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Logic

I have two question abouts logic. In occidental thought, logic generally is presented as bi-polar (no pun intended); however, I am not quite sure of the 'correct' formulation. Is it: either "A is true," or "not A is true" or is it: Either "A is true," or "A is not true." and if you could expand on the distinction a little perhaps? More interesting, to me, is that I have heard the assertion that there also exists a "quadrant" logical system that also works. Either: "A is true" or " Not A is true" or "Neither A nor Not A are true" or "Both A nor Not A are true" This latter form of logic seems to work for things like polarization of light, for example (if you have a vertical polarizer in front of a horizontal polarizer, no light gets through; but if you insert a polarizer at a 45 degree angle in between the two, some light gets through). I can think of some other examples as well; I prefer to hear your responses.
Accepted:
October 14, 2010

Comments

Richard Heck
October 18, 2010 (changed October 18, 2010) Permalink

Classical logic (at least, understood from the perspective of classical semantics) rests in part upon the so-called "law of bivalence". This is usually formulated as: Every formula is either true or false. To put it slightly differently, we can begin with the idea (which emerges from Frege) that the "semantic value" of a formula is its truth-value. Then classical semantics involves two claims: (i) every formula has exactly one truth-value; (ii) there are only two truth-values, Truth and Falsity. This formulation does not involve any reference to "not", so the contrast between "not: A is true" and "A is not true" is avoided.

There are several sorts of alternatives to this. The one you mention can be presented in several different ways. One involves retaining (ii) but dropping (i) and, indeed, not replacing (i) with anything. So, on this view, a formula can have zero, one, or two truth-values. Logics can be built on this idea, and they "work" in the sense that one can rigorously present them, study them, etc. Whether they "work" in any other sense—e.g., whether it is even possible to think in the way they would prescribe—is philosophically contentious.

I confess that I don't understand the bit about polarization. Alternative logics have been proposed to deal with some of the oddities of quantum mechanics, but these are very different.

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