Indicative conditional
Encyclopedia
In natural language
Natural language
In the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written...

s, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional
Material conditional
The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q is logically equivalent to the negative compound: not . A material conditional compound itself is often simply called a conditional...

, an indicative conditional does not have a stipulated definition. The philosophical literature on this operation is broad, and no clear consensus has been reached.

Discrepancies between the material conditional and the indicative conditional

The material conditional does not always function in accordance with everyday if-then reasoning. Therefore there are drawbacks with using the material conditional to represent if-then statements.

One problem is that the material conditional allows implications to be true even when the antecedent is irrelevant to the consequent. 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 following: 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, i.e., relevant logic.

For a related problem, see vacuous truth
Vacuous truth
A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...

.

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
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

.

A further problem 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 logic
Paraconsistent logic
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...

s.

Psychology and indicative conditionals

Most behavioral experiments on conditionals in the psychology of reasoning have been carried out with indicative conditionals, causal conditionals, and counterfactual conditionals. People readily make the modus ponens
Modus ponens
In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...

 inference, that is, given if A then B, and given A, they conclude B, but only about half of participants in experiments make the modus tollens
Modus tollens
In classical logic, modus tollens has the following argument form:- Formal notation :...

 inference, that is, given if A then B, and given not-B, only about half of participants conclude not-A, the remainder say that nothing follows (Evans et al, 1993). When participants are given counterfactual conditionals, they make both the modus ponens
Modus ponens
In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...

 and the modus tollens
Modus tollens
In classical logic, modus tollens has the following argument form:- Formal notation :...

inferences (Byrne, 2005).
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