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7.1 Tautologies and Taut-Falsities

Now that you’ve learned some of BOOL’s syntax and semantics, we can start using it to study logical truth and validity.

In this section we use BOOL to show that some sentences are logically true or logically false: they are necessarily true or false because of logic.

We’ll start by doing a truth table you are familiar with.

Now think about what that means: Pv~P is true no matter what.

The atomic sentence P might be true or false, but the complex sentence Pv~P can’t possibly be false.

After all, the rows of the truth table are exhaustive: they cover every possible value. Since BOOL is bivalent, there are only two possible values.

That means we’ve discovered a logical truth with BOOL: Pv~P. In English: Pia is guilty or not guilty.

Additionally, this truth table shows or proves that Pv~P is logically true: the truth table conclusively demonstrates that that sentence can’t be false.

There are many types of logical truth. Pv~P is guaranteed to be true because of the meaning of the truth-functional connectives. We will call logical truths like that tautologies.

Tautology: a sentence that is logically true because of the truth-functional connectives.

Sometimes “tautology” is used in English for any sort of obvious or necessarily true sentence, but in this book we will use the word in a specific way: a tautology is logical truth that depends just on truth-functional connectives.

Tautology: a sentence with all Ts in its truth table.

Our tool for studying truth-functional connectives is truth tables, so here is an equivalent way of defining tautologies: they are sentences whose truth functions are all Ts.

Law of the Excluded Middle: Pv~P

Pv~P is sometimes called the Law of the Excluded Middle, because it says either Pia is guilty or not guilty, so there’s no third option or middle ground in between.

Pv~P is the simplest tautology, but there are many more. The skill we want you to acquire is to take any sentence and assess whether it is a tautology. In order to do that, the first thing you need to do is compute its truth function. Then you just see if it has all Ts.

If a sentence isn’t a tautology, that means it has at least one F.

Taut-falsities: a sentence with all Fs in its truth function.

If it has all Fs, then it is the opposite of a tautology: it is logically false because of the truth functional connectives. We will call those sentences tautological falsities, or taut-falsities for short.

Tautologically Contingent: a sentence with at least one T and one F.

If a sentence has at least one T and one F, then it isn’t necessarily true or false. Our name for that is contingent.

Technically we should call it “tautologically contingent”, since it is contingent insofar as the truth-functional connectives are concerned. We’ll just call it “contingent” for now, and clarify this point once you’ve learned other types of logic.

Using this terminology, let’s assess those two sentences.

Now see if you can figure out which of these sentences is tautologically false.