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logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the ...
of
modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
. It is logically equivalent to the
material conditional The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol \to is interpreted as material implication, a formula P \to Q is true unless P is true and Q is false. M ...
of
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
, combined with the
necessity Necessary or necessity may refer to: Concept of necessity * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by a particular moment * Necessary and sufficient condition, in l ...
operator from
modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
. For any two
proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...
s ''p'' and ''q'', the
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
''p'' → ''q'' says that ''p'' materially implies ''q'' while \Box (p \rightarrow q) says that ''p'' strictly implies ''q''. Strict conditionals are the result of
Clarence Irving Lewis Clarence Irving Lewis (April 12, 1883 – February 3, 1964) was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logician, he later branched into epis ...
's attempt to find a conditional for logic that can adequately express
indicative conditional In natural languages, an indicative conditional is a conditional sentence such as "If Leona is at home, she isn't in Paris", whose grammatical form restricts it to discussing what could be true. Indicatives are typically defined in opposition to c ...
s in natural language. They have also been used in studying Molinist theology.


Avoiding paradoxes

The strict conditionals may avoid paradoxes of material implication. The following statement, for example, is not correctly formalized by material implication: : If Bill Gates graduated in medicine, then Elvis never died. This condition should clearly be 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 (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
using material implication leads to: : Bill Gates graduated in medicine → Elvis never died. This formula is true because whenever the antecedent ''A'' is false, a formula ''A'' → ''B'' is true. Hence, this formula is not an adequate translation of the original sentence. An encoding using the strict conditional is: : \Box (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 to be a correct translation of the original sentence.


Problems

Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems with consequents that are necessarily true (such as 2 + 2 = 4) or antecedents that are necessarily false. 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: : \Box (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, although it does not seem that the original sentence should be. A similar situation arises with 2 + 2 = 5, which is necessarily false: : If 2 + 2 = 5, then Bill Gates graduated in medicine. Some logicians view this situation as indicating that the strict conditional is still unsatisfactory. Others have noted that the strict conditional cannot adequately express
counterfactual conditional Counterfactual conditionals (also ''contrafactual'', ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be h ...
s, and that it does not satisfy certain logical properties. In particular, the strict conditional is transitive, while the counterfactual conditional is not. Some logicians, such as
Paul Grice Herbert Paul Grice (13 March 1913 – 28 August 1988), usually publishing under the name H. P. Grice, H. Paul Grice, or Paul Grice, was a British philosopher of language who created the theory of implicature and the cooperative principle ( ...
, 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
relevance logic Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, b ...
to supply a connection between the antecedent and consequent of provable conditionals.


Constructive logic

In a constructive setting, the symmetry between ⥽ and \Box is broken, and the two connectives can be studied independently. Constructive strict implication can be used to investigate interpretability of
Heyting arithmetic In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it. Axiomatization Heyting arithmetic can be characterized jus ...
and to model arrows and guarded
recursion Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in m ...
in computer science.


See also

* Corresponding conditional *
Counterfactual conditional Counterfactual conditionals (also ''contrafactual'', ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be h ...
*
Dynamic semantics Dynamic semantics is a framework in logic and natural language semantics that treats the meaning of a sentence as its potential to update a context. In static semantics, knowing the meaning of a sentence amounts to knowing when it is true; in dyna ...
* Import-Export *
Indicative conditional In natural languages, an indicative conditional is a conditional sentence such as "If Leona is at home, she isn't in Paris", whose grammatical form restricts it to discussing what could be true. Indicatives are typically defined in opposition to c ...
*
Logical consequence Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more stat ...
*
Material conditional The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol \to is interpreted as material implication, a formula P \to Q is true unless P is true and Q is false. M ...


References


Bibliography

*Edgington, Dorothy, 2001, "Conditionals," in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell. *For an introduction to non-classical logic as an attempt to find a better translation of the conditional, see: ** Priest, Graham, 2001. ''An Introduction to Non-Classical Logic''. Cambridge Univ. Press. *For an extended philosophical discussion of the issues mentioned in this article, see: ** Mark Sainsbury, 2001. ''Logical Forms''. Blackwell Publishers. * Jonathan Bennett, 2003. ''A Philosophical Guide to Conditionals''. Oxford Univ. Press. {{Formal semantics Conditionals Logical connectives Modal logic Necessity Formal semantics (natural language)