Net set logical reasoning - Critical Thinking

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Logical Reasoning
Goalfinder Classes: CBSE NET 2016 - Paper 1
Total number of Pages: 86
Portion covered till CBSE NET December 2015
Classes
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Logical Reasoning
Goalfinder Classes: CBSE NET 2016 - Paper 1
Table of Content
Reasoning 1
Premise 2
Conclusion 2
Exercise on Premise and Conclusion 4
Argument 5
Valid argument 6
Invalid argument 6
Sound / Unsound argument 6
Exercise Arguments 9
Reasoning 10
Inductive / Deductive Reasoning and Arguments 11
Cogent and Uncogent Arguments 12
Exercise on Inductive and Deductive Arguments 13
Exercise: Sound /Unsound, Valid/ Invalid Arguments 15
Definitions 16
AEIO Forms 20
Exercise - Proposition Translation 24
What are the Main Types of Reasoning? 28
Deductive reasoning 28
Reductive Reasoning 29
Abductive reasoning 29
Inductive reasoning 31
Exercise1: Inductive or Deductive 35
Exercise 2 (Inductive and Deductive) 38
Exercise 2 (Inductive and Deductive) 39
Syllogism (Deductive reasoning) 41
1. Categorical Reasoning 42
2. Hypothetical Reasoning (if- then) 42
3. Disjunctive Reasoning (Either P or Q) 43
Circular reasoning 44
Some types of inductive reasoning: 45
Generalization reasoning 45
Causal reasoning 45
Analogical reasoning 45

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Predictive Conjecture or Analogical Predicting 46
Exercise 1 on Reasoning 47
Exercise 2 on Reasoning 49
Proposition 51
Square of Opposition 51
Exercise: Contradictory, contrary, subcontrary, subaltern 55
Applied Definitions 56
Lexical Definition 56
Stipulative Definition 57
Precising Definition 57
Theoretical Definitions 58
Persuasive Definitions 59
Exercise 1 Definitions 61
Exercise 2 Definitions 62
Assertion and Argument 63
Transitivity, Symmetricity, Reflexivity and Equivalence 64
Transitivity 64
Symmetricity 64
Reflexivity 64
Equivalence 65
Venn Diagram 66
A. EAE-1 68
B. AAA-1 69
C. AII-3 70
D. AII-2 71
E. EAO-4 71
Fallacies in Arguments 73
Answers to exercises 75

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Sample - Logical Reasoning
It is an important section in the CBSE Net exam as almost 3-4 questions come in this section, mastering
this section enables one to ensure 6-8 marks in Paper 1.
Premise
Premise: Some common premise-flags are the words because, unless, since, given that, and for. These
words usually come right before a premise. Here are some examples:
Premise Indicators
since as shown by may be inferred from
because In as much as may be deduced from
for as indicated by in view of the fact that
as the reason is that given that
follow from
Unless
for the reason that
unless
granted that
For
Example Premise:
(Therefore) Your car needs a major overhaul, for the carburetor is shot. (for the … is premise)
Given that euthanasia is a common medical practice, (hence) the state legislatures ought to legalize it
and set up some kind of regulations to prevent abuse.
Because euthanasia is murder, (so) it is always morally wrong.
(So) We must engage in constructive action, because India needs rebuilding.
Since politics is a hotly contested issue in this country, (thus) nobody should force his opinion about it
on anyone else.
Conclusion
Some common conclusion-flags are the words thus, therefore, hence, it follows that, so, and
consequently.
Conclusion Indicators
therefore which shows that accordingly
hence which means that then
thus which entails that consequently
so which implies that we may infer
that
ergo which allows us to infer
Subsequently
I conclude that
Consequently

