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Once logic has been formalised, it's an easy move to ask whether certain axioms can be altered. One of these is that logic is two-valued. This has been done and we have formal multi-valued logic.

Ordinary language sustains an idea of truth & falsity which fits in with the usual logic. Is there some way of understanding the truth of a multi-valued logic in ordinary language?

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Is there some way of understanding the truth of a multi-valued logic in ordinary language?

Sure; but first we have to specify which particular multi-valued logic we are speaking about.

Some multi-valued logics have a value for "unknown"; this is fairly intuitive, as there are plenty of things that we cannot say are true or false, but simply don't know.

Some multi-valued logics attempt to assign levels of truth; some things are more true than others. This can be useful with sorites problems (Is a particular man bald? Yes, but less bald than some other men...) and with problematic presuppositions (Is the current king of France bald?)

Of course, this is all orthogonal to the question in the title; the question of whether mathematical objects are figments of the imaginations of mathematicians depends on whether one is a Platonist, an Intuitionist, a Fictionalist, etc., and the case of multi-valued logic is not special in this regard.

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Two of the most important logics which are not (necessarily) two-valued are Boolean algebras and probability theory. (I'm well aware that even a Boolean algebra must be supplemented by additional "instructions" to turn it into a multi-valued logic). Both look like bad examples, because Boolean algebras don't seem to add anything new over classical logic, and probability theory doesn't seem to fit into the formal scheme of multi-valued logic. On the positive side, already a two-dimensional Boolean algebra can be turned into an instructive example of a multi-valued logic with interesting interpretations, where nearly no property of classical logic has to be sacrificed. Probability theory on the other hand seems to be really useful for many "real world problems", and the product Boolean algebra over the sample space can help to clarify the relation to multi-valued logic (clarifying in a certain sense why probability theory doesn't fit directly into the scheme of multi-valued logic).

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  • "[P]robability theory doesn't seem to fit into the formal scheme of multi-valued logic" --- in what way would you say that it fails? Perhaps in that its truth value would often be construed to be subjective? This is indeed a problem that I would have with it, but is this also what you have in mind? Despite this shortcoming, probability theory actually is a strong contender for a practical, multi-valued refinement of boolean logic. Jul 19, 2012 at 10:09
  • @NieldeBeaudrap The subjectivity is not the problem. The problem is that a probability value is only "incomplete information". The formal scheme of multi-valued logic contains among others the truth degree functions which interpret the propositional connectives (conjunction=and, disjunction=or, implication, negation=not and equivalence). The probability of (not A) is a function of the probability of A, namely P(not A)=1-P(A), but the probability of (A and B) is not a function of the probability of A and the probability of B in the same sense, because P(A and B)=P(A|B)*P(B)=P(A)*P(B|A). Jul 19, 2012 at 17:36
  • So, you would rule out probability as a form of logic per se because you cannot infer the precisely value of conjunctions and disjunctions from those of the conjuncts and disjuncts; that's fair enough. I might be inclined only to require that the logical operations obey obvious monotonicity properties, which are indeed satisfied by probability; but the fact that we don't have functionality without complete information of the universal distribution is I suppose a serious problem for the usual objectives of logic. Jul 19, 2012 at 17:49
  • @beaudrap: what are the obvious monoticity properties? Jul 20, 2012 at 5:33
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This is a bit of a circular question. What do we actually mean by 'truth', especially when referring to using notions of truth in reasoning about the world?

An instructive exercise is to ask why we should think that boolean logic as we use it today is 'true'. When we get down to it, it seems to me that when we say that it is 'true', we actually mean that it is useful: we find it to be a reliable tool with which to do reasoning, and we find that whenever we make a mistake, the error can be traced to the ideas to which we applied logic, and not to the logic itself. That is to say: it seems a mode of reasoning which is as good as the information upon which it acts — it is subject to the maxim of Garbage In, Garbage Out, but it is not so brittle or sensitive to error that it is useless. This prompts us to trust it.

If this is true of boolean logic, could another more subtle logic also prove similarly useful, or even more useful? Of course, boolean logic forces us to make precise distinctions when asking questions, which is potentially useful. But when considering the world around us, we run into famous problems such as the Sorites paradox. These paradoxes can be described as paradoxes of language rather than logic, but the distinction is perhaps illusuory: just as we are encouraged to ask whether a pile of sand is "a heap" or is not "a heap", we ask ourselves whether the Sorites itself indicates "a weakness of language" or "a weakness of logic". Because logic is only as useful as the language which is fed to it, and language only as useful for reasoning as the logic applied to it, the real question is how best to repair the flaw in the Language/Logic system, if we decide that it is worthwhile.

But the real teaser is not the Sorites, and artificial distinctions in colloquial language, but advances in science. What ought we to say about Newtonian Mechanics — is it true? Well, it's useful enough for most purposes. It's not badly wrong. It's not correct enough to high enough precision for all purposes, but you can build bridges, skyscrapers, and lunar expeditions relying on just Newtonian mechanics and a bit of crude material science. And what about the Earth being round? If by 'round' you mean "a sphere", you should account for the bulging at the equator due to the centrifugal force; and if you mean "an oblate spheroid", you're still neglecting the variation in elevation. These descriptions are practical approximations to a very complicated 'truth', that is, a complicated state of affairs; and yet they're not what we would describe as "horribly wrong". They're just not 'perfectly' right; you can get closer to the truth by a more detailed description.

So, if you're principally interested in ordinary language (which I'd be careful about — is cold a thing that can seep into your bones, does the Sun really rise and fall in the sky?), I would say that we already implicitly recognise that multivalued logic is more useful — 'truer' — than boolean logic. It's just that when we demand precision, we don't know how to move forward except to replace the complicated messy notion of truth with the idealised boolean one, and hope that degrees of error are numerically measurable, so that we can capture the truth (and our uncertainty about it) in quantity rather than verity.

The only question is whether this is the best way of doing things. I rather hope that mathematics yields a multivalued logic which has greater practical power than boolean logic. And although it would be a formidable task to do so, I don't believe that it is a priori impossible.

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