LOGIC
For, the study of philosophy is not undertaken to know what
men feel, but to know what is the truth of things" -Aquinas

Why Study Logic and learn Critical
Thinking?
In republican nation, whose citizens are to be led by reason and persuasion and
not by force, the art of reasoning becomes of the first importance – Thomas
Jefferson
• Logic allows us to improve the quality of the argument we use
• It improves the our ability to evaluate the arguments of others
• It improves the our ability to communicate more clearly and efficiently
• It teaches us how reasoning can go wrong and what is needed for reasoning
to be structured
• It teaches us to develop emotional and intellectual distance between yourself
and idea – whether your own or others – in order to better evaluate their
truth, validity, and reasonableness

What is Logic?
• Logic may be defined as the organized body of knowledge, or science, that
evaluates arguments
• It is the study of methods and principles used to distinguish good reasoning
from bad reasoning
• We all encounter arguments in life: reading books & newspapers, hear them
on television, and when communicating with friends, science & religion,
ethics & law,
• Hence the aim of logic is to develop a system of methods and principles that
we may use as criteria for evaluating the arguments of others and as guides
in constructing arguments of our own

• Whatever the subject or content of an argument, the logician is interested in
its form and quality
• i.e. does the argument do what it claims to do? Does its conclusion logically
infer from its premises?
• Let us now look at What is an argument and its components:

Arguments, Propositions, Premises, & Conclusions
• An argument does not mean a verbal fight with someone but rather:
• An argument is a group of statements (or propositions), one or more of
which (the premises) are claimed to provide support for, or reasons to
believe, one of the others (the conclusion).
• E.g.
All film stars are celebrities. (Premise)
Halle Berry is a film star. (Premise)
Therefore, Halle Berry is a celebrity. (Conclusion)
• All arguments fall into two groups:
• Good Arguments: Those in which the premises really do support the conclusion
• Bad Arguments: Those in which the premises do not support the conclusion though they
claim to e.g.
Some film stars are men. (Premise)
Cameron Diaz is a film star. (Premise)
Therefore, Cameron Diaz is a man (Conclusion)

• The purpose of logic, as the science that evaluates arguments, is thus to
develop methods and techniques that allow us to distinguish good arguments
from bad.
• To understand arguments fully we must first understand its basic components
that are propositions (statements), inference, premises, and conclusions:

Basic concepts to note about Arguments
• Arguments are made up of statements (propositions) which are the building
the building blocks of arguments
• A Statement (proposition) is declarative sentence that is either true or false
• E.g. Chocolate truffles are loaded with calories (TRUE)
• Political candidates always tell the complete truth (FALSE)
• Tiger Woods plays golf and Maria Sharapova plays tennis (Since both parts are true
hence the whole statement is TRUE)
• Therefore a statement asserts that something is (or is not) the case
• Any proposition can be affirmed or denied

• Truth Value: The Truth or Falsity of a statement is known as the Truth
Value of that proposition
• Thus if a proposition is true its Truth Value is ‘True’ and if a proposition is false its truth
Value is ‘False’
• The Truth or Falsity of a proposition may not be known but it must either
be true or false
• E.g. the Truth Value of the statement that ‘There is life on some other planet in our galaxy’ is
not known but it must either be true or false
• Unlike statements many sentence cannot be said to be either true or false
• Where is Kalam? (question)
• Let’s go to a movie tonight. (proposal)
• I suggest you get contact lenses (suggestion)
• Turn off the TV right now (command)
• Fantastic! (exclamation)

• Sentences are part of some language but propositions are not part of any
language:
• Today is Monday (English)
• Aaj Peer hai (Urdu)
• Nan Peer dai (Pushto)
All the three sentences above mean the same thing (proposition)
• The same sentence may make different statements if the context changes
• E.g. The statement ‘The largest country in the world is the third most populous country in
the world’ was true for USSR but is now not true for Russia

• Simple Proposition: that makes only one assertion
• E.g. It is raining outside
• Compound Proposition: A proposition containing two or more simple
propositions
• E.g.
• Conjunctive Proposition … if true all of its components must be true
• Pakistan is located in South-East Asia and it is an Islamic State
• Disjunctive (or Alternate) Proposition … if true at least one of its components must be true
• Either Imran Khan will win the national elections or he will not win
Part 1
Part 2
Part 1
Part 2

• Hypothetical or Conditional (If Then) Proposition … False only when the antecedent
is true consequent is false
• It takes the form of:
If ________________ Then _________________
• E.g. If God did not exist, it would be necessary to invent him (Voltaire)
• More on Hypothetical conditions and their role in arguments latter on …
Antecedent
Consequent
Antecedent
Consequent

• Inference: In arguments when we link propositions by affirming (or
reaching) one proposition (conclusion) on the basis of one or more other
propositions (premises) we say an inference has been drawn
• The statements that make up an argument are divided into one or more
premises and one and only one conclusion.
• Example of an argument consisting of one premise:
No one was present when life first appeared on earth (P). Therefore any statement
about origins of life on earth should be considered a theory, not a fact. (Conclusion)
This is the simplest form of an argument containing one premise and one conclusion

Premises
Conclusion
Claimed
Evidence
What is claimed to follow
from the evidence
Inference drawn
Argument

• Sometimes both the premise and the conclusion may be stated in the same
sentence:
• Since it turns out that all humans are descendants from a small number of African ancestors
in our recent evolutionary past, believing in profound differences between humans is as
ridiculous as believing in a flat earth
• Sometimes the conclusion of an argument may precede its premise:
• Smoking should be banned in public places. After all passive smoking can cause cancer.
• You can’t separate peace from freedom because no one can be at peace if they don’t have
freedom
• These are all simple propositions. Most arguments are complicated as they
contain compound propositions with several intricately related components.
But whether simple or complex, arguments contain a group of propositions
with one conclusion and the other are premises to support the conclusion.

• Consider the hypothetical proposition:
• If life evolved on Mars in the early period of its history when it had an atmosphere and
climate similar to earth, then it is likely that life evolved on countless other planets scientists
believe to exist in our galaxy
There is no argument here. No inference is drawn in this passage. The proposition asserts only
that if the former is true then the latter must be true. But both could very well be false.
• On the other hand the following passage containing IF-Then statement is an argument
If country A is developing nuclear weapons, then country A is a threat to world peace.
Country A is developing nuclear weapons.
Therefore, country A is a threat to world peace.
Antecedent
Consequent

If our borders are porous, then terrorists can enter the country at will.
If terrorists can enter the country at will, then all of us are less secure.
Therefore, if our borders are porous, then all of us are less secure.
• The relation between conditional statements and arguments may now be
summarized as follows:
1. A single conditional statement is not an argument.
2. A conditional statement may serve as either the premise or the conclusion (or both)
of an argument.
• Conditional statements are especially important in logic because they express the relationship
between necessary and sufficient conditions.
• A is said to be a sufficient condition for B whenever the occurrence of A is all that is needed for
the occurrence of B. For example, being a dog is a sufficient condition for being an animal.
• On the other hand, B is said to be a necessary condition for A whenever A cannot occur without
the occurrence of B. Thus, being an animal is a necessary condition for being a dog.

If X isa dog,then X isan animal. (beinga dogis sufficient for beingananimal)
If X isan animal,then X may or may not be a dog (not sufficient further evidence isneededto determine)
If X isnot an animal,then X isnot a dog. (beingan animalis necessaryfor beinga dog)
• Dog in the box example:
Dog
Animal
Animal
Dog?
Cat?
Racoon?

