Deduction

• Deductive arguments, deduction, deductive logic all means thesame thing.– They are different ways of referring to the same style of

reasoning• Deduction is just one mode of reasoning, but it is a formidable form

of reasoning. Deductive reasoning is common in the ‘imperialdisciplines.

• Deductive reasoning is prized because deductive arguments havethe highest standards of cogency.

– Of all the modes of reasoning, deductive arguments have the strongestrelationship between the premises

• Recall: we reserve the term sound to describe good deductivearguments.

Recall: Validity

• In day-to-day life, we talk about a valid driver’s license, or the validation of software serial number.

• Validity is a technical term in deductive logic.• If an argument has an appropriate deductive form, then we say

that the argument is valid.• Recall: deductive logic is just one mode of reasoning, though it is

a very powerful form of reasoning.

• A deductive argument is either “valid” or “invalid”. • When we say that an argument is invalid we mean that we are rejecting

the argument because it doesn’t have the right deductive form.• When we say this, we don’t care about the content of the premises.

• That is, we don’t care what the premises are about.

Recall: Validity (more)

• If an argument has a valid form and the premises are true, then the conclusion of the argument must be true.– The conclusion is deductively entailed.

• But, an argument may be valid (have the right form or structure) but not be true. It’s premises could be false.

• Important: Committing to the validity of an argument, does not necessarily commit you to the truth conclusion.– Likewise, asserting the invalidity of an argument, does not necessarily

commit you to rejecting the conclusion. (You might accept the conclusion, but reject the form of the argument given for the conclusion.)

Validity IllustratedP1: If it’s whirly-gig then smack-gurgle.P2: This is a whirly-gig.C1: This is also a smack-gurgle.

• This is a valid argument. We recognize the form as being valid. Allarguments of this form are valid:

P1: If P then Q.P2: PC1: Therefore, Q

• But, not all arguments of this form necessarily produce true conclusions. Inthis case, we don’t know if the premises are true or false.• We only know that argument is valid. We know nothing about its truth!

An example of validity• Premise: All apples are fruit.• Premise: All fruit contain antioxidants.• Conclusion: All apples contain antioxidants.

• Premise: All aardvarks have long noses.• Premise: Anything with a long nose has an acute sense of smell.• Conclusion: All aardvarks have an acute sense of smell.

• Logical form:• All As are Bs.• All Bs are Cs.• Therefore, all As are Cs.

A Valid But Untrue SyllogismConsider this syllogism again:

P1: All men are purple.P2: Socrates is a man.C1: Socrates is purple.

This syllogism also has the form:P1: All S are PP2: M is SC1: M is P

Recall: Truth and Validity are different things. Validity guarantees (and only guarantees) that if the premises are true the conclusion will be true.

In this case, P1 is false. So, the conclusion is false. Nonetheless, the form of the syllogism is valid.

Important!Validity and Truth are Different.

Example 1:P1: All men are mortal.P2: Socrates is a man.C1: Socrates is mortal.(Valid and True)

Example 2:P1: All men are purple.P2: Socrates is a man.C1: Socrates is purple.(Valid but false)

Common Form:All S are PM is an S.Therefore, M is P

Note: Example 1 and Example 2 have exactly the same form or structure of argument. This is one valid form of deductive argument.

Valid Deductive Arguments are Truth Preserving

Look at Example 2 again:P1: All men are purple.P2: Socrates is a man.C1: Socrates is purple.(Valid but false)

• What went wrong? Clearly, P1 is false. It is not true that “All men are purple.”

• Hypothetically, if P1 were true, then C1 would be true too!

• Valid deductive arguments are truth preserving.– That’s a fancy way of saying that: in valid deductive arguments are machines the

gobble up true premises and split out true conclusions.

Why we ♥ deduction …• In deductive arguments, the premises entail the

conclusion.

• This is because valid deductive arguments are truth preserving.

• And so, it is not possible for the premises of a deductive argument to be true and the conclusion false.

• In more detail:If (1) the premises are true,And (2) the form of the deductive argument is valid,Then (3) the conclusion must be true

… always and forever which is why ‘imperial disciplines’ love deduction!

Deductive Logic• Often when people say “logic” they mean “deductive logic”

– There are other logics, notably, inductive logic.

• Deductive logic is studies the valid forms of deductive argument.– It looks at the forms of argument that preserve truth.– It looks at ways of testing arguments to see if they are valid (truth

preserving).

• Important: Form NOT Content!– Deductive logic is concerned purely with forms of argument, not

the content.– It is not concerned with truth. It is only concerned with figuring

out the arguments that preserve truth.

Scope of Course Concern

• There are many types of deductive logic.– We will be looking briefly at one: categorical logic– The aim is to understand what deductive arguments look likes so

you know when you are dealing with a situation involving deductive entailment.

• Shameless advertisement: If you want deductive logic, take PHIL2210 and then PHIL3110.

