, is called the consequent. A conditional is considered true when the antecedent and consequent are both true or if the antecedent is false. When the antecedent is false, **the truth value of the consequent does not matter; the conditional will always be true**.

## Is true in all cases except when the antecedent is true and the consequent is false?

The “⊃” symbol is called the “horseshoe” and it represents what is called the “material conditional.” **A material conditional** is defined as being true in every case except when the antecedent is true and the consequent is false.

## Why is an implication true if the premise is false?

We could say that a false premise implies that the implication is true exactly when the conclusion is true, but that would be odd because then **the premise doesn’t do anything**. We could say that a false premise implies the implication is true exactly when the conclusion is false, but eww.

## What makes a conditional statement true?

A conditional statement is a statement that can be written in the form “If P then Q,” where P and Q are sentences. For this conditional statement, P is called the hypothesis and Q is called the conclusion. Intuitively, “If P then Q” means that **Q must be true whenever P is true**.

## When can we say that a conditional statement are always true?

Though it is clear that a conditional statement is false only when the hypothesis is true and the conclusion is false, it is not clear why **when the hypothesis is false**, the conditional statement is always true. To try to explain why this is this case, we consider another example. Example 1.3.

## Under what condition or conditions is an implication be false?

An implication is the compound statement of the form “if p, then q.” It is denoted p⇒q, which is read as “p implies q.” It is false only **when p is true and q is false**, and is true in all other situations.

## Can false premises lead to a true conclusion?

**False premises can lead to either a true or a false conclusion even in a valid argument**. In these examples, luck rather than logic led to the true conclusion.

## Why is a conditional statement true if the hypothesis is false?

Hypotheses followed by a conclusion is called an If-then statement or a conditional statement. This is read – if p then q. **A conditional statement is false if hypothesis is true and the conclusion is false**. The example above would be false if it said “if you get good grades then you will not get into a good college”.

## What is a true/false statement?

A true-false statement is **any sentence that is either true or false but not both**. A negation of a statement has the opposite meaning of a truth value. A negations is written as ~p. If we call the statement: cucumbers are green, p then: p: cucumbers are green – this statement is true.

## What is the antecedent of this conditional statement?

For propositions P and Q, the conditional sentence P⟹Q P ⟹ Q is the proposition “If P, then Q. ” **The proposition P** is called the antecedent, Q the consequent. The conditional sentence P⟹Q P ⟹ Q is true if and only if P is false or Q is true.

## When the antecedent is true and the component is false?

A conditional asserts that if its antecedent is true, its consequent is also true; **any conditional with a true antecedent and a false consequent must be false**. For any other combination of true and false antecedents and consequents, the conditional statement is true.

## What makes a conditional false is a false antecedent and a false consequent?

If one component statement in a conjunction is false, the conjunction is false. The statement is TRUE because a conjunction is true only when both component statements are true. **A conditional statement is false only when the consequent is true and the antecedent is false.**

## What is a conditional statement that is false but has a true inverse?

Negating both the hypothesis and conclusion of a conditional statement. For example, the inverse of “If it is raining then the grass is wet” is “If it is not raining then the grass is not wet”. Note: As in the example, **a proposition** may be true but its inverse may be false.

## When p is false and q is true then p or q is true?

A second style of proof is begins by assuming that “if P, then Q” is false and derives a contradiction from that. In the truth tables above, there is only one case where “if P, then Q” is false: namely, P is true and Q is false.

IF…., THEN….

P | Q | If P, then Q |
---|---|---|

F | T | T |

F | F | T |

## Are inverse statements always true?

If the statement is true, then the contrapositive is also logically true. **If the converse is true, then the inverse is also logically true**. If two angles are congruent, then they have the same measure.

Converse, Inverse, Contrapositive.

Statement | If p , then q . |
---|---|

Inverse | If not p , then not q . |

Contrapositive | If not q , then not p . |