Recognizing Arguments
• Every argument is a structured cluster of propositions but, not every structured cluster
of propositions is an argument
• E.g.
• Leonardo da Vinci understood described and illustrated the principles of hydrodynamics, gross
anatomy, physics and astronomy, He invented the helicopter, the armoured tank, and the submarine.
He painted like an angel and despite being phobic about deadline wrote often and well. In addition,
according to Vasari, he was drop dead gorgeous. And he generated all the near magical
accomplishments from a behind a curtain of personal discretion so dense and insulating that no
historian or psychologist has been able to pull it aside to reveal the person behind the personage.
• There is NO Argument here!
• Therefore for a structured cluster of propositions to be an argument two conditions
must be fulfilled:
1. At least one of the statements must claim to present evidence or reasons … the premise
2. There must be a claim that the alleged evidence supports or implies something—that is, a
claim that something follows from the alleged evidence or reasons …. the conclusion

• The first condition expresses a factual claim, and deciding whether it is
fulfilled often falls outside the domain of logic.
• This second condition expresses what is called an inferential claim which
says that something supports or implies something, or that something follows
from something.
• E.g.
Since Imran Khan is the president of Pakistan
Therefore he can bring the desired change in the country
Factual claim (we are not concerned whether it is fulfilled or not)
Inferential claim
(we are concerned with this)

Explicit & Implicit Inferential Claims
• Inferential claim can be Explicit or Implicit
• Explicit Inferential claims are usually asserted by premise indicator and
conclusion indicators:
• Premise Indicators:
• Conclusion Indicators:
Seeing that tortured prisoners will say anything just to relieve the pain. Consequently, torture
is not a reliable method of interrogation.
since in that seeing that as indicated by may be inferred from for the reason that
because as in as much as for given that owing to
therefore accordingly entails that wherefore we may conclude
hence thus it must be that it follows that consequently
for this reason implies that we may infer so as a result

• Implicit Inferential Claims:
Some arguments contain no indicators. With these, the reader/listener must ask such
questions as: What single statement is claimed (implicitly) to follow from the others?
What is the arguer trying to prove? What is the main point in the passage? The answers
to these questions should point to the conclusion.
• The space program deserves increased expenditures in the years ahead (c). Not only does the
national defence depend on it (p1), but the program will more than pay for itself in terms of
technological spinoff s (p2). Furthermore, at current funding levels the program cannot fulfil its
anticipated potential (p3)
No premise or conclusion indicators are used in the above passage but it is a valid
argument in which the inferential claims are made implicitly

Not every structured proposition is an argument!
• Simple Non-inferential passages: Those that lack a claim that anything is
being proved e.g.:
• A Warning
• A piece of advice
• A statement of belief or opinion
• A report: to provide information on something
• Illustrations: to show what something means or how something is done
• Conditional Statements: as discussed before
• An Explanation: that aims to shed light on some event or phenomenon

Explanations and Arguments
Premises
Conclusion
Claimed
Evidence
What is claimed to
follow from the
evidence
Inference
drawn
Explanations
Explanandum
Claimed to
shed light on
Accepted Fact
To distinguish explanations from arguments, identify the statement that is either the explanandum
or the conclusion (usually this is the statement that precedes the word “because”). If this statement
describes an accepted matter of fact, and if the remaining statements purport to shed light on this
statement, then the passage is an explanation.
E.g.
The sky appears blue from the earth’s surface because light rays from the sun are
scattered by particles in the atmosphere.

Exercise
I. Each of the following passages contains a single argument. Using the letters “P” and “C,” identify the
premises and conclusion of each argument, writing premises first and conclusion last:
1. Titanium combines readily with oxygen, nitrogen, and hydrogen, all of which have an adverse effect on
its mechanical properties. As a result, titanium must be processed in their absence.
4. When individuals voluntarily abandon property, they forfeit any expectation of privacy in it that they
might have had. Therefore, a warrantless search or seizure of abandoned property is not unreasonable
under the Fourth Amendment.
7. It really does matter if you get enough sleep. We need sleep to think clearly, react quickly, and create
memories. Studies show that people who are taught mentally challenging tasks do better after a good
night’s sleep. Other research suggests that sleep is needed for creative problem solving.
10. Punishment, when speedy and specific, may suppress undesirable behaviour, but it cannot teach or
encourage desirable alternatives. Therefore, it is crucial to use positive techniques to model and reinforce
appropriate behaviour that the person can use in place of the unacceptable response that has to be
suppressed.
16. Radioactive fallout isn’t the only concern in the aftermath of nuclear explosions. The nations of planet
Earth have acquired nuclear weapons with an explosive power equal to more than a million Hiroshima
bombs. Studies suggest that explosion of only half these weapons would produce enough soot, smoke,
and dust to blanket the Earth, block out the sun, and bring on a nuclear winter that would threaten the
survival of the human race.

1. P: Titanium combines readily with oxygen, nitrogen, and hydrogen, all of which have an adverse effect on its
mechanical properties.
C: Titanium must be processed in their absence.
4. P: When individuals voluntarily abandon property, they forfeit any expectation of privacy in it that they might
have had.
C: A warrantless search and seizure of abandoned property is not unreasonable under the Fourth
Amendment.
7. P1: We need sleep to think clearly, react quickly, and create memories.
P2: Studies show that people who are taught mentally challenging tasks do better after a good night’s sleep.
P3: Other research suggests that sleep is needed for creative problem solving.
C: It really does matter if you get enough sleep.
10. P1: Punishment, when speedy and specific, may suppress undesirable behaviour.
P2: Punishment cannot teach or encourage desirable alternatives.
C: It is crucial to use positive techniques to model and reinforce appropriate behaviour that the person can
use in place of the unacceptable response that has to be suppressed.
16. P1: The nations of planet earth have acquired nuclear weapons with an explosive power equal to more than a
million Hiroshima bombs.
P2: Studies suggest that explosion of only half these weapons would produce enough soot, smoke, and dust
to blanket the earth, block out the sun, and bring on a nuclear winter that would threaten the survival
of the human race.
C: Radioactive fallout isn’t the only concern in the aftermath of nuclear explosions.

Deductive and Inductive Arguments
• Every argument makes an inferential claim … the claim that its premises
provide grounds for the truth of its conclusion.
• But there are two very different ways in which the conclusion is supported
by its premises:
• Hence we have two different classes of arguments: Deductive arguments
and Inductive Arguments
• Whether an argument is Deductive or Inductive depends upon the strength
of the inferential claim that the argument makes

Deductive Arguments
• Deductive Arguments make a claim that its conclusion is supported by its premises
conclusively.
• Which means a deductive argument is an argument incorporating the claim that it is
impossible for the conclusion to be false given that the premises are true.
• Deductive arguments are those that involve necessary reasoning.
• E.g.
Socrates is a man
All men are mortal
Therefore, it necessarily follows that Socrates is mortal

Inductive Arguments
• In inductive arguments no claim of conclusiveness is made
• Which means even if the premises of inductive arguments are true they do
not support its conclusions with certainty
• That is an inductive argument is an argument incorporating the claim that it
is improbable that the conclusion be false given that the premises are true.
• E.g.
Most actors are celebrities
Xyz is an actress
Therefore xyz is a celebrity

Validity, Truth, Soundness, Strength, Cogency
• Remember we talked about the Truth value of a proposition which means a
proposition is either true or false
• Similarly a deductive argument is either Valid or Invalid.
• Valid deductive argument: is an argument in which it is impossible for the
conclusion to be false given that the premises are true
• That is if a conclusion of a deductive argument does follow from its premises
with strict necessity then that deductive argument is valid
• E.g.
All television networks are media companies.
NBC is a television network.
Therefore, NBC is a media company.

Media Company
Television
Company NBC
Valid

• Invalid deductive argument: is a deductive argument in which it is possible
for the conclusion to be false given that the premises are true.
• In these arguments the conclusion does not follow with strict necessity from
the premises, even though it is claimed to.
• E.g.
All banks are financial institutions.
Wells Fargo is a financial institution.
Therefore, Wells Fargo is a bank.

