One example of such a statement is “if Tokyo is in France, then the Eiffel Tower is in Bolivia”. Such statements are considered **vacuous truths**, because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent.

## What happens when the antecedent is false?

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**.

## Can a conjunction be true even if it has a false conjunct?

Conjuncts: the statements that are combined in a conjunction (ex. Mary has blue hair and Tom has purple hair); **a conjunction is true only if both its conjuncts are true, but false otherwise**.

## 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.

## What makes a conditional false?

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”. If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional.

## What do you call this statement which can only be true if both statements are true or false?

**A conjunction** is true only if both statements that form the conjunction is true. If we have two statements that are joined by “or” we have a disjunction.

## What makes a disjunction true?

A disjunction is true **if any one of the statements in it is true**. Here the statement p is true and q is false. So, the disjunction p∨q is true. The statement q is false and r false.

## What is syllogism law?

In mathematical logic, the Law of Syllogism says that if the following two statements are true: (1) If p , then q . (2) If q , then r . Then we can derive a third true statement: (3) If p , then r .

## What do we call a premise with an if-then statement?

If–then arguments , also known as **conditional arguments or hypothetical syllogisms**, are the workhorses of deductive logic. They make up a loosely defined family of deductive arguments that have an if–then statement —that is, a conditional—as a premise. The conditional has the standard form If P then Q.

## Is a compound statement that is always true?

**A tautology** is a compound statement that is always true.

## What do you call a statement that is always false?

**Contradiction**: A statement form which is always false.

## What is a statement that is either true or false?

**Proposition** is simply a statement that is either true or false, has no variables involved. But predicates can take variables, and once we replace the variable by a constant, it becomes a proposition.

## What do you call a statement that is always true?

**Tautology**: A statement that is always true, and a truth table yields only true results.

## Which is the false statement?

A lie is a statement that is known to be untrue and is used to mislead. A false statement is **a statement that is untrue but not necessarily told to mislead**, as a statement given by someone who does not know it is untrue.

## What is proposition statement that is always false?

A proposition has only two possible values: it is either true or false. We often abbreviate these values as T and F, respectively. Given a proposition p, we form another proposition by changing its truth value.

2.1: Propositions.

p | ¯p |
---|---|

T | F |

F | T |