# Strict implication calculus

A logical calculus based on a strict implication, i.e. on a logical operation along the lines of "if …, then …" . For a strict implication, the so-called "paradoxes of (material) implication" are completely or partially removed: A false statement implies any statement, and a true statement is implied by any statement.

A strict implication calculus aims to reflect the link in meaning between the premise and the conclusion of a conditional proposition. A whole series of strict implication calculi exists (Lewis calculi, Ackermann calculi, and others), which are distinguished from each other by the fact that formulas that can be inferred in some of them cannot be inferred in others (for example, in Lewis calculi the "paradoxes of implication" are only partially removed, whereas they are completely removed in Ackermann calculi). A strict implication calculus has close links with the formalization of modal expressions ( "it is possible" , "it is impossible" , "it is necessary" , etc.); in some calculi, strict implication is expressed through modalities (cf. Modality), while in others, modalities are expressed through strict implication.

#### References

[1] | R. Feys, "Modal logics" , Gauthier-Villars (1965) |

[2] | C.I. Lewis, C.H. Langford, "Symbolic logic" , Dover, reprint (1959) |

[3] | W. Ackermann, "Begrundung einer strengen Implikation" J. Symbolic Logic , 21 : 2 (1956) pp. 113–128 |

#### Comments

Another group of attempts to avoid "paradoxes of implication" , such as the disjunctive syllogism $A\land(\neg A\lor B)\Rightarrow B$, goes by the name of relevant logic, [a1]. The Lewis calculi S1–S5 of [1] are also known as the Lewis survey systems, [a3].

#### References

[a1] | J. Norman (ed.) R. Sylvan (ed.) , Directions in relevant logic , Kluwer (1989) |

[a2] | G.E. Hughes, M.J. Cresswell, "An invitation to modal logic" , Methuen (1972) |

[a3] | R. Wójcicki, "Theory of logical calculi" , Kluwer (1988) pp. 154ff |

**How to Cite This Entry:**

Strict implication calculus.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Strict_implication_calculus&oldid=31826