Financial Institution
Banks Wells Fargo
Dotted line indicates that it may or may not be the case
Invalid

• Important point to note about validity of an argument: validity is something that is
determined by the relationship between premises and conclusion. The question is not
whether the premises and conclusion are true or false, but whether the premises
support the conclusion.
Therefore a deductive argument that has false premises and/or a false
conclusion can still be valid:
All automakers are computer manufacturers.
United Airlines is an automaker.
Therefore, United Airlines is a computer manufacturer.
Similarly a deductive argument with all true premises and conclusion can be
invalid
All banks are financial institutions.
Wells Fargo is a financial institution.
Therefore, Wells Fargo is a bank.
False
False
False
VALID
Argument
True
True
True
INVALID
Argument

Deductive Args Valid Invalid
True Premises
True Conclusion
All Primates have hearts
All humans are primates
Therefore all humans have hearts
If I owned all the currency in the reserve bank of Pakistan, then I
would be wealthy
I do not own all the currency in the reserve bank of Pakistan
Therefore I am not wealthy
True Premises
False Conclusion
If Imran Khan owned all the currency in the state bank of
Pakistan, then he would be wealthy
Imran Khan does not own all the currency in the state bank of
Pakistan
Therefore Imran Khan is not wealthy
False Premises
True Conclusion
All insects are primates
All humans are insects
Therefore all humans are primates
All primates have horns
All humans have horns
Therefore all humans are primates
False Premises
False Conclusion
All bipeds have horns
All snakes are bipeds
Therefore all snakes have horns
All primates have horns
All humans have horns
Therefore all primates are humans
The empty cell shows that any deductive argument having true premises
and a false conclusion is must be invalid
Sound

• A sound argument is a deductive argument that is valid and has all true
premises
• An Unsound argument is a deductive argument that is invalid, has one or
more false premises, or both
Sound Argument Valid Argument
All True
Premises

• Just as deductive arguments are valid or invalid, inductive arguments are strong or
weak
• Strong Inductive Argument: is an inductive argument in which it is improbable that
the conclusion be false given that the premises are true
• In such arguments, the conclusion does in fact follow probably from the premise but
not with certainty as is the claim made in deductive arguments. No such claim is
made in inductive arguments.
• E.g.
All dinosaur bones discovered to this day have been at least 50 million years old.
Therefore, probably the next dinosaur bone to be found will be at least 50 million years old.
E.g. 2
All meteorites found to this day have contained salt.
Therefore, probably the next meteorite to be found will contain salt.

• Weak inductive argument is an argument in which the conclusion does not
follow probably from the premises, even though it is claimed to
• E.g.
Mr Ali is from Pakistan and he is dark skinned
Mr Imran is also from Pakistan
Therefore probably Mr Imran will also be dark skinned
Through experience we know that is not true therefore it is a weak inductive
argument
E.g.
During the past fifty years, inflation has consistently reduced the value of the
American dollar.
Therefore, industrial productivity will probably increase in the years ahead.
Discuss why inductive arguments are important in everyday life
No
relation

Inductive Args Strong Weak
True Premises
Probably true
Conclusion
All previous US presidents were
older than 40
Therefore, probably the next US
president will be older than 40
A few US presidents were lawyers
Therefore, probably the next US president will probably be older
than 40
True Premises
Probably false
Conclusion
A few US presidents were unmarried
Therefore, probably the next US president will unmarried
False Premises
Probably true
Conclusion
All previous US presidents were TV
debaters
Therefore, probably the next US
president will be a TV debater
A few US presidents were dentists
Therefore, probably the next US president will be a TV debater
False Premises
Probably false
Conclusion
All previous US presidents died in
office
Therefore, probably the next US
president will die in office
A few US presidents were dentists
Therefore, probably the next US president will be a dentist
Cogent

Note on deductive and inductive arguments
• A deductive argument is either valid or invalid. There is no middle ground.
• No new information added to a deductive argument can make it more valid or more invalid.
• The strength or weakness of inductive argument is a matter of degree. To be considered strong,
an inductive argument must have a conclusion that is more probable than improbable. Otherwise
it’s a weak argument.
• But as opposed to deductive arguments the inductive arguments can become more stronger or
weaker by introducing new information.
• E.g.
Most actors are celebrities
Anna Das is an actor
Therefore, Anna Das is a celebrity
Most actors are celebrities
Anna Das is a theatre actor
Most theatre actors are not celebrities
Therefore, Anna Das is a celebrity
Strong Inductive argument becomes
weak inductive argument
By adding new
information

Most actors are celebrities
Anna Das is an actor
Therefore, Anna Das is a celebrity
Most actors are celebrities
Anna Das is an actor
Anna Das is also a cabinet minister
Therefore, Anna Das is a celebrity
By adding new
information
Strong Inductive argument becomes
even stronger inductive argument
Similarly we an make an argument stronger by adding more
information

• A cogent argument is an inductive argument that is strong and has all true
premises. Also, the premises must be true in the sense of meeting the total
evidence requirement.
• An Uncogent argument is an inductive argument that is weak, has one or
more false premises, fails to meet the total evidence requirement, or any
combination of these.
Cogent
Argument
Strong Argument
All True
Premises

Summary of how to evaluate a deductive and inductive
arguments
• For both deductive and inductive arguments, two separate questions need to be answered: (1) Do the
premises support the conclusion? (2) Are all the premises true?
• To answer the first question we begin by assuming the premises to be true.
• Then, for deductive arguments we determine whether, in light of this assumption, it necessarily
follows that the conclusion is true. If it does, the argument is valid; if not, it is invalid.
• For inductive arguments we determine whether it probably follows that the conclusion is true. If it
does, the argument is strong; if not, it is weak.
• For inductive arguments we keep in mind the requirements that the premises actually support the
conclusion and that they not ignore important evidence.
• Finally, if the argument is either valid or strong, we turn to the second question and
determine whether the premises are actually true. If all the premises are true, the argument
is sound (in the case of deduction) or cogent (in the case of induction).
• All invalid deductive arguments are unsound, and all weak inductive arguments are uncogent.

Statement True
False
Group of
Statements
Arguments
Non-Arguments
Deductive
Inductive
Deductive
Arguments
Valid
Invalid
(all are unsound)
Sound
Unsound
Inductive
Arguments
Strong
Weak
(all are uncogent)
Cogent
Uncogent

Exercise
I. The following arguments are deductive or Inductive. Determine whether each is valid or invalid/ strong or
weak. Furthermore, determine whether the argument is sound or unsound/ cogent or uncogent.
1. Since Moby Dick was written by Shakespeare, and Moby Dick is a science fiction novel, it follows that
Shakespeare wrote a science fiction novel.
4. The longest river in South America is the Amazon, and the Amazon flows through Brazil. Therefore, the
longest river in South America flows through Brazil.
10. Every province in Canada has exactly one city as its capital. Therefore, since there are thirty provinces in
Canada, there are thirty provincial capitals.
1. The grave marker at Arlington National Cemetery says that John F. Kennedy is buried there. It must be the
case that Kennedy really is buried in that cemetery.
7. People have been listening to rock and roll music for over a hundred years. Probably people will still be
listening to it a year from now.
10. Coca-Cola is an extremely popular soft drink. Therefore, probably someone, somewhere, is drinking a
Coke right this minute.

1. Valid, unsound; false premises, false conclusion.
4. Valid, sound; true premises, true conclusion.
10. Valid, unsound; false premise, false conclusion.
1. Strong, cogent; true premise, probably true conclusion.
7. Strong, uncogent; false premise, probably true conclusion.
10. Strong, cogent; true premise, probably true conclusion.

