Strict conditional - AbsoluteAstronomy.com

In logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, a strict conditional is a 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...

that is acted upon by the necessity operator from 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...

. For any two propositions

and

, the formula

says that

materially implies

while

says that

strictly implies

. Strict conditionals are the result of Clarence Irving Lewis

Clarence Irving Lewis

Clarence Irving Lewis , usually cited as C. I. Lewis, was an American academic philosopher and the founder of conceptual pragmatism. First a noted logician, he later branched into epistemology, and during the last 20 years of his life, he wrote much on ethics.-Early years:Lewis was born in...

's attempt to find a conditional for logic that can adequately express indicative conditional

Indicative conditional

In natural languages, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional, an indicative conditional does not have a stipulated definition...

s. Such a conditional would, for example, avoid the paradoxes of material implication

Paradoxes of material implication

The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic. One of these paradoxes is the paradox of entailment....

. The following statement, for example, is not correctly formalized by material implication.

If Bill Gates had graduated in Medicine, then Elvis never died.

This condition is clearly false: the degree of Bill Gates has nothing to do with whether Elvis is still alive. However, the direct encoding of this formula in classical logic

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

using material implication lead to:

Bill Gates graduated in Medicine

Elvis never died.

This formula is true because a formula

is true whenever the antecedent

is false. Hence, this formula is not an adequate translation of the original sentence. Strict conditions are encodings of implications in modal logic attempting A different encoding is:

(Bill Gates graduated in Medicine

Elvis never died.)

In modal logic, this formula means (roughly) that, in every possible world in which Bill Gates graduated in Medicine, Elvis never died. Since one can easily imagine a world where Bill Gates is a Medicine graduate and Elvis is dead, this formula is false. Hence, this formula seems a correct translation of the original sentence.

Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems. The following sentence, for example, is not correctly formalized by a strict conditional:

If Bill Gates graduated in Medicine, then 2 + 2 = 4.

Using strict conditionals, this sentence is expressed as:

(Bill Gates graduated in Medicine

2 + 2 = 4)

In modal logic, this formula means that, in every possible world where Bill Gates graduated in medicine, it holds that 2 + 2 = 4. Since 2 + 2 is equal to 4 in all possible worlds, this formula is true.

Some logicians view this situation as paradoxical, and to avoid it they have created counterfactual conditionals. Others, such as Paul Grice

Paul Grice

Herbert Paul Grice , usually publishing under the name H. P. Grice, H...

, have used conversational implicature to argue that, despite apparent difficulties, the material conditional is just fine as a translation for the natural language 'if...then...'. Others still have turned to relevant logic to supply a connection between the antecedent and consequent of provable conditionals.

The corresponding conditional of an argument

Argument

In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...

(or derivation) is a material conditional

Material conditional

whose antecedent

Antecedent (logic)

An antecedent is the first half of a hypothetical proposition.Examples:* If P, then Q.This is a nonlogical formulation of a hypothetical proposition...

is the conjunction

Logical conjunction

In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

of the argument's (or derivation's) premise

Premise

Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...

s and whose consequent

Consequent

A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then".Examples:* If P, then Q.Q is the consequent of this hypothetical proposition....

is the argument's conclusion.

See also