Elementary Logic FAQ
I. Arguments
Validity and Soundness
Valid and Invalid Argument Forms
Consistency and Entailment
Good Arguments
Arguments and Explanations
II. Sentential logic
SL Formalization
III. Predicate logic
MPL Formalization
Logical properties in MPL
I.
Arguments
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.
Good
arguments
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.
Arguments and Explanations
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.
Sentential Logic
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 "(B↔A)".
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.
Predicate Logic
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".
Logical properties in MPL
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.
Last modified 3 January 2011