Chapter One, Tutorial Three
Possibility, Deductive Logic, Inductive Thinking

We've defined the notion of an argument's "validity" (i.e., its being valid) in terms of possibility:

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

But the earlier, rough formulation may still seem more helpful: an argument is valid if its conclusion is inescapable given the premises. So, why do we give the official definition in terms of possibility?

The reason is for exactness; logical concepts need to be precise. But "inescapability" is a bit vague. We will take it to mean "the conclusion must be true given the premises". But what does this talk about what must be true mean? Answer: what must be true is what couldn't possibly be otherwise.

Hmm...is this getting us anywhere? Or just running around in semantic circles? Well, all this "semantics" will prove most helpful as we come to better understand possibility: We will see in the coming chapters that we can spell out exactly what logicians mean by "possible". Then we will define many concepts (like "valid") in terms of possibility. Remember this as you go through the course – it will help make sense of the proceedings!

For now we need to get the beginnings of an idea about how "possibility" will be used here. There are at least two ways to think about possibility:

  1. 'Possible' meaning 'open' and opposed to 'settled by the facts': In this sense, the possible is what still might be the case given what is settled. Here "settled" comes to something like "determined given what one knows, what has happened so far in time, or what our laws of nature require".
    Examples:

    I know that George (without the "W") Bush, the 41st president, served exactly one term as US president. So, it's not possible that he was the longest serving president. That's settled. But, it remains possible, given what I know as I write this, that he is the only Republican in the Twentieth Century to serve exactly one term. I doubt it, but I haven't checked. My knowledge doesn't settle the matter.

    Similarly, we say that it's impossible for Bush 41 to live unassisted on Jupiter (given the limitations of human biology). It's settled then, by natural law, that he is not living unassisted on Jupiter; there's no possibility of it.

  2. 'Possible' meaning 'Logically Allowed': This is the more inclusive concept of possibility. In this sense, the possible is what might or could have occurred or happened given only constraints of language.
    Examples:

    George (without the "W") Bush could have served two terms. We know that he didn't but believe that he might have if only he'd shown more sensitivity to the common person's needs. The possible scenario is worth consideration in a study of recent political history. Likewise, it's logically allowed (though not allowed by laws of nature) that Bush travel at speeds greater than that of light. Our language allows us to think about such a scenario (In fact, Einstein did think about such scenarios in order to show that they can't be a part of nature.)

Logic is not about the actual course of history or our knowledge of it; rather, logic is about language. We will be most interested in the possibilities allowed by language: what could have turned out to be given only constraints of language. Call this "logical possibility". Possibility of type 2 (immediately above) is, then, of interest to us.

Recapitulation:


We'll just write "possible" after this tutorial. The only type of possibility of interest will be logical possibility.


Click on the one sentence discussing possibility which is of type 2 – logical possibility:

  1. It's not possible that there are only five planets; scientists have discovered that the Earth is a planet and that there are at least two planets which cannot be observed without high powered telescopes.
  2. If, as many believe, our universe is fated to undergo infinite expansion, then it is not possible for life to exist forever.
  3. It's not possible that something be both a red giant star and small bit of galactic dust.