Question

Having trouble doing this

In this problem you can use either the natural deduction system or the Hilbert-style proof system

(a) Give a deduction. You may not use any meta-rules such as the Deduction Theorem or completeness.

$\vdash (\alpha \rightarrow \neg\beta)\rightarrow(\beta \rightarrow \neg\alpha)$

b) Give a deduction or show that a deduction exists, i.e. you may use meta-rules such as hypothetical syllogism or the deduction theorem (but not completeness.)

$\vdash (\alpha \rightarrow (\alpha \rightarrow \beta)) \rightarrow (\alpha \rightarrow \beta)$

Answer 1

proportional rightarrow beta pr not d v not beta Im not beta v not 2 com not beta rightarrow not (not 2). Not beta rightarrow d (D . N). therefore (proportional rightarrow not beta) rightarrow (not beta rightarrow proportional). (proportional rightarrow (proportional rightarrow beta)) rightarrow (proportional rightarrow beta) pr not proportional v (proportional rightarrow beta) Im not proportional v (not proportional v beta) Im (not proportional v not proportional) v beta Asso not proportional v beta Id proportional rightarrow beta Hence the result.