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Reasoning
Inductive reasoning makes broad generalizations from specific cases or observations.
 In this process of reasoning, general assertions are made based on past specific pieces of
evidence.
 This kind of reasoning allows the conclusion to be false even if the original statement is true.
 For example, if one observes a college athlete, one makes predictions and assumptions about
other college athletes based on that one observation.
 Scientists use inductive reasoning to create theories and hypotheses.
In opposition, deductive reasoning is a basic form of valid reasoning.
 In this reasoning process a person starts with a known claim or a general belief and from there
asks what follows from these foundations or how will these premises influence other beliefs.
 In other words, deduction starts with a hypothesis and examines the possibilities to reach a
conclusion.
 Deduction helps people understand why their predictions are wrong and indicates that their
prior knowledge or beliefs are off track.
 An example of deduction can be seen in the scientific method when testing hypotheses and
theories.
 Although the conclusion usually corresponds and therefore proves the hypothesis, there are
some cases where the conclusion is logical, but the generalization is not.
 For example, the statement, “All young girls wear skirts. Julie is a young girl. Therefore, Julie
wears skirts,” is valid logically but does not make sense because the generalization from the
original statement is not true.
The syllogism is a form of deductive reasoning in which two statements reach a logical conclusion.
 With this reasoning, one statement could be “Every A is B” and another could be “This C is A”.
 Those two statements could then lead to the conclusion that “This C is B”.
 These types of syllogisms are used to test deductive reasoning to ensure there is a valid
hypothesis.
Another form of reasoning is called abductive reasoning.
 This type is based on creating and testing hypotheses using the best information available.
 Abductive reasoning produces the kind of daily decision-making that works best with the
information present, which often is incomplete.
 This could involve making educated guesses from observed unexplainable phenomena.
 This type of reasoning can be seen in the world when doctors make decisions about diagnoses
from a set of results or when jurors use the relevant evidence to make decisions about a case.

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Inductive / Deductive Reasoning and Arguments
Two types of arguments:
1. deductive–are intended to be valid, there is no gray area between validity and invalidity
2. inductive–are not intended to be valid, there is a gray area between strong and weak
Deductive Inductive
Introduction
(from Wikipedia)
Deductive reasoning, also called deductive
logic, is the process of reasoning from one
or more general statements regarding what
is known to reach a logically certain
conclusion.
Inductive reasoning, also called induction
or bottom-up logic, constructs or evaluates
general propositions that are derived from
specific examples.
Arguments
Arguments in deductive logic are either
valid or invalid. Invalid arguments are
always unsound. Valid arguments are sound
only if the premises they are based upon
are true.
Arguments in inductive reasoning are
either strong or weak. Weak arguments
are always uncogent. Strong arguments are
cogent only if the premises they are based
upon are true.
Validity of
conclusions
Conclusions can be proven to be valid if the
premises are known to be true
(Correctness).
Conclusions may be incorrect even if the
argument is strong and the premises are
true.
Cogent : (of an argument or case) clear, logical, and convincing

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AEIO Forms
Aristotelian Four-fold Classification of Categorical Propositions
Aristotle classified categorical proposition in four, based on Quality and Quantity distribution:
The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and
gave them standard forms (now often called A, E, I, and O) This is based on the Latin affirmo (I affirm),
referring to the affirmative propositions A and I, and nego (I deny), referring to the negative
propositions E and O.
Subject category is named S and the predicate category is named P (A predicate is the completer of a
sentence. A simple predicate consists of only a verb, verb string, or compound verb. The predicate of
"The boys went to the zoo" is "went to the zoo." In case of proposition predicate means something that
is affirmed or denied of the subject in a proposition in logic)
A propositions, or universal affirmatives take the form: All S are P.
E propositions, or universal negations take the form: No S are P.
I propositions, or particular affirmatives take the form: Some S are P.
O propositions, or particular negations take the form: Some S are not P.
Distributivity of proposition
The two terms (subject and predicate) in a categorical proposition may each be classified as distributed
or undistributed. If ‘all’ members of the term's class are affected by the proposition, that class is
distributed; otherwise it is undistributed.
Every proposition therefore has one of four possible distribution of terms.
A form
An A-proposition distributes the subject to the predicate, but not the reverse. Consider the following
categorical proposition: "All dolphins are mammals". All dolphins are indeed mammals but it would be
false to say all mammals are dolphins. Since all dolphins are included in the class of mammals, " dolphins
" is said to be distributed to "mammals". Since all mammals are not necessarily dolphins, "mammals" is
undistributed to " dolphins ".
example of an E-proposition:
S
P
S P
S P
S
P