AnalysingArguments
Arguments in everyday life are more complex:
• the number of premises may vary
• the order of the premises may vary
• premises may be repeated using different words
• premises may be left out etc.
We can use two techniques for analysing arguments:
1. Paraphrasing
2. Diagraming

1. Paraphrasing
• Consider the following example:
Archimedes will be remembered when Aeschylus is forgotten, because
languages die but mathematical ideas do not
• Now this single sentence can be analysed as follows by bringing
forward the unstated premises and inferences:
1. Languages die
2. The plays of Aeschylus are in a language
3. So the work of Aeschylus will eventually die
4. Mathematical ideas are permanent and therefore never die
5. The work of Archimedes was with mathematical ideas
6. So the work of Archimedes will not die
Therefore Archimedes will be remembered when Aeschylus is forgotten

• Another example of paraphrasing:
Geese are migratory waterfowl, so they fly south for winter
This argument has a missing premise:
Migratory waterfowl fly south of winter
The argument can now be rephrased as:
All Geese are migratory waterfowl
All migratory waterfowls are birds that fly south for winter
Therefore, all geese are birds that fly south for winter

2. Diagramming
• Logical analysis of arguments found in editorials, essays, letters etc.
is very difficult because they involve
• Non-arguments: such as explanations, advise, illustrations, statements
of opinions etc.
• Lengthy arguments involve complex arrangements of sub-arguments
that feed into the main argument.
• Some argumentative passages involve completely separate strands of
argumentation that lead to separate arguments.
• Diagraming helps facilitate these extended arguments by assigning
numerals to various statements in a passage and use arrows to
represent the inferential link

Consider all these various forms:
The contamination of underground aquifers represents a pollution
problem of catastrophic proportions. Half the nations dirking water,
which comes from these aquifers, is poisoned by chemical wastes
dumped into the soil for generations.
This arguments is diagrammed as:
This diagram says that the statement , the premises, support
statement , the conclusion
1
2
1
2
2
1

The selling of human organs such as heart, kidney, and cornea should be
outlawed. Allowing human organs to be sold will inevitably lead to a situation
where the only the rich can afford transplants. This is so, because whenever
something scarce is bought and sold as a commodity, the price always goes
up. The law of supply and demand requires it.
The argument is drawn as follows:
1
2
3
4
4
3
2
1
Vertical Pattern

The selling of human organs such as heart, kidney, and cornea should be
outlawed. If this practise is allowed to foothold people in dire financial straits
will start selling their own organs to pay for their bills. Alternatively, those will
criminal bent will take to killing healthy young people and selling their organs on
the market. In the financial analysis, buying and selling human organs comes
close to buying and selling of life itself.
The diagram says that statement the conclusion, is supported by premises
Indipendantly
1
2
3
4
1
2 3 4
Horizontal Pattern
1
2 3 4

Getting poor people off welfare rolls require that we modify their behaviour
pattern. The vast majority of people on welfare are high school dropouts,
single parents, or people who abuse alcohol and drugs. These behaviour
patterns frustrate any desire poor people may have to get a job and improve
their condition in life.
Statement 2 and 3 do not support the conclusion 1 independently. But when
combined they do support the conclusion. Premises 2 and 3 are dependant
upon each other for their support of the conclusion 1.
1
2
3
1
2 3
Conjoint
Premises

Dropping out of school and bearing children outside of marriage are two of
the primary causes of poverty in this country. Therefore, to eliminate poverty
we must offer incentives for people to get high school diplomas. Also, we
must find some way to encourage people to get married before they start
having children
Statement 1, the premise, supports both statements 2 and 3, the conclusions
1
2
3
1
2 3
Multiple
Conclusions

Complex Arguments
Government mandates for zero-emission vehicles won’t work because
only electric cars qualify as zero-emission vehicles, and electric cars won’t
sell. They are too expensive, their range of operation is too limited, and
recharging facilities are not generally available.
1 2
3
4 5
6
1
2 3
4 5 6
Example taken from
newspaper editorial

Rhinos in Kenya are threatened with extinction because poachers are
killing them for their horn. Since the rhino has no natural predators, it
does not need its horn to survive. Thus there should be an organized
program to capture rhinos in the wild and remove their horn. Such a program
would eliminate the incentive of the poachers.
1 2
3 4
5
6
1
2 3
4
5
6
Example from a letter
to the editor

Skating is a wonderful form of exercise and relaxation, but today’s
rollerbladers are a growing menace and something should be done to
control them. Rollerbladers are oblivious to traffic regulations as they
breeze through red lights and skim down the wrong way on one-way
streets. They pose a threat to pedestrians because a collision can cause
serious injury. Rollerbladers are even a hazard to shopkeepers as they
zoom through stores and damage merchandise.
1 2
3
4 5
6
7 8
9 10
11
1
23
4
5 610
7
811
9
is merely an
introductory
sentence
Example from magazine
article

We can expect small changes to occur in the length of our calendar year for an
indefinite time to come. This is true for two reasons. First, the rotation of the earth
exhibits certain irregularities. And why is this so? The rotation of any body is
affected by its distribution of mass, and the earth’s mass distribution is continually
subject to change. For example, earthquakes alter the location of the tectonic plates.
Also, the liquid core of the earth sloshes as the earth turns, and rainfall
redistributes water from the oceans. The second reason is that the motion of the
tides causes a continual slowing down of earth’s rotation. Tidal motion produces
heat, and the loss of this heat removes energy from the system.
1
2 3
4 5
6
7
8 9
10
11
12
1
2
3
4
5 6
10
7 8
11
9
12
Are premise indicators and do
not add any support

Exercise
The following arguments were abstracted from newspaper articles, editorials, and letters to the editor. Use the
method presented in this section to construct argument patterns. If a statement is redundant or plays no role in
the argument, do not include it in the pattern.
1 The conditions under which many food animals are raised are unhealthy for humans. 2 To keep these
animals alive, large quantities of drugs must be administered. 3 These drugs remain in the animals’ flesh and
are passed on to the humans who eat it.
1 When parents become old and destitute, the obligation of caring for them should be imposed on their
children. 2 Clearly, children owe a debt to their parents. 3 Their parents brought them into the world and cared
for them when they were unable to care for themselves. 4 This debt could be appropriately discharged by
having grown children care for their parents.
1 There is a lot of pressure on untenured college teachers to dumb down their courses. 2 Administrators tend to
rehire teachers who bring in more money, and 3 teachers who dumb down their classes do precisely this. Why?
Because 4 easier classes attract more students, and 5 more students means more money for the school.

CATEGORICAL PROPOSITIONS
• A proposition (statement) that relates two classes or categories is called a categorical
proposition
• By a class we mean a collection of all objects having some specified characteristics in
common
• Such as ‘humans’ or ‘doctors’ or ‘junk food’ or ‘school cafeterias’ or ‘happy endings’ or
‘magazine’ or ‘students learning logic’ etc.
• Therefore a categorical proposition is one which relates two classes, one denoted by
subject term ‘S’ and the other by the predicate term ‘P’ , by asserting that the whole or
part of the class ‘S’ is included in or excluded from class ‘P’

• Therefore there are exactly 4 ways in which categorical propositions relate two classes:
1. All of one class ‘S’ are included all of the other class ‘P’:
• E.g. The class of all dogs is included in the class of all mammals
2. All of one class ‘S’ are excluded from all of the other class ‘P’: i.e. they have no members in
common
• E.g. The class of triangles is excluded from the class of circles
3. Part of the subject class ‘S’ is included in the predicate class ‘P’:
• E.g. The class of all chess players is partially included in the class of females
4. Part of the subject class ‘S’ is excluded from the predicate class ‘P’:
• E.g. The class of all animals is partially excluded from the class of mammals
• Hence we have exactly 4 kinds of categorical propositions:

The 4 kinds of standard form of categorical
propositions
• A. - Universal Affirmative
• All S are P
• All politicians are liars
• E. - Universal Negative
• No S are P
• No politicians are liars
• I. - Particular Affirmative
• Some S are P
• Some politicians are liars
• O. - Particular Negative
• Some S are not P
• Some politicians are not liars

Further points to note about categorical propositions
1. The four categorical propositions A, E, I, O are the building blocks of the deductive
arguments
2. Quantifiers: The words “all,” “no,” and “some” are called quantifiers because they
specify how much of the subject class is included in or excluded from the predicate
class.
3. In formal deductive logic ‘some’ means ‘at least’ one
4. The letters S and P stand respectively for the subject and predicate terms
5. The words “are” and “are not” are called the copula because they link (or “couple”)
the subject term with the predicate term.
6. “subject term” and “predicate term” do not mean the same thing in logic that “ subject”
and “predicate” mean in grammar

• For example consider the following example:
All members of the Pakistan Medical Association are people holding degrees
from recognized academic institutions.
• This standard-form categorical proposition is analysed as follows:
quantifier: all
subject term: members of the Pakistan Medical Association
Copula : are (because they link or couple the subject and predicate)
predicate term: people holding degrees from recognized academic institutions
Quantifier (subject term) copula (predicate term)
All s are p
Some s are p
No s are p
Some s are not p

Exercise
• Some executive pay packages are insults to ordinary workers.
• Some preachers who are intolerant of others’ beliefs are not television
evangelists.
• All oil-based paints are products that contribute to photochemical smog.
• No stressful jobs are occupations conducive to a healthy lifestyle.

