MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION
MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION
24 Robert Goldblatt“true” or “false”, having a world as its first argument, and having individuals asthe remaining arguments when n ≠ 0. A value system relative to U is a functionS assigning a member of A to each individual variable, and an n-place intensionalpredicate to each n-place predicate variable. The notion of a formula being trueor false for the universe U, the world M and the value system S — or morebriefly for UMS — is defined in the expected way for the non-modal connectivesand quantifiers, including quantifiers binding predicate variables. For modalizedformulas Lp and Mp it is declared thatLp is true for UMS iff for every world M ′ of U, p is true for UM ′ S;Mp is true for UMS iff for some world M ′ of U, p is true for UM ′ S.A formula is valid in the universe U if it is true for UMS for every world M andvalue system S of U.Bayart used the notation ä, I, ë for a Gentzen sequent, with ä (the antecedent)and ë (the consequent) being finite sequences of formulas, and I a separatingsymbol. The sequent is true in UMS if some member of ä is false or else somemember of ë is true. He adopted the axiom schema ¨p, I, ¨p and a system of twentyfivededuction rules, showing in [1958] that all deducible sequents are valid in alluniverses. There are four modal rules, allowing the introduction of the modalitiesL and M into antecedents and consequents:p, ä, I, ëLp, ä, I, ëp, ä, I, ëMp, ä, I, ëä, I, ë, pä, I, ë, Lpä, I, ë, pä, I, ë, Mp .The last two rules are subject to the restriction that any formula appearing in äor ë must be “couverte”, meaning that it is formed from formulas of the types Lqand Mq using only the non-modal connectives and quantifiers. Such a formula hasthe same truth value in UMS and UM ′ S for all worlds M, M ′ .The [1959] paper proved the completeness of this sequent system for validity incertain quasi-universes obtained by allowing predicate variables to take values ina restricted class of intensional predicates. From this it was shown that the firstorder fragment of the system is complete for validity in all universes. The methodused was subsequently generalised in [Cresswell, 1967] to obtain a completenesstheorem for the relational semantics of a first order version of the modal logic T(see section 5.1).It is worth recording Bayart’s explanation of why the set of worlds of a universeU = A, B is essential to this theory. He considered the possibility of dispensingwith B, requiring a value system S to interpret an n-place predicate variable asan extensional predicate (i.e. a truth-valued function on A n ), and modelling thenecessity modality by declaring thatLp is true of US iff p is true of US ′ for every value system S ′ .He noted that this interpretation fails to validate the formula∃y L(bx ∨ ¬by)
Mathematical Modal Logic: A View of its Evolution 25(where b is a unary predicate variable), a formula that is valid according to theabove semantics. His explanation of the flaw in this alternative approach is thatit gives Lp the same meaning as the universal closure of p (i.e. ∀v 1 · · · ∀v n p, wherev 1 , . . . , v n are the free variables of p), and confuses necessity with validity.4.4 Meredith, Prior and GeachArthur Prior [1967, p. 42] wrote thatIn some notes made in 1956, C. A. Meredith related modal logic to what hecalled the ‘property calculus’.This material was made available by Prior as a one-page departmental mimeograph[Meredith, 1956] which was published much later in the collection [Copeland,1996a]. Its basic idea was to express modal formulas in the first-order language ofa binary predicate symbol U, beginning with the following definitions, in which Land M are connectives for necessity and possibility (but the other notation is thatof this paper rather than the original Polish):(¬p)a = ¬(pa)(p → q)a = (pa) → (qa)(Lp)a = ∀b(Uab → pb)Possible axioms for U are then listed:(Mp)a = (¬L¬p)a = ∃b(Uab ∧ pb).1. Uab ∨ Uba2. Uab → (Ubc → Uac)3. Uab → (Ucb → Uac)4. Uaa5. Uab → Uba,and it is noted that “1 gives 4”; “3, 4 give 5”; and “3, 5 give 2”. The notesare written in this telegraphic style with no interpretation of the symbolism, butpresumably “pa” may be read “a has property p”.It is stated that quantification theory alone allows the derivation of( )L(p → q) → (Lp → Lq) a,and then formal deductions are given of (Lp → p)a using 4; of (Lp → LLp)a using2; of (MLp → Lp)a using 2 and 5; and of ∀apa from (Lp)a using 1 and 5. Theconclusion is as follows:Thus 1, or 4, gives T; 1, 2 or 4, 2 gives S4; 1, 3 or 4, 3 gives S5; and 1, 3 (butnot 4, 3) gives the equivalence of the above (Lp)a with the usual S5 (Lp)a,i.e. ∀apa.
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- Page 26 and 27: 26 Robert GoldblattPrior’s articl
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- Page 38 and 39: 38 Robert Goldblattnormal (“queer
- Page 40 and 41: 40 Robert Goldblattinterpreting for
- Page 42 and 43: 42 Robert GoldblattDiodorean interp
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- Page 46 and 47: 46 Robert Goldblatt6.1 Incompletene
- Page 48 and 49: 48 Robert Goldblatttions: every nor
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- Page 52 and 53: 52 Robert Goldblattof the monadic s
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- Page 68 and 69: 68 Robert Goldblatt[Hennessy and Li
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Mathematical Modal Logic: A View of its Evolution 25(where b is a unary predicate variable), a formula that is valid according to theabove semantics. His explanation of the flaw in this alternative approach is thatit gives Lp the same meaning as the universal closure of p (i.e. ∀v 1 · · · ∀v n p, wherev 1 , . . . , v n are the free variables of p), and confuses necessity with validity.4.4 Meredith, Prior and GeachArthur Prior [1967, p. 42] wrote thatIn some notes made in 1956, C. A. Meredith related modal logic to what hecalled the ‘property calculus’.This material was made available by Prior as a one-page departmental mimeograph[Meredith, 1956] which was published much later in the collection [Copeland,1996a]. Its basic idea was to express modal formulas in the first-order language ofa binary predicate symbol U, beginning with the following definitions, in which Land M are connectives for necessity and possibility (but the other notation is thatof this paper rather than the original Polish):(¬p)a = ¬(pa)(p → q)a = (pa) → (qa)(Lp)a = ∀b(Uab → pb)Possible axioms for U are then listed:(Mp)a = (¬L¬p)a = ∃b(Uab ∧ pb).1. Uab ∨ Uba2. Uab → (Ubc → Uac)3. Uab → (Ucb → Uac)4. Uaa5. Uab → Uba,and it is noted that “1 gives 4”; “3, 4 give 5”; and “3, 5 give 2”. The notesare written in this telegraphic style with no interpretation of the symbolism, butpresumably “pa” may be read “a has property p”.It is stated that quantification theory alone allows the derivation of( )L(p → q) → (Lp → Lq) a,and then formal deductions are given of (Lp → p)a using 4; of (Lp → LLp)a using2; of (MLp → Lp)a using 2 and 5; and of ∀apa from (Lp)a using 1 and 5. Theconclusion is as follows:Thus 1, or 4, gives T; 1, 2 or 4, 2 gives S4; 1, 3 or 4, 3 gives S5; and 1, 3 (butnot 4, 3) gives the equivalence of the above (Lp)a with the usual S5 (Lp)a,i.e. ∀apa.
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