What does it mean for two statements to be logically equivalent?

What does it mean for two statements to be logically equivalent?

Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution for their. statement variables.

How do we recognize logically equivalent conditional statements?

A conditional statement is logically equivalent to its contrapositive. Suppose a conditional statement of the form “If p then q” is given. The inverse is “If ~p then ~q.” Symbolically, the inverse of p q is ~p ~q.

What is logical equivalence examples?

Now, consider the following statement: If Ryan gets a pay raise, then he will take Allison to dinner. This means we can also say that If Ryan does not take Allison to dinner, then he did not get a pay raise is logically equivalent.

What does it mean if a statement is logically equivalent?

In logic and mathematics, statements and are said to be logically equivalent if they are provable from each other under a set of axioms, or have the same truth value in every model. The logical equivalence of and is sometimes expressed as , , , or. , depending on the notation being used.

Can two false sentences be logically equivalent?

No two false sentences are logically equivalent. A pair of equivalent sentences must both be false at the same time if they are false at all. Page 43. Focus on exercise sets 1 and 5.

Are logically equivalent statements consistent?

No two statements that are logically equivalent are (mutually) contradictory. Each pair of logically-equivalent statements is inconsistent if and only if they are both self-contradictory. Each pair of logically-equivalent statements is consistent if and only if neither is self-contradictory.

Are all true sentences logically equivalent?

Furthermore, by definition, two sentences (or propositions) are logically equivalent if and only if they have the same truth values (no matter what truth values their atomic constituents, if any, have). So, because tautologies always have the same truth value (namely, true), they are always logically equivalent.

What is the relationship between two equivalence tautological equivalence and logical equivalence?

Sentences that have the same truth value in every possible circumstance are logically equivalent. Logically equivalent sentences whose equivalence is due to the meanings of the truth functional connectives they contain are tautologically equivalent.

How do you determine if two statements are logically equivalent?

The following table depicts how two statements that are logically equivalent correlate with one another, whether both are true, one is true and one is false, or if both are false. As long as both statements have the same meaning or coverage, then this table holds true and the statements will always be connected and be logically equivalent.

How do you prove that two things are equivalent?

In general, there are two ways to show that two things are equivalent. You could use logical reasoning, or a truth table. Method 1: logical reasoning For example, you could say (for a smaller case):

Is it possible to prove a logical equivalency?

We have seen that it often possible to use a truth table to establish a logical equivalency. However, it is also possible to prove a logical equivalency using a sequence of previously established logical equivalencies.

How do you prove logical equivalence with a truth table?

We have seen that it often possible to use a truth table to establish a logical equivalency. However, it is also possible to prove a logical equivalency using a sequence of previously established logical equivalencies. For example, P → Q is logically equivalent to ⌝ P ∨ Q.