Categorical Logic

• Aristotle (384-322BCE), an Ancient Greek philosopher, showed that some good arguments have common shapes, forms or patterns.

• Aristotle thought of logic as a tool or organon for inquiry.

• We call Aristotle’s tool categorical logic, or syllogistic logic, or simply syllogism.

Four Categorical Statements

1) Universal Affirmative – All S is P“All men are mortal”“Every man is an island”

2) Universal Negative – No S is P“No conservatives are fiscally responsible.”“No penguins are flying birds.

3) Particular Affirmative – Some S is P“Some guinea pigs have long hair.”“Some Tories are good people”

4) Particular Negative – Some S is not P“Some people are not generous”“Some rocks are not sedimentary.”

Variables

• The four forms of the syllogism are: All S are P; No S are P; Some S are P; Some S are not P.

• We use ‘S’ and ‘P’ to stand for the specific content of the statements. – We do this because we are only interested in the form of the

statements.– The use of variables helps us achieve generality.

• We say that ‘S’ and ‘P’ are variables.– The specific content of S and P may change or vary.

Categories

• We use ‘S’ to stand for the subject of the statement (what the statement is about) and ‘P’ to stand for the predicate of the statement (what is begin said about the subject.)– The choice of ‘S’ and ‘P’ as variables is entirely

conventional. We could use any letter or other symbol

• Categorical logic is the logic of categories.– The subject and predicate will always be categories.

• Any category can include only one thing, five things, or everything. Likewise, a category can exclude only one thing, five things, or everything.

Examples

• All robins are birds. – Every thing in the category robin is a member of the category

birds.

• No elephants are reptiles.– Everything in the category elephant is not a member of the

category reptile.

• Some mammals are marsupials.– Some of the things in the category of mammals are members of

the category or marsuplials.

• Some animals are not viviparous.– Some of the things that are animals are not things that are

viviparous.

Categorical Form

• We make many assertions about categories, but most of the time the sentences that we utter are not exactly in categorical form.

• Nevertheless, it is still plausible to represent their content in the categorical form. – That is, we can translate many different kinds of

sentences into sentences with one of the four categorical forms.

Everyday sentences can be translated into categorical sentences and categorical sentences can be translated into standard form …

• Every robin is a bird.All robins are birds.All S are P

• Every elephant is not a reptile.No elephants are reptiles. No S are P

• There are a few mammals that are marsupialsSome mammals are marsupials. Some S are P

• You can find animals that are not viviparous.Some animals are not viviparous. Some S are not P

Translation into Categorical Form

(1) Stylistic variants: many different sentences can assert exactly the same categorical relation. – “All Texans are ornery”, “Every single Texan is ornery”, “Any

Texan is ornery”, “There is no Texan who is not ornery.”

(2) ‘A’ and ‘The’ are ambiguous between a particular and a universal (or generic) sense.

– “A lion is a mammal” makes a different claim that “A lion just escaped from the zoo”

(3) ‘Any’ is generally universal, but is particular when inside the scope of a ‘not’.

– Compare ‘Amy can solve any problem’ and ‘Amy cannot solve any problem’.

(4) ‘Only’: Like ‘all’, ‘only’ asserts a universal affirmative, but it asserts the opposite inclusion relation.

– ‘Only Ss are P’ translates into Categorical form as ‘All Ps are S’.

Syllogism

• Syllogisms are a basic form of argument.• Syllogisms have three lines; two premises and one

conclusion.

• You see lots of syllogisms. Here are two you have seen in the course:

P1: All men are mortal.P2: Socrates is a man.C1: Socrates is mortal.

P1: All three-sided figures are triangles.P2: This figure has three sides.C1: This figure is a triangle.

Anatomy of a Syllogism• A syllogism consists of three sentences. Two premises and one

conclusion.• Each sentence is made up of two terms.

– Every term occurs twice in a syllogism.

• Count the terms in the following syllogismP1: All S are MP2: No M are PC1: No S are P

• Each unique term that makes up the syllogism has a particular name.– The predicate, P, of the conclusion is the major term.– The subject, S, of the conclusion is the minor term.– The term that appears in both premises but not the conclusion is

the middle term.

Translating Syllogisms into Categorical Form

P1: All men are mortal.P2: Socrates is a man.C1: Socrates is mortal.

P1: All three-sided figures are triangles.P2: This figure has three sides.C1: This figure is a triangle.

P1: All S are PP2: M is SC1: M is P

P1: All S are PP2: M is SC1: M is P

• These syllogisms have different contents but the same logical form! They are valid for the same reason.

Translate Syllogism into Categorical Form

“No lawyers are golfers, but some golfers are doctors. Therefore, no lawyers are doctors.”

No L are G.Some G are D.Therefore, no L are D.

Translation“Only doctors are golfers, and any golfer has good turf shoes. So, all doctors have good turf shoes.”

All G are D.All G are S.Therefore, all D are S.

Venn Diagrams

• In Symbolic Logic (1881), English mathematician John Venn (1834-1923) invented a way to pictorially represent categorical statements, and to test syllogisms for validity.

