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We have a calculus $\vdash$ in the set $\mathcal{F}\{\vee\}$ of propositional formulas with the signature $\{\vee\}$. It has the following four unary Hilbert-style rules:

$$ \begin{align} (1)\ \alpha/\alpha\vee\beta,\quad (2)\ \alpha\vee\alpha/\alpha,\quad (3)\ \alpha\vee\beta/\beta\vee\alpha,\quad (4)\ \alpha\vee(\beta\vee\gamma)/(\alpha\vee\beta)\vee\gamma \end{align} $$

I am supposed to prove the following:

$$ \alpha\vdash\beta \Rightarrow \alpha\vee\gamma\vdash\beta\vee\gamma $$

I don't know how to do this at all. I don't know how to even start to make statements about $\alpha\vee\gamma$, as I haven't been given any tautologies acting as a logical axiom scheme, and applying the four rules of inference to $\alpha\vdash\beta$ only lets me make further conclusions as to what we can derive from $\alpha$, but not from $\alpha\vee\gamma$.

This is a part of exercise 4 in chapter 1.6 of A Concise Introduction to Mathematical Logic by Wolfgang Rautenberg. The whole exercise is to prove completeness of the calculus $\vdash$, and the solutions manual states that the first step is to prove the property above.

Edit: Can I do the following?

$$ \begin{align} \alpha\vee\gamma&\vdash(\alpha\vee\gamma)\vee\beta&(1)\\ &\vdash\alpha\vee(\gamma\vee\beta)&(4)\\ \end{align} $$

And if yes, how do I get rid of the $\alpha$? Can I use that $\alpha\vdash\beta\vee\gamma$? How and why?

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1 Answer 1

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Hint: I don't know about Hilbert-style, but in ordinary propositional logic, you could start with the assumption that $A\implies B$, then a 2nd assumption $B \lor C$ which will give you two cases to consider.

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