We want our logic of declarative sentences to be precise. But what does this mean? We can help clarify how we might pursue this by looking at sentences in a natural language that are perplexing, apparently because they are not precise. Here are three.
Tom is kind of tall.
When Karen had a baby, her mother gave her a pen.
This sentence is false.
We have already observed that an important feature of our declarative sentences is that they can be true or false. We call this the “truth value” of the sentence. These three sentences are perplexing because their truth values are unclear. The first sentence is vague, it is not clear under what conditions it would be true, and under what conditions it would be false. If Tom is six feet tall, is he kind of tall? There is no clear answer. The second sentence is ambiguous. If “pen” means writing implement, and Karen’s mother bought a playpen for the baby, then the sentence is false. But until we know what “pen” means in this sentence, we cannot tell if the sentence is true.
The third sentence is strange. Many logicians have spent many years studying this sentence, which is traditionally called “the Liar”. It is related to an old paradox about a Cretan who said, “All Cretans are liars”. The strange thing about the Liar is that its truth value seems to explode. If it is true, then it is false. If it is false, then it is true. Some philosophers think this sentence is, therefore, neither true nor false; some philosophers think it is both true and false. In either case, it is confusing. How could a sentence that looks like a declarative sentence have both or no truth value?
Since ancient times, philosophers have believed that we will deceive ourselves, and come to believe untruths, if we do not accept a principle sometimes called “bivalence”, or a related principle called “the principle of non-contradiction”. Bivalence is the view that there are only two truth values (true and false) and that they exclude each other. The principle of non-contradiction states that you have made a mistake if you both assert and deny a claim. One or the other of these principles seems to be violated by the Liar.
We can take these observations for our guide: we want our language to have no vagueness and no ambiguity. In our propositional logic, this means we want it to be the case that each sentence is either true or false. It will not be kind of true, or partially true, or true from one perspective and not true from another. We also want to avoid things like the Liar. We do not need to agree on whether the Liar is both true and false, or neither true nor false. Either would be unfortunate. So, we will specify that our sentences have neither vice.
We can formulate our own revised version of the principle of bivalence, which states that:
Principle of Bivalence: Each sentence of our language must be either true or false, not both, not neither.
This requirement may sound trivial, but in fact it constrains what we do from now on in interesting and even surprising ways. Even as we build more complex logical languages later, this principle will be fundamental.
Some readers may be thinking: what if I reject bivalence, or the principle of non-contradiction? There is a long line of philosophers who would like to argue with you, and propose that either move would be a mistake, and perhaps even incoherent. Set those arguments aside. If you have doubts about bivalence, or the principle of non-contradiction, stick with logic. That is because we could develop a logic in which there were more than two truth values. Logics have been created and studied in which we allow for three truth values, or continuous truth values, or stranger possibilities. The issue for us is that we must start somewhere, and the principle of bivalence is an intuitive way and—it would seem—the simplest way to start with respect to truth values. Learn basic logic first, and then you can explore these alternatives.
This points us to an important feature, and perhaps a mystery, of logic. In part, what a logical language shows us is the consequences of our assumptions. That might sound trivial, but, in fact, it is anything but. From very simple assumptions, we will discover new, and ultimately shocking, facts. So, if someone wants to study a logical language where we reject the principle of bivalence, they can do so. The difference between what they are doing, and what we will do in the following chapters, is that they will discover the consequences of rejecting the principle of bivalence, whereas we will discover the consequences of adhering to it. In either case, it would be wise to learn traditional logic first, before attempting to study or develop an alternative logic.
We should note at this point that we are not going to try to explain what “true” and “false” mean, other than saying that “false” means not true. When we add something to our language without explaining its meaning, we call it a “primitive”. Philosophers have done much to try to understand what truth is, but it remains quite difficult to define truth in any way that is not controversial. Fortunately, taking true as a primitive will not get us into trouble, and it appears unlikely to make logic mysterious. We all have some grasp of what “true” means, and this grasp will be sufficient for our development of the propositional logic.