An important and useful principle for understanding a language is the difference between syntax and semantics. “Syntax” refers to the “shape” of an expression in our language. It does not concern itself with what the elements of the language mean, but just specifies how they can be written out.
We can make a similar distinction (though not exactly the same) in a natural language. This expression in English has an uncertain meaning, but it has the right “shape” to be a sentence:
Colorless green ideas sleep furiously.
In other words, in English, this sentence is syntactically correct, although it may express some kind of meaning error.
An expression made with the parts of our language must have correct syntax in order for it to be a sentence. Sometimes, we also call an expression with the right syntactic form a “well-formed formula”.
We contrast syntax with semantics. “Semantics” refers to the meaning of an expression of our language. Semantics depends upon the relation of that element of the language to something else. For example, the truth value of the sentence, “The Earth has one moon” depends not upon the English language, but upon something exterior to the language. Since the self-standing elements of our propositional logic are sentences, and the most important property of these is their truth value, the only semantic feature of sentences that will concern us in our propositional logic is their truth value.
Whenever we introduce a new element into the propositional logic, we will specify its syntax and its semantics. In the propositional logic, the syntax is generally trivial, but the semantics is less so. We have so far introduced atomic sentences. The syntax for an atomic sentence is trivial. If P is an atomic sentence, then it is syntactically correct to write down
By saying that this is syntactically correct, we are not saying that P is true. Rather, we are saying that P is a sentence.
If semantics in the propositional logic concerns only truth value, then we know that there are only two possible semantic values for P; it can be either true or false. We have a way of writing this that will later prove helpful. It is called a “truth table”. For an atomic sentence, the truth table is trivial, but when we look at other kinds of sentences their truth tables will be more complex.
The idea of a truth table is to describe the conditions in which a sentence is true or false. We do this by identifying all the atomic sentences that compose that sentence. Then, on the left side, we stipulate all the possible truth values of these atomic sentences and write these out. On the right side, we then identify under what conditions the sentence (that is composed of the other atomic sentences) is true or false.
The idea is that the sentence on the right is dependent on the sentence(s) on the left. So the truth table is filled in like this:
|Atomic sentence(s) that compose the dependent sentence on the right||Dependent sentence composed of the atomic sentences on the left|
All possible combinations of truth values of the composing atomic sentences
|Resulting truth values for each possible combination of truth values of the composing atomic sentences|
We stipulate all the possible truth values on the bottom left because the propositional logic alone will not determine whether an atomic sentence is true or false; thus, we will simply have to consider both possibilities. Note that there are many ways that an atomic sentence can be true, and there are many ways that it can be false. For example, the sentence, “Tom is American” might be true if Tom was born in New York, in Texas, in Ohio, and so on. The sentence might be false because Tom was born to Italian parents in Italy, to French parents in France, and so on. So, we group all these cases together into two kinds of cases.
These are two rows of the truth table for an atomic sentence. Each row of the truth table represents a kind of way that the world could be. So here is the left side of a truth table with only a single atomic sentence, P. We will write “T” for true and “F” for false.
There are only two relevant kinds of ways that the world can be, when we are considering the semantics of an atomic sentence. The world can be one of the many conditions such that P is true, or it can be one of the many conditions such that P is false.
To complete the truth table, we place the dependent sentence on the top right side, and describe its truth value in relation to the truth value of its parts. We want to identify the semantics of P, which has only one part, P. The truth table thus has the final form:
This truth table tells us the meaning of P, as far as our propositional logic can tell us about it. Thus, it gives us the complete semantics for P. (As we will see later, truth tables have three uses: to provide the semantics for a kind of sentence; to determine under what conditions a complex sentence is true or false; and to determine if an argument is good. Here we are describing only this first use.)
In this truth table, the first row combined together all the kinds of ways the world could be in which P is true. In the second column we see that for all of these kinds of ways the world could be in which P is true, unsurprisingly, P is true. The second row combines together all the kinds of ways the world could be in which P is false. In those, P is false. As we noted above, in the case of an atomic sentence, the truth table is trivial. Nonetheless, the basic concept is very useful, as we will begin to see in the next chapter.
One last tool will be helpful to us. Strictly speaking, what we have done above is give the syntax and semantics for a particular atomic sentence, P. We need a way to make general claims about all the sentences of our language, and then give the syntax and semantics for any atomic sentences. We do this using variables, and here we will use Greek letters for those variables, such as Φ and Ψ. Things said using these variables is called our “metalanguage”, which means literally the after language, but which we take to mean, our language about our language. The particular propositional logic that we create is called our “object language”. P and Q are sentences of our object language. Φ and Ψ are elements of our metalanguage. To specify now the syntax of atomic sentences (that is, of all atomic sentences) we can say: If Φ is an atomic sentence, then
is a sentence. This tells us that simply writing Φ down (whatever atomic sentence it may be), as we have just done, is to write down something that is syntactically correct.
To specify now the semantics of atomic sentences (that is, of all atomic sentences) we can say: If Φ is an atomic sentence, then the semantics of Φ is given by
Note an important and subtle point. The atomic sentences of our propositional logic will be what we call “contingent” sentences. A contingent sentence can be either true or false. We will see later that some complex sentences of our propositional logic must be true, and some complex sentences of our propositional logic must be false. But for the propositional logic, every atomic sentence is (as far as we can tell using the propositional logic alone) contingent. This observation matters because it greatly helps to clarify where logic begins, and where the methods of another discipline ends. For example, suppose we have an atomic sentence like:
Force is equal to mass times acceleration.
Igneous rocks formed under pressure.
Germany inflated its currency in 1923 in order to reduce its reparations debt.
Logic cannot tell us whether these are true or false. We will turn to physicists, and use their methods, to evaluate the first claim. We will turn to geologists, and use their methods, to evaluate the second claim. We will turn to historians, and use their methods, to evaluate the third claim. But the logician can tell the physicist, geologist, and historian what follows from their claims.