• Every category is represented by a circle. We overlap the circles, so that we can represent inclusion and exclusion relations between categories.

One Category

One Category

Two Categories

Lawyers Golfers

Two Categories

Lawyers Golfers

Lawyers who are also golfers.

Golfersonly

Lawyers only

Picturing Categorical Statements

• We can use Venn diagrams to picture categorical statements and syllogisms.

• We use Venn diagrams to picture logical space.

• “[In Venn diagrams] logical space is represented in circles and parts of circles. To indicate that there is nothing in an area of logical space, we shade in the area. To indicate that there is something, we put an x in the space.” [216]

Universal Affirmative

“All lawyers are golfers.”“All S are P.”

P“golfers”

S“lawyers”

Universal Negative

“No lawyers are golfers.”“No S are P.”

P“golfers”

S“lawyers”

Particular Affirmative

“Some lawyers are golfers.”“Some S are P.”

X

P“golfers”

S“lawyers”

Particular Negative

X

“Some lawyers are not golfers.”“Some S are not P.”

S“lawyers”

P“golfers”

Venn Diagrams and Syllogisms• We can use Venn diagrams to picture syllogisms, AND

to check whether or not the syllogism is valid or invalid.

• Consider the syllogism:All S are PNo P is MTherefore, no S is M.

• Every categorical statement has two terms, and so we pictured categorical statements with two circles, one for each term.

• Since a syllogism has three terms, we will need three circles to picture the syllogism, again one for each term.

Venn Diagram for a Syllogism

M

PS

An Example

• Consider the syllogism:– All liberals are Keynesians.– No Keynesians are monetarists.– Therefore, no liberal is a monetarist.

• The syllogism has the form:All S are MNo M are PTherefore, no S are P.

• We can picture this syllogism in a Venn diagram.

P1: All S are MP2: No M are PC1: No S are P

S P

M

** P1: All S are M **P2: No M are PC1: No S are P

S P

M

** P1: All S are M **P2: No M are PC1: No S are P

S P

M

P1: All S are M** P2: No M is P **

C1: No S is P

S P

M

Validity and Invalidity

• In valid deductions, the conclusion states no more information than has already been stated in the premises.– This is why valid deductive arguments are truth preserving.

They simply preserve whatever truth happens to have been stated in the premises.

• If the conclusion states more than has been stated in the premises, then the conclusion is invalid.

• To see if a conclusion is valid or invalid, we simply check to see if the information in the conclusion is already pictured in the Venn diagram.

– If the information is already on the diagram, then the syllogism is valid.– If the information is not already on the diagram – if something new has been

stated in the conclusion – then the syllogism is invalid.

P1: All S are MP2: No M is P

** C1: No S is P **

S P

M

Valid! The area in green picturing “No S is P” was already shaded black and red. So, all the information in the conclusion is information already in the premises.

Another Example

P1: Some people who smoke also drink.P2: Some people who drink take risks.C1: Some people who smoke take risks.

P1: Some S are MP2: Some M are PC1: Some S are P

** P1: Some S are M **P2: Some M are PC1: Some S are P

S P

M

X

** P1: Some S are M **P2: Some M are PC1: Some S are P

S P

M

XX

P1: Some S are M ** P2: Some M are P **

C1: Some S are P

S P

M

XX X

P1: Some S are MP2: Some M are P

** C1: Some S are P **

S P

M

XX X

X (!)

Invalid! Note that the X marked with an exclamation point was not information included in either of the premises.

Like forms have like properties of validity and invalidity

• We began with examples that included content. We replaced the content with variables.– This allows us to achieve generality.– The specific content of an argument does not affect its validity or

invalidity.• From the first example, all syllogisms are valid if they have the form:

P1: All S are MP2: No M are PP3: No S are P.

• From the second example, all syllogisms are invalid if they have the form:P1: Some S are MP2: Some M are PC1: Some S are P

• Keep in mind, these are just specific examples. There are, of course, many other valid syllogism forms. We will need to test each to see if it is a valid or invalid form.

Last Example

P1: All fuzzies are dolittles.P2: Some dolittles are mugwumps.C1: No fuzzies are mugwumps.

P1: All S are MP2: Some M are PC1: No S are P

• We are going to do this example in a single diagram.

P1: All S are MP2: Some M are PC1: No S are P

S P

M

XX

Invalid! Note the shading and the X in the very center section indicates a contradiction.

Summary of Some Key Points

• All categorical statements have one of four forms. Each has a unique Venn diagram with two circles.

• We can use a Venn diagram with three circles to represent a syllogism and test whether the syllogism is valid or invalid.

• Remember that validity and invalidity has nothing to do with truth and falsity.

• Valid arguments preserve truth. In a valid argument, if the premises are true, then the conclusion will be true.

• Valid deductive arguments provide the strongest relationship between premises and conclusion. Such arguments are sound; i.e., possess the highest standard of cogency.