Thursday, December 1, 2011

Non-triviality of conditionals

Here's a rough start of a theory of non-triviality of conditionals.

A material conditional "if p then q" is trivially true provided that (a) the only reason that it is true is that p is false or (b) the only reason that it is true is that q is true or (c) the only reasons that it is true are that p and q are true.

A subjunctive conditional "p □→ q" is trivially true provided that (a) the only reason that it is true is that p and q are both true or (b) the only reason that it is true is that p is impossible or (c) the only reason that it is true is that q is necessary or (d) the only reasons that it is true are that p is impossible and q is necessary.

For instance, "If it is now snowing in Anchorage, then it is now snowing in the Sahara" understood as a material conditional is trivially true, because the falsity of the antecedent (I just checked!) is the only reason for the conditional to be true. The contrapositive "If it not now snowing in the Sahara, then it is not now snowing in Anchorage" is trivially true, since it is true only because of the truth of the consequent. On the other hand, "If I am going to meet the Queen for dinner tonight, I will wear a suit" is non-trivially true. It is true not just because its antecedent is false--there is another explanation.

Likewise, "Were horses reptiles, then Fermat's Last Theorem would be false" and "Were Fermat's Last Theorem false, horses would be mammals" are "Were I writing this, it would not be snowing in Anchorage" are trivially true, in virtue of impossibility of antecedent, necessity of consequent and truth of antecedent and consequent, respectively. But "Were horses reptiles, either donkeys would be reptiles or there would no mules" is non-trivally true--there is another explanation of its truth besides the impossibility of antecedent, namely that reptiles can't breed with mammals and mules are the offspring of horses and donkeys.


Dan Johnson said...

Hi Alex,

Regarding the material conditional, it sounds to me like your conditions for triviality amount to saying: the material conditional is trivially true if the only reason that it is true is that its components meet the truth-conditional requirements for it to be true. (I think you listed all the lines of the truth-table on which a material conditional is true in your conditions for its being trivially true.)

It almost suggests that the important thing about the truth of material conditionals can't be its meeting of the truth-conditions. Does that sound right? That feels sort of weird. I'd like your thoughts on that.

Alexander R Pruss said...

Thanks, Dan, for your help. I am ashamed to say I hadn't noticed that I listed all the lines.

I am inclined to abandon what I said about the material conditional. But I won't abandon. Instead, let me build on it.

Write the material conditional as:
~p or q.

Consider now an explanation E of the truth of the material conditional. Sometimes every such explanation explains the material conditional by explaining only one or both of the two disjuncts. If so, then as long as there is an explanation of the truth of the conditional, we say that we have triviality.

Sometimes, however, there is an explanation that does not explain the obtaining of any of the two disjuncts, but instead explains the disjunction as a whole. In this case, we have non-triviality.

We can give a similar theory of the triviality of disjunctions. Say that a disjunction "p or q" holds trivially provided that it has an explanation and every explanation of the disjunction is an explanation of one or more of the disjuncts. Many disjunctions, like that the moon is round or cows give milk, appear to be trivially true.

But some disjunctions, such as that p or ~p, or that 123 is even or odd, are non-trivially true.

Thus, one way to explain p or ~p is by explaining p or by explaining ~p. But there is another explanation of the disjunction, and that's by invoking the law of excluded middle and/or its proof from double-negation elimination. The latter explanation explains does not explain p and does not explain ~p.

Similarly, while the simplest way to explain that 123 is even or odd is to say that it's odd, another way to explain it is to note that 123 is an integer and every integer is even or odd. The latter explanation is an explanation of 123 being even or odd but is not an explanation of 123 being even and is not an explanation of 123 being odd.

We might also distinguish between metaphysical triviality and epistemic triviality. Say that a disjunction is metaphysically trivial if it is trivial in the above way. Say that it is epistemically trivial if one has reason to accept it and each of one's reasons to accept it is a reason to accept one or both disjuncts. Say that it is epistemically non-trivial if there is a reason to accept it that isn't a reason to accept a disjunct. We can then do the same thing with the indicative conditionals.

