Indicative conditional

This article should be merged with Conditional

In logical calculus of mathematics, logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). The operator is denoted using an left-arrow "→".

The hypothesis is sometimes also called necessary condition for the conclusion, while the conclusion may be called sufficient condition for the hypothesis.

It is defined using the following truth table:

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

In the case that the hypothesis is true, the result is the same as conclusion. Otherwise, the whole statement is true regardless the value of conclusion.

The same binary relation is called inclusion (for sets), subsumption (for concepts), or implication (for propositions). This relation might also be represented by the sign < because it has formal properties analogous to those of the mathematical relation less than or more exactly $\leq$, especially the relation of not being symmetrical.

In the conceptual interpretation, when $a$ and $b$ denote concepts, the relation $a\in b$ signifies that the concept $a$ is subsumed under the concept $b$; that is, it is a species with respect to the genus $b$. From the extensive point of view, it denotes that the class of $a$'s is contained in the class of $b$'s or makes a part of it; or, more concisely, that "All $a$'s are $b$'s". From the comprehensive point of view it means that the concept $b$ is contained in the concept $a$ or makes a part of it, so that consequently the character $a$ implies or involves the character $b$. Example: "All men are mortal"; "Man implies mortal"; "Who says man says mortal"; or, simply, "Man, therefore mortal".

In the propositional interpretation, when $a$ and $b$ denote propositions, the relation $a\Rightarrow b$ signifies that the proposition $a$ implies or involves the proposition $b$, which is often expressed by the hypothetical judgement, "If $a$ is true, $b$ is true"; or by "$a$ implies $b$"; or more simply by "$a$, therefore $b$". We see that in both interpretations the relation may be translated approximately by "therefore".

Remark. -- Such a relation is a proposition, whatever may be the interpretation of the terms $a$ and $b$.

Consequently, whenever a $\Rightarrow$ relation has two like relations (or even only one) for its members, it can receive only the propositional interpretation, that is to say, it can only denote an implication.

A relation whose members are simple terms (letters) is called a primary proposition; a relation whose members are primary propositions is called a secondary proposition, and so on.

From this it may be seen at once that the propositional interpretation is more homogeneous than the conceptual, since it alone makes it possible to give the same meaning to the copula in both primary and secondary propositions.