sufficient But Not Necessaryo Sufficient And Necessaryo Condition “ B ” may arise in the presence of “ A ” , but not exclusively so Condition “ B ” may not arise in the absence of “ A ” On closer inspection the Conjunction is superfluous…Necessity certainly entails Sufficiency ( redundant ) whereas Sufficiency is something far less restrictive than Necessity “ A ” is not the only thing that might occasion “ B ” No “ A ” … No “ B ”
  • 26. © Art Traynor 2011 Logic Explanation “why” something is the case… Non-Arguments Hurley Section 1.2, ( Pg. 17 ) Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) can be characterized into distinct classifications A group of Statements, comprised of at least two salients purporting to describe or otherwise account for an event or phenomena Explanation The Explandum , describing the event or phenomenann The Explanans , providing the causal or precipitating conditionsnn Not asserting to offer proof, as in argument Argument Explanation conclusion explanandum more obvious less obvious less obvious more obvious proof elucidation An Argument however may also serve as an Explanation answering “ why ” something is the case as well as offering proof that it is the case… Explanation + Proof = Argument
  • 27. © Art Traynor 2011 Logic Explanation “why” something is the case… Non-Arguments Hurley Section 1.2, ( Pg. 17 ) Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) can be characterized into distinct classifications A species of proto-Argument Explanation Like Conditionals , an Explanation proceeds by Inferencenn Where the Explanans exhibit proof the Explanation will satisfy ASCnn i.e. : Explanation + Proof = Argument Problematic Indicatorsnn Becauseo May ambiguously preface either a Premise or an Explanans Thuso May ambiguously preface either an Conclusion or an Illustration
  • 28. © Art Traynor 2011 Logic Explanation “why” something is the case… Non-Arguments Hurley Section 1.2, ( Pg. 17 ) Argument Recognition An Argument can be difficult to recognize  Non-Arguments ( NonArg ) can be characterized into distinct classifications A species of proto-Argument Explanation Like Conditionals , an Explanation proceeds by Inferencenn Where the Explanans exhibit proof the Explanation will satisfy ASCnn i.e. : Explanation + Proof = Argument Problematic Indicatorsnn Sinceo Promise to Preface your Premise… In its Temporal sense is merely a species of narrative ( i.e. Report ? ) In its Logical sense it can preface a Premise
  • 29. © Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ?  Deduction  Induction Evaluation Standards
  • 30. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True That a Conclusion does not follow with Necessity from the Premises does not conclusively indicate that an Argument is Inductive  It could be a poorly executed Deductive Argument Necessarily… Certainly… Deductive Indicia Absolutely… Definitely… Problematic Indicatorsnn Musto Can ambiguously indicate either Probability or Necessity
  • 31. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Necessarily… Certainly… Deductive Indicia Absolutely… Definitely…  Morphism The Aristotelian convention holds that Deduction proceeds canonically from Generalized class Premises to a Conclusion directed toward Particular class elements General → Particular General → General Particular → Particular Deduction conclusion statement statement statement premises statement General Particular This need not necessarily be the case as it is valid for the argument to assume the following alternate Morphisms Canonic Particular → General This is easy to recall as “ Deduct ” suggests a subtractive morphism
  • 32. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Necessarily… Certainly… Deductive Indicia Absolutely… Definitely…  Form Certain Formal constructions indicate Class inclusion in Deduction Mathematics Definitions Categorical Syllogism Hypothetical Syllogism Disjunctive Syllogism Deduction conclusion statement statement statement premises statement General Particular
  • 33. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Necessarily… Certainly… Deductive Indicia Absolutely… Definitely…  Form Certain Formal constructions indicate Class inclusion in Deduction 1 Mathematics Exclusive of subject matter lodged in Probability Theorynn Beware of the inaptly named “ Mathematical Induction ” it is in point of fact Deductive nn
  • 34. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Defined… Means… Definitional Indicia Connotes… Tantamount…  Form Certain Formal constructions indicate Class inclusion in Deduction 2 Definitions Identifies the Term by non-tautological correspondencesnn Isolates the Term as a proper subset from its constituent correspondencesnn Terminologynn Examplesnn Maximal parsimonyo Maximal syntactic generalityo Trivialo Superficialo
  • 35. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True All… No… C-Syll Indicia Some…  Form Certain Formal constructions indicate Class inclusion in Deduction 3 Categorical Syllogism All professors are academicians Syllogism Structure Criteria ( SSC ) 1. Exactly two Premises And An Argument consisting of exactly 2. A singular Conclusion Example: No academicians are headhunters Therefore , no professors are headhunters conclusion premises Essentially a notion of Transitivity