We can now give the standard Gricean story about the paradoxes of material implication and material disjunction (i.e., disjunction). There is no point to asserting indicative conditionals or disjunctions that clearly hold only metaphysically and epistemically trivially--you'd be more informative if you simply asserted the true disjunct(s), and yet you'd be just as brief.

I haven't thought through the case where you have one triviality without the other.

There is a somewhat analogous concept for a conjunction. We can say that the conjunction "p and q" holds coincidentally provided that every explanation of the conjunction consists of one explanation of p and a different explanation of q.

Heath White said...

The upshot appears to be that, if we are interested in non-trivial truths, then truth-functionality is no help in understanding them. Rather, insight needs to come from generalizations discovered elsewhere and elsehow: that every proposition is either true or false, every integer is even or odd, mules come from donkeys and horses and reptiles can’t breed with mammals; etc.

Another way to put this is to say that non-trivial material conditionals are also non-material conditionals.

That in turn means, I think, that the methods of classical logic are of no help in gaining non-trivial knowledge. Russell would be disappointed but that seems about right to me.

Alexander R Pruss said...

Sounds like it. Easy come, nothing much gained.

By the way, there may be a counterexample to my Gricean claims . Suppose that A and B are independent events, each of which is moderately likely to happen, but not likely enough to justify belief that it will happen. If A and B are just below the justificatory line, their disjunction will be just above the justificatory line, and by my account above, the disjunction is epistemically non-trivial. But the following seems clear: "if ~A, then B" is a paradoxical thing to say.

Dan Johnson said...

Alex and Heath,

This is really cool. Allow me to restate (mostly for my own benefit) what Heath said: the truth-conditions are merely necessary conditions for the truth of the material conditional. But the conditional's merely being true is trivial; what is important about the conditional is not that it is true but something else, like (if Alex is right) the explanation of its truth.

This has the very uncomfortable consequence, though, that the truth of the material conditional just doesn't, by itself, matter to us. What matters is why it is true. So merely coming to KNOW material conditionals grants us merely trivial knowledge; what we need to get something valuable is UNDERSTANDING of the material conditional, which will involve knowing its explanation.

Strange but cool.

Alexander R Pruss said...

I wouldn't say that the truth of the conditional doesn't matter in itself. The conditional could be metaphysically trivial but epistemically non-trivial. In such a case it could license useful inferences.

Consider the metaphysically trivial conditional that if it's snowing in Anchorage right now, then it's snowing in the Sahara. It wouldn't be epistemically trivial if some being that knew all about the weather revealed the conditional to me without revealing the truth values of the the antecedent or consequent. And since I have independent grounds for thinking it probably isn't snowing in the Sahara, I learn by modus tollens that it's not snowing in Anchorage. This is all useful stuff.

Probably, we won't get much use out of a conditional that is epistemically trivial, though.

Mike Almeida said...

why don't you have as a triviality condition for subjunctives that both p and q are necessary? would you say that a subjunctive is trivially true when the p entails q? And trivially false when p entails ~q? It incidentally does not look trivially true that if there were a largest prime then pigs would have wings. Lewis (86:1.6) says about these that "...they're not things we'd want to say", but he stops short of saying they're false, as it seems to me they are. He bites the bullet and claims, without feeling the need to explain, that they too are trivially true.

Alexander R Pruss said...


p's being necessary doesn't help explain why the subjunctive is true: q's being explanatory does all the work. So it falls under (c).

Alexander R Pruss said...

It's worth noticing, by the way, that the metaphysical triviality of a conditional can be a contingent matter. Thus, when the match is in ordinary circumstances, "If the match is struck, it will catch on fire" is a non-trivially true conditional, but in the absence of oxygen, when the match is not struck, it is a trivially true conditional.

By the way, when I say that something is a trivially true conditional, I mean it is trivially true as a conditional. It's not enough if it's trivially true for some other reason. Thus, in my sense, tautologies like "if p, then p" are non-trivially true conditionals, because there is an explanation of the conditional that doesn't explain the falsity of the antecedent or the truth of the consequent.