Deductive Argument Forms

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Introduction and Two Patterns in Natural Language

As you may recall, the goal of a deductive inference is to affirm some conclusion with absolute certainty. Two things have to happen for this goal to be realized in a particular deductive argument. First, the premises have to true. Second, the logical form of the premises has to have a property called validity. If successful deductive argument has true premises and a valid structure. An argument with these two features is called sound.

Let's set the question of the truth of the premises aside for now so that we can focus on the logical forms typically found in deductive arguments. These forms are useful for structuring your own thought or for seeing logical form in things you read or hear. A simple form, called Modus Ponens, involves one premise which establishes an "if . . . then" relationship between two claims and then, a second premise, which asserts the claim in the "if" clause of the first premise. If you have premises with this form, you can infer consequent of the first conditional as the conclusion.

For example,

P1. If Tim is a new student, then he should expect to feel confused.

P2. Tim is a new student

C: Tim should expect to feel confused.

Another pattern involving conditionals requires the same first premise, a conditional, and then, for the second premise, you deny the consequent of the conditional in the first premise. If you have this structure, you can infer the negation of the antecendent of the conditional in the first premise.

For example,

P1. If wishes were horses, then pigs would fly.

P2. Pigs can't fly.

C: Wishes aren't horses.

Formalizing Deductive Logical Structure

These two arguments in our example both follow deductive valid patterns. Since we focusing on the patterns (or logical structure) of the premises, it might help to abstract from the natural language (English, in this case) at work in the premises. After all, when we talk about structure, we don't really care about the content of the claims. So, if we replaced claims with capital letters, the first example looks like this:

P1. A --> B

P2. A

C: B

And the second pattern looks like this:

P1. A --> B

P2. Not B

C: Not A

Because these are valid patterns, if the premises are true, the conclusion of the argument must be true. Recall that validity is the property of deductive arguments that guarantees that the premises will never be true while the conclusion is false. Within logic (see " Propositional Logic") this is explained in terms of the truth table for the argument. Here, we simply introduce the patterns which the tables demonstrate to be valid.

Basic Logical Forms

Modus Pollens

1. P --> Q

2. P

C. Q

Modus Tollens

1. P --> Q

2. ~Q

C. ~P

Disjunctive Syllogism

1. P v Q

2. ~P

C. Q

Hypothetical Syllogism

1. P --> Q

2. Q --> R

C. P --> R

Double Negation

1. ~~P

C: P

The first two patterns, Modus Ponens and Modus Tollens, come up alot in reconstruction. One of the most basic ways to get an argument going is to connect two claims in a conditional (if ... then) structure.

Disjunctive Syllogism involves affirming that two claims are in an "either or" relationship. In the second premise, you deny the truth of one of the claims and then conclude the other. Again, this is a pattern you will see within rationales as you become more sensitive to argumentative patterns.

Hypothetical Syllogism may remind you of the "transitive property" in math (for example, if 3 is greater than 2 and 2 is greater than 1, then 3 is greater than 1). In this case, it simply affirms that "chain reasoning" is valid.

While this is a basic set of patterns, there are others. The choice of valid argument patterns within a deductive logic depends in part upon the goals of the logician. Our own goals in noticing logical patterns is more practical -- to be able to identify the patterns in reflective contexts as a means of offering. For this reason, we have chosen a basic set of patterns that are roughly intuitive.

Combining the Patterns

In one sense, valid deductive argument patterns are not great mysteries. For example, disjunctive syllogism says, in effect, that if you can narrow down the possibilities for truth about something to two, "either one or the other," then show that one claim is false (it's negation is true), then you can infer the other one. That's disjunctive syllogism. This should be pretty intuitive, so you might wonder how valuable it is to notice these forms in our argumentation. Part of the answer is the same as the reason why it is important to make missing premises explicit in reconstructions -- sometimes it's important to state things that are assumed or seem obvious because seeing these claims helps us ask critical questions about. Likewise, when you make the logical structure of a piece of speech or writing clear, you can step back and examine that structure more clearly than when it is implicit. Maybe the deductive structure is too strong for the intent of the speaker. Or maybe, by making the logical form clear, you can determine that the truth of one of the premises requires more scrutiny. There are also a couple of formal fallacies to look out for, such as:

Formal Fallacies

Denying the Antecedent

1. P --> Q

2. ~P

C. ~Q

Affirming the Consequent

1. P --> Q

2. Q

C. P

Undistributed Middle

1. P --> Q

2. R --> Q

C. P --> R