Question

I have a table of the rules of inference in propositional logic. Among the entries are an Associative and a Commutative. The Associative rule is expressed with disjunction, but the commutative is expressed with conjunction. Obviously, in terms of permutations of syntax, it seems there should be both an Associativity of Conjuction and Disjunction as two distinct expressions as well as Commutativity of Conjunction and Disjunction as two distinct expressions. And yet in the table, there is not.

Is it historical? Might it be a space constraint? Did the author have a preference? Is it understood that some rules of inferences are preferential as canonical primitives in computation? Is there a canonical taxonomy of sentential rules of inference? Is there some unknown unknown relevant to my question?

What would be the rationale for excluding two of the four possible syntactical configurations in the resource I have listing the rules of inference?

Answer 1

The exclusion of certain syntactical configurations in the rules of inference table is likely due to the lack of practical applications or implications in propositional logic. The included rules are the most commonly used and effective for deriving valid conclusions.

Step-by-step explanation:

The rationale for excluding the Associativity of Conjunction and Disjunction as well as the Commutativity of Conjunction and Disjunction in the resource you have is likely due to the fact that these rules do not have significant practical applications or implications in the context of propositional logic. The rules of inference included in the resource are the ones that are widely used and have been shown to be effective for deriving valid conclusions from propositional logic statements. It is important to note that the rules of inference in propositional logic are not exhaustive and there might be alternative rules that can be used, but the ones included in the resource are the most commonly used ones.