An Example

Let x = 1.

Square both sides:  x2 = 1

Subtract 1 from both sides:  x2 – 1= 0

Factor:  (x+1)(x-1) = 0

Divide both sides by (x-1): x+1 = 0

Substitute the value of x:  1 + 1 = 0

Conclusion:  2 = 0.

Wason Selection Task

Consider the conditional:  If a card has a vowel on one side, there is an even number on the other side.  What is the minimum number of cards you would have to turn over to determine whether that statement is true of the four cards (a-d) below?  Which cards would they be?

 E 4 5 C

a                                 b                               c                          d

A PARADOX IN SEMANTIC THEORY

Convention (T):  (i) If p, then [p] is true; (ii) If [p] is true, then p.  (where 'p' is a substitutional variable that takes sentences as values, and '[p]' stands for the quotation of the sentence substituted for 'p')

Suppose also that for any statement p, either [p] is true or [-p] is true.  [LEM]

Suppose also that for any statement p, it is not the case that both [p] is true and [-p] is true.  [LNC]

Consider the statement:  (1) This sentence is not true.

(1) is equivalent to:

(2) "This sentence is not true" is not true.

Let us write out the equivalence:

(3) This sentence is not true iff "This sentence is not true" is not true.

By LEM, Either (1) is true or (1) is not true.

Suppose (1) is true.  That is, suppose:

(4) "This sentence is not true" is true.

Then from Convention (T), one can infer (1) itself:

(1) This sentence is not true.

And from (1) and (3) one can infer (2), which contradicts (4).

Suppose (1) is not true.  That is, suppose:

(5) "This sentence is not true" is not true.

From (5) and (3), one can infer (1).  From (1) and Convention (T) one can infer:

(6) "This sentence is not true" is true.

(6) contradicts (5).  So either supposition, that (1) is true or that (1) is not true, leads to a contradiction.