# Deductive Argument Forms

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## Contents

## Deductive Forms in *Natural* Language

The goal of a deductive inference is to affirm the truth of 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*, which is discussed below. A successful deductive argument has true premises and a valid structure. An argument with these two features is called *sound*. In a sound deductive argument, the conclusion must be true.

To see deductive form, we will set the question of the truth of the premises aside, temporarily. The truth of the premises is a function of both their form and content, but we are focusing on logical form, because our immediate goal is to see structures that either occur in the rationales we read and hear or which we could use to organize an argument that we are trying to reconstruct. These forms are also useful for structuring your own thought.

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,

Premise 1: If Tim is a new student, then he should expect to feel confused.

Premise 2: Tim is a new student

Conclusion: 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 (the part after "then") of the conditional in the first premise. If you have this structure, you can infer the negation of the *antecedent* 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.

or,

P1. If the tests are negative, we have nothing to worry about.

P2. We have something to worry about.

C: The test are not negative.

## Formalizing Deductive Logical Structure

These two arguments in our example both follow deductive valid patterns. Since we are focusing on the patterns (or logical structure) of the premises, it might help to abstract from the specific natural language (English, in this case) in the premises. After all, when we talk about structure, we do not really care about the content of the claims. So, if we replaced claims with capital letters, use an arrow "-->" to represent the "If...then" structure, "~" to mean negation, then the first example looks like this:

P1. A --> B

P2. A

C: B

And the second pattern looks like this:

P1. A --> B

P2. ~ B

C: ~ A

What follows is a basic set of logical forms which you should find useful in organizing some of your thoughts and in seeing deductive logical structure in the "field".

## Basic Logical Forms

### Modus Ponens

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 a lot in reconstruction. One of the most basic ways to get an argument going is to connect two claims in a conditional (if ... then) structure and then to show that either the first part of the conditional is true (affirmed as a premise), or that the last part of the conditional is false (it's negation is affirmed as a premise).

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 a good reconstruction of the rationales in our own or others' thinking. For this reason, we have chosen a basic set of patterns that are roughly intuitive.**

## Validity

Validity is defined in the following way:

- validity
- A property of deductive arguments which have the sort of logical structure which guarantees that if the premises are true, then the conclusion will be. This is a "conditional guarantee" of the truth of the conclusion, since the premises must still be true for the conclusion to follow.

With our list of Basic Logical Forms, we have given you five examples of valid logical structures. Each of the inferences in these patterns carries the "conditional guarantee" of validity. That means that *if* the premises are true, then the conclusion must be true. The confusing part of the definition of validity is it's reference to a "conditional guarantee". Validity itself just refers to the formal properties of the structure of the argument. But if you have this structure *and* true premises, you may be assured of the truth of the conclusion.

An argument that has a valid structure and true premises is called a *sound* argument.

Consider the following true / false problems to test your understanding of validity:

True or False:

- In a valid argument, the conclusion is always true.
- In a valid argument, it is impossible for the premises to be true while the conclusion is false.
- If you have an argument with true premises and a true conclusion, it must be valid.

### Using the terms "true" and "valid"

In everyday discussion, it is common to say that someone has a "valid point" or that their argument is "true." Having a valid point normal just means that you have a point that is worthy of consideration. If you give an argument and someone says, "That's true," they probably mean that they agree with the truth of your conclusion. Now that we have a technical definition of validity which only applies to only to deductive arguments and which specifically focuses of structure, we should distinguish it from this everyday usage. It is more precise to say that *truth* and *falsity* are properties of claims and that *validity* is a property of logical structure in deductive argument.

## Formal Fallacies

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 them. 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.

Focusing on form also allows us to keep an eye out for a couple of formal fallacies. Pattens that look valid but are not.

### 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