Consider the following truth table, which serves to define the logical connective ⇔,
P | Q || P⇔Q T | T || T T | F || F F | T || F F | F || T
According to the above truth table, the logical connective ⇔ is defined as the binary operation which takes as its arguments two statements
Q, and produces another statement which has truth value "true" when both the truth values of
Q are the same, and "false" otherwise.
⇔ in the above way, one can prove (using truth tables) that the statement
P ⇔ Q is equivalent to (that is, it has the same truth table as) the statement
(P ⇒ Q) ∧ (Q ⇒ P). In showing these statements
P ⇔ Q and
(P ⇒ Q) ∧ (Q ⇒ P) are "equivalent" using their truth tables, we are invoking a "truth functional" notion of equivalence.)
However, does there exist a more rudimentary notion of equivalence, separate from the truth table definition of equivalence? I can try to suggest why this might be desirable. For example, it is commonplace to consider the statement
3>2 "equivalent" to the statement
3-2>0, not only because both statements are true (and hence have the same truth values, allowing us to say
(3>2)⇔(3-2>0) is "true" using the above definition of
⇔), but also in that they are saying the same thing in terms of their content or meaning! Therefore, we might want to pick a new symbol such as
≡ and say (in a stronger sense) that
On the other hand, consider the statements
4+6=10. Since both statements are true, we could write (using the definition of
⇔ above) that
(3>2)⇔(4+6=10), which seems counter-intuitive. Or, for another example, suppose the statements "the sky is blue" and "the grass is green" are both always true. Then since both statements have the same truth value, we could again write (using the definition of
⇔ above) that
"the sky is blue" ⇔ "the grass is green" is true.
Here's my point: Although in the examples above we've exhibited truth-functional equivalence, we probably wouldn't think of these statements as "equivalent" in terms of their meaning, since they are saying completely different things. E.g., the content of
4+6=10 are unrelated to each other.
Am I missing something here? Is there a more rudimentary notion of equivalence, separate from the truth table definition? Should we use the symbols
⇔ differently, perhaps reserving
≡ for statements equivalent in their content or meaning (like
3-2>0) and not merely for statements such as
Thanks for your thoughts!