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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 should because they are logically equivalent. Roughly, that is, they mean the same thing. 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 no row making one sentence true and the other false. If and only if there is no such row, the members of the pair are logically equivalent.
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.