# Deductive reasoning

Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.[1]

Deductive reasoning goes in the same direction as that of the conditionals, and links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.

Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from specific cases to general rules, i.e., there is epistemic uncertainty. However, the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning.

Deductive reasoning differs from abductive reasoning by the direction of the reasoning relative to the conditionals. Deductive reasoning goes in the same direction as that of the conditionals, whereas abductive reasoning goes in the opposite direction to that of the conditionals.

## Simple example

An example of a deductive argument:

1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.

The first premise states that all objects classified as "men" have the attribute "mortal". The second premise states that "Socrates" is classified as a "man" – a member of the set "men". The conclusion then states that "Socrates" must be "mortal" because he inherits this attribute from his classification as a "man".

## Reasoning with modus ponens, modus tollens, and syllogism

### Modus ponens

Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary form of deductive reasoning. A single conditional statement is made, and a hypothesis (${\displaystyle P}$) is stated. The conclusion (${\displaystyle Q}$) is then deduced from the statement and the hypothesis. The most basic form is listed below:

1. ${\displaystyle P\rightarrow Q}$  (conditional statement)
2. ${\displaystyle P}$  (hypothesis stated)
3. ${\displaystyle Q}$  (conclusion deduced)

In deductive reasoning, we can conclude ${\displaystyle Q}$ from ${\displaystyle P}$ by using modus ponens. However, if the conclusion (${\displaystyle Q}$) is given instead of the hypothesis (${\displaystyle P}$) then there is no definitive conclusion.

The following is an example of an argument using modus ponens in the form of an if-then statement:

1. If an angle satisfies 90° < ${\displaystyle A}$ < 180°, then ${\displaystyle A}$ is an obtuse angle.
2. ${\displaystyle A}$ = 120°.
3. ${\displaystyle A}$ is an obtuse angle.

Since the measurement of angle ${\displaystyle A}$ is greater than 90° and less than 180°, we can deduce that ${\displaystyle A}$ is an obtuse angle. If however, we are given the conclusion that ${\displaystyle A}$ is an obtuse angle we cannot deduce the premise that 90° < ${\displaystyle A}$ < 180°. It might be true that other angles outside this range are also obtuse.

### Modus tollens

Modus tollens (also known as "the law of contrapositive") states that, in a true conditional, if the conclusion is false, then the hypothesis must be false also. Reasoning with modus tollens goes in the opposite direction to that of the conditional; therefore this is a form of abductive reasoning. The law of modus tollens is derived by combining contraposition and modus ponens. The general expression for modus tollens is the following:

1. ${\displaystyle P\rightarrow Q}$.
2. ${\displaystyle \lnot Q}$.
3. Therefore, we can conclude that ${\displaystyle \lnot P}$.

The following are examples:

1. If it is raining, then there are clouds in the sky.
2. There are no clouds in the sky.
3. Thus, it is not raining.

### Law of syllogism

The law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:

1. ${\displaystyle P\rightarrow Q}$
2. ${\displaystyle Q\rightarrow R}$
3. Therefore, ${\displaystyle P\rightarrow R}$.

The following is an example:

1. If Larry is sick, then he will be absent.
2. If Larry is absent, then he will miss his classwork.
3. Therefore, if Larry is sick, then he will miss his classwork.

We deduced the final statement by combining the hypothesis of the first statement with the conclusion of the second statement. We also allow that this could be a false statement. This is an example of the transitive property in mathematics. The transitive property is sometimes phrased in this form:

1. ${\displaystyle A=B}$.
2. ${\displaystyle B=C}$.
3. Therefore, ${\displaystyle A=C}$.

## Validity and soundness

Argument terminology

Deductive arguments are evaluated in terms of their validity and soundness.

An argument is “valid” if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be “valid” even if one or more of its premises are false.

An argument is “sound” if it is valid and the premises are true.

It is possible to have a deductive argument that is logically valid but is not sound. Fallacious arguments often take that form.

The following is an example of an argument that is “valid”, but not “sound”:

1. Everyone who eats carrots is a quarterback.
2. John eats carrots.
3. Therefore, John is a quarterback.

The example’s first premise is false – there are people who eat carrots who are not quarterbacks – but the conclusion would necessarily be true, if the premises were true. In other words, it is impossible for the premises to be true and the conclusion false. Therefore, the argument is “valid”, but not “sound”. False generalizations – such as "Everyone who eats carrots is a quarterback" – are often used to make unsound arguments. The fact that there are some people who eat carrots but are not quarterbacks proves the flaw of the argument.

In this example, the first statement uses categorical reasoning, saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning – also known as term logic – was developed by Aristotle, but was superseded by propositional (sentential) logic and predicate logic.

Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is “valid”, it is possible for the conclusion to be false (determined to be false with a counterexample or other means).

## History

Aristotle started documenting deductive reasoning in the 4th century BC.[2]