MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION
MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION
78 Robert GoldblattGrothendieck generalised the notion of a topology on a set to that of a topologyon a category, by generalising the notion of an open covering of a set. He usedthis as a basis on which to formulate sheaf theory. F. William Lawvere and MilesTierney showed that the theory could be developed axiomatically by starting witha topos E having a morphism j : Ω → Ω, called a topology on E, satisfyingproperties that allow the construction of a certain sub-topos of “j-sheaves”. Thepair (E, j) will be called a site. The axioms for j are categorical versions of therequirement that an operation on a lattice bemultiplicative : j(x · y) = jx · jy,idempotent : j(jx) = jx, andinflationary : x ≤ jx.In the address at which he first announced this new theory Lawvere [1970] statedthatA Grothendieck “topology” appears most naturally as a modal operator ofthe nature “it is locally the case that”.Intuitively, a property holds locally at a point x of a topological space if it holdsat all points “near” to x, or throughout some neighbourhood of x. Alternatively,a property holds locally of an object if it is covered by open sets for each of whichthe property holds. For example a locally constant function is one whose domainis covered by open sets on each of which the function is constant.Define a local operator 67 on a Heyting algebra H to be any operation j that ismultiplicative, idempotent and inflationary, and call the pair A = (H, j) a localalgebra. The general theory of these algebras has been studied by Donald Macnab[1976; 1981], who showed that local operators can be alternatively defined by thesingle equation(x ⇒ jy) = (jx ⇒ jy).Any local algebra is a candidate for modelling a modal logic based on the intuitionisticcalculus IPC. Since j is multiplicative and has j1 = 1, this will be anormal logic when ✷ is interpreted as j, but there has been some uncertaintyas to whether a modality modelled by j is of universal or existential character.Note that a local operator has a mixture of the properties of topological interiorand closure operators. It fulfills all of the axioms of an interior operator exceptIx ≤ x, satisfying instead the inflationary condition which is possessed by closureoperators. But topological closure operators are additive (C(x + y) = Cx + Cy), aproperty not required of j.Let J be the set of all modal propositional formulas satisfied by all local algebraswith 1 designated. The proof theory and semantics (algebraic, relational,neighbourhood, topos-theoretic) of this logic was investigated in [Goldblatt, 1981]where the symbol ∇ was used in place of ✷ and interpreted as a “geometric”67 Also known in the literature as a “nucleus”.
Mathematical Modal Logic: A View of its Evolution 79modality. It was shown that J can be axiomatised by adding to the axioms andrules for IPC the three axioms∇(p → q) → (∇p → ∇q)∇∇p → ∇pp → ∇p.The last axiom allows derivation of the rule from α infer ∇α. There are a numberof alternative axiomatisations of J, one of which is to add to IPC the axioms(p → q) → (∇p → ∇q)∇∇p → ∇p∇⊤.As Macnab’s characterisation of local operators suggests, J can also be specifiedby the single axiom(p → ∇q) ↔ (∇p → ∇q).In the presence of classical Boolean logic, the middle axiom ∇∇p → ∇p in thefirst group is deducible from the other two, and the logic becomes the ratheruninteresting system K+(p → ∇p) whose only connected validating frames arethe two one-element frames S • and S ◦ (see section 6.1). But in the absence of thelaw of excluded middle we have a modal logic with many interesting models. Inparticular it has relational models based on structures S = (K, ≤, ≺) which refinethe Kripke semantics for IPC. Here ≤ is a partial ordering of K and ≺ is a binaryrelation interpreting ∇ as a universal quantifier in the familar way:M |= x ∇α iff M |= y α for all y such that x ≺ y.To ensure that M(∇α) is ≤-increasing it is required that x ≤ y ≺ z impliesx ≺ z. The logic J is characterised by the class of such frames in which ≺ is asubrelation of ≤ that is dense in the sense that x ≺ y implies ∃z(x ≺ z ≺ y).There is a canonical frame S J of this kind that characterises J, and the logic alsohas the finite model property with respect to such frames. In addition there is acharacterisation of J by neighbourhood frames (K, ≤, N) (see 5.3), where N x is afilter in the lattice of ≤-increasing subsets of K, and the following conditions hold:x ≤ y implies N x ⊆ N y ,{y : x ≤ y} ∈ N x ,{y : U ∈ N y } ∈ N x implies U ∈ N x .If ∇α is defined to be the formula ¬¬α, then the axioms of J become theorems ofIPC. Lawvere [1970] observed thatThere is a standard Grothendieck topology on any topos, namely doublenegation, which is more appropriately put into words as “it is cofinally thecase that”.
- Page 28 and 29: 28 Robert GoldblattThis proposal be
- Page 30 and 31: 30 Robert Goldblattwhere α ′ is
- Page 32 and 33: 32 Robert Goldblattif α is atomic
- Page 34 and 35: 34 Robert GoldblattHintikka gives a
- Page 36 and 37: 36 Robert Goldblattbetween worlds a
- Page 38 and 39: 38 Robert Goldblattnormal (“queer
- Page 40 and 41: 40 Robert Goldblattinterpreting for
- Page 42 and 