Logic Lesson 4: Types of Deductive Arguments

There will be some Greek words here, but don’t panic – you don’t have to know them by heart to understand logic. Much of logic was developed by Aristotle, and the Greek names have been kept. But the concepts are simple. So don’t let the Greek bother you.

There are some terms that are used in logic which you already use yourself:

And;
Or;
If … then;
If and only if.

These are connectives and are a part of deductive arguments.

Deductive arguments are based on propositions, and these propositions are demonstrations of the relationships of two (or more) categories. The categories we will use here are category Q, and category P. First we will list the relationships in words, and then in Venn diagrams.

Categorical Propositions

All Q is P; This means that all of the Q category is a subset of the P category.



No Q is P; Q is completely outside and independent of the p category.



Some Q is P; Part of the Q category is inside the P category.



Some Q is not P; Part of the Q category is outside of the P category.



These categories have been labeled A, E, I, O for analysis which we will attack later. The labels are shown here:

A. All Q is P; This means that all of the Q category is a subset of the P category.

E. No Q is P; Q is completely outside and independent of the p category.

I. Some Q is P; Part of the Q category is inside the P category.

O. Some Q is not P; Part of the Q category is outside of the P category.



Deductive Reasoning

Deduction works from a known valid category to try to determine the validity of a second category. An example we have already used is this:

Premise 1: If P is valid, then Q is valid; (This is a proposition, not an argument).

Premise 2: P is valid; (This is an assertion of truth or "what is").

Conclusion: Therefore, Q is valid. (This is a conclusion).

The argument contains three elements: Two premises consisting of a proposition and an assertion; and the conclusion.

The proposition, we can see, is the A proposition but stated a little differently: It says that the state of category Q is demonstrated by the state of P, or, the validity of Q is a subset of the validity of P. The Venn diagram again shows how this maps out, logically and categorically:


From the diagram it is easy to see that if P is valid, then Q must be valid also, because Q is part of P.

If the premises of the deductive argument are valid, then the conclusion cannot be other than valid: the conclusion is absolutely a function driven only by the premises. Also, one of the premises, the assertion, is claimed to be factual or True. It is the Truth of this premise that guarantees the Truth of the proposition – if the argument follows the proper format. In this case it is P which is asserted to be True.

This format is also written as follows:

If P, then Q
P;
Q.

Argument Failures

An argument fails if it doesn’t follow a proper format. An argument fails if the proposition is not really the case (i.e., the categories aren’t really related in the fashion shown in the proposition). And the argument fails if the assertion is not actually true or valid. Remember that it requires an overarching truth (or law, as in a law of nature, or an undeniably valid axiom) in order to validate the proposition.

Deductive Argument Cases

These cases are also known as Basic Rules of Inference, because the conclusion is inferred [note 1]from the premises. To change things up a little, the letters are reversed; don't get hung up on a letter always meaning the same thing - letters are just symbols, randomly attached to categories. Make your own Venn diagrams for these on scratch paper; it's really quite simple.

1. Modus Ponens

If Q, then P;
Q;
P.

Recognize this? We’ve been using it in the examples.

2. Modus Tollens

If Q, then P;
~ Q; (the tilde, ~, just means “not”, or a negation; here we have: ~Q = not Q)
~ P.

3. Hypothetical Syllogism

If Q, then P, and If P, then S;
Q;
S

This is also written in a more familiar form:

If Q, then P;
If P, then S;
If Q, then S.

So the validity of Q guarantees the validity of S.

4. Disjunctive Syllogism

Either Q Or P is true;
~P;
Q.

5. Constructive dilemma

[If Q, then P] And [If S, then T];
Q Or S;
P Or T.

6. Absorption

If Q then P;
Therefore, If Q, then (Q And P).

7. Simplification

P And Q;
P.

8. Conjunction

P;
Q;
P And Q.

9. Addition

P;
P Or Q.

As always, ask any question, make any comment.

Notes
Note 1: In logic the term inference has a more concrete meaning than it might have in general usage. In logic, inference means that the "conclusion definitely follows from...", assuming that the argument structure is correct and the premises are true and meaningful. This is slightly different from common usage, where inference can also mean to extrapolate meaning improperly from premises.