"Logic"
is a set of formal rules for examining ideas in order to determine whether they
are true. Logic can be divided into two broad methods: deductive logic and
inductive logic.
Deductive logic reasons from statements
about reality, called premises, to
conclusions. Two or more premises may be combined in a syllogism to reach a conclusion.
Deductive
logic is often illustrated with the following example of a syllogism:
Premise
1: All men are mortal
Premise
2: Socrates is a man.
Conclusion:
Therefore, Socrates is mortal.
If the conclusion must follow from the
premises - that is, if it is impossible for the premises to be true and the
conclusion to be false - then the argument is valid. The above syllogism is valid because it's impossible for
Socrates to be immortal if he is a man and if all men are mortal. Note that an
argument being valid doesn't necessarily mean that its conclusion is true, only
that it follows from the premises. It's possible to have a valid argument
that's false:
Premise
1: All men are fish.
Premise
2: Socrates is a man.
Conclusion:
Therefore, Socrates is a fish.
The
conclusion follows from the premises, so it's a valid argument. But premise one
is false, so the conclusion is also false. It's important to remember that
"valid" doesn't mean "true." Valid refers to the structure
of an argument, not the truth of its premises or conclusion.
To
get conclusions that are true, an argument must be sound: it must both have true premises and be valid. Sound
arguments always yield conclusions that are true.
Common
mistakes when constructing syllogisms are Affirming
the Consequent and Denying the
Antecedent. In an if-then statement, such as "If you are a man, then
you are mortal," the antecedent follows "if," and the consequent
follows "then."
Affirming the Consequent:
Premise
1: All men are mortal
Premise
2: My fish is mortal.
Conclusion:
Therefore, my fish is a man.
This
argument affirms the consequent. Premise one says that if you are a man, then
you are mortal. The argument mistakenly takes premise one to also say that if
you are mortal, then you are a man. But, "If you are a man, then you are
mortal," is not the same as, "If you are mortal, then you are a
man." The mistake occurs when one takes the truth of the consequent, the
"then" part of the statement, to mean that the antecedent, the
"if" part of the statement, is also true. The "then" part
of premise one says that "you are mortal," and this is true of my
fish. The mistake is in thinking that the "if" part of premise one,
"you are a man," is also true of my fish. It is a mistake because
being mortal is an attribute of being a man, but being a man is not an attribute
of being mortal. The consequent, being mortal, is dependent on the antecedent,
being a man, but the antecedent is not dependent on the consequent. If you are
a man, you must be mortal, but lots of things are mortal, so being mortal
doesn't mean that you must be a man.
Denying the Antecedent:
Premise
1: All fish are mortal
Premise
2: Socrates is not a fish.
Conclusion:
Therefore, Socrates is not mortal.
This
argument denies the antecedent. Premise one says that if you are a fish, then
you are mortal. The argument mistakenly takes premise one to also say that if
you are not a fish, you are not mortal. The mistake occurs when one takes the
denial or falsehood of the antecedent to mean that the consequent is also
false. "If you are a fish, then you are mortal," doesn't imply that,
"If you are not a fish you are not mortal." It is a mistake because
while if the antecedent is true, the consequent must be true, the falseness of
the antecedent tells us nothing at all about the consequent. Knowing that
something is not a fish tells us nothing at all about whether it's mortal.
There may be many things that can cause the consequent. All we know is that the
particular one described in the antecedent is not true. If you are a fish, then
you are mortal, but lots of things are mortal, so not being a fish doesn't mean
you're not mortal.
Inductive logic is a method of reasoning in which an
argument's premises show that the conclusion is probably true. Unlike deductive logic, where if the argument is
sound, the conclusion must be true,
in inductive logic even when the argument is sound, there is a possibility that
the conclusion is false. Deductive logic takes the form, A and B, therefore C,
while inductive logic takes the form, A and B, so probably C, too.
Induction
is used to make predictions about what is likely to be true based on our
previous knowledge. Deduction reasons from the universal to the particular,
from statements about all men to a statement about Socrates, a particular man.
Induction extrapolates from the particular to the universal. We might look at
Socrates and inductively come to conclusions about all men. The more evidence
that points towards a conclusion, the stronger the inductive argument is and
the more likely it is to be true. If we look at a hundred men, we can make
better predictions about all men than if we only look at Socrates. Anything we
see in Socrates might be peculiar to him, but if it is shared by a hundred
other men, it is more likely that it is shared by all men. If we look at a million
men, and the all share a feature, it is even more likely that it is shared by
all men.
I
might make an inductive argument about the probability of my friend sharing his
lunch:
1. Every time
I've told my friend that I'm hungry, he offered to share his lunch.
2. If I tell
him I'm hungry now, he'll probably offer to share his lunch.
This
is an inductive argument because the conclusion, while probable, is not
certain. Perhaps today my friend is especially hungry, or perhaps I've offended
him in some way, and so today he won't offer to share. But given that every
other time I've told him I'm hungry, he offered to share his lunch, it's likely
that he'll offer to share this time, too.
The
greater my experience, the more probable it is that my conclusion is correct.
If my friend has offered to share two or three times, then his generosity in
those instances may not be indicative of his usual behavior. If he has shared
with me every day for the past year, then I can be pretty sure that he will
share today, too.
Most
of science relies on inductive logic. If something is tested and we get the
same result over and over, then we can be reasonably confident that we will
always get that result. If it is tested by different people in different
conditions and they also get the same result, then our confidence grows
stronger. It's unlikely that the conclusion is false, but there is always the
possibility that our conclusion will be overturned by new evidence that shows
our inference was mistaken.
