Introduction to Deduction

As we explain in the Introduction to Induction and Deduction, an argument is inductive if its major premise is based on observation or experience, and deductive if its major premise is based on a rule, law, principle, or generalization. In general, there are two distinct ways of expressing a deductive argument: as a syllogism, or as a conditional. Any deductive argument can be expressed as either a syllogism or a conditional, though some arguments may seem to lend themselves more naturally to one form or the other. Similarly, tests for the validity of syllogisms and conditionals may appear quite different, but do essentially the same thing.

Syllogisms: The major premise of a syllogism states that something, Y, is or is not true for all or part of some group, X; the minor premise affirms or denies that some group or individual, Z, is part of X; and the argument then concludes whether that thing Y (from the major premise) is true or not true for that group or individual Z (from the minor premise). One form of a syllogism can be expressed by the following paradigm:

All X are Y
Z is X
Therefore, Z is Y

Consider the following example:

Everyone in class today received instructions for writing the essay. Mandia was in class today. Therefore, Mandia received instructions for writing the essay.
You might think that "everyone in class today received instructions for the essay" sounds like an observation, but it is a generalization: no observer is identified, and no process of observation is recounted. By using a generalization, we focus attention more directly on the truth of an assertion (and less on the manner of its verification); this is especially effective when the generalization is widely accepted, or when there is strong evidence to support it.

We can restate the argument as follows:

[Major:] "Receiving instructions" is true for all of the group "in class today."
[Minor:] "Mandia" is a member of the group "in class today."
[Conclusion:] "Receiving instructions" is true for "Mandia."
Notice that, twice, the phrase in the original example, "received instructions for writing the essay," became in the restatement, "receiving instructions." There are two reasons for this. First, a restatement of an argument should eliminate or shorten unnecessary terms, to make the argument more comprehensible. Here, we shortened "instructions for writing the essay" to "instructions"; if significant, the phrase's original form can be resubstituted in the conclusion.

Second, in order to avoid confusion, it is always best to use a state-of-being verb (for example, forms of the verb "to be") in the restatement of an argument, and convert the original verbs to other parts of speech. In this case, "received" has become a participial phrase, "receiving instructions," that functions as a noun.

Conditionals: The other common form of a deductive argument, a conditional, expresses that same reasoning in a different way. The major premise is, If something is true of P, then something is true of Q. The minor premise either affirms that it is true of P, or denies that it is true of Q. In the former case, the argument concludes that the something is true of Q; in the latter, that something is not true of P. One form of a conditional is expressed by the following paradigm:

If P then Q
P
Therefore, Q

The above example could be given in the form of a conditional as follows:

If Mandia was in class today, he received instructions for writing the essay. Mandia was in class. Therefore, he received instructions for writing the essay.

In the form of the paradigm above, this conditional can be restated as follows:

[Major:] If "in class" is true, then "received instructions" is true.
[Minor:] "In class" is true.
[Conclusion:] "Received instructions" must be true.

Notice that a conditional seems to use only two terms (P and Q), while a syllogism uses three (X, Y, and Z). But the third term is actually there. In our example, it is Mandia who is "in class," and Mandia who "received instruction."


Summary. Consider this example:

Jerzy claims that all his test scores have been good, and so his course grade should be good, too.
We can express that argument as a syllogism or a conditional:
Syllogism:

All good tests get good grades.

Jerzy's are good tests.


Therefore, Jerzy gets a good grade.
--or--

Conditional:

If good tests, then good grades.

Good tests.


Therefore, good grade.
These two arguments reach the same conclusion, and their minor premises are similar, but their major premises appear to be rather different. In fact, "All good tests get good grades" and "If good test then good grade" are just two ways of expressing a relationship between good test scores and good course grades.


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