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Standard Approach to Basic Modal Logics

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Natural Deduction, Hybrid Systems and Modal Logics

Part of the book series: Trends in Logic ((TREN,volume 30))

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Abstract

In this Chapter we focus on the class of non-axiomatic systems that are called standard in the sense of keeping intact all the machinery of suitable systems for classical logic. Extensions are obtained by means of additional modal rules. This group covers modal extensions of standard Gentzen SC, Hintikka-style modal TS, and some ND systems.

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Notes

  1. 1.

    In fact Zeman’s work contains formalizations of the whole Lewis’ family S1–S5.

  2. 2.

    We mean the family of TS’s called implicit by Goré [117].

  3. 3.

    In fact, also a system of destructive resolution of [94] may be included in this group – it will be introduced later in Chapter 7 (Section 7.4.2).

  4. 4.

    Corcoran in fact applied horizontal, instead of vertical, manner of displaying derivations, but this is only a slight departure of no real importance.

  5. 5.

    In fact, Fitch did not realize that he formalized T; he claimed that his rules gives “almost S2”.

  6. 6.

    We mean soundness profs which are based on the transformation of every formula into a sequent containing this formula in the succedent and the record of active assumptions in the antecedent – cf. introductory remarks in Section 2.6.

  7. 7.

    It is in fact a rule which preserves normality even in monotonic logics – cf. the next section.

  8. 8.

    Small differences concern the rule of \(\forall \) introduction which is an inference rule in [104,105] but proof construction rule in KMGP, the formulation of rules for free logic, and the fact that Garson does not apply KM apparatus of show-lines and boxes but Fitch’s bars. Also Garson defines modal reiteration only for K and add axioms for extensions.

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  1. Dept. Logic, University of Lódz, Kopcinskiego 16/18, 90-232, Lódz, Poland

    Dr. Andrzej Indrzejczak

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Indrzejczak, A. (2010). Standard Approach to Basic Modal Logics. In: Natural Deduction, Hybrid Systems and Modal Logics. Trends in Logic, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8785-0_6

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