  • 36. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True  Form Certain Formal constructions indicate Class inclusion in Deduction 4 Hypothetical Syllogism If Tony is convicted of a felony, then he will go to jail Syllogism Structure Criteria ( SSC ) 1. Exactly two Premises And An Argument consisting of exactly 2. A singular Conclusion Example: If he goes to jail, then his wife will divorce him Therefore , if Tony is convicted of a felony , then his wife will divorce him conclusion premises Essentially a notion of Transitivity A Syllogism wherein the Premises include a Conditional
  • 37. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.2, ( Pg. 25 ) Deduction An Argument is Valid only if its Conclusion follows Necessarily from its Premises Nothing to do with Taxes !  Presuming the Truth of the Premises , it is impossible for the Conclusion to not be True Either… Or… Dj-Syll Indicia  Form Certain Formal constructions indicate Class inclusion in Deduction 5 Disjunctive Syllogism Either the battery is charged or the car won’t start Syllogism Structure Criteria ( SSC ) 1. Exactly two Premises And An Argument consisting of exactly 2. A singular Conclusion Example: The battery is not charged Therefore , the car won’t start conclusion premises Essentially a notion of Transitivity A Syllogism wherein the Premises include a Disjunction
  • 38. © Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ?  Deduction  Induction
  • 39. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.3, ( Pg. 25 & 27 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Induction conclusion statement statement statement premises statement Particular General  Induction has been conventionally ( i.e. historically ) regarded to proceed from Particular Premises to a more generalized Conclusion The Conclusion Content conspires to “ go beyond ” the Premises Content 
  • 40. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.3, ( Pg. 25 & 27 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Induction conclusion statement statement statement premises statement Particular General  Morphism The Aristotelian convention holds that Induction proceeds canon- ically from Particularized class Premises to a Conclusion directed toward the entirety of the Generalized class elements or Set This need not necessarily be the case as it is valid for the argument to assume the following alternate Morphisms Particular → General General → General Particular → Particular Canonic General → Particular
  • 41. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 26 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction Prediction Analogy Inductive Generalization Authority Signs Casual Inference Induction conclusion statement statement statement premises statement Particular General
  • 42. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 1 Prediction Premises assert Temporal Claims about the state of an Ontological system nn Conclusion avers a speculative Claim about the resulting state of the Ontological system at some point ensuing nn Induction conclusion statement statement statement premises statement Particular General
  • 43. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 1 Prediction There is an implicit Claim that the state of an Ontological system is parameterized by time nn The Claim however is not deterministic and thus its Truth Value is necessarily associated with a Probability distribution nn Induction conclusion statement statement statement premises statement Particular General
  • 44. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 2 Analogy Premises assert a Claim of Similarity ( i.e. Congruence ? ) between one Ontological system ( OntSys ) and another nn The Analogy is implicitly postulated to obtain from a Claim about a better known Ontsys exhibiting a similar Claim about a less familiar OntSys nn Induction conclusion statement statement statement premises statement Particular General
  • 45. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 2 Analogy Premises assert a Claim of Similarity ( i.e. Congruence ? ) between one Ontological system ( OntSys ) and another nn Conclusion avers a speculative Inference about the consequent claim of the lesser known OntSys by reference to the consequent claim of the better-known OnsSys nn Induction conclusion statement statement statement premises statement Particular General
  • 46. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 3 Inductive Generalization Premises assert Claims about a Quantitative or Qualitative state of some attribute characteristic of a sample or Subset of an OntSys nn The inductive proposition suggests that the pre- valence or state of this characteristic attribute in the sample or Subset will exhibit in similar fashion in some larger population or Superset of the Ontsys toward which the Claim is directed nn Induction conclusion statement statement statement premises statement Particular General
