This may sound obvious but I think it was worth mentioning. :)
As we've been discussing them, sequences of symbols like "¬(p ∧ q)
" are not actually sentences. This is because, although we have given meaning to the connectives, we have not given meaning to the sentence letters.
A sequence of symbols like "¬(p ∧ q)
" is said to be a schema. (The plural of "schema" is "schemata".) They are forms that sentences can take. The kind of formal logic that studies the truth relationships between compound sentences based on the meaning of truth-functional sentential connectives is called truth-functional logic or propositional logic.
It is a kind of logic on its own, but it is also a fragment of first-order logic (which is what we've been doing, and which is the kind of logic that modern set theory is built on -- to propositional logic, FOL adds things like names, predicates, variables of quantification, quantifiers, function symbols, definite descriptions, and a built-in notion of identity through a =
predicate).
Schemata are sometimes called propositional formulas, and sentence letters are sometimes called propositional variables.
The notion of an interpretation of a schema is important, but there are actually two things that can mean.
In the strictest sense, an interpretation of a truth-functional schema is an assignment of actual sentences to its sentence letters.
However, sometimes what one means by an interpretation is merely the ascription of truth values to its sentence letters.
Two truth-functional schemata are said to be equivalent when they are true under all the same interpretations.
As a simple example, p
is equivalent to p
, but it is not equivalent to q
.
There are interpretation of p
and q
where they have the same truth value. But they do not have the same truth value under all interpretations. In contrast, p
has the same truth value as itself under all interpretations.
Likewise, ¬(p ∨ q)
has the same truth value as ¬p ∧ ¬q
under all interpretations.
Another phrase for equivalent, in this context, is logically equivalent. For truth-functional schemata, there is no ambiguity in speaking of equivalence--that is, there is nothing that it can mean to say truth-functional schemata are equivalent other than that they are logically equivalent--but in other contexts it is possible for confusion to arise.
I recommend now expanding
that truth table with columns for the biconditional between the each pair of equivalent schema, that is, with a column for: