Elementary Logic FAQ

Validity and soundness

Q1: An argument is valid, if when all the premises are true the conclusion is true. What if the premises are inconsistent? What if it is impossible for all the premises to be true at the same time? Is the argument still valid?
A1: Yes. An argument with inconsistent premises is valid, regardless of what the conclusion is. If an argument has inconsistent premises, then it is impossible for all the premises to be true at the same time; hence it is impossible for all the premises to be true while the conclusion is false.

Q2: If the conclusion of an argument is tautological, does that means that the argument is valid?
A2: Yes. An argument with a tautological conclusion is valid, regardless of what the premises are. If the conclusion is a tautology, then there is no possible situation where the conclusion is false. Hence there is no possible situation where the premises are true while the conclusion is false.

Q3: If every argument with a tautological conclusion is valid, then why does it say in A02.7 that this argument is invalid?
(Premise) All cows are mammals.
(Conclusion) Therefore, the sun is larger than the moon.
Isn’t the conclusion a tautology?
A3: This argument is invalid since there is a possible situation where all cows are mammals, but the sun not to be larger than the moon. The conclusion is true, but the conclusion is not a tautology; there is a possible situation where the sun is not larger than the moon.

Q4: "It is either raining or not raining. Therefore there is no largest prime number." Why is this a valid argument as the premise and the conclusion are talking about completely different things?
A4: An argument is valid if there is no possible situation where the premises are true and the conclusion is false. Since it is necessarily true that there is no largest prime number, there is no possible situation where the conclusion is false. So there is not possible situation where the premises are true and the conclusion is false. Thus, the argument is valid.
You might think that this is strange. For it doesn't really seem as if the conclusion follows from the premises. So you might wonder whether our definition of "valid argument" is a good way to make precise the idea of a conclusion following from premises. That is a good thing to wonder about. Philosophers and logicians have thought about this a great deal. This issue is beyond the scope of this course. (If you are curious, you might read this article.)

Q5: I know that all sound arguments must have true conclusions because such arguments are by definition valid with all true premises. On the other hand, if an argument is valid and has a true conclusion, does it follow that it is sound?
A5: No. A valid argument may have a true conclusion even if not all its premises are true. For instance:
(Premise) If 4 is a prime number, then 5 is a prime number.
(Premise) 4 is a prime number.
(Conclusion) Therefore, 5 is a prime number.

Q6: "All arguments are either valid or unsound." Is this statement true?
A6: Yes. If the statement is false, there there is at least one argument that is both sound and invalid. But a sound argument is valid, so no argument is both sound and invalid. So the statement is true.

Valid and Invalid Argument Forms

Q7: Is "P or Q, P, Therefore not-Q" a valid argument form?
A7: No, if "P or Q" means "P or Q (or both)". This inclusive "or". This is a valid argument form if "P or Q" means "P or Q (but not both)". This is exclusive "or". Note that in this course we will assume the "or" is inclusive "or", unless explicitly stated otherwise.

Q8: Are arguments of the form "denying the antecedent” (or "affirming the consequent") necessarily invalid?
A8: No. For instance, the following argument is valid and sound, even though it has the form of denying the antecedent:
(Premise) If x is a positive integer that is only divisible by itself and 1, then x is a prime number.
(Premise) 4 is not a positive integer divisible only by itself and 1.
(Conclusion) Therefore 4 is not a prime number.”

Consistency and Entailment

Q9: If a statement X entails another statement Y, does it follow that there must be a possible situation in which both X and Y are true at the same time?
A9: No. If X entails Y, it only follows that there is no possible situation where X is true and Y is false at the same time. Even if X entails Y, it may still be that X and Y can never be true at the same time, for instance, if X is an inconsistent statement.

Q10: If statement X entails another statement Y, does it follow that whenever X is false, Y must also be false?
A: No, it does not.

Q11: If P is inconsistent with Q, then whenever P is true, Q is false. But
if P is consistent with Q, does it follow that Q is true whenever P is true?
A11: No. When P is consistent with Q, it only follows that it is logically possible for both P and Q to be true at the same time. However, it does not follow that whenever P is true Q must also be true. For instance: "Peter is male" is consistent with "Peter hates John." However, it is not the case that whenever Peter is male, Peter hates John.