  • 47. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 3 Inductive Generalization Premises assert Claims about a Quantitative or Qualitative state of some attribute characteristic of a sample or Subset of an OntSys nn Conclusion avers a speculative Inference that some larger universe or Superset populating the sample or Subset will exhibit the characteristic attribute in similar fashion in that larger Set nn Induction conclusion statement statement statement premises statement Particular General
  • 48. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 4 Authority Premises assert Claims arising from the expertise of a learned or informed figure or found in a canonical text addressing the OntSys nn Conclusion avers a speculative Inference that the validity of the resulting claim is thus free from mistake , error , or prevarication nn Induction conclusion statement statement statement premises statement Particular General
  • 49. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 5 Signs Premises assert a Claim arising from the declarative or admonitory exhortation of a symbolic representation nn A “sign” is not usually produced by the phenomena it signifies nn Induction conclusion statement statement statement premises statement Particular General
  • 50. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 5 Signs Premises assert a Claim arising from the declarative or admonitory exhortation of a symbolic representation nn Conclusion avers a speculative Inference urging as an advisory or precaution that certain action be taken or avoided nn Induction conclusion statement statement statement premises statement Particular General
  • 51. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 6 Causality Premises assert a Claim proceeding from knowledge of a Cause to knowledge of its Effects or vice-versa nn Sometimes conflated with Arguments based on Signs , as an Effect can often be interpreted as a “sign” of a Cause… nn Induction conclusion statement statement statement premises statement Particular General An Effect , unlike a Sign , is evolved from its correlative cause ( i.e. a Sign does not produce an effect ) nn
  • 52. © Art Traynor 2011 Logic Taxonomy Arguments Hurley Section 1.2, ( Pg. 28 ) Induction An Argument is Valid if its Conclusion follows with Probability from its Premises  Presuming the Truth of the Premises , it is possible , but not probable that the Conclusion is not True Probable… Improbable… Inductive Indicia Plausible… Implausible… Likely… Unlikely… Reasonable to conclude… Form Certain Formal constructions indicate Class inclusion in Induction 6 Causality Premises assert a Claim proceeding from knowledge of a Cause to knowledge of its Effects or vice-versa nn Induction conclusion statement statement statement premises statement Particular General Conclusion avers a speculative Inference positing a nexus between Cause and Effect that is less than wholly deterministic nn
  • 53. © Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation 3 Character of the Link between Premise & Conclusion Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ? Inferential Morphism: G ⋀ P ↔ P ⋀ G  Morphism General → General Particular → Particular General → Particular Particular → General
  • 54. © Art Traynor 2011 Logic Classification Criteria Arguments Classification Criteria 1 Indicators 2 Form of Argumentation 3 Character of the Link between Premise & Conclusion Evaluate ASC Satisfaction: Argument or Non-Argument Do we have an Argument ? Deductive or Inductive If we have an Argument, what Form does it assume ? Inferential Morphism: G ⋀ P ↔ P ⋀ G  Morphism Particular → General General → General Particular → Particular General → Particular
  • 55. © Art Traynor 2011 Logic Statement Arguments Hurley Section 1.3, ( Pg. 25 & 27 )Quantification Addresses the inherent Cardinality of An Argument is Valid if its Conclusion follows with Probability from its Premises A Statement that asserts a Claim about one or more members of a Class or Elements of a Set  Particular Statement A Statement that asserts a Claim about every or all members of a Class or Elements of a Set  General Statement
  • 56. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 34 )Validity Every Deductive Argument is either Valid or Invalid  An Argument in which the Conclusion follows necessarily from the Premises If the Premises are assumed True , it is impossible that the Conclusion be false   An Invalid Deductive Argument is thus one in which the Conclusion does not follow with necessity from the premises If the Premises are assumed True , it is possible that the Conclusion be false   There is thus no middle ground between Validity and Invalidity A Binary State  There are no Arguments that are “ almost ” Valid or Invalid
  • 57. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 34 )Validity Every Deductive Argument is either Valid or Invalid  There is thus no middle ground between Validity and Invalidity A Binary State  There are no Arguments that are “ almost ” Valid or Invalid  There is only an indirect relation between Validity and Truth If the Premises are assumed True , the Conclusion is True on the basis of that assumption alone !   For an Argument to be Valid it is not necessary that either the Premises or the Conclusion be True ! It is sufficient for Validity to obtain merely IF the Premises are assumed True , the Conclusion is rendered True on the basis of that assumption alone ! 