• Quantifier: some; subject term: executive pay packages; copula: are;
predicate term: insults to ordinary workers.
• Quantifier: some; subject term: preachers who are intolerant of others’ beliefs;
copula: are not; predicate term: television evangelists.
• Quantifier: All; Subject term: Oil based paints; Copula: are Predicate term:
products that contribute to photochemical smog
• Quantifier: No; Subject Term: Stressful jobs; Copula: are; Predicate term:
occupations conducive to healthy life style

Quality, Quantity and Distribution
• Quality and quantity are attributes of categorical propositions. To understand these
concepts consider the following example:
Proposition Meaning in class notation
All S are P. Every member of the S class is a member of the P class;
that is, the S class is included in the P class.
No S are P. No member of the S class is a member of the P class; that
is, the S class is excluded from the P class.
Some S are P. At least one member of the S class is a member of the P
class.
Some S are not P. At least one member of the S class is not a member of the
P class.

Quality:
The quality of a categorical proposition is either affirmative or negative depending on
whether it affirms or denies class membership. Accordingly, “All S are P” and “Some S
are P” have affirmative quality, and “No S are P” and “Some S are not P” have negative
quality. These are called affirmative propositions and negative propositions,
respectively.
Quantity:
The quantity of a categorical proposition is either universal or particular, depending on
whether the statement makes a claim about every member or just some member of the
class denoted by the subject term. “All S are P” and “No S are P” each assert something
about every member of the S class and thus are universal propositions. “Some S are
P” and “Some S are not P” assert something about one or more members of the S class
and hence are particular propositions.
Imp note:
‘All’ and ‘No’ imply universal quantity wile ‘Some’ imply particular quantity.
But categorical propositions have no “qualifier”. In universal propositions the quality is
determined by the quantifier, and in particular propositions it is determined by the copula.

Distribution:
• Unlike quality and quantity, which are attributes of propositions, distribution is an attribute of the
terms (subject and predicate) of propositions.
• A proposition distributes a term if it refers to all members of the class designated by that term;
otherwise it is undistributed. I.e. If a term is distributed it says something about every member of that
term.
A: All S are P ……. S is distributed and P is not
E: No S are P ……. Both S and P are distributed
I: Some S are P …… Neither S nor P is distributed
O: Some S are not P ….. P is distributed and S is not
P
S
P
S
*S
P
*S
P

Proposition Letter name Quantity Quality Terms
distributed
All S are P A universal affirmative S
No S are P E universal negative S and P
Some S are P I particular affirmative none
Some S are not P O particular negative P

Exercise
I. For each of the following categorical propositions identify the letter name,
quantity, and quality. Then state whether the subject and predicate terms are
distributed or undistributed.
• No vampire movies are films without blood.
• Some Chinese leaders are not thoroughgoing opponents of capitalist
economics
• Some hospitals are organizations that overcharge the Medicare program.
II. Change the quality but not the quantity of the following statements.
• All drunk drivers are threats to others on the highway.
• Some CIA operatives are not champions of human rights.

III. Change the quantity but not the quality of the following statements.
• All owners of pit bull terriers are people who can expect expensive lawsuits.
• Some residents of Manhattan are not people who can afford to live there.
IV. Change both the quality and the quantity of the following statements.
• All oil spills are events catastrophic to the environment.
• Some corporate lawyers are not people with a social conscience.

I.
• E proposition, universal, negative, subject and predicate terms are distributed.
• O proposition, particular, negative, subject term undistributed, predicate term
distributed.
• I proposition, particular, affirmative, subject and predicate terms undistributed.
II.
• No drunk drivers are threats to others on the highway.
• Some CIA operatives are champions of human rights.
III.
• Some owners of pit bull terriers are people who can expect expensive lawsuits.
• No residents of Manhattan are people who can afford to live there.
IV.
• Some oil spills are not events catastrophic to the environment.
• All corporate lawyers are people with a social conscience.

Drawing Immediate Inferences
• Immediate inference is one where the conclusion is drawn directly from only one
premise
E.g.
If it is true that: (A) All horses have 4 legs
Therefore, it is true that (I) some horses have 4 legs
If it is true that: (A) All horses have 4 legs
Therefore, it is false that: (E) no horses have 4 legs
• There are a number of relations amongst the 4 propositions that allow us to draw
immediate inferences some directly and some through manipulation
• These relationships gives us solid grounds for a great deal of reasoning we do in
everyday life
• Let us have a look at these relationships

Conversion, Obversion, and Contraposition
• Conversion, Obversion, and Contraposition are operations that can be performed on a categorical
proposition, resulting in a new statement that may or may not have the same meaning and truth value as
the original statement.
• They are used for drawing immediate inferences
• Venn diagrams are used to determine how the two statements relate to each other.
Conversion
• It is the simplest of the three operations and is obtained by switching the subject term with the predicate
term of a categorical proposition
• For example, if the statement “No foxes are hedgehogs” is converted, the resulting statement is “No
hedgehogs are foxes.”
• This new statement is called the converse of the given statement.
• To see how the four types of categorical propositions relate to their converse, compare the following sets of
Venn diagrams:

X
X
X
Given Statement Form Converse .
All S are P All P are S
No S are P No P are S
X
Some S are P Some P are S
Some P are not SSome S are not P
PS
PS PS

S P
Switch
(Copula)(Quantifier)
Conversion
We can see from above that Conversion is not valid for A & O statements and applying conversion to
them will yield new statements. But Conversion is valid for I & E statements so converting them yields the
same truth value as the original statements. Therefore through conversion we can get the following valid
immediate inferences:
No A are B
Therefore, No B are A
Some A are B
Therefore, some B are A
Converting A & O statements, on the other hand, commits Fallacy of Illicit Conversion:
All cats are animals (T) Some animals are not dogs (T)
Therefore, All animals are cats (F) Therefore, Some dogs are not animals (F)

Obversion
Obversion requires two steps:
(1) changing the quality (without changing the quantity), and
(2) replacing the predicate with its term complement.
Class compliments and Term Compliment
Non-Dogs Dogs

S P(Copula)(Quantifier)
Obversion
Change
Quality
Replace
with term
compliment

X
X
X
Given Statement Form Obverse .
All S are P No S are non-P
No S are P All S are non-P
Some S are P Some S are not non-P
Some S are non-PSome S are not P
PS PS
PS
PS
X

Obverse of all statements is equivalent to their original statements and holds true for all:
All A are B. Some A are B.
Therefore, no A are non-B Therefore, some A are not non-B.
e.g. All horses are animals Some trees are maples
No horses are non-animals Some trees are not non-maples
No A are B Some A are not B
Therefore, all A are non-B Therefore, some A are non-B.
e.g. No cats are dogs Some humans are not women
All cats are non-dogs Some humans are non-women

Contraposition
Like Obversion, contraposition requires two steps:
1. Switching the subject and predicate terms and
2. Replacing the subject and predicate terms with their term complements.
S P
Switch & replace with
(Copula)(Quantifier)
term compliment

X
X
X
Given Statement Form Contrapositive .
All S are P All non-P are non-S
No S are P
X
Some S are P
Some S are not P
PS PS
PS
PS
PS
No non-P are non-S
Some non-P are non-S
Some non-P are not non-S

