Proving a conditional by assuming the antecedent

Question

My question is about proving the conditional statement true in propositional logic or math. In proving conditionsal statements, a lot of proofs assume the antecedent is true and then show that the consequent must be true from that.

Let's say the conditional is: If P, then Q. What does it mean in the truth table, for example, when P is true and Q is true. On the wiki page Conditional Proof it says "Thus, the goal of a conditional proof is to demonstrate that if the CPA [conditional proof assumption] were true, then the desired conclusion necessarily follows. The validity of a conditional proof does not require that the CPA [conditional proof assumption] to be true, only that if it were true it would lead to the consequent." So we assume P is true- but it potentially might not be- to prove the conditional is true. And the conditional is still true even when P turns out to be false. So when we are using truth tables to define the conditional statement "If P, then Q", what does it mean where it says P is true? We don't know that P is true we just assumed it.

I guess I assumed that it was shown that P is a true proposition and Q is a true proposition, therefore "If P, then Q" is a true proposition.

Answer 1

1. So when we are using truth tables to define the conditional statement "If P, then Q", what does it mean where it says P is true?

   We don't know that P is true we just assumed it.

   As you pointed out, the truth table for the logical operator/connective $\to$ is providing its definition. Notice that this definition doesn't consider the interpretation (assigned meaning) of the antecedent and consequent. Think of $\to$ as a truth-function (like any mathematical function), which returns a single, clearly-defined output (either T or F) when given any allowable input (which here is any of its four possible input combinations).

   This definition is neutral to the truth value of $P;$ we never had to assume that $P$ is true.

2. In proving conditional statements, a lot of proofs assume the antecedent is true and then show that the consequent must be true from that.

   I guess I assumed that it was shown that P is a true proposition and Q is a true proposition, therefore "If P, then Q" is a true proposition.

   Consider the following symbolisation key:
   $\quad P_1:$ Pigs are animals.
   $\quad P_2:$ Pigs are canines.
   $\quad Q_1:$ Quadrants contain right angles.
   $\quad Q_2:$ Quadrants contain obtuse angles.

   Then, for each interpretation (choice of $P_i$ and $Q_j$), the statement $P_i\to Q_j$ has a definite truth value.

   If, in a particular interpretation, it is known in advance that $P_i\to Q_j$ is a true statement, then $Q$ is true under the assumption/hypothesis that $P$ is true.

3. When making logical arguments (e.g., in Mathematics), $P$ not an atomic statement like the above, but actually a conjunction of premises each of which is a quantified compound statement. Similarly for $Q.$

   When, under a theory's axioms, the statement $P\to Q$ is logically true (i.e., true regardless of interpretation), then we call the statement a theorem. An example of a valid argument is “$P$ is true, and $P\to Q$ is a theorem; therefore $Q$ is true”.