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As announced in another thread, I created Hilbert system.
However, I am a bit confused about exactly how a Hilbert system formalizes mathematical practice. In particular, how does it formalize hypothetical reasoning? When I want to prove a theorem like “If $A$ and $B$ then $C$”, I start out by assuming $A$ and $B$ and trying to prove $C$. I know how to formalize this in natural deduction: I start a derivation with $A$ and $B$ at the top, and when I’ve gotten to $C$ then I apply implies-intro, cross out the $A$ and $B$, and conclude $A\to B\to C$. And in a type theory or sequent calculus, I am trying to prove a hypothetical sequent $A,B\vdash C$, after which I apply implies-intro again to get $A\to B\to C$. But in a system where the only rules are about deducing “global” theorems, how do I formalize the hypothetical-reasoning method of proving an implication?
Does this help? Example 1 shows that even $A \to A$ requires some work. Once you have the deduction theorem, section 2, things get easier.
Hmm, so the answer is that Hilbert-style proofs do not directly formalize informal proofs, but proving something Hilbert-style requires trickery that has nothing to do with how we actually write proofs?
In particular, the deduction theorem fails for the substructural paraconsistent/relevance logics that I’m currently trying to understand: e.g. with a Hilbert meaning of $\vdash$ we have $A,B\vdash A\&B$, but we don’t have $A\to (B\to A\&B)$ since $\to$ is the linear/relevant implication (a.k.a. $\multimap$). I’m trying to understand how paraconsistent/relevance logicians can “think” in these logics, given that they generally present them only as a Hilbert system.
My recollection of the point of Hilbert systems was that they’re not for proving formally what is proved informally, but rather to prove things about a deductive system. With so few (often 1) rules of inference, it’s easier to prove that something cannot be derived.
I though natural deduction was set up to contrast with this approach to resemble how we reason naturally. Wikipedia says:
Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system)
and Gentzen
First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a “calculus of natural deduction”.
If that’s the case, then it seems backwards if Hilbert systems came first historically. What good would it be to prove that something cannot be derived in a formal system if you don’t know yet that that formal system has anything to do with ordinary reasoning?
One only needs establish the strength of a system relative to other systems that are known to able to represent ordinary reasoning.
There’s an intricate history to tell here, which I ought to know more about. Hilbert systems first appear in book form, I think, in
Hilbert, David and Ackermann, Wilhelm, 1928, Grundzüge der theoretischen Logik, Berlin: Springer.
But perhaps had already appeared in
Hilbert, David, 1918, “Prinzipien der Mathematik”, Lecture notes by Paul Bernays. Winter-Semester 1917/18. Typescript. Bibliothek, Mathematisches Institut, Universität Göttingen.
Anyway, it’s certainly post-Russell and Whitehead.
Okay, so although natural deduction came after Hilbert systems, there might have been other systems prior to Hilbert’s that also represented ordinary reasoning more directly?
That still doesn’t answer my real question of how people can describe a new logic with just a Hilbert system and then proceed to reason informally in it without first giving an equivalent natural deduction or sequent calculus, but it’s useful to know.
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