# Proving a conditional by assuming the antecedent

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.

• It's not clear to me whether you are asking about proving $Q$ or about proving "if $P$ then $Q$". Oct 3, 2021 at 8:49
• What does truth value of P mean in proving the conditional. It can't mean P is a true proposition because we just assumed P was true. We didn't prove it true. In fact it can be a false proposition, but we can still assume it's true in proving "If P, then Q" is a true proposition. So what does it mean when P has a true value in the truth table? Oct 3, 2021 at 14:05
• Let $P$ be "$x>3$". Let $Q$ be "$x>2$". From $P$, we can deduce $Q$, so it is true that "if $P$, then $Q$". This makes no assumption about the truth of $P$. If $x$ turns out to be $2.5$, or $1$, it is still true that "if $P$, then $Q$", even though $P$ happens to be false in these cases. Oct 4, 2021 at 1:59
• Not sure if this is what you are getting at, but if you know only that P is false and that P implies Q, you cannot infer anything about the truth value of Q, i.e. whether Q is true or false. Oct 4, 2021 at 16:30

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”.