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

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

One problem is that the logical conditional is that it 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 logicRelevant logic also called relevance logic is any of a family of non-classical substructural logics that impose certain restrictions on implication. It is generally, but not universally, called relevant logic by Australian logicians, and relevance logic b.

For a related problem, see vacuous truth.

Another issue is that the logical conditional is only designed to deal well 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 logical 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.

5 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: Control flow, Counterfactual conditional , Strict conditional

Logic

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