Brave New Logic

In classical formal logic, every statement is either true or false: those which are false are precisely those which are not true. In the early 20th century however, constructivist mathematicians wanted to see how far they could get without this “law of the excluded middle” and began to develop new intuitionistic logics in which some statements are true, others false, and the rest neither true nor false. Though at first glance this may seem more mystical than mathematical, many years later, intuitionism remains the focus of a reasonable amount of serious scientific and philosophical interest.

Far less well-known is its eccentric younger cousin: paraconsistent logic. In most versions of this, the middle is again excluded, so each statement must be either true or false, but now some are allowed be both true and false. In all other systems this type of contradiction would spell immediate meltdown, but paraconsistent logic is built to cope with it: it is inconsistency-tolerant.

 

Though this avant-garde logic is intriguing, surprisingly coherent, and has applications in both the foundations of mathematics, and in computer science, the problem is that it seems to lack a natural model. Some people have suggested, not entirely persuasively, that it might be used to handle wave/particle duality in quantum physics, or to provide a means of resolving self-referential conundrums, such as the liar paradox.

I’d suggest that the best model for this peculiar new logic is New Labour: paraconsistent to the core, it both is, and isn’t a Labour government.

For an example of paraconsistency in policy-making, take the recently announced plans for the labelling of homeopathic medicines, which may read “Contents: 100% water. This product can be used in the treatment of lung-cancer”. Which of course it can, despite all the evidence saying that it can’t.

A paraconsistent approach also provides neat resolutions to the major questions of the New Labour era. For instance: Did British foreign policy cause the 7/7 bombings? Well no, obviously it didn’t. And yes it did.

Because it is specifically designed to withstand it, this whirl of contradictions needn’t entail the collapse of the whole party machine, or even indicate any problem whatsoever. The system is simply functioning as it was intended to. New Labour is inconsistency-tolerant.

9 comments
  1. And there, in a Teabag, one might say, you have the difference between mathematical logic and human endeavours.

  2. Doormat said:

    For those who still have even a spark of curiosity in them (so, not Duff then) I can recommend Wikipedia for more info on paraconsistent logics. Although Larry’s link to New Labour political philosophy (possible a contradiction in terms) holds, I think, much promise of further study…

  3. G. Tingey said:

    Erm, no.

    Side 1 – 1: “The statement on the other side of this card is true”
    Side 2 – 2: “The statement on the other side of this card is true”

    Err …

    Answer: The stements on this card are not dterminable (or decidable).

    What they are trying to say (I think) is a restatement of something like Godel’s theorem (should be an umlaut in there, I know) …

    Ther are undecidable propositions.
    There are cases where a certain outcome cannot be arrived at.

    It is important, oddly enough, but it CAN make your brain hurt.

  4. Duff – are you suggesting that mathematical logic can develop without human endeavour?

    G Tingey – What they are trying to say (I think) is a restatement of something like Godel’s theorem

    Erm, no. It has been suggested that paraconsistent logic might be used to in relation to Gödel’s theorems, I don’t know how far anyone’s got with that. But the two are essentially different.

  5. G. Tingey said:

    And statement 2 should have finished “false” – careless with the cut-&-paste there.
    Oops.

    True / false / undecidable / undecided (?)
    True If and Flase If – with differing values for If also as in ..If and only if, or if plus other comstraint – a bit like the diferences between necessary and sufficient conditions.
    etc.
    As I said, it can make your head hurt.

  6. G. Tingey

    Just to be clear, decidability is a separate issue, having to do with whether you could (in theory) program a computer to decide on the truth/falsity of a given statement.

    And your two-sides of the card thing is a variation of the liar paradox (“this statement is false”) which I mentioned in the post.

  7. “Contents: 100% water. This product can be used in the treatment of lung-cancer”. Which of course it can, despite all the evidence saying that it can’t.

    I may be missing the point, but I don’t see there’s a problem with that unless the label also tries to suggest that the treatment’s likely to have any effect. And, of course, I’m sure water is used in the treatment of lung cancer — washing down pills and so forth:

    GLENDOWER

    I can call spirits from the vasty deep.

    HOTSPUR

    Why, so can I, or so can any man;
    But will they come when you do call for them?

  8. NS – my fault, didn’t express myself at all clearly. Imagine this – a bottle of liquid called something medicinal and/or “natural” – sounding. It’s on sale in a chemist’s shop, and isn’t cheap. Somewhere on the bottle is an official (state-sanctioned) label which reads: “This product can be used in the treatment of lung-cancer.” Elsewhere on the packaging, in far smaller print, is written “Contents: 100% water.”

    I’m sure water is used in the treatment of lung cancer

    This is why the statement is true. However the intended implication, and the one which most people are likely to understand – that the product has some medicinal property in regard to lung-cancer – is false.

  9. Yes, well Larry this is all very clever, but does it take us any nearer solving the conundrum of your mum?