Q12: If statement X does not entail statement Y, does it follow that whenever X is true, Y is false?
A12: No. If X does not entail Y, it only follows that there is at least one possible situation where X is true and Y is false. Even so, it may turn out that X and Y are both actually true. For instance, “LeBron James is a good basketball player” does not entail “LeBron James plays for the Cleveland Cavaliers”, even though both statements were true in 2007.

Q13: If an argument contains a hidden assumption, does that mean that it is not a good argument?
A13: No. Inductively strong arguments are invalid, and hence contain hidden assumptions the adding of which will turn them into valid arguments. But some inductively strong arguments are good arguments.

Q14. What is the difference between an argument and an explanation?
A14: Someone who gives an explanation is not trying to state reasons to believe that something is true. Instead, the point of a giving an explanation can be to provide understanding--- to help someone understand why something is true. For example, suppose Ethel says, ``Why is Herman so happy?" and Nora replies ``Herman is happy because he just won the lottery." Here Nora is explaining to Ethel why Herman is happy; Nora is trying to help Ethel understand why Herman is happy. Nora is giving an explanation, not making an argument. Nora is not trying give Ethel reasons to believe that Herman is happy. Nora is not trying to convince Ethel that Herman is happy. Ethel already believes that Herman is happy.
Suppose instead that Nora says ``Herman is not at home'', and Ethel replies ``No, Herman is at home, because he is either at his office or at home, and I just called his office and he is not there.'' Here Ethel is giving an argument. The conclusion of Ethel's argument is that Herman is at home. Ethel is giving reasons to believe that Herman is at home: Herman is either at home or at his office, and Herman is not at his office.
The same indicator word can be used both in arguments and in explanations. For example we have just seen the word "because" used both in an argument and in an explanation. "Since" is another word that can be used in both cases.

II.

SL formalization

Q14: Do "(A&B)" and "(B&A)" count as two WFFs or just one?
A14: These are two different WFFs. However, they are logically equivalent. The same goes for "(AVB)" and "(BVA)", "(A
B)" and "(BA)".

Q15: Should I formalize "Neither P nor Q" as "~(PVQ)" or "(~P&~Q)"?
A15: Both formalizations are acceptable and are logically equivalent.

Q16: Should I formalize "If P, then R if Q" as "((P&Q)→R)" or "(P→(Q→R))"?
A16: Both formalizations are acceptable and are logically equivalent. But notice that "(P→(Q→R))" better preserves the structure of the original statement.

Q17: Should we use the entailment symbol "
" even if the sequent is invalid?
A17: Yes. In this course we are using the symbol "
" as part of a sequent, even if the sequent is invalid. For example, "A (A & ~A)" is a sequent, though an invalid one.

III.

MPL Formalization

Q18: Shouldn't "If something is good then it is expensive" be formalized like this: "
x(Gx→Ex)" ?
A18: No. In many cases the best way to formalize the word "something" is to use an existential quantifier.
But not always. An appropriate formalization of "If something is good then it is expensive" is "
x(Gx→Ex)".
"
x(Gx→Ex)" would be an appropriate formulation of "There is something that is either expensive or not good".

Q19: What is the meaning of an "element" of a domain of quantification?
A19: An element of a domain is member of that domain, i.e. something that it is in that domain. For example, if the domain is the set of all human beings, then the President of France is an element of that domain.

Q20: Given only that the MPL WFF "(Ga&Gb)" is true under an interpretation, does it follow that the domain of that interpretation contains at least two elements?
A20: No. That might be an interpretation where "a" and "b" refer to the same element of the domain.

Q21: In MPL05.4, it says that φ entails ψ if and only if φ and ~ψ are inconsistent. Why?
A21: If φ entails ψ, then there is no interpretation under which φ is true and ψ is false. So there is no interpretation under which φ is true and ~ψ is true. So φ and ~ψ are inconsistent. If φ and ~ψ are inconsistent, then there is no interpretation under which φ is true and ~ψ is true. So there is no interpretation under which φ is true and ψ is false. So φ entails ψ.