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Tutorial Three
Further Deductive Concepts

We defined the notion of validity in terms of possibility. Here's that definition again:

An argument is valid just in case it is not possible that its conclusion be false while its premises are all true.

For a valid argument, then, the conclusion is inescapable given the conclusion.

Again...the concept of logical possibility:

The possibility in question here is sometimes called logical possibility. Logical possibility is about what might have happened in some possible world, about how things could have been (even if actual matters that have become settled now preclude it).

So, for example, it is logically possible that George W. Bush never became US president. Even though we know that he did, our language allows us to consider a possible, but counterfactual situation in which Gore won in the Supreme Court, votes were recounted and Bush was declared the loser.

There are lots of other uses of the word "possible". You might say: "I'm sure that G.W.Bush didn't lose; it's just not possible that I'm mistaken." This is an epistemic sense of possibility. But our logical possibility is different; it's a semantic conception.

Why make so much of the notion of logical possibility? The reason is that there are a number of deductive concepts that can be defined in terms of this one notion. So, possibility will allow some unification. Also, the symbolic languages that we develop later allow us to be very clear about what possibility amounts to in the context of their use.

Perhaps the most important deductive concept after validity is that of a logical truth:

A sentence is logically true just in case it is not possible for that sentence to be false.

So, for example, "All Irish males are male" is a logical truth. So, is "Each triangle has three sides".

Further Definitions

An analytic truth is a sentence true in virtue of its meaning.
A necessary truth is a sentence true in all possible worlds--this is what we call a "logical truth.
A tautology is a sentence true in virtue of its form.
Finally, an a priori sentence is one which can be known true prior to experience.

It's worth noticing that some of the easiest examples of logical truths are also analytic, necessary, tautologous, and a priori: "all dogs are dogs" or "it's raining or it's not".

But one might think that "water is made of H₂0" is an example of a necessary truth (how could real water be anything else!) but seems neither a priori, analytic, nor tautologous.

Sometimes these logical truths are called "analytic" or "necessary truths" or "tautologies" or even "a priori". But such labels have slightly different definitions — see the sidebar just above — so their equivocation with logical truth is controversial. (Only as first approximation should you identify these notions.)

Sometimes the notion of necessary truth is given a symbolization: 'S' (read "box-S") symbolizes "it is necessary that S". We won't get to this "modal logic" in what follows.

Now, click on each of the following which IS a logical truth.