# Questions tagged [hilbert-calculus]

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.

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### Hilbert Transform Calculation of a pulse

Can anybody help me with the Hilbert transform of the following pulse: $$p(t) = \frac{8B\rho \cos(2\pi Bt+2\pi B\rho t)+\frac{\sin(2\pi Bt-2\pi B\rho t)}{t}}{\pi\sqrt{2B}(1-64(B\rho t)^2)}$$ I tried ...
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### How to prove ⊢ (B → ¬B) → ¬B using an informal deduction

So I need to prove ⊢ (B → ¬B) → ¬B I can use the Modus ponus (MP) rule, and deduction theorem (DT). And I have these 3 axioms: α → (β → α) --- (A1) (α → (β → γ)) → ((α → β) → (α → γ)) --- (A2) (¬β →...
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### Axiomatic proofs in propositional logic

I have to use these three axioms (A1) $P \to (Q \to P)$ (A2) $(P \to (Q \to R)) \to ((P \to Q) \to (P \to R))$ (A3) $(\neg Q \to \neg P) \to ((\neg Q \to P) \to Q)$ along with Modus Ponens to ...
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### What is the characteristic equation of this linear ODE?

I am trying to solve the following linear ODE for $y(x)$: $$y^{\prime\prime\prime}+y^{\prime\prime}+\mathcal{H}[y^{\prime\prime}]+y^\prime-cy=0$$ subject to the boundary conditions $y\rightarrow0$ ...
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### Classical Propositional Logic and Axioms

We define a new proof system N over the connectors: {∨,¬} For every α and β- 𝐴1: (𝛼 ∨ (𝛽 ∨ (¬𝛼))) (axiom) Deductions: 𝑀𝑃1: if we have 𝛼, 𝛽 then we can deduce (¬(¬(α∨β))) 𝑀𝑃2: if we have (...
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### can we a prove ⊢ (α → α) → (α → α)

The system L0 is defined as follows: Axioms: A1 (α → (β → α)) A2 2. (α → (β → γ) → ((α → β) → (α → γ)) A3 ((¬β → ¬α) → ((¬β → α) → β)) In one of my problem sheets, I am told that I am allowed to ...
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### Does the deduction theorem hold in Q?

The proof of the deduction theorem (for a system including Hilbert Calculus) that I am familiar with uses modus ponens to prove the result one way, and mathematical induction the other way. Robinson'...
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### 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 ...
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### When does $(\square P \land \square Q) \to \square (P \land Q)$ hold?

If all axioms of classical propositional calculus hold and we work in modal logic that is at least K (ie. extremely weak), it is trivial to show $\square(P \land Q) \to (\square P \land \square Q)$. ...