Deductive reasoning

https://en.wikipedia.org/wiki/Deductive_reasoning

Deduction is the process of reasoning that starts with premises and proceeds to reach a logically certain conclusion. If all premises are true and the rules of inference were strictly followed, then the conclusion is necessarily true.

Deductive reasoning goes in the same direction as that of logical conditionals, and links premises with conclusions. Therefore, deduction is an "if-then" procedure (like everything else, really): "assuming the following premises are true, then the following conclusion follows".

Examples of deduction

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."

An example of an argument using deductive reasoning leading to erroneous conclusion: "All policemen are instruments [of the law]". "The piano is an instrument". Therefore, "all policemen are pianos".

Modus ponens

Modus ponens (also known as affirming the antecedent or the law of detachment) is the primary deductive rule of inference.

Modus ponens applies to arguments that have as first premise a conditional statement, P -> Q, and as second premise the antecedent, P, of the conditional statement. It obtains the consequent, Q of the conditional statement as its conclusion.

P -> Q    P
----------- (MP)
     Q

In PLs, in Haskell, this would correspond to having a function from a to b and having an arg a; then you can obtain b by applying f to a.

mp :: (a -> b) -> a -> b
mp f x = f x

However, the antecedent P cannot be obtained in a similar way, that is as the conclusion from the premises of the conditional statement, P -> Q, and the consequent Q; such an argument commits the logical fallacy of affirming the consequent.

Modus tollens

Modus tollens, or the law of contrapositive, is a deductive rule of inference.

It validates an argument that has as premises a conditional statement, p -> q, and the negation of the consequent, ¬q, and as conclusion the negation of the antecedent, ¬p.

In contrast to modus ponens, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following:

P -> Q   ¬Q
----------- (MT)
    ¬P

p: it is raining q: the sky is cloudy

If p then q.                        p -> q
It is not the case that q.          ¬q
------------------------------      ------
Thus, it is not the case of p.      ¬p

Law of syllogism

In proposition logic 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:

P -> Q    Q -> S
---------------- (HS)
     P -> S