  • 58. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 34 )Validity Every Deductive Argument is either Valid or Invalid  There is thus no middle ground between Validity and Invalidity A Binary State  There are no Arguments that are “ almost ” Valid or Invalid  There is only an indirect relation between Validity and Truth  For an Argument to be Valid it is not necessary that either the Premises or the Conclusion be True !  Just as a False Premise and a False Conclusion does not preclude Validity , so too a True Premise and True Conclusion does not vouchsafe Validity ! The probative test for Validity is that the Conclusion follow with Necessity from the Premises  If it is possible that the Conclusion may be False , a Deductive Argument fails Validity 
  • 59. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 34 )Validity Every Deductive Argument is either Valid or Invalid  There is thus no middle ground between Validity and Invalidity A Binary State  There are no Arguments that are “ almost ” Valid or Invalid  There is only an indirect relation between Validity and Truth  For an Argument to be Valid it is not necessary that either the Premises or the Conclusion be True !  Just as a False Premise and a False Conclusion does not preclude Validity , so too a True Premise and True Conclusion does not vouchsafe Validity !  The Truth Value of the Premises and Conclusion thus tell us nothing , cannot tell us anything , about the Validity of a Deductive Argument Any Deductive Argument exhibiting True Premises and a False Conclusion is thus necessarily Invalid 
  • 60. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Every Deductive Argument is either Valid or Invalid Premises Conclusion T T T F Validity F T F F ? Invalid ? ? Soundness A Deductive Argument that is Valid and exhibits True Premises is regarded as a Sound Argument  In the absence of the satisfaction of these two conjunctive conditions the Deductive Argument is rendered Unsound
  • 61. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Every Deductive Argument is either Valid or Invalid Soundness A Deductive Argument that is Valid and exhibits True Premises is regarded as a Sound Argument  In the absence of the satisfaction of these two conjunctive conditions a Deductive Argument is rendered Unsound All the Premises must be True Satisfying ASC , a Sound Argument will always feature a True Conclusion  Deductive Argument Structure Criteria ( DASC ) 1. If the Premises are assumed True Then An Argument ( Satisfying ASC ) for which 2. The Conclusion follows Necessarily A Sound Argument is a species of Argument featuring additional structure… i.e.: True Premises Sound Argument = Valid Argument + True Premises The Deductive analogue of a Cogent Inductive Argument
  • 62. © Art Traynor 2011 Logic Induction Arguments Hurley Section 1.4, ( Pg. 36 )Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak  Strong Inductive Argument The Conclusion follows with probability presuming True premises The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum Induction ( Strong ) conclusion True True premises True ?Probable This barrel contains 100 apples Example: Eighty apples randomly chosen were confirmed ripe Therefore , probably all 100 apples are ripe Particular General
  • 63. © Art Traynor 2011 Logic Induction Arguments Hurley Section 1.4, ( Pg. 36 )Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak  Weak Inductive Argument The Conclusion does not follow with probability presuming True premises The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum Induction ( Weak ) conclusion True True premises True ?Not Probable This barrel contains 100 apples Example: Three apples randomly chosen were confirmed ripe Therefore , probably all 100 apples are ripe Particular General
  • 64. © Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak  Inductive Arguments can be strengthened or weakened by modifications to their Premises  Strength or Weakness are only indirectly related to Truth and Falsity Induction ( Less Weak ) statement Induction ( Weak ) True True True ?Not Probable Induction ( Less Strong ) statement Induction ( Strong ) True True True ?Probable Probable True ? True ? True True True True Not Probable
  • 65. © Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak Example: This barrel contains 100 apples Eighty apples randomly chosen were confirmed ripe Therefore , probably all 100 apples are ripe This barrel contains 100 apples Eighty apples randomly chosen were confirmed ripe Therefore , probably all 100 apples are ripe Induction ( Less Strong ) statement Induction ( Strong ) True True True ?Probable Probable True ? True True One unripe apple found earlier had been removed  Inductive Arguments can be strengthened or weakened by modifications to their Premises
  • 66. © Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak Example: Of the many U.S. Presidents Therefore , probably the next American President will be a Federalist Induction ( Strong ) False False False ?Probable  Strength or Weakness are only indirectly related to Truth and Falsity The vast majority have been Federalists conclusion premises A Strong IA ( SIA ) cannot render a Conclusion probably True If these False Premises however were considered True , a True Conclusion may probably follow
