# 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:

1. A is true.
2. If A is true, then B is true.
3. Therefore, B is true.

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

1. If Q is true, then R is true.
2. R is not true.
3. Therefore, Q is not true.

The Internet Encyclopedia of Philosophy says concerning deductive reasoning:

 “ A deductive argument is an argument that is intended by the arguer to be deductively valid, that is, to provide a guarantee of the truth of the conclusion provided that the argument's premises are true. This point can be expressed also by saying that, in a deductive argument, the premises are intended to provide such strong support for the conclusion that, if the premises are true, then it would be impossible for the conclusion to be false. An argument in which the premises do succeed in guaranteeing the conclusion is called a (deductively) valid argument. If a valid argument has true premises, then the argument is said also to be sound. All arguments are either valid or invalid, and either sound or unsound; there is no middle ground, such as being somewhat valid. Here is a valid deductive argument: It's sunny in Singapore. If it's sunny in Singapore, then he won't be carrying an umbrella. So, he won't be carrying an umbrella. The conclusion follows the word "So". The two premises of this argument would, if true, guarantee the truth of the conclusion. However, we have been given no information that would enable us to decide whether the two premises are both true, so we cannot assess whether the argument is deductively sound. It is one or the other, but we do not know which. If it turns out that the argument has a false premise and so is unsound, this won't change the fact that it is valid. [1] ”

## 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:

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