How to reduce predicate logic into propositional logic?

Question

I read that predicate logic reduces into propositional logic. However, I couldn't find anything online that explains the process. Do you know where I could find an explanation?

Thank you.

Answer 1

Full predicate logic (with polyadic predicates) cannot be reduced to propositional logic, but monadic predicate logic can. You can see a sketch of this reduction, e.g., here.

Answer 2

No, if I understand you correctly (and a lot depends on what is meant by "reduction"), what you suggest is generally impossible. Predicate logic has greater expressive ability than propositional logic (and, as it happens, propositional logic is usually included in what is called the "propositional fragment" of first-order logic, which is propositional logic with some new things I explain below). The reason for this is that while propositional logic says nothing about logical form of an argument or set of arguments beyond propositional variables and connectives, first-order logic (with or without identity) goes further: introduces quantifiers (for saying how much of something there is) and predicates and variables, constants (saying which properties hold of which objects). And from this base you can build functions (as a special kind of two-place predicate), etc. You can't do any of this in propositional logic. All you can do is observe the general argumentative structure between propositions (which, again, are not yet analysed into subject-predicate form).

Answer 3

I should add that there is a theorem of Herbrand which states that an arbitrary formula $\phi$ of FOL is valid iff it can be converted to a disjunction of instances of the quantifier free part of $\phi$, which suggests a kind of reduction to the propositional notion of tautology, but the kind of full scale reduction (like, to propositional variables) I think you intend is out of the question, because of the conversion into the disjunction will always involve first-order notions.