Logical conditional

In propositional calculus, or logical calculus in mathematics, the 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 a right-arrow "→". The hypothesis is sometimes also called sufficient condition for the conclusion, while the conclusion may be called necessary condition for the hypothesis.

It must be emphasized that in logic there is no single interpretation of the conditional; it represents a variety of closely-related concepts, which have more specific names and often separate symbols (such as ⇒ and ⊃).

Contents

1 Conditional statements

2 Material conditional

3 Derivation from axioms

4 Discrepancies between the material conditional and everyday if-then reasoning

5 Connection with other concepts

6 See also

7 External Links

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 are false. 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.

Material conditional

The truth value of expressions involving the logical conditional is often (but by no means always) defined by the following truth table:

p	q	p ⊃ q
T	T	T
T	F	F
F	T	T
F	F	T

This particular conditional is called material implication or the material conditional, and is properly represented by the symbol ⊃ rather than the less specific symbol →.

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:

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)
Thus (p and q) implies p → q (Conditional proof)

The second row as follows:

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).
Thus (p and ¬q) implies ¬(p → q) (Conditional proof)

The third row as follows:

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)
Thus (¬p and q) implies (p → q)

And the fourth row as follows:

Suppose ¬p and ¬q. Under this assumption:
1. ¬p (by Conjunction elimination)
2. Suppose ¬q. Under this assumption:
  1. ¬p (Copying from above)
3. Thus ¬q → ¬p (Conditional proof)
4. ¬q → ¬p and p → q are logically equivalent.(See Contrapositive.)
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.

Discrepancies between the material conditional and everyday if-then reasoning

Many people find that the material conditional does not always function in accordance with their intuitive ideas about if-then reasoning.

One problem is that the material conditional allows implications to be true even when the antecedent and the consequent have no logical connection. For example, it's commonly accepted that the sun is made of gas, on one hand, and that 3 is a prime number, on the other. The standard definition of implication allows us to conclude that: since the sun is made of gas, 3 is a prime number. This is arguably synonymous to the follow: the sun's being made of gas makes 3 be a prime number. Many people intuitively think that this is false, because the sun and the number three simply have nothing to do with one another. Logicians have tried to address this concern by developing alternative logics, e.g. relevant logic.

For a related problem, see vacuous truth.

Another issue is that the material conditional is not designed to deal with counterfactuals and other cases that people often find in if-then reasoning. This has inspired people to develop modal logic.

Another trouble is that the material conditional is such that P AND ¬P → Q, regardless of what Q is taken to mean. That is, a contradiction implies that absolutely everything is true. Logicians concerned with this have tried to develop paraconsistent logics.

Connection with other concepts

The logical conditional, and particularly the material conditional, is closely related to inclusion (for sets), subsumption (for concepts), or implication (for propositions). It also has formal properties analogous to those of the mathematical relation less than or more exactly <math>\leq<math>, especially the relation of not being symmetrical.

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

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

Consequently, whenever a <math>\Rightarrow<math> 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.