# Deduction

A **deduction** in formal logic is a way of proving a proposition. Specifically, when a conclusion is inferred from premises or facts, it is said to "follow" (*like the water downstream to the spring source*), from previously stated propositions via certain logical rules.

Deduction is used most often in geometry, but also has its place in philosophy and law. Deduction is often un-used in science, as deduction requires that one have certainty of truth of both the major and minor premises—something science is unwilling to make claims about.

The normal contrast to deduction is induction, whereby the two are posed as distinct opposites. However, they are not so distinct as to be alien to each other. Rather, induction and deduction have a single common root from which they necessarily are operationally distinguished: the mind. So, in regard to formal logic in this root sense, a merely mechanical or electronic simulator of genuine mental operations does not possess either any actual deductive or actual inductive logic.

The most famous formal deductive rule goes as follows:

- A is true.
- If A is true, then B is true.
- Therefore, B is true.

Another type of deduction, known as "proof by contradiction", is:

- If Q is true, then R is true.
- R is not true.
- Therefore, Q is not true.

## Truth Values in Sentential Logic

T implies T, True T implies F, False F implies F, True F implies T, True

The final two are rooted in the fact that the consequent following from an antecedent that is not true says nothing about the truth or falsity of the statement, it must be *assumed* to be true.

Aristotle codified the rules of syllogistic deduction two millennia ago in Ancient Greece. He created 256 forms of syllogisms relating to groups of things. For example, an AAA1 syllogism:

- All men are mortal.
- Socrates is a man.
- Therefore, Socrates is mortal.

See also: