This question is related to my previous question on forms of statements, here: Rigorous definition of the set of forms of a propositional formula. Consider the propositional formula $(p \land p)$, where $p$ is a propositional atom. It is of the form $A$, and it is also of the form $(A \land A)$. But, I am unclear as to whether it is also of the form $(A \land B)$. Is $(A \land B)$ indeed one of the forms of the formula $(p \land p)$? Of course, we could ask an analogous question with a formula like $((p \land p) \land q)$, or $((p \land p) \land p)$. Basically, I want a precise definition of the forms of a statement, that would clarify my confusion.
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Two equivalent ways of formalizing a proof system (specifically: propositional calculus): 1) with axioms in the language: $p \to (q \to p)$ and a rule for Substitution, and 2) with schemata: $\varphi \to (\psi \to \varphi)$. – Mauro ALLEGRANZA Apr 13 '22 at 08:01
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See also the post Substitution in Propositional calculus – Mauro ALLEGRANZA Apr 13 '22 at 08:04
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The Answer given to that linked question will give you the proper definition to answer this question.
Consider: can you substitute sentences for $A$ as well as for $B$ such that $A \land B$ becomes $p \land p$? Answer: Yes. Just substitute $p$ for $A$ and substitute $p$ for $B$. So yes, $ p \land p$ is of the form $A \land B$ … meaning that $A \land B$ is a form of $p \land p$.
I suppose you are worried about the fact that we substituted the same statement $p$ for two different statement variables $A$ and $B$. But remember that $A$ and $B$ are just that: variables. So they could have the same value, just as in arithmetic the values of $x$ and $y$ in the expression $x + y$ can be the same.
Bram28
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