- #1
PWiz
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Homework Statement
Provide a complete formal proof that ## \vdash ((A \rightarrow B) \rightarrow C)
\rightarrow (B \rightarrow C)##.
Homework Equations
I am only allowed to use modus ponens and these four 'sentential logic' axioms:
A1 ## \neg \alpha \rightarrow (\alpha \rightarrow \beta)##
A2 ##\beta \rightarrow (\alpha \rightarrow \beta)##
A3 ##(\alpha \rightarrow \beta) \rightarrow ((\neg \alpha \rightarrow \beta) \rightarrow \beta)##
A4 ##(\alpha \rightarrow (\beta \rightarrow \gamma )) \rightarrow ((\alpha \rightarrow \beta) \rightarrow (\alpha \rightarrow \gamma ))##
The Attempt at a Solution
I have no idea where to begin. I'm thinking about using axiom 2, but I don't know how I would proceed from there. The problem would become easy if I was allowed to 'add an antecedent', but I am not allowed to directly do that. Any help is appreciated. Please note that I am a new student to logic, and I have only studied zero-order logic until now, so kindly provide hints that I will be able to understand. (The homework is due tomorrow!)