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.

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 […]

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 […]

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} […]

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, […]

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 […]

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 🙁

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 […]

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