Logic Primer 3: Thinking Logically

Logical thinking is a process. As long as the rules are not broken, the thought process will bring good conclusions. Now we will look at logical syllogisms.

The most basic logical structure is the syllogism. The syllogism is a deductive argument consisting of premises and a conclusion.1

It should be noted from the outset that for each of the following syllogisms presented, pages and pages could be (and have been) written with much more detail, explanation, and exceptions. What has been presented here is only a cursory glance at each one and should be treated as such. The reader is encouraged to delve into a more systematic textbook to explore fully.

A categorical syllogism is composed of two unconditional statements that lead deductively to an unconditional conclusion. An example of a categorical syllogism is as follows:

1. All cats are mammals.
2. Fuzzy is a cat.
3. Therefore, Fuzzy is a mammal.

The categorical syllogism has various forms and moods, which will not be detailed here, but the basic form simply entails two statements leading to a conclusion.

Hypothetical syllogisms take the form of a hypothetical statement. This syllogism has the word “IF” at its core. The hypothetical proposition uses the word if to make a conditional statement: if one state of affairs is true, then another state of affairs will follow. The first hypothetical syllogism is the Modus Ponens, structured like this:

If P, then Q.
P.
Therefore, Q.

Modus ponens means “way of affirmation” in Latin because it affirms the antecedent of the first proposition. One form of the cosmological argument takes the form of modus ponens:

If a contingent being exists, then a necessary being must exist as its cause.
A contingent being exists.
Therefore, a necessary being must exist as its cause.2

The other hypothetical syllogism is called Modus Tollens, which means “the way of denial.” This form of syllogism denies the consequent (the “then Q” part of the first statement). It is structured like this:

If P, then Q.
Not Q.
Therefore, Not P.

Disjunctive Syllogisms are either/or sentences. One statement is made with two alternatives, of which only one can be true.3 The disjunctive syllogism looks like this:

Either P or Q.
Not Q.
Therefore, P.

The way the disjunctive syllogism works requires for one alternate to be denied for the other one to be true. It is a fallacy to affirm one alternate to eliminate the other, because it is possible for them both to be true. Geisler and Brooks offer an excellent example of this fallacy found in Bertrand Russell’s book Why I am not a Christian:

Life was caused either by evolution or by design.
Life was caused by evolution.
Therefore, it was not caused by design (so there is no reason to posit God).

Geisler and Brooks explain: “This approach commits the formal fallacy of affirming one alternate. Even if the minor premise were true, the conclusion would not follow. For it is possible that both are true; that is, that evolution is designed.”4

The conjunctive syllogisms take the form of “both…and” statements. Here is the form:

Both P and Q are true.
Therefore, P.
Therefore, Q.

The conjunctive syllogism is fairly straightforward. Both terms in the first statement are separated and can be affirmed individually.

The Dilemma form of syllogism takes two hypothetical syllogisms and weds them with a disjunction. Here is what the dilemma looks like:

(If P, then Q) and (If R, then S).
P or R.
Therefore, Q or S.

The mathematician Pascal presented a dilemma with this syllogism:

If God exists, I have everything to gain by believing in him.
And if God does not exist, I have nothing to lose by believing in him.
Either God does exist or he does not exist.
Therefore, I have everything to gain or nothing to lose by believing in God.5

The final syllogism presented here is the Sorites. This comes from a Greek word meaning “heap.” The premises are stacked together in a heap to come to a final conclusion. An example:

All A are B……………or……………If A then B
All B are C……………or……………If B then C
All C are D……………or……………If C then D
Therefore, all A are D.…..or…..Therefore, if A then D.

That is a basic look at basic logical syllogisms.

Here are some resources that will get you started:

Audio resource:
– Critical Thinking Audio Course

Helpful book:
– Introduction to Logic by Harry Gensler

Some web Sites on logic:
– Philosophy Pages logic index
– Atheism Analyzed looks at atheism from a logical perspective.

Tomorrow we will take a look at language.

1 Geisler & Brooks, p. 194.
2 Ibid., p. 61.
3 In a weak disjunction both may be true.
4 Ibid., p. 66.
5 Ibid., p. 69.