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Chapter Three, Tutorial Three Back in chapter one, we introduced various logical notions trading on the concept of possibility. For example, we said that a sentence is logically true if and only if ______? If and only if what??? Select the correct way to fill in the blank: |
Good!
Our definition is this:
A sentence is logically true if and only if it could not possibly be false.
Now, we may do a truth table to see what is and is not possibly false. Here's an example:
| R | R | v | ~ | R | |
| T | T | T | F | T | |
| F | F | T | T | F |
Notice that the main connective of 'Rv~R' is the wedge. That is why the column under this connective is highlighted. It contains Rv~R's truth values.
For the sentence at hand, there are only the two possibilities: 'R' either true or false. But, as the table indicates, either way Rv~R' is true. Thus, we can tell that Rv~R' is logically true: there is no way for it to be false!
This does make sense: Rv~R' could symbolize "It's raining or it's not". This English statement is obviously true (at a place and time), but it's never an interesting meteorological fact! Rather, "It's raining or it's not" can be seen to be true just as a matter of how we use our language. (You don't have to look outside to ascertain its truth!)
We will call a sentence of SL "logically true in SL" just in case there is no truth value assignment (roughly, no row of a truth table) making it false.
(Notice how close this is to the chapter one definition. There "logically true" means: no possibility makes a sentence false. Now we just say what the possibilities are in the context of SL: truth value assignments.)
It should be clear how we test to see if a sentence is logically true in SL:
To test for logical truth in SL, one simply constructs a truth table for the sentence in question and checks to see that this sentence is not false in any row. (Remember, you are looking for truth values only under the main connective of the sentence.) If the sentence is never false in any row, then it is logically true. Otherwise it is not.
Now, let's apply this thinking. Which of the following are logically true in SL? (Do truth tables – on paper, in your head, or using the Web Widget for tables – to see/guess which of the following is false on no row of its truth table.)