  • 67. © Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak Example: During the past fifty years Therefore , industrial productivity will probably increase in the years ahead  Strength or Weakness are only indirectly related to Truth and Falsity Inflation has consistently reduced the value of the American dollar conclusion premises Induction ( Weak ) True True True ?Not Probable The Truth of the Premises and probable Conclusion cannot render a weak IA ( WIA ) Strong
  • 68. © Art Traynor 2011 Logic Induction Arguments Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak Induction ( Weak ) True True False ?Not Probable Any SIA featuring True Premises but with a probably False Conclusion is Ipso Facto rendered into a WIA  Strong Inductive Argument ( SIA ) The Conclusion follows with probability presuming True premises  Weak Inductive Argument ( WIA ) The Conclusion does not follow with probability presuming True premises  Inductive Arguments can be strengthened or weakened by modifications to their Premises  Strength or Weakness are only indirectly related to Truth and Falsity
  • 69. © Art Traynor 2011 Logic Induction Arguments Hurley Section 1.4, ( Pg. 36 )Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak  The Conclusion follows with probability presuming True premises  The Conclusion does not follow with probability presuming True premises The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum  Inductive Arguments can be strengthened or weakened by modifications to their Premises  Strength or Weakness are only indirectly related to Truth and Falsity Strong Inductive Argument ( SIA ) Weak Inductive Argument ( WIA )
  • 70. © Art Traynor 2011 Logic Induction Arguments Hurley Section 1.4, ( Pg. 36 )Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum  Strength or Weakness are only indirectly related to Truth and Falsity Induction conclusion statement statement statement premises statement Particular General
  • 71. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 38 ) Premises Conclusion T Probably T T Validity F F ? Weak ? ? Probably F Probably T Probably F Induction An Argument is Valid if its Conclusion follows with Probability from its Premises Inductive Arguments are either Strong or Weak The relative Strength of an Inductive Argument is thus not a Binary State , but more akin to a continuum  Strength or Weakness are only indirectly related to Truth and Falsity
  • 72. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 38 ) Cogency A Inductive Argument that is Strong and exhibits True Premises is regarded as a Cogent Argument  In the absence of the satisfaction of these two conjunctive conditions an Inductive Argument is rendered Uncogent All the Premises must be True Satisfying ASC , a Cogent Argument will always feature a Conclusion that is True with Probablity  Inductive Argument Structure Criteria ( IASC ) 1. If the Premises are assumed True Then An Argument ( Satisfying ASC ) for which 2. The Conclusion Probably follows A Cogent IA ( CIA ) is a species of Argument featuring additional structure… i.e.: True Premises Cogent Argument = Strong Argument + True Premises Induction An Argument is Valid if its Conclusion follows with Probability from its Premises The Inductive analogue of a Sound Deductive Argument
  • 73. © Art Traynor 2011 Logic Classification Arguments Hurley Section 1.4, ( Pg. 38 ) Statements Deductive Arguments FalseTrue Truth Value InvalidValid Validity Unsound Sound Unsound Inductive Arguments WeakStrong Strength Uncogent Cogent Uncogent
  • 74. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The validity of an Argument arises from the Form or Structure of the Argument The Argument is valid because of the way in which the terms of the Premises are arranged We can substitute any terms we choose in the form of the argument and obtain a Valid Argumentnn Depending on what terms we substitute for “ A ” , “ B ” , and “ C ” the Conclusion will sometimes evaluate to True or False truth values nn Example: All Adlers are Bobkins Therefore , all Adlers are Crockers All Bobkins are Crockers conclusion premises All A are B All B are C All A are CParticular Essentially Transitive! General Valid Deduction
  • 75. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The validity of an Argument arises from the Form or Structure of the Argument The Argument is valid because of the way in which the terms of the Premises are arranged We can substitute any terms we choose in the form of the argument and obtain a Valid Argumentnn Nevertheless if we assume the Premises True we obtain a True Conclusionnn Example: All Adlers are Bobkins Therefore , all Adlers are Crockers All Bobkins are Crockers conclusion premises All A are B All B are C All A are CParticular Validity is not affected by Truth ! General Valid Deduction