• As seen from the above Venn diagrams of all the 4 propositions only the
contrapositives of A and O statements give the same truth value as original statements
and hence immediate inferences for them are valid:
All dogs are animals
Therefore, All non-animals are non-dogs
• Whereas the contrapositives of E and I statements are not equivalent to their original
statements and therefore commit the fallacy of Illicit Contraposition:
E: No dogs are cats. (True)
Therefore, no non-cats are non-dogs. (False)
I: Some animals are non-cats. (True)
Therefore, some cats are non-animals. (False)

Summary of Conversion, Obversion, & Contraposition
Given Statement Converse Truth Value
E: No A are B. No B are A. Same truth value as given
I: Some A are B. Some B are A. statement
A: All A are B. All B are A. Undetermined truth value
O: Some A are not B. Some B are not A.
Given Statement Obverse Truth Value
A: All A are B. No A are non-B.
E: No A are B. All A are non-B. Same truth value as given statement
I: Some A are B. Some A are not non-B.
O: Some A are not B. Some A are non-B.
Given Statement Contrapositive Truth Value
A: All A are B. All non-B are non- A. Same truth value as given
O: Some A are not B. Some non-B are not non- A. statement
E: No A are B. No non-B are non- A. Undetermined truth value
I: Some A are B. Some non-B are non- A.

Exercise Part I
Exercises 1 through 6 provide a statement, its truth value in parentheses, and
an operation to be performed on that statement. Supply the new statement and
the truth value of the new statement:
Given statement Operation New statement Truth Value
1. No A are non-B. (T) conv.
2. Some A are B. (T) contrap.
3. All A are non-B. (F) obv.
4. All non-A are B. (F) contrap.
5. Some non-A are not B. (T) conv.
6. Some non-A are non-B. (T) obv.

Key Part I
Given statement Operation New statement Truth Value
1. No A are non-B. (T) conv. No non-B are A T
2. Some A are B. (T) contrap. Some non-A are non-B Undermined
3. All A are non-B. (F) obv. No A are B F
4. All non-A are B. (F) contrap. All non-B are A F
5. Some non-A are not B. (T) conv. Some B are not non-A Undermined
6. Some non-A are non-B. (T) obv. Some non-A are not B T

Exercise Part II
Exercises 7 through 12 provide a statement, its truth value in parentheses, and a new
statement. Determine how the new statement was derived from the given statement and
supply the truth value of the new statement.
Given statement Operation New statement Truth Value
7. No non-A are non-B. (F) No B are A.
8. Some A are not non-B. (T) Some A are B.
9. All A are non-B. (F) All non-B are A.
10. No non-A are B. (F) All non-A are non-B.
11. Some non-A are not B. (T) Some non-B are not A.
12. Some A are non-B. (F) Some non-B are A.

Key Part II
Given statement Operation New statement Truth Value
7. No non-A are non-B. (F) Contrap. No B are A. Undermined
8. Some A are not non-B. (T) Obv. Some A are B. T
9. All A are non-B. (F) Conv. All non-B are A. Undermined
10. No non-A are B. (F) Obv. All non-A are non-B. F
11. Some non-A are not B. (T) Contrap. Some non-B are not A. T
12. Some A are non-B. (F) Conv. Some non-B are A. F

The Traditional Square of Opposition
A E
I O
T
F
T
F
Contrary
Subcontrary
Subalternation
Subalternation

The four relations in the traditional square of opposition may be characterized as follows:
Contradictory = opposite truth value
Contrary = at least one is false (not both true)
Subcontrary = at least one is true (not both false)
Subalternation = truth flows downward, falsity flows upward
The Contradictory relation expresses opposition thus:
If A is true = O is false If A is false = O is true
If O is true = A is false If O is false = A is true
Same relations hold between E and I
A: All dogs are animals. (T)
O: Therefore, it is false that some dogs are not animals. (F)
I: Some humans are aliens (F)
E: Therefore, it is true that no humans are aliens (T)

The Contrary relation expresses only partial opposition between A & E propositions …. Both
statements cannot be true at the same time: at least one is false
If A is true = then E is false If E is true = then A is false
But
If A is false = then E can either be true or false …. Thus E is undetermined
If E is false = then A can either be true or false …. Thus A is undetermined
The following statements can both be false but both cannot be true so if one is true the other is false:
India will win this match against Pakistan
Pakistan will win this match against India
A All cats are animals (T)
E Therefore, it is false that no cats are animals (F)
E No cats are dogs (T)
A All cats are dogs (F)
A All animals are cats (F)
E No animals are cats (F)

The Subcontrary relation also expresses only partial opposition between I & O propositions
…. But here Both statements cannot be false at the same time : at least one is True
If I is False = then O is true If O is false = then I is true
But
If I is true = then O can either be true or false …. Thus O is undetermined
If O is true= then I can either be true or false …. Thus I is undetermined
The following statements can both be true but both cannot be false so if one is false the other is true:
Some diamonds are precious stones
Some diamonds are not precious stones
I Some cats are dogs (F)
O Some cats are not dogs (T)
O Some cats are not animals (F)
I Some cats are animals (T)

The Subalternation relation expresses corresponding relation between the A & I as well as
between E & O propositions
Thus the A proposition: All spiders are 8 legged creatures (T) Has a corresponding I proposition: Some
spiders are 8 legged creatures (T)
Similarly the E proposition: No whales are fishes (T) Has the following O proposition: Some whales are
not fishes (T)
 Only the Truth flows from top to bottom thus:
If A is true = I is true & If E is true = then O is true
But if A is false = I can be true or false & If E is false = then O can be true or false
 Only the Falsity flows from bottom to top thus:
If I is False = then A is false & If O is false = then E is false
But if I is True = then A can be true or false & If O is True = then E can be true or false

Formal Fallacies
• Fallacy is a type of argument that seems to be correct; but that proves on examination
not to be so.
• There are two types of fallacies:
• Formal fallacies are those that can be determined by the form of a categorical syllogism
• Informal fallacies are those that can only be determined by the content of the argument and cannot be
determined merely by its form
Exercise Cover from Book

Categorical Syllogisms
A Syllogism is a deductive argument consisting of three propositions: of which two are
premises and one is conclusion
Both the premises and the conclusion are one of the A, E, I, O propositions
Eg.
No Politicians are professors
Some doctors are professors
Therefore, Some doctors are not politicians
• The above syllogism is called categorical because the propositions are categorical
• Which means they relate two categories and the syllogism as a whole relates three
categories
• These categories are called terms:

• Each of the three terms in a categorical syllogism has its own name depending on its
position in the argument.
• The major term, by definition, is the predicate of the conclusion, and
• the minor term is the subject of the conclusion.
• The middle term, which provides the middle ground between the two premises, is the
one that occurs once in each premise and does not occur in the conclusion.
• Thus, for the argument just given, the major term is ‘politicians’, the minor term is
‘doctors’, and the middle term is ‘professors’
• The premise that has the major term is called the ‘Major Premise’, the premise that has
the minor term is called the ‘Minor Premise’

A standard-form categorical syllogism is one that meets the following four conditions:
1. All three statements are standard-form categorical propositions i.e. A, E, I, O
2. Each term is used in the same sense throughout the argument.
3. The major premise is listed first, the minor premise second, and the conclusion last.
Note: to put a categorical syllogism into standard form:
First identify the conclusion
From that identify the major term and the minor terms
Based on the above identify the relative major and minor premises
Then put the major premise first, minor premise second, and the conclusion last in the
following format:
1st: Quantifier __________ copula ________
2nd: Quantifier __________ copula ________
3rd: Quantifier __________ copula ________
Major premise (containing major term)
Minor premise (containing minor term)
Conclusion
Minor term Major term

Mood and Figure
The mood of a categorical syllogism consists of the letter names of the propositions that make
it up.
No painters are sculptors. (E)
Some sculptors are artists. (I)
Therefore, some artists are not painters. (O)
The mood of this standard-form categorical proposition is EIO
The figure of a categorical syllogism is determined by the location of the two occurrences of
the middle term in the premises. Four different arrangements are possible:
Figure 1
M P
MS
PS
Figure 2
MP
MS
PS
Figure 3
M P
M S
PS
Figure 4
MP
M S
PS

