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Logical 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 sufficient condition for the conclusion, while the conclusion may be called necessary condition for the hypothesis.

Table of contents

1 Conditional statements 2 Truth table 3 Derivation from axioms 4 Connection with other concepts

Conditional statements

A conditional statement, or simply a conditional for short, is an "if-then" statement, written in the form: 'if P, then Q'. Here, 'P' is the antecedent (the "if" part of the statement) and 'Q' is the consequent (the "then" part). For example, in "If you give me ten dollars, then I will be your best friend," the claim "you give me ten dollars" is the antecedent of the conditional, and "I will be your best friend" is the consequent.

In traditional logic, a statement if A then B is true if and only if either A is false or B is true, or both. There have been attempts in areas such as modal logic to find a formal definition that is closer to the 'intuitive' meaning: in the traditional logic interpretation "If it is raining now, then I am a unicorn." is true provided it is not raining now.

Truth table

The truth value of expressions involving → is often defined by the following truth table:

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

Derivation from axioms

This table needn't be taken as "the definition of →", however, because its contents can also be derived from the axioms of the propositional calculus.

We can derive the first row as follows:

1. Suppose p and q. Under this assumption:
   1. q (by Conjunction elimination)
   2. Suppose p. Under this assumption:
      1. q (copying 1.1)
   3. Thus p → q (Conditional proof)
2. Thus (p and q) implies p → q (Conditional proof)

The second row as follows:

1. Suppose p and ¬q. Under this assumption:
   1. ¬q (by Conjunction elimination)
   2. Suppose p → q. Under this assumption:
      1. q (Modus ponens)
      2. ¬q (Copying from above)
      3. q and ¬q (Conjunction introduction)
   3. Since this is a contradiction, then ¬(p → q).
2. Thus (p and ¬q) implies ¬(p → q) (Conditional proof)

The third row as follows:

1. Suppose ¬p and q. Under this assumption:
   1. q (by Conjunction elimination)
   2. Suppose p. Under this assumption:
      1. q (Copying from above)
   3. Thus p → q (Conditional proof)
2. Thus (¬p and q) implies (p → q)

And the fourth row as follows:

1. Suppose ¬p and ¬q. Under this assumption:
   1. ¬q (by Conjunction elimination)
   2. Suppose p. Under this assumption:
      1. ¬q (Copying from above)
   3. Thus p → ¬q (Conditional proof)
2. Thus (¬p and ¬q) implies (p → q)

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.

Connection with other concepts

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 , especially the relation of not being symmetrical.

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

In the propositional interpretation, when and denote propositions, the relation signifies that the proposition implies or involves the proposition , which is often expressed by the hypothetical judgement, "If is true, is true"; or by " implies "; or more simply by ", therefore ". 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 and .

Consequently, whenever a 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.

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See also: Counterfactual conditional

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