Question Concerning Validity and Logical Arguments

Question

I have some questions about the study of logical reasoning and arguments.

Is it true that no two false sentences are logically equivalent? For example, "A square has 5 sides" and "I have one biological parent". I could not think of counterexamples to this.

Also, would every inconsistent set of sentences (that is to say, sentences that contradict each other) includes a logically false sentence in SL?

Lastly, can a valid argument with a false conclusion have true premises? I'm almost certain that this one is plausible since a valid argument does not require the truth of the conclusion.

Answer 1

1. To say of two sentences that they are logically equivalent is usually understood to mean something like they are true under the same interpretations, or they share the same models. To use less technical language, there is no possible way for one of them to come out true and the other false. Any two contradictions are logically equivalent, since there is no way for either to come out true. So, "water is a metal and water is not a metal" is logically equivalent to "the Pope is Chinese and the Pope is not Chinese". But sentences that are contingently false are not logically equivalent, so "water is a metal" is not logically equivalent to "the Pope is Chinese". It would be possible for one to be true and the other false.

2. A set of inconsistent sentences does not need to include a logically false sentence, i.e. a contradiction. The set { A and not B, B and not C, C and not A} is inconsistent, but no one sentence is a contradiction. We could say that the set as a whole entails a contradiction.

3. A valid argument with true premises cannot have a false conclusion. Validity guarantees that true premises lead to a true conclusion. If the conclusion is false and the argument is valid then at least one premise is false.

If you are talking about more advanced kinds of logic, there might be some exceptions to the above, but I am assuming you are asking about elementary logic.