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What are the Main Types of Reasoning?
Deductive reasoning
Deductive = Premise  Conclusion,
from a general rule to a specific conclusion): (Mathematicians
adopt this approach)
This theory of deductive reasoning – also known as term logic
– was developed by Aristotle, but was superseded by
propositional (sentential) logic and predicate logic.
– Allows us to draw conclusions that must hold given a set of
facts (premises)
There were 20 persons originally (premise)
There are 19 persons currently (premise)
Therefore, someone is missing (conclusion)
Here if the premises were true, then the conclusion would
certainly also be true.
• You have tickets to a game
• You agree to meet Bill and Mary at the corner of road or at the seats.
– If you see Mary on the corner of road, you expect to see Bill as well.
– If you do not see either of them at the corner, you expect to see them at the seats when you
get to the stadium.
Deductive reasoning is a basic form of valid reasoning. Deductive reasoning, or deduction, starts out
with a general statement, or hypothesis, and examines the possibilities to reach a specific, logical
conclusion. The scientific method uses deduction to test hypotheses and theories.
In deductive reasoning, if something is true of a class of things in general, it is also true for all
members of that class. For example, "All men are mortal. Manoj is a man. Therefore, Manoj is mortal."
For deductive reasoning to be sound, the hypothesis must be correct. It is assumed that the premises,
"All men are mortal" and "Manoj is a man" are true. Therefore, the conclusion is logical and true.
An example of deductive reasoning can be seen in this set of statements: Every day, I leave for work in
my car at eight o’clock. Every day, the drive to work takes 45 minutes I arrive to work on time. Therefore,
if I leave for work at eight o’clock today, I will be on time.
The deductive statement above is a perfect logical statement, but it does rely on the initial premise
being correct. Perhaps today there is construction on the way to work and you will end up being late.
Reasoning
Deductive Reductive
AbductiveInductive
Syllogism
Categorical
Hypothetical
Disjunctive
Causal
Analogical

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This is why any hypothesis can never be completely proved, because there is always the possibility for
the initial premise to be wrong.
Syllogism (Deductive reasoning)
a form of reasoning in which a conclusion is drawn from two given or assumed propositions (premises);
In its earliest form, defined by Aristotle, from the combination of a general statement (the major
premise) and a specific statement (the minor premise), a conclusion is deduced.
Major premise: All mortals die.
Minor premise: All men are mortals.
Conclusion: All men die.
Here, the major term is die, the minor term is men, and the middle term is mortals. Again, both
premises are universal, hence so is the conclusion.
In syllogism, the two statements — a major premise and a minor premise — reach a logical conclusion.
For example, the premise "Every A is B" could be followed by another premise, " C is A." Those
statements would lead to the conclusion "Thus C is B." Syllogisms are considered a good way to test
deductive reasoning to make sure the argument is valid.
The law of syllogism takes two conditional statements and forms a conclusion by combining the
hypothesis of one statement with the conclusion of another.
Here is the general form:
1. P → Q
2. Q → R
3. Therefore, P → R.
The following is an example:
1. If Larry is sick, then he will be absent.
2. If Larry is absent, then he will miss his classwork.
3. If Larry is sick, then he will miss his classwork.
There are three regular patterns of syllogisms:
 Categorical
 Hypothetical
 Disjunctive
A C
B

