Lecture 6: Validity for TFL arguments & Entailment

Any valuation makes every sentence letter either true or false
The truth-functional nature of the connectives then guarantees that

a valuation also determines the truth value of every sentence

Using Truth Tables

To determine whether an argument is valid or invalid:

Show that:

Either there is no valuation, or there is at least one valuation that makes all premises T and the conclusion F

1. Determine the sentence letters occurring in the argument.

2. Generate all possible valuations, i. e., assignments of T and F to the sentence letters.

3. Determine the truth values of the premises and the conclusion.

4. Check each valuation (i. e., line in the truth table) for

whether it makes all premises T and the conclusion F:

5. Can we find such a valuation?

• Yes: The argument is invalid.

• No: The argument is valid

Modus ponens

A ! B

) B

Modus tollens

A ! B

¬B

) ¬A

Hypothetical syllogism

A ! B

B ! C

) A ! C

Argument by cases

A _ B

A ! C

B ! C

) C

Constructive dilemma

A _ B

A ! C

B ! D

) C _ D

The last example has 4 sentence letters, so we need 24 = 16 lines in the truth table

Invalid argument examples:

Denying the conjunct

¬(A ^ B)

¬B

) A

Affirming the consequent

(A ! B)

) A

Denying the antecendent

(A ! B)

¬A

) ¬B

Fallacy of the undistributed middle

(A ! B)

(C ! B)

) (A ! C )

Entailment

Instead of saying that the argument A1 , ... , An ) B is valid,

we can also express that as a relation between sentences:

A1 , ... , An entail B,

also written as

A1 , ... , An |= B.

The term ‘entailment’ and the symbol ‘ |= ’ are standard in logic.

We can now reformulate the earlier definitions:

Definition

The sentences A1 , ... , An entail the sentence B

iff no valuation makes all of A1 , ... , An true and B false.

Alternatively, sentences A1 , ... , An entail a sentence B

iff every valuation either makes at least one of A1 , ... , An false or makes B true.

In this case we write A1 , ... , An ✏ B