Tutorial Five
Possibility, Semantics, Truth Tables
In our symbolic language we can spell out just what possibility amounts to.
Why is that a big deal? Well, this will allow us easy application of the deductive concepts, all of which are defined in terms of possiblity. It will also allow us to precisely say what various sentences of our language mean. So, it is a pretty big deal.
Here's the idea. Think about a very simple sentence.
Wa&~Wb
Agnes will but Bob will not attend law school. Let's think about what this means.
First, the '~' means negation or "not". So, modeling our semantics on English, a sentence like '~Wb' is true if 'Wb' is false. Yes? And, of course, if 'Wb' is true, then '~Wb' is false. (This is just to say that "Bob won't attend law school" is true if and only if "Bob will attend law school" if false.)
Now, what we've just agreed to about Bob and about '~Wb', is true about any negation. In order to put the point generally, we need to be able to describe any negation. To do so we'll write "~X" to as a stand in for any negation, that is any negated sentence. Then the general point is straigtforward:
~X is true iff X is false.
We can even put this in terms of a table ('T' represents truth; 'F' represents falsehood):
X  ~X  
possibility one:  T  F 
possibility two:  F  T 
Variables work as standins or blanks to be filled in. They allow us to think in general terms. You are familiar with variables from elementary algebra:
x+y = y+x.
But what number is x? y? No particular number, of course. They are standins for any number, i.e., they range over all numbers.
Similarly, our X and Y are variables ranging over all sentences of our symbolic language. They are tools just used to describe the semantics of our symbolic language. Thus they are not a part of this symbolic language but are really a part of English. It is common to say that English is the "metalanguage", the language used to talk about the symbolic languge. Then X and Y are "metavariables" of English.
We use metavariables in order to mention sentences of our symbolic language. Similarly, in English we use quote marks to mention sentences of the symbolic langauge. Example:
'~Wa' is a negation.
In the above English statement I am only mentioning a symbolic logic sentence (the negation) not using it, not asserting it. The use/mention distinction is important in logic and philosophy.
In the next tutorial, we will complete the development of our symbolic language. At that time we will add variables that are part of that language.
You see, there are two possiblities for the truth value of X: either true or false. And if X is true, then it's negation is false (possibility one). And vice versa (possibility two). Make sure you read the table horizontally, from left to right.
Similarly for '&': A sentence like 'Wa&Wb' ("Anges and Bob will attend law school") is true just in case both components, both conjuncts, are true. That is, it's true if and only if both "Agnes will attend" and "Bob will attend" are true. Make sure this seems obvious to you...yes?...then notice that the next table expresses the idea generalized, that a conjunction X&Y is true just in case both X and Y are true:
X  Y  X&Y  
possibility one:

T  T  T 
possibility two:

T  F  F 
possibility three:

F  T  F 
possibility four:

F  F  F 
Make sure you see why there are exactly four possibilities that are relevant here, that there are exactly four possible ways to assigning truth values to X and Y.
We can give truth tables for each of our connectives. But first let's apply what we have at the moment:
Assuming that 'Wa' is true, what is the truth value of 'Wa&~Wa'?