  • 76. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The validity of an Argument arises from the Form or Structure of the Argument The Argument is valid because of the way in which the terms of the Premises are arranged We can substitute any terms we choose in the form of the argument and obtain a Valid Argumentnn A False Conclusion however can never arise from True Premises within an otherwise Valid form nn Example: All Adlers are Bobkins Therefore , all Adlers are Crockers All Bobkins are Crockers conclusion premises All A are B All B are C All A are CParticular Validity is not affected by Truth ! General Valid Deduction
  • 77. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Premises Conclusion T T T F Validity F T F F V I V Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Validitynn The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Deductive Argument Structure Criteria ( DASC ) 1. If the Premises are assumed True Then An Argument ( Satisfying ASC ) for which 2. The Conclusion follows Necessarily A Sound Argument is a species of Argument featuring additional structure… i.e.: True Premises V
  • 78. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The validity of an Argument arises from the Form or Structure of the Argument The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Example: All Adlers are Bobkins Therefore , all Adlers are Crockers All Crockers are Bobkins conclusion premises All A are B All C are B All A are CParticular The Form fails Transitivity… The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Validity is not Commutative… General Invalid Deduction
  • 79. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Example: conclusion premises All A are B All C are B All A are CParticular The Form fails Transitivity… The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Validity is not Commutative… This is problematic however because it is mathematically valid ! All Adlers are Bobkins Therefore , all Adlers are Crockers All Crockers are Bobkins General Invalid Deduction
  • 80. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Example: All Cats are Animals Therefore , all Cats are Dogs All Dogs are Animals conclusion premises All C are A All D are A All C are D General Particular The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o This substitution however yields a manifestly False Conclusion The Class Equivalency here is valid by Union but does not obtain as Intersection Invalid Deduction
  • 81. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Example: All Cats are Animals Therefore , all Cats are Earthlings All Animals are Earthlings conclusion premises All C are A All A are E All C are EParticular The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Class Equivalency is somehow implicated in the Argument Form General Valid Deduction
  • 82. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Premises Conclusion T T T F Validity F T F F I I I I Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o The possibility of Invalidity violates the DASC Deductive Argument Structure Criteria Validate or Vitiate Invalid Deduction All A are B All C are B All A are CParticular General
  • 83. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Premises Conclusion T T T F Validity F T F F Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o Deductive Argument Structure Criteria ( DASC ) 1. If the Premises are assumed True Then An Argument ( Satisfying ASC ) for which 2. The Conclusion follows Necessarily A Sound Argument is a species of Argument featuring additional structure… i.e.: True Premises I I I I
  • 84. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Premises Conclusion T T T F Validity F T F F Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn The mere arrangement of terms can render a Valid form Invalid …o If we assume the Premises True , it is not Necessarily the case that the Conclusion is True ! o I I I I An Argument is Invalid if and only if its form allows for a substitution instance having True Premises and a False Conclusion
  • 85. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.4, ( Pg. 36 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn If an Argument is Invalid, it will exhibit an Invalid Form ( e.g. Formally Invalid )o However if the Form is Invalid (e.g. Formally Invalid ) , there is at least one conjunctive Substitution instance featuring True Premises and False Conclusion o An Argument is Invalid if and only if its form allows for a substitution instance having True Premises and a False Conclusion Premises Conclusion T T T F Validity F T F F I I I I
  • 86. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn If an Argument is Invalid, it will exhibit an Invalid Form ( e.g. Formally Invalid )o However if the Form is Invalid (e.g. Formally Invalid ) , there is at least one conjunctive Substitution instance featuring True Premises and False Conclusion o An Argument is Invalid if and only if its form allows for a substitution instance having True Premises and a False Conclusion Otherwise every substitution featuring True Premises would exhibit a True Conclusion , as required by the Valid Deductive Argument ( VDA ) form o Conversely if a substitution were to yield an Argument featuring True Premises and exhibiting a False Conclusion , that substitution violates VDA Structure Criteria ( VDASC ) and is thus evaluates Invalid 
