Chapter Two, Tutorial Three

Semantics and Syntax for SL

A semantics is a definition of meaning for a language. But 'meaning' itself has a number of meanings. Here we fix on a simple idea of meaning: truth conditions.

To understand, say, the meaning of

Sam is riding a horse

is to know under what conditions this statement would be true. That is, one would need to know who Sam is, what riding is, and what a horse is. Then one would know the meaning well enough to understand just when it would be true.

So, we'll give meaning in terms of truth conditions. Let's start with about the easiest case: conjunctions.

Conjunction

So, one way to think about meaning is in terms of truth conditions. We shall apply this idea to molecular sentences of SL (without names). Begin with our simple example

A&B

symbolizing the English "Agnes and Bob will attend law school."
Under what conditions is this true? Obviously it's true when and only
when both 'A' and 'B' are true. This doesn't sound too interesting yet,
but be patient!

First, we will say that any conjunction of SL is like this (not just 'A&B' but also 'C&T', 'F&~G', 'F&(T>U)', etc.): each is true just in case both its conjuncts are true. We will put this general point as follows:

1a. *Any* sentence of SL of the form

&

is __true__ if and only if both and are true.

The idea is that you can fill in the box with any true sentence of SL and the oval with any other true sentence of SL and the result, &, is also true. So, for example:

Let's just fill in for our two placeholders, box and oval.

If we let 'A' stand for "The author of the Logic Café is president of the US" and 'B' stand for "Kobe Bryant is male", then we know that ~A is true and that B is true.

So, 1a tells us that

is __true__ if and only if both are true.

So, because the sentences filling in the box and oval are both true, our definition tells us that '~A&B' is true.

Then what about 'A&~B', this is true? false?

Yes.

'A&~B' is false because both of its conjuncts are false. We could write this as follows:

The first conjunct -- in the box -- says that I'm US president. So, it's false. The second says that Kobe is not male. So, it's also false as the picture describes. Then *applying 1a*, we see that our conjunction is not true.

That's it! The final result is that the conjunction, "A&~B" is false.

But this is awfully complicated. We need a better way.

Our and are rather large place-holders. Something smaller will help us give a compact statement of truth conditions for all of SL. But, you'll need to remember that our compact replacement are just that: they are place-holders too.
So, instead of the box and oval we'll use...special metavariables.

"Metavariables"...quite a term. But a metavariable is really just a placeholder like our box or oval. So, don't be put off. But, for the sake of saving space, we'll write 'P' instead of box and 'Q' instead of oval. Just remember: 'P' and '*Q*' are just like box and oval: placeholders.

Now we can rewrite 1 in clearer fashion:

1b. Any sentence P&Q of SL is true if and only if both P and Q are true. Otherwise P&Q* *is false.

...that 'P' and '*Q*' are have a different shape and color. This is our way of saying that we are talking about any sentences P and Q.

'P' and 'Q' are not *particular* sentences of SL. Instead they each act as stand-ins or placeholders for an arbitrary sentence of SL. We will call them variables or metavariables.

Similarly, 'x' and 'y' in 'x+y = y+x' do not stand for any*particular* numbers. Rather, these variables provide a way to talk about any pair of numbers.

OK, time to get down to business defining semantics.Similarly, 'x' and 'y' in 'x+y = y+x' do not stand for any

Truth Tables

There is one more tool we need for clearer semantics: the truth
table. A truth table will just *re-express* our definition
1.

When P and Q
are both true, then P&Q
is **true**. We've already said that, of course.

...if P is true, but Q is false, then P&Q is **false**. (Because the only way for P&Q to be true is for both P and Q to be true – that's what 1 says.)

We are cashing out what 1 amounts to.

The results so far can can be summarized in a table:

If P is true and Q is true | then... | 'P&Q' is true |

If P is true but Q is false | then... | 'P&Q' is false |

Better, we can simplify this table
as follows:

P | Q | P&Q | |

possibility one: | T | T | T |

possibility two: | T | F | F |

Here each *row* just means that if P and Q have the truth values given in that row, then P&Q has the truth value shown there in the far right column of that row. Each row, then, represents a possible assignment of truth values to P and Q

Have you noticed anything missing?

There are two more rows to give for our table. We have neglected a row mentioning the possibility that P and Q are both false: