In logic a Hilbert calculus, sometimes called Hilbert system, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for propositional and first-order logic.
Questions tagged [hilbert-calculus]
121 questions
16
votes
2 answers
Difference between Logical Axioms and Rules of Inference
What's the difference between Logical Axioms and Rules of Inference? In my understanding, both are ordered pairs of formulas which are used to reach a conclusion through syllogisms.
My questions
Can both be formalized in a language?
Are both…
Incognito
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11
votes
1 answer
Relationship between sequent calculus and Hilbert systems, natural deduction, etc
I am trying to learn the basics of logic and I'm confused on how these proof systems work together. The big ones I see are Hilbert style, and then Gentzen style which includes natural deduction, and sequent calculus. I also see "intuitionistic…
Brandon L
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11
votes
3 answers
Difference between Gentzen and Hilbert Calculi
What is the difference between Gentzen and Hilbert Calculi?
From my understanding of Rautenberg's Concise Introduction to Mathematical Logic, Gentzen calculus is based on sequents and Hilbert calculus, on tautologies.
But isn't every Gentzen…
Incognito
- 1,195
9
votes
6 answers
What is the motivation for the axioms for Propositional Calculus in Mendelson's "Introduction to Mathematical Logic"?
On pp. 26-27 of his Introduction to Mathematical Logic (5th edition), Elliott Mendelson writes:
If $\mathscr{B}$, $\mathscr{C}$, and $\mathscr{D}$ are wfs of $\mathrm{L}$, then the following are axioms of $\mathrm{L}$:
(A1) $(\mathscr{B}…
kjo
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8
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1 answer
Hilbert style proof systems vs Natural deductions: Some naive questions
I have a couple of naive questions about Hilbert calculus / Hilbert-style deductive system and its relation it philosophically/conceptionally opposite "natural deduction" style based proof systems.
Recall, that Hilbert style deductive system mainly…
user267839
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5
votes
3 answers
Derive by modus ponens $[A\rightarrow(B\rightarrow C)]\rightarrow[(A\rightarrow B)\rightarrow(A\rightarrow C)]$
How could I derive by modus ponens the formula $$[A\rightarrow(B\rightarrow C)]\rightarrow[(A\rightarrow B)\rightarrow(A\rightarrow C)]$$ from, and just from, the following axiom schemata?
$(A\lor A)\rightarrow A$.
$A\rightarrow(A\lor B)$.
$(A\lor…
Fitzcarraldo
- 358
5
votes
2 answers
strong completeness of a formal system
Given a formal system $D$ where the axioms are the same as in Hilbert system for propositional logic and the inference rule is $$\frac{a\rightarrow b, \quad a\rightarrow \neg b}{\neg a}$$ I need to answer:
Is the system sound (if $\vdash_D \varphi$…
CforLinux
- 393
5
votes
2 answers
Hilbert System Logical Axiom 1 follows from Axioms 2 and 3
I'm reading Wikipedia and it lists the first four logical axioms that allow (together with modus ponens) for the manipulation of logical connectives.
$\phi \to \phi $
$\phi \to \left(\psi \to \phi \right)$
$\left(\phi \to \left(\psi \rightarrow \xi…
jet457
- 619
5
votes
1 answer
How to prove $((A \to B) \to A) \to A$ using Lukasiewicz's axioms, MP and deduction theorem?
This is an exercise from A.G. Hamilton's Logic for Mathematicians, section 2.1, p. 36. I have tried to do this for 10 long years, since 2010. Unsuccessful.
Exercise 3: Using the deduction theorem for $L$, show that the
following wfs. are…
João Alves Jr.
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5
votes
2 answers
What is the probability of randomly generating a tautology?
Suppose we randomly generate a classical Hilbert propositional calculus formula $F$ with $n$ variables, using the following method:
$F = x_i$ for each of $i \leq n$ with probability $\frac{1}{n+2}$.
$F = \neg F_1$, where $F_1$ is generated…
Chain Markov
- 16,012
5
votes
2 answers
Derive $P \to \neg \neg P$ in a structure with not and implies
We can define an abstract system with the following three axiom schemes that define $\to$ and $\lnot$ as follows:
ax1. $P\to(Q\to P)$
ax2. $(\lnot Q \to \lnot P)\to(P\to Q)$
ax3. $(P\to(Q\to R))\to((P\to Q)\to(P\to R))$
And any logical expressions…
Abhimanyu Pallavi Sudhir
- 3,492
4
votes
1 answer
Proving $(A\rightarrow (B\rightarrow C))\rightarrow ((A\rightarrow B)\rightarrow (A\rightarrow C))$ in Hilbert/Ackermann axiomatic system
I am working upon exercises from "Introduction to Mathematical Logic" by Mendelson chapter 1.6.
Given a formal system L1 : $\vee$ and $\neg$ are the primitive connectives. We use $B\rightarrow C$ as an abbreviation for
$\neg B\vee C$. We have four…
user4035
- 415
4
votes
1 answer
Is this Hilbert proof system complete?
Note: This post considers propositional logic, with $\to$, $\bot$ as the base connectives, $\neg \phi$ is an abbreviation for $\phi\to \bot$.Consider a usual Hilbert-style proof system(with modus-ponens as the sole inference rule) with the…
Vivaan Daga
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4
votes
2 answers
Axiomatic derivation - what does instancing an axiom practically entail?
I'm stunned by the chapter in my coursebook (which is in Dutch, so please advice if I am mistranslating any of the terms) about deriving from a system of axioms and derivation rules. The exercise is to prove the following:
$$p \vdash_S q \rightarrow…
KeizerHarm
- 185
- 5
4
votes
1 answer
Hilbert style proof of double negation introduction and reductio ab adsurdum
I'm trying to prove:
$\phi\to\neg\neg\phi$
$(\neg\phi\to\neg\psi)\to((\neg\phi\to\psi)\to\phi)$
Using these axioms with modus ponens and the deduction theorem:
A1: $\phi\to(\psi\to\phi)$
A2:…
Matt Dickau
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