Articles of formal proofs

How to prove this sequent using natural deduction?

How do I prove $$S\rightarrow \exists xP(x) \vdash \exists x(S\rightarrow P(x))$$ using natural deduction? Just an alignment of which axioms or rules that one could use would be much appreciated.

Formal deductions on Hilbert system

I have proved {(α→β),(β →γ)} ⊢ (α→γ) } using formal deductions using Modus Ponens and the three axiom of H2 : A1: A -> (B-> A) A2: (A-> (B->C)) -> ((A-> B) -> ( A -> C)) A3: (( ¬ A) -> ( ¬B)) -> ( B -> A) However now my professor is asking […]

Natural deduction proof / Formal proof : Complicated conclusion with no premise

Find a formal proof for the following: $\vdash [(\neg p \land r)\rightarrow (q \lor s )]\longrightarrow[(r\rightarrow p)\lor(\neg s \rightarrow q)]$ As you can see. No premise to use. We have to use assumptions and eliminate them. This also means no using equivalent formulas. Because we can easily end the problem by finding a shorter and […]

natural deduction: introduction of universal quantifier and elimination of existential quantifier explained

Currently, I am dealing with the calculus of natural deduction by Gentzen. This calculus gives us rules to manipulate so-called sequents. Definition. If $\phi_1,\dots, \phi_n,\phi$ are formulas, then $\phi_1\dots\phi_n\vdash\phi$, often abbreviated by $\Gamma\vdash\phi$, is called a sequent. Can somebody please explain to me the following two inference rules? Introduction of the universal quantifier. $$ \begin{array}{c} […]

proving $ (A \rightarrow B \vee C )\rightarrow((A\rightarrow B) \vee (A\rightarrow C))$

I’m looking for a way to prow $ (A \rightarrow B \vee C )\rightarrow((A\rightarrow B) \vee (A\rightarrow C))$ from the following axioms and rules $$\vdash A \rightarrow A$$ $$\vdash A \wedge B \rightarrow A$$ $$\vdash A \wedge B \rightarrow B$$ $$\vdash A \rightarrow A \vee B $$ $$\vdash B \rightarrow A \vee B $$ $$A, […]

Prove $( \lnot C \implies \lnot B) \implies (B \implies C)$ without the Deduction Theorem

The issue is Exercise 1.47 (d) in Elliot Mendelson’s “Mathematical Logic”. The exercise is to prove $(\lnot C\implies\lnot B)\implies(B\implies C)$ by using the three axioms $(A1,A2,A3)$ without using the Deduction theorem (and without any hypothesis). The axioms are: $A1: B\implies(C\implies B)$ $A2: (B\implies(C\implies D))\implies((B\implies C)\implies(B\implies D))$ $A3: (\lnot C\implies\lnot B)\implies((\lnot C\implies B)\implies C)$ The only […]

Proof of transitivity in Hilbert Style

We can use the following axioms: $$\begin{align} &A\to(B\to A)&\tag{A1}\\ &[A\to(B\to C)]\to[(A\to B)\to(A\to C)]&\tag{A2}\\ &(\lnot A\to\lnot B)\to(B\to A)&\tag{A3} \end{align}$$ We need to prove: $$A\to B, B\to C\vdash A\to C$$ The hint is to use The Deduction Theorem. I can’t for the love of me figure it out, please help 🙁

Calculus of Natural Deduction That Works for Empty Structures

Currently, I am dealing with the calculus of natural deduction by Gentzen. This calculus gives us rules to manipulate so-called sequents. Definition. If $\Gamma$ is a set of formulas and $\phi$ a formula, then $\Gamma\vdash\phi$ is called a sequent. The rules of this calculus of natural deduction are: Hypothesis. $$ \begin{array}{c} \hline \Gamma\vdash\phi \end{array}\text{, where […]