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I'm a beginner at proof writing and I'm confused about what I can use as truth to prove a statement of the form $((\varphi \to \psi) \land \chi) \implies \sigma,$ where $\varphi, \psi, \chi$ and $\sigma$ are distinct formulas.

I'm omitting the contents of the formulas because this is something that has been confusing me for different statements.

I know for sure that I can use $\chi,$but I'm not sure what I can use from $(\varphi \to \psi)$. Can I consider the contents of $\psi$ as truth to derive $\sigma$? Can I use $\varphi$ as well?

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    $\begingroup$ If you can establish $\varphi$, then you can use $\psi$. You can also use the fact that $\varphi\to \psi$ is equivalent to $\lnot \varphi \lor\psi$. So if you can establish $\sigma$ from $\lnot \varphi$ and then also from $\psi$ (all while being able to freely use $\chi$), then you’re done. $\endgroup$
    – spaceisdarkgreen
    Commented May 23, 2023 at 23:45
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    $\begingroup$ Just a quick note regarding my edit: notice that φ ≠ ψ ≠ χ ≠ σ does not mean that the four formulae are distinct from one another, since for example φ = χ is not precluded. $\endgroup$
    – ryang
    Commented May 24, 2023 at 1:21

1 Answer 1

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The formula $$\big((P\to Q) \land R\big) \to S$$ is equivalent to each of the following

  • $\Big((\lnot P\land R) \to S \Big) \quad\land\quad \Big((Q\land R) \to S\Big)$
  • $R\to \Big( (\lnot P \to S ) \quad\land\quad (Q \to S) \Big).$
  • $(R \land \lnot S) \to (P \land \lnot Q).$

Does that help?

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  • $\begingroup$ Just to be sure: “$\rightarrow$“ here means “implies”? $\endgroup$
    – Simón Flavio Ibañez
    Commented May 24, 2023 at 2:49
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    $\begingroup$ @SimónFlavioIbañez For simplicity, I'm just using material implication throughout. The OP can then adapt the substance of this answer to their requirement. $\endgroup$
    – ryang
    Commented May 24, 2023 at 2:51

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