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Precising Definition
Exact meaning of a word. This definition gives a quantitative measure of an existing word. A precision
definition takes a word that is normally vague (e.g., lite, low-income, or middle-aged) and gives it a clear,
precisely defined meaning. Most terms used in legal, scientific, or medical settings require precise
meanings. For example,
Example : Light (or lite) foods, according to USDA standards, are foods that contain at least one-third
fewer calories than comparable products
“One horsepower” is now defined precisely as “the power needed to raise a weight of 550 pounds by
one foot in one second”—calculated to be equal to 745.7 watts.
Meter: A meter is the internationally accepted unit of measure for distance. Originally it was defined, by
stipulation, as one ten-millionth of the distance from one of the earth’s poles to the equator, and this
was represented by a pair of carefully inscribed scratches on a metal bar made of platinum-iridium, kept
in a vault near Paris, France. However, scientific research required more precision. A “meter” is now
defined, precisely, as “the distance light travels in one 299,792,458th of a second.” Building on this, a
“liter” is defined precisely as the volume of a cube having edges of 0.1 meter.
A precise definition is required in case of For example, is a sport utility vehicle (SUV) a car or a light
truck? The fuel economy standards and pollution controls applied to “light trucks” are more lenient
than those applied to “cars,”
Dead: A precise definition of who should be considered “dead” is required. (Death was once defined as
the cessation of heartbeat (cardiac arrest) and of breathing, but the development of CPR and prompt
defibrillation have rendered that definition inadequate because breathing and heartbeat can sometimes
be restarted. Events which were causally linked to death in the past no longer kill in all circumstances;
without a functioning heart or lungs, life can sometimes be sustained with a combination of life support
devices, organ transplants and artificial pacemakers.)
Planet: How, for example, should we define the word “planet”? For many years it was believed with
little controversy, and all children were taught, that planets are simply bodies in orbit around the sun
and that there are nine planets in the solar system—of which the smallest is Pluto, made of unusual
stuff, with an unusual orbit, and most distant from the Sun. But other bodies, larger than Pluto and
oddly shaped, have been recently discovered orbiting the sun. Are they also planets? Why not? Older
definitions had become conceptually inadequate. An intense controversy within the International
Astronomical Union (IAU), still not fully resolved, has recently resulted in a new definition of “planet,”
according to which there are only eight planets in our solar system. And now a new category, “dwarf
planet” (for bodies such as Pluto, Ceres, and Eris) has been defined.
Needed were definitions that would accommodate new discoveries as well as old, while maintaining a
consistent and fully intelligible account of the entire system. Such definitions (not as simple as we might
like) were adopted by the IAU in 2006. A planet is “a celestial body that, within the Solar System, (1) is in
orbit around the Sun; and (2) has sufficient mass for its self-gravity to overcome rigid body forces so that

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Equivalence
A relation R is an equivalence iff R is transitive, symmetric and reflexive. For example, identical is an
equivalence relation: if x is identical to y, and y is identical to z, then x is identical to z; if x is identical to y
then y is identical to x; and x is identical to x.
"Is congruent to" is an equivalence relation because it is reflexive (all things are congruent to
themselves), symmetric (if some x is congruent to some y, then that y is congruent to that x), and
transitive, (if x is congruent to y and y is congruent to z then x is congruent to Z.
Example:
(i) “is a child of” is irreflexive, asymmetric, intransitive.
No person is a child of him/herself
 For all people, if x is a child of y, then y is not a child of x.
 if x is a child of y a
 and y a child of z, then for no person is x a child of z.
(ii) “is a brother of” is irreflexive, non-symmetric, nontransitive.
No person is a brother of him/herself
 If x is a brother of y, then y may be x’s brother.
 However, if x is a brother of y and y is a female, y is not x’s brother.
 if John is a brother of Joe and Joe of Bill, then John is a brother of Bill.
 However, if John is a brother of Joe and Joe of John, John is not a brother of himself.
(iii) “is a descendent of” is irreflexive, asymmetric, transitive.
No person is a descendent of themselves.
 For no pair of people x,y is x a descendent of y and y of x.
 If x is a descendent of y and y of z, then x is a descendent of z.
(iv) “is an uncle of” is nonreflexive, nonsymmetric, nontransitive.
John can be an uncle an uncle of himself if he marries his aunt.
 If John is Joe’s uncle, Joe is not John’s uncle.
 But if John is John’s uncle, then John is John’s uncle, therefore not asymmetric.
 Ditto for nontransitive

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