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Could someone explain clearly why predicate logic is said to be "semidecidable"? I know that the set of invalid formulas of a predicate language is not effectively numerable, which implies that predicate logic is not decidable (a set of expressions is decidable iff both it and its relative complement - in this case the set of invalid formulas - are effectively numerable). Instead, the set of valid formulas of a predicate language is effectively numerable. Does "semidecidable" simply means that the set of valid formulas of predicate logic is effectively numerable?

A further question: which are the main "reasons" for the undecidability of predicate logic?

Thanks a lot for your help.

Fish

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Yes, it just means that the set of valid formulas of predicate logic are effectively enumerable. The algorithm is simple: search through all possible sequences of sentences, and check each one to see if it is a proof using the standard deductive steps; if so, enumerate the conclusion of the proof into your set.

It's hard to say what the "cause" is - mathematical phenomena have proofs, not causes. But the key reason for the undecidability is that predicate logic is too powerful; it's powerful enough to describe the algorithm you might try to use, so it can circumvent it.

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  • $\begingroup$ +1. To the OP, a neat instance of predicate logic describing algorithms (and one which seems surprisingly unknown!) is Trakhtenbrot's theorem. $\endgroup$ – Noah Schweber Mar 6 '17 at 23:42
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Does "semidecidable" simply means that the set of valid formulas of predicate logic is effectively numerable?

Yes, "semidecidable," "effectively numerable," "recursively enumerable," and "computably enumerable" are all synonyms.

Now, as to the phrase "predicate logic is semidecidable": usually when we refer to a logic as being semidecidable, what we mean is that the inference problem - "does $p\vdash q$?" - is semidecidable (the set of pairs $(p, q)$ such that $p\vdash q$ is semidecidable). For strong enough logics, though, $p\vdash q$ iff $\vdash p\implies q$, so the logic is semidecidable iff the set of validities is. (For an example where this is not true, take e.g. a logic that doesn't even have a symbol for implication - like equational logic, which might sound artificially weak but is actually in certain ways the "right" logic for universal algebra.)

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  • $\begingroup$ The statement that $p \vdash q$ implies $\vdash p \implies q$ is the deduction theorem, but this theorem is not strong enough to guarantee that the set of validities is semidecidable: the Godel Completeness theorem is needed for that. In the Trakhtenbrot theorem example given above Godel Completeness does not hold, and the set of validities is not semi-decidable! $\endgroup$ – Roy Simpson Feb 5 at 12:11
  • $\begingroup$ @RoySimpson Quite right, it seems I was conflating "validity" (= true in every model) with "tautology" (= provable from the empty theory). $\endgroup$ – Noah Schweber Feb 5 at 12:41
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It is said to be semi-decidable because while it is not (fully) decidable, there is an algorithm that correctly classifies all the valid first-order logic statement as valid ... but this algorithm will go into an infinite loop when presented with certain invalid statements. And yes, this is the same as saying that the set of valid formulas of predicate logic is effectively numerable.

The fact that there is such an algorithm is closely related to the completeness of first-order logic, first proved by Kurt Godel in 1929.

One way to see that first-order logic is undecidable (meaning that there is no algorithm that correctly classifies all valid first-order logic statement as valid and correctly classifies all invalid first-order logic statement as invalid) is by looking at Alan Turing's work, and in particular the Halting problem:

It turns out that you can describe any halting machine and its behavior using first-order logic. Therefore, if first-order logic is decidable, we can solve the halting problem: For any machine M and input I we can describe using first-order logic that machine and input (using a big-ass, but still finite statement $S_{M,I}$), and we can also create a statement $H$ of what it would mean for that machine to halt. We then use the First-order logic decision procedure to decide whether $S_{M,I} \rightarrow H$ is valid or not: if it is valid, then the machine M with input I halts, and if it is invalid, then it does not halt. But since we know the Halting problem is unsolvable, the decision problem for first-order logic (what David Hilbert called the Entscheidungsproblem) is likewise unsolvable.

As to the 'cause' of all this ... one issue is of course that to see whether a first-order logic formula is valid is to see whether it is true in all possible interpretations ... but there infinitely many interpretations. That is not a proof (and indeed there are many times we handle infinities just fine), but if there were only finitely many interpretations, then it would be decidable just fine, so this can be seen as a 'cause' of sorts.

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  • $\begingroup$ Thanks! So, concerning the "cause", can you say that the problem is that you can have infinitely many combinations between domains and interpretation functions? $\endgroup$ – Fishermansfriend Mar 7 '17 at 11:10
  • $\begingroup$ Infinite models can sometimes be difficult to check too, I suppose. $\endgroup$ – Fishermansfriend Mar 7 '17 at 11:11
  • $\begingroup$ @Fishermansfriend Yes, infinities of both kind are part of the problem, because if everything was finite, then everything would be decidable. In propositional logic, for example, there are only finitely many possible interpretations given any finite set of statements of finite length, so propositional logic is decidable. But infinity is only part of the problem. Between Reese's and my answer: first-order logic is 'powerful enough' to describe the algorithms that can check something about 'themselves'.... and that leads to inherent contradictions if you assume they are 'all-powerful'. $\endgroup$ – Bram28 Mar 7 '17 at 12:22

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