E.g.
No painters are sculptors. (E)
Some sculptors are artists. (I)
Therefore, some artists are not painters. (O)
The mood of the above standard-form categorical proposition is EIO and the figure is 4
Hence the form of the syllogism is EIO-4
After the categorical proposition is put into standard form then its validity or invalidity
maybe determined by mere inspection of its form.
1
2 3
4

Since there are 4 kinds of categorical propositions and there are three categorical
propositions in a categorical syllogism, there are 64 possible moods (4 × 4 × 4 = 64). And
since there are four different figures, there are 256 different forms of categorical
syllogisms (4 × 64 = 256).
Of these 256 possible categorical syllogism only 15 are (unconditionally) valid and the
rest are invalid:
Figure 1 Figure 2 Figure 3 Figure 4
AAA EAE IAI AEE
EAE AEE AII IAI
AII EIO OAO EIO
EIO AOO EIO

Exercises I
The following syllogisms are in standard form. Identify the major, minor, and middle
terms, as well as the mood and figure of each.
1. All neutron stars are things that produce intense gravity.
All neutron stars are extremely dense objects.
Therefore, all extremely dense objects are things that produce intense gravity.
2. No insects that eat mosquitoes are insects that should be killed.
All dragonflies are insects that eat mosquitoes.
Therefore, no dragonflies are insects that should be killed.
3. No environmentally produced diseases are inherited afflictions.
Some psychological disorders are not inherited afflictions.
Therefore, some psychological disorders are environmentally produced diseases.
4. No people who mix fact with fantasy are good witnesses.
Some hypnotized people are people who mix fact with fantasy.
Therefore, some hypnotized people are not good witnesses.

II
II. Put the following syllogisms into standard form, using letters to represent the terms, and
name the mood and figure.
1. No Republicans are Democrats, so no Republicans are big spenders, since all big spenders
are Democrats.
4. Some insects that feed on milkweed are not foods suitable for birds, inasmuch as no
monarch butterflies are foods suitable for birds and all monarch butterflies are insects that feed
on milkweed.
5. No illegal aliens are people who have a right to welfare payments, and some migrant
workers are illegal aliens. Thus, some people who have a right to welfare payments are migrant
workers.
6. Some African nations are not countries deserving military aid, because some African nations
are not upholders of human rights, and all countries deserving military aid are upholders of
human rights.
7. All pranksters are exasperating individuals, consequently some leprechauns are
exasperating individuals, since all leprechauns are pranksters.

III
Reconstruct the syllogistic forms from the following combinations of mood and figure.
OAE-3 EIA-4 AII-3 IAE-1 AOO-2 AAA-1 OEA-4
IV.
Construct the following syllogisms.
1. An EIO-2 syllogism with these terms: major: dogmatists; minor: theologians; middle: scholars who
encourage free thinking.
2. An unconditionally valid syllogism in the first figure with a particular affirmative conclusion and these
terms: major: people incapable of objectivity; minor: Supreme Court justices; middle: lockstep
ideologues.
3. An unconditionally valid syllogism in the fourth fi gure having two universal premises and these
terms: major: teenage suicides; minor: heroic episodes; middle: tragic occurrences.
4. A valid syllogism having mood OAO and these terms: major: things capable of replicating by
themselves; minor: structures that invade cells; middle: viruses.
5. A valid syllogism in the fi rst fi gure having a universal negative conclusion and these terms: major:
guarantees of marital happiness; minor: prenuptial agreements; middle: legally enforceable
documents.

Explain the whole process of proving validity or invalidity of a
standard form categorical syllogism through Venn-Diagrams
1
2
3
4
5 6 7

Pointers for using Venn Diagrams
1. Marks (shading or placing an X) are entered only for the premises. No marks are
made for the conclusion.
2. If the argument contains one universal premise, this premise should be entered first
in the diagram. If there are two universal premises, either one can be done first.
3. When entering the information contained in a premise, one should concentrate on the
circles corresponding to the two terms in the statement. While the third circle cannot
be ignored altogether, it should be given only minimal attention.
4. When inspecting a completed diagram to see whether it supports a particular
conclusion, one should remember that particular statements assert two things. “Some
S are P” means “At least one S exists and that S is a P”; “Some S are not P” means
“At least one S exists and that S is not a P.”
5. When shading an area, one must be careful to shade all of the area in question.
Examples:

6. The area where an X goes is always initially divided into two parts. If one of these
parts has already been shaded, the X goes in the non-shaded part. Examples:
---------------------------
If one of the two parts is not shaded, the X goes on the line separating the two parts.
Examples:
-----------------------------
This means that the X may be in either (or both) of the two areas—but it is not known
which one.
7. An X should never be placed in such a way that it dangles outside of the diagram,
and it should never be placed on the intersection of two lines.

Illustrations
• EAE-2 (valid)
• AEE-1 (invalid)
• IAI-4 (valid)
• AOO-2 (valid)
• IAI-1 (invalid)
• AOO-1 (invalid)
• AAA-1 (valid)
• OIO-1 (invalid)

Syllogistic Rules and Formal Fallacies
• Valid syllogisms confirm to certain rules.
• If any of these rules are violated a specific fallacy is committed and the syllogism is
invalid.
• Since the fallacies is due to the form of the syllogism therefore it is called formal
fallacies (as opposed to informal fallacies discussed latter)
• These rules can be used to cross-check with Venn-diagram technique discussed
above.
• Of the five rules presented in this section, the first two depend on the concept of
distribution, the second two on the concept of quality, and the last on the concept of
quantity.

Statement Type Terms Distributed
A Subject
E Subject, predicate
I None
O Predicate
Rule 1: The middle term must be distributed at least once.
Fallacy: Undistributed middle.
Example:
All sharks are fish.
All salmon are fish.
All salmon are sharks.
Rule 2: If a term is distributed in the conclusion, then it must be distributed in a premise.
Fallacies: Illicit major; illicit minor.
Examples: All horses are animals.
Some dogs are not horses.
Some dogs are not animals.
All tigers are mammals.
All mammals are animals.
All animals are tigers.

Rule 3: Two negative premises are not allowed.
Fallacy: Exclusive premises.
Example:
No fish are mammals.
Some dogs are not fish.
Some dogs are not mammals.
Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires
a negative premise.
Fallacy: Drawing an affirmative conclusion from a negative premise.
or
Drawing a negative conclusion from affirmative premises.
Examples: All crows are birds.
Some wolves are not crows.
Some wolves are birds.
All triangles are three-angled polygons.
All three-angled polygons are three-sided polygons.
Some three-sided polygons are not triangles.

