Good Math/Bad Math

Wednesday, May 24, 2006

Logic Fun: Intuitionistic Logic

It's time for a bit of a side-track from the series of articles about lambda calculus. Models of lambda calculus are based on something called intuitionistic logic, so I'm going to write a bit about that. It's also quite interesting in its own right.

Syntactically, intuitionistic logic looks the same as first order predicate logic. But the meanings of statements in it are often quite different. The basic idea behind intuitionistic logic is that some of the statements that you can make in propositional logic are too strong. In intuitionistic logic, a statement is only true if there is a proof that it is true; a statement that something is false means that that statement cannot be proved - so the negative statement "not P" can be read as an affirmative statement: "There is a proof that there is no proof for P".

This has the implication that many inferences that we're used to in propositional logic are not accepted in intuitionistic logic:
  • In propositional logic, the statement "X or not X" is a tautology, that is a statement that is always true: the statement means that either X is true, or X is not true. In intuitionistic logic, it is not: the statement "X or not X" means that we can prove that X is true or we can prove that X is false. X is true in intuitionistic logic if and only if we have a proof that X is true; X is false if and only if we can prove that there is no proof for X. If we do not have a proof for either the truth or the falsehood of X, then "X or not X" is not a true statement: it's possible that X is true, but we don't have a proof for it.
  • In propositional logic, we can often infer statements like "exists X : P(x)", even though we don't know what x is. In intuitionistic logic, to infer that statement, it must be possible to demonstrate at least one value for which it's true. Similarly, in order to prove that a universal statement is false, we need to be able to show a concrete counterexample that demonstrates its falsehood.
  • In both propositional and intuitionistic logic, the statement that "P -> not not P" is true. But in intuitionistic logic, "not not P -> P" is not true. Since "not P" means "We can prove that there is no proof of P", the fact that P is true (i.e., that we have a proof that P is true) means that we can prove that "not P" is false - because the fact that there is a proof for P provides a concrete counterexample to the statement that there is no proof for P - so "not not P", which says "There is no proof that there is no proof for P" is proven true.

    But going the other direction: the fact that we can't prove that there is no proof for P does not imply that there is a proof for P. Being unable to prove that there is no proof for something is a much weaker statement that saying that that thing is true.
One way to think of intuitionistic logic is to think of it as if there were actually three values that can be assigned to a predicate: it can be in one of two definite states (proved true or proved false) or it can be in an indefinite state (unproven). If a predicate P is unproven, then we can assert neither that P is true nor that P is false. But if the predicate is in an indefinite state, and we can show a proof, then we can move it to a definite state: if we can find a way to provide a proof of P, then it becomes true; or if we can show that it's unprovable (generally through a counterexample), then it becomes false.

The formal semantics of intuitionistic logic are more complicated than the semantics of predicate logic. They tend to involve things like lattice theory. In another post, I'll talk about one way of describing the semantics of intuitionistic logic called Kripke Semantics. Kripke semantics is useful not just for intuitionistic logic, but for a lot of other things, like temporal logics, modal logics, and model checking. (Wikipedia describes another version of the semantics of intuitionistic model based on something called Heyting Algebra, but I'm not familiar with it.)

5 Comments:

  • ------
    One way to think of intuitionistic logic is to think of it as if there were actually three values that can be assigned to a predicate: it can be in one of two definite states (proved true or proved false) or it can be in an indefinite state (unproven)
    ------

    This is true even in classical logic so long as you are looking at things syntactically and not semantically. The belief that everything is true or false assumes an underlying "true" model (how neo-Platonist of us). If we step back in classical logic and look at the deductive closure of a theory, we get much the same thing. Given a sentence/forumla and a theory, either we can deduce the sentence from the theory, deduce its negation, or deduce neither.

    The advantage of intuitionism is that the deducibility of a formula is decidable even for incomplete theories (theories for which "neither" is a possibility for some sentences). That is why intuitionistic logic is the logic of choice for automated theorem provers.

    By Anonymous Walker, at 2:30 PM  

  • walker:

    There's a bit more to intuitionism than that. Intuitionism provides a framework for a constructive approach to theorem proving - so you don't just get proofs of theorems, you get concrete examples.

    For example, the programming language Prolog is based largely on intuitionistic logic; the way prolog works is to negate a statement that you want an example of; then it searches for a counter-example of that negation - giving you an enumeration of values for which a statement is true.

    The constructive properties of intuitionistic logic are as important as its decidability properties for theorem proving systems.

    By Blogger MarkCC, at 2:37 PM  

  • I find the concept of negative proofs in math fascinating, since in other sciences, you typically can't prove a negative.

    Got any quick links on how they proved it's impossible to "square the circle" for me? The wikipedia entry I read on it mentioned that squaring the circle would require finding the algebraic value of pi, but pi is "transcendental".

    By Blogger Bronze Dog, at 2:44 PM  

  • -------
    There's a bit more to intuitionism than that. Intuitionism provides a framework for a constructive approach to theorem proving - so you don't just get proofs of theorems, you get concrete examples.
    -------

    Oh, I am not disputing that intuitionism has other uses. My claims were solely directed to the paragraph that I quoted (i.e. the trinary state claim). In that regard, the advantage of intuitionism is decidability.

    But I am not sure what you mean by concrete examples (other than what I get in classical logic). Even in classical logic, for a deducible sentence I can get the concrete proof sequence, simply by enumerating proof steps and applying cut rules. The difference is that when there is no proof of deducibility, I can manifest a concrete Kripke model that demonstrates there is no proof. Perhaps that is what you mean.

    By Anonymous Walker, at 3:53 PM  

  • walker:

    You worked out the answer to your question by yourself. With normal predicate logic, I can show the proof as a chain of inference steps; when a statement is false, I can find the reasoning that demonstrates its falsehood. But with intuitionistic logic, if I build up a Kripke model and follow things through, I wind up with a specific counterexample that shows that the proof is impossible.

    In terms of programming, that means a lot. There are languages, like prolog, that theoretically work by letter the programmer describe a solution using intuitionistic logic; then the interpreter/compiler finds the solution by *negating* the description, and searching for counter-examples that show that the negation is unprovable. Those counter-examples are the solutions that the programmer described. So you can get a kind of concrete result from intuitionistic reasoning that you can't from the more general predicate inference.

    By Blogger MarkCC, at 9:45 PM  

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