If in propositional logic one is trying to simplify a formula by evaluating its subformula, would it be considered an abuse of notation to actually substitute the bits $\{0,1\}$ in for the formula, to say something like "$0\wedge 1\equiv 0$" or "$0\wedge 1=0$".

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    $\begingroup$ I’m not entirely sure what you have in mind; could you give an actual example? $\endgroup$ – Brian M. Scott Jul 21 '13 at 0:23
  • $\begingroup$ @BrianM.Scott Say A and B have truth value $1$ and I want to evaluate $\neg (A\wedge B)$, then I could say $\neg(A\wedge B)\equiv \neg( 1\wedge 1)\equiv \neg 1 \equiv 0$. This way I am consistently aware of the truth values of the formula A and B, with out having to look back to what there truth values are. $\endgroup$ – Ethan Jul 21 '13 at 0:27
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    $\begingroup$ If you’re working in a setting in which truth values are $0$ and $1$, and the connectives have been defined as operations on $\{0,1\}$, then such a calculation would be fine. $\endgroup$ – Brian M. Scott Jul 21 '13 at 0:28

Judging by the comments it seems that what you are trying to do is fine. I think that it would be slightly clearer to use some predetermined symbols for "True" and "False", e.g. $\top$ and $\perp$ or $\Bbb T$ and $\Bbb F$.

This makes it easier to understand that you're talking about truth values.

Of course, if the context is clear that $0$ and $1$ are truth values, then writing those is fine as well.

Let me add my usual advice, be sure to write something like this at the start:

Given a proposition $A$ and an assignment $\sigma$, we shall write $1$ if $A$ is true in that assignment, and $0$ otherwise.

This way you can hint the reader how you are going to abuse them. Or the notation.

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    $\begingroup$ Or use the words 'true' and 'false' (as in $\;A \land \lnot A \equiv \textrm{false}\;$), as Dijkstra c.s. do. $\endgroup$ – Marnix Klooster Jul 21 '13 at 13:44

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