• Note: As a result of the interaction of these first four rules, it turns out that no
valid syllogism can have two particular premises.
Rule 5: If both premises are universal, the conclusion cannot be particular.
Fallacy: Existential fallacy.
Example: All mammals are animals.
All tigers are mammals.
Some tigers are animals.
• Before deciding if a syllogism breaks Rule 5 make sure it does not break other 4 rules.
• Discuss Aristotelian vs. Boolean standpoints

Propositional Logic
• In Categorical Propositions and Categorical Syllogism letters represented terms
• The fundamental elements in Propositional Logic are whole statements (propositions)
• And those Statements are represented by letters, and these letters are then combined
by means of the operators to form more-complex symbolic representations.
• Remember Simple statement is one that does not contain any other statement as a
component
• Whereas Compound statement is one that contains at least one simple statement as a
component. E.g.:
• It is not the case that Al Qaeda is a humanitarian organization. It is not the case that A
• Dianne Reeves sings jazz, and Christina Aguilera sings pop. D and C
• Either people get serious about conservation or energy prices will skyrocket. Either P or E
• If nations spurn international law, then future wars are guaranteed. If N then F
• The Broncos will win if and only if they run the ball. B if and only if R

Operator Name Logical function Used to translate
∼ tilde negation not, it is not the case that
• dot conjunction and, also, moreover
∨ wedge disjunction or, unless
⊃ horseshoe implication if . . . then . . . , only if
≡ triple bar equivalence if and only if
It is not the case that A. ∼A
D and C. D • C
Either P or E. P ∨ E
If N then F. N ⊃ F
B if and only if R. B ≡ R
Disjuncts, conjuncts, Conditional statement representing material implication,
Antecedent, consequent, Bi-conditional statement representing material equivalence,

Negation (The Tilde)
• the tilde is always placed in front of the proposition it negates. All of the other operators
are placed between two propositions.
• Also, unlike the other operators, the tilde cannot be used to connect two propositions.
Thus, G ∼ H is not a proper expression.
• But the tilde is the only operator that can immediately follow another operator. Thus, it
would be proper to write G • ∼H.
• The tilde is used to negate a simple proposition, but it can also be used to negate a
compound proposition—for example ∼(G • F).
• In this case the tilde negates the entire expression inside the parentheses.
• These statements are all negations. The main operator is a tilde.
• ∼ B
• ∼(G ⊃ H)
• ∼[ (A ≡ F) • (C ≡ G) ]

• The main operator is the operator that has as its scope everything else in the
statement.
H • (J ∨ K), ∼(K • M), K ⊃ ∼(L • M),
• The dot symbol is used to translate such conjunctions as “and”, “also”, “but”,
“however”, “yet”, “still”, “moreover”, “although”, “nevertheless”, and “both”:
• Tiffany sells jewellery, and Gucci sells cologne. T • G
• Tiffany sells jewellery, but Gucci sells cologne. T • G
• Tiffany sells jewellery; however, Gucci sells cologne. T • G
• Tiffany and Ben Bridge sell jewellery. T • B
• These statements are all conjunctions. The main operator is a dot.
• K • L
• (E v F) • (G v H )
• [ (R ⊃ T ) v (S ⊃ U) ] • [ (W ≡ X ) v (Y ≡ Z ) ]

• The wedge symbol is used to translate “or”, “unless”, and “either”
• Aspen allows snowboards or Telluride does. A ∨ T
• Either Aspen allows snowboards or Telluride does. A ∨ T
• Aspen allows snowboards unless Telluride does. A ∨ T
• Unless Aspen allows snowboards, Telluride does. A ∨ T
• These statements are all disjunctions. The main operator is a wedge.
• C v D
• (F • H) v (K • L)
• [S • (T ⊃ U) ] v [X • (Y ≡ Z ) ]
• Note that A ∨ T is logically equivalent to T ∨ A.
• Also T • G is logically equivalent to G • T.

• The horseshoe symbol is used to translate “if . . . then . . . ,”, “only if”, “in case”,
“provided that”, “given that”, “on condition that”, and “implies that”
• The function of “only if” is, in a sense, just the reverse of “if”. For example, the
statement “You will catch a fish only if your hook is baited” does not mean “If your
hook is baited, then you will catch a fish.”
• If it meant this, then everyone with a baited hook would catch a fish. Rather, the
statement means “If your hook is not baited, then you will not catch a fish” which is
logically equivalent to “If you catch a fish, then your hook was baited”.
• To avoid mistakes in translating “if” and “only if” remember this rule:
The statement that follows “if” is always the antecedent, and the statement that
follows “only if” is always the consequent.
Thus, “C only if H” is translated C ⊃ H, whereas “C if H” is translated H ⊃ C.
• If Purdue raises tuition, then so does Notre Dame. P ⊃ N
• Notre Dame raises tuition if Purdue does. P ⊃ N
• Purdue raises tuition only if Notre Dame does. P ⊃ N
• Cornell cuts enrolment provided that Brown does. B ⊃ C
• Cornell cuts enrolment on condition that Brown does. B ⊃ C
• Brown’s cutting enrolment implies that Cornell does. B ⊃ C

• In translating conditional statements, it is essential not to confuse antecedent with
consequent. The statement A ⊃ B is not logically equivalent to B ⊃ A.
• H ⊃ J
• (A v C ) ⊃ (D • E )
• [K v (S • T ) ] ⊃ [F v (M • O) ]
Necessary and Sufficient Conditions
• Event A is said to be a sufficient condition for event B whenever the occurrence of A
is all that is required for the occurrence of B.
• On the other hand, event A is said to be a necessary condition for event B whenever
B cannot occur without the occurrence of A.
• E.G.: Flu = Feeling miserable (sufficient) and air = alive (necessary)
• SUN mnemonic S ⊃ N
• Hilton’s opening a new hotel is a sufficient condition for Marriott’s doing so. H ⊃ M
• Hilton’s opening a new hotel is a necessary condition for Marriott’s doing so. M ⊃ H

• The triple bar symbol is used to translate the expressions “if and only if” and “is a
sufficient and necessary condition for”:
• JFK tightens security if and only if O’Hare does. J ≡ O
• JFK’s tightening security is a sufficient and necessary condition for O’Hare’s doing so. J ≡ O
• Analysis of the first statement reveals that J ≡ O is logically equivalent to (J ⊃ O) •
(O ⊃ J).
• The statement “JFK tightens security only if O’Hare does” is translated J ⊃ O, and
“JFK tightens security if O’Hare does” is translated O ⊃ J. Combining the two
English statements, we have (J ⊃ O) • (O ⊃ J), which is just a longer way of
writing J ≡ O.
• These statements are all bi-conditionals (material equivalences). The main
operator is a triple bar.
• M ≡ T
• (B v D) ≡ (A • C )
• [K v (F ⊃ I ) ] ≡ [L • (G v H) ]

Whenever more than two letters appear in a translated statement, we must use parentheses,
brackets, or braces to indicate the proper range of the operators.
Prozac relieves depression and Allegra combats allergies, or Zocor lowers cholesterol. (P • A) ∨ Z
Prozac relieves depression, and Allegra combats allergies or Zocor lowers cholesterol. P • (A ∨ Z)
Either Prozac relieves depression and Allegra combats allergies or Zocor lowers cholesterol. (P • A) ∨ Z
Prozac relieves depression and either Allegra combats allergies or Zocor lowers cholesterol. P • (A ∨ Z)
Prozac relieves depression or both Allegra combats allergies and Zocor lowers cholesterol. P ∨ (A • Z)
Prozac relieves depression and Allegra or Zocor lowers cholesterol. P • (A ∨ Z)
If Merck changes its logo, then if Pfizer increases sales, then Lilly will reorganize. M ⊃ (P ⊃ L)
If Merck’s changing its logo implies that Pfizer increases sales, then Lilly will reorganize. (M ⊃ P) ⊃ L
If Schering and Pfizer lower prices or Novartis downsizes, then Warner will expand production. [(S • P) ∨ N] ⊃ W

• Do not confuse these three statement forms:
A if B B ⊃ A
A only if B A ⊃ B
A if and only if B A ≡ B
Further Examples:
It is not the case that K or M ∼K ∨ M
Not both S and T ∼(S • T)
By De Morgan’s rule ∼(S • T) is equivalent to ∼S ∨ ∼T
Similarly:
Not either S or T ∼(S ∨ T)
Which by De Morgan’s rule is equivalent to ∼S • ∼T.

Further illustrations:
• Megan is not a winner, but Kathy is ∼ M • K
• Not both Megan and Kathy are winners ∼ (M • K)
• Either Megan or Kathy is not a winner ∼ M ∨ ∼ K
• Both Megan and Kathy are not winners ∼ M • ∼ K
• Not either Megan or Kathy is a winner ∼ (M ∨ K)
• Neither Megan nor Kathy is a winner ∼ (M ∨ K)
Notice the function of “either” and “both”:
• Not either A or B. ∼( A v B )
• Either not A or not B. ∼A v ∼B
• Not both A and B. ∼( A • B )
• Both not A and not B. ∼A • ∼B