Deductive Entailment

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4 properties of Deductive Inference

- Non-ampletive
- All or nothing: All true premises and true conclusion (valid/cogent) or false
- Truth Preserving: premises all true. conclusion 100% true
- Erosion Proof:

Categorical Syllogism

Deductive Argument: conclusion is inferred from two premises.

- contains 3 terms
- each term occurs in exactly 2 inferences

Universal Affirmative

(A) All S are P

Universal Negative

(E) No S are P

Particular Affirmative

(I) Some S are P

Particular Negative

(O) Some S are not P

Square of Opposition

An arrangement of the 4 Categorical forms.

Contradictory, contrary, sub-contrary

(of a given statement)

All S is P(A) contradictory to Some S is not P(O)

No S is P(E) contradictory to Some S is P(I)

Contrary

(A) contrary to (E) : can’t both be true, but can both be false

ex:

All flowers are blue = false

No flowers are blue = false

sub-Contrary

(I) sub-contrary to (O) : can both be true but can’t both be false

Some flowers are blue = true

Some flowers are not blue = true

Distributed

(A) subject term = distributed. Predicate term = not.

(E) BOTH subject term and Predicate term is distributed

(I) NEITHER S or P is distributed

(O) subject term = not. Predicate term = distributed.

stereotyping

Problem with Universal Affirmative (A)

- putting them into categories and making universal judgments about all or most members of the category.

ex: All Canadians like snow.

Major term

(predicate term) Term that appears in the predicate position in the conclusion of the syllogism. 1. major term 2. minor term 3. conclusion

Minor term

(subject term)

Term that appears in the subject position in the conclusion of a syllogism

- 2nd premise

Middle term

Term that occurs in both premises of the syllogism and not the conclusion.

- enables us to logically deduce the premise to the conclusion.
- MUST be distributed in at least ONE premise

Fallacy of the undistributed middle

Fallacy committed when the middle term of the syllogism is not distributed in at least 1 premise.

Mood of syllogism

The letters A E I O make the mood

ex: 1. No heroes are cowards

2. Some soldiers are cowards

3. Therefore, some soldiers are heroes

Mood: E I I

4 figures

(1)M-P

S-M

therefore S-P

(2) P-M

S-M

Therefore S-P

(3) M-P

M-S

therefore S-P

(4) P-M

M-S

therefore S-P

Rules for testing Deductive Validity in a Categorical Syllogism

Rule #1: For a syllogism to be valid, the middle term must be distributed in at least one premise

Rule #2: For a syllogism to be valid, no term can be distributed in the conclusion unless that term is also distributed in at least one premise

Rule #3: For a syllogism to be valid, at least one premise must be affirmative

Rule #4: For a syllogism to be valid, if it has a negative conclusion, it must have a negative premise. And if it has one negative premise, it must also have a negative conclusion.

Rule #5: If a syllogism has two universal premises, it cannot have a particular conclusion and be valid.

Steps for the Categorical Syllogism

- Put into form
- identify major term, minor term and middle term
- Identify if each line is A E I or O
- Identify what’s distributed
- Identify the mood
- Apply the rules. Is it valid or nah?

Laughter is the best medicine

form:

All things that are laughter are things that are the best medicine

What is a deductively valid argument?

If the premises are true. The conclusion also must be true.

types:

- affirming the antecedent

- Denying the consequence

- Hypothetical syllogism

- Disjunctive syllogism

What is a deductively invalid argument?

if the premises are true but the conclusion is false.

types:

- Denying the antecedent

- Affirming the consequence

Affirming the antecedent

(deductively valid argument) (modus ponens) 1. if P, then Q 2. P 3. therefore, Q

Denying the consequence

(deductively valid argument) (modus tollens) 1. If P, then Q 2. not P 3. therefore, not Q

Hypothetical syllogism

(deductively valid argument)

Disjunctive syllogism

(deductively valid argument)

- Either P or Q
- Not P
- therefore Q.

Denying the antecedent

(deductively invalid argument)

affirming the consequence

(deductively invalid argument)