Chapter Three, Tutorial Four Consider this pair of sentences: Bob will attend law school only if he does well on the LSAT's.
(B>W) Don't these sentences sound like about the same thing? They Let's try to recall just how we defined logical equivalence back in chapter one. Two members of a pair of sentences are logically equivalent if and only if ______? If and only if what??? Select the correct way to fill in the blank: |

Good!

Our definition is this:

The two members of a pair of sentences are logically equivalent if and only if it is not possible for one of the pair to be true while the other is false.

Now, in SL we can give a much more precise account of logical equivalence. You guessed it: this will be done by way of truth value assignments and truth tables.

As usual, we talk about possibility by way of truth value assignments. So, all the possibilities for a pair of sentences can be summarized in terms of the rows of a truth table. We make the definitions and table tests much like those of the last chapter:

The two members of a pair of sentences are logically
equivalent *in SL* if and only if there is no truth value assignment
making one member of the pair true and the other false.

Notice that this definition of logical equivalence *in SL* is just like
the general definition of logical equivalence except that in SL we can say
exactly what is possible in terms of truth value assignments.

Now, for our truth table test.

To test for the logical equivalence in SL, one simply constructs
* ONE* truth table for whatever pair of sentences is in question
and checks to see that there is

Next, let's apply this test to the example which started this page.

Bob will attend law school only if he does well on the LSAT's.
(B>W)

If Bob doesn't do well on the LSAT's, he will not attend law school. (~W>~B)

We said that these sentences are logically equivalent. Now we can prove it. We simply do a single truth table for the pair (as symbolized).

B | W | B | > | W | ~ | W | > | ~ | B | ||

T | T | T | T | T | F | T | T | F | T | ||

T | F | T | F | F | T | F | F | F | T | ||

F | T | F | T | T | F | T | T | T | F | ||

F | F | F | T | F | T | F | T | T | F |

Hmm...this looks like a mess at first. But it's just a single table for both
'B>W' and for '~W>~B'.
The two columns under the main connective of each sentence are highlighted.
Nothing else of the mess matters. So, all you need to do is notice that *on
each row, the two sentences have the same truth value*.

For example, in row one both are true (both yellow columns have 'T' in row one, i.e., both sentences are true in the first possibility). In row two both are false. And so on. The two yellow columns and, so, the truth values of the pair coincide exactly. They are equivalent.

Now you try one. Consider the following table very much like the earlier one.

B | W | W | > | B | ~ | W | > | ~ | B | ||

T | T | T | T | T | F | T | T | F | T | ||

T | F | F | T | T | T | F | F | F | T | ||

F | T | T | F | F | F | T | T | T | F | ||

F | F | F | T | F | T | F | T | T | F |

This table differs because it includes 'W>B' rather than 'B>W'. Let's try to see what this table means. Which of the following is true of the above table? Click on all correct answers.

- The table may be used to test for logical equivalence of the pair 'W>B' and '~W>~B'.
- The table is not quite correct. Under 'W' in the sentence 'W>B' one finds a column: "TFTF". But it should be "TTFF".
- The table is correct. Under 'W' in the sentence 'W>B' one finds a column: "TFTF". Only under 'B' should there be "TTFF".
- 'W>B' and '~W>~B' are logically equivalent because they are both true in the first row.
- 'W>B' and '~W>~B' are NOT logically equivalent because they have different truth values in rows two and three.