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:

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

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.

  1. All living things are alive.
  2. No whale is a non-whale.
  3. Up until today, I had climbed only one 5.12 rock pitch.