  • 87. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Otherwise every substitution featuring True Premises would exhibit a True Conclusion , as required by the Valid Deductive Argument ( VDA ) form o Conversely if a substitution were to yield an Argument featuring True Premises and exhibiting a False Conclusion , that substitution violates VDA Structure Criteria ( VDASC ) and is thus evaluates Invalid  This constructive Invalidation of an otherwise Valid Deductive form admits an alternate, direct definition of Invalidity ( i.e. a means to Prove Invalidity )  Counter Example Method ( CEM )o The Argument is thus Proven Invalid ! A Deductive form of Argumentation is Isolated – Evaluated by Inspection A Substitution is constructed featuring True Premises and a False Conclusion
  • 88. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 45 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o The Argument is thus Proven Invalid ! A Deductive form of Argumentation is Isolated – Evaluated by Inspection A Substitution is constructed featuring True Premises and a False Conclusion Example: Since some employees are not social climbers We may conclude that some VP’s are not Social Climbers And all Vice Presidents are Employees conclusion premises → Some E are not S Therefore Some V are not S All V are E
  • 89. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o The Argument is thus Proven Invalid ! A Deductive form of Argumentation is Isolated – Evaluated by Inspection A Substitution is constructed featuring True Premises and a False Conclusion Example: conclusion premises → Some E are not S Therefore Some V are not S All V are E All V are E Therefore Some V are not S Some E are not S BUT…Some E not S are Not V i.e. We lose some “ E ” along the way here… i.e. There are some “ E ” that are “ Not S ” and “ Not V ”
  • 90. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 45 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises → Some Animals are not Mammals Therefore Some Dogs are not Mammals All Dogs are Animals E = Animals S = Mammals V = Dogs Particular General Invalid Categorical Syllogism
  • 91. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 45 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises Some Animals are not Mammals Therefore Some Dogs are not Mammals All Dogs are Animals Particular General Invalid Deductive Argument 1. True Premises 2. False Conclusion Invalid Categorical Syllogism
  • 92. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises → If the Government imposes import restrictions Therefore since the Government will not impose import Restrictions ( then ) the price of Automobiles will rise If G then P Not G Therefore, not P Particular General Invalid Hypothetical Syllogism It follows that the price of Automobiles will not rise This is an Invalid Deductive Argument ( IDA ) because the price of Automobiles may rise notwithstanding Government inaction
  • 93. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises → If G then P Therefore , not P Not G If G then P Not G Therefore, not P Particular General Invalid Hypothetical Syllogism This is an Invalid Deductive Argument ( IDA ) because the price of Automobiles may rise notwithstanding Government inaction
  • 94. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Example: conclusion premises→ If Abraham committed suicide, Therefore , Abraham Lincoln is not Dead Then Abraham Lincoln is dead G = Abraham Lincoln committed suicide P = Abraham Lincoln is dead Particular General Invalid Hypothetical Syllogism This is an Invalid Deductive Argument ( IDA ) because the price of Automobiles may rise notwithstanding Government inaction Abraham Lincoln did not commit suicide
  • 95. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o conclusion premises If the Government imposes import restrictions Therefore since the Government will not impose import Restrictions ( then ) the price of Automobiles will rise Particular General Invalid Hypothetical Syllogism It follows that the price of Automobiles will not rise The Form of the Argument is revealed by substitution ( i.e. Counter Example ) to be invalid – one exhibiting True Premises and a False Conclusion If Abraham committed suicide, Therefore , Abraham Lincoln is not dead Then Abraham Lincoln is dead Abraham Lincoln did not commit suicide ↔
  • 96. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 46 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o conclusion premises If the Government imposes import restrictions Therefore since the Government will not impose import Restrictions ( then ) the price of Automobiles will rise Particular General Invalid Hypothetical Syllogism It follows that the price of Automobiles will not rise Applying CEM to a Conditional, it is best to construct a Substitution expressing a Necessary hypothetical nexus: e.g. as between suicide and death If Abraham committed suicide , Therefore , Abraham Lincoln is not dead Then Abraham Lincoln is dead Abraham Lincoln did not commit suicide ↔
