I am confused on the usage of the word "interpretation" and/or "model" when it comes to propositional logic versus first-order logic because there are so many conflicting / unclear notions that I would like to clear up.
My current understanding is as follows and I would like any corrections / elaborations on where I am confused:
I am using the definition of a "model" being an interpretation that satisfies a formula / set of formulas. In the case of a theory we can treat its axioms as a set of formulas, and so a "model of a theory" meaning any interpretation that satisfies the set of axioms of that theory.
In propositional logic an "interpretation" is an arbitrary assignment of true/false values to all the atomic propositions in the alpha set. For example $p_0 = T, p_1 = F, p_2 = T, p_3 = F, …$ and so on. This tells us which row of any given truth table we should look at when evaluating the "truth value" of any fixed proposition.
But then in first-order logic, it seems like an "interpretation" is no longer some specific assignment of values to non-logical terms, but rather entire number systems like "the natural numbers," which would also be a "model" of, for instance, peano arithmetic, which satisfies its axioms.
Why is this? Why wouldn't we say "boolean variables" model propositional logic then? Why wouldn't we say some specific assignment of values satisfy first-order logic / PA / etc?
Why is interpretation seemingly being used differently in both cases? If an interpretation is what we'd call the specific T/F assignments in propositional logic, then what do we call the choice of a boolean system in the first place?
And more of a side question, but then what of propositional logic systems like natural deduction which have no axioms at all? What models "satisfy" it if there's no set of axioms to represent the theory?
Best Answer
In both propositional and predicate logic, the truth value of a formula is always either true or false, once an interpretation has been given. The set $\{true, false\}$ is not something you choose; it is a fixed part of how the logic works.
However, in predicate logic, formulas are not everything there is. Predicate logic also has terms, which are expressions that can be the arguments of relation symbols. (For example, in the language of arithmetic $2>3$ or $5=x+2$ are formulas; $2\cdot 3$ or $x+2$ are terms).
An interpretation in predicate logic tells you
In propositional logic there are no terms, no functions, and predicates. All of the atomic formulas are propositional letters. Seen from the predicate-logic end we can view a propositional letter as a "predicate symbol" that takes no operands. Thus, if we apply the above sense of interpretation, such as symbol should be represented either by the set $\{()\}$ that contains the (unique) tuple of length 0, or by the empty set.
But this corresponds to a choice of whether the propositional letter is true or false -- so an "interpretation" for propositional logic is effectively the same as a map from the propositional letters to $\{true, false\}$. All we need to do is to write $true$ and $false$ instead of $\{()\}$ and $\varnothing$.
Since there are no terms, there is no need for an interpretation to specify which kind of values the terms would have if there were any.