  • 97. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Only useful for Proving Arguments Invalid The only arrangement of Truth & Falsity that Proves anything in Deductive Argumentation is True Premises with a False Conclusion  Premises Conclusion T T T F Validity F T F F V I V V
  • 98. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Invaliditynn Counter Example Method ( CEM )o Only applicable to Deductive Argumentation Only useful for Proving Arguments Invalid A VDA featuring True Premises and a True Conclusion Proves nothing The only arrangement of Truth & Falsity that Proves anything in Deductive Argumentation is True Premises with a False Conclusion  The relative Strength or Weakness of Inductive Argumentation is only tenuously dependent on the Form of the Argument  No method that relates exclusively to the form of an Inductive Argument can be used to prove the Argument Weak 
  • 99. © Art Traynor 2011 Logic Deduction Arguments Hurley Section 1.5, ( Pg. 42 )Validity Every Deductive Argument is either Valid or Invalid  The validity of an argument has nothing to do with its specific subject matter The Argument is valid because of the way in which the terms of the Premises are arranged Formal Validitynn For a Deductive Argument, Subject Matter is irrelevant to a determination of Validity Deductive Argument Form InvalidValid Subject MatterIrrelevant Germaine
  • 100. © Art Traynor 2011 Logic Extended Arguments Arguments Hurley Section 1.6, ( Pg. 49 )Extended Arguments ( ExArg ) A composite Argument, satisfying ASC comprised of : one or more Standards of Evaluation , ( i.e. Deduction and/or Induction ) and possibly featuring numerous Non-Arguments ( NonArg ) The process by which extraneous subject matter is eliminated from the ExArg , and Premises and a Conclusion are isolated / identified Warning Advice Non-Arguments Belief / Opinion Description Report Exposition Illustration Conditional Explanation Hurley Section 1.2, ( Pg. 25 ) Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises Hurley Section 1.2, ( Pgs. 11 – 18 )  Logical Analysis A Conclusory Statement must be associated in a well-ordered fashion to its Premises by positing its Inferential Nexus ( InfNex ) 
  • 101. © Art Traynor 2011 Logic Extended Arguments Arguments Hurley Section 1.6, ( Pg. 49 )Extended Arguments ( ExArg ) A composite Argument, satisfying ASC comprised of : one or more Standards of Evaluation , ( i.e. Deduction and/or Induction ) and possibly featuring numerous Non-Arguments ( NonArg ) Warning Advice Non-Arguments Belief / Opinion Description Report Exposition Illustration Conditional Explanation Hurley Section 1.2, ( Pg. 25 ) Argument Structure Criteria ( ASC ) 1. Evidentiary Premises And A Set of Statements composed of 2. A single Conclusion Arising Logically from the Premises Hurley Section 1.2, ( Pgs. 11 – 18 )  Logical Analysis Inferential Nexus ( InfNex ) Vertical Inferential Nexus ( VIN )nn Horizontal Inferential Nexus ( HIN )nn The process by which extraneous subject matter is eliminated from the ExArg , and Premises and a Conclusion are isolated / identified
  • 102. © Art Traynor 2011 Logic Extended Arguments Arguments Hurley Section 1.6, ( Pg. 49 )Extended Arguments ( ExArg ) A composite Argument, satisfying ASC comprised of : one or more Standards of Evaluation , ( i.e. Deduction and/or Induction ) and possibly featuring numerous Non-Arguments ( NonArg ) Warning Advice Non-Arguments Belief / Opinion Description Report Exposition Illustration Conditional Explanation Hurley Section 1.2, ( Pgs. 11 – 18 )  Logical Analysis Inferential Nexus ( InfNex ) Vertical Inferential Nexus ( VIN )nn The process by which extraneous subject matter is eliminated from the ExArg , and Premises and a Conclusion are isolated / identified conclusion premises The contamination of underground aquifers represents a pollution problem of catastrophic proportions Half the nation’s drinking water , which comes from these aquifers , is being poisoned by chemical wastes dumped into the soil for generations ↔ 1 2 [ if ] Half the nation’s drinking water , comes from aquifers , being poisoned by chemical wastes dumped into the soil for generations 2 [ Then ] contamination of underground aquifers represents a pollution problem of catastrophic proportions 1
  • 103. © Art Traynor 2011 Logic Extended Arguments Arguments Hurley Section 1.6, ( Pg. 49 )Extended Arguments ( ExArg ) A composite Argument, satisfying ASC comprised of : one or more Standards of Evaluation , ( i.e. Deduction and/or Induction ) and possibly featuring numerous Non-Arguments ( NonArg ) Warning Advice Non-Arguments Belief / Opinion Description Report Exposition Illustration Conditional Explanation Hurley Section 1.2, ( Pgs. 11 – 18 )  Logical Analysis Inferential Nexus ( InfNex ) Vertical Inferential Nexus ( VIN )nn The process by which extraneous subject matter is eliminated from the ExArg , and Premises and a Conclusion are isolated / identified conclusion premises ↔ [ if ] Half the nation’s drinking water , comes from aquifers , being poisoned by chemical wastes dumped into the soil for generations 2 [ Then ] contamination of underground aquifers represents a pollution problem of catastrophic proportions 1 2 1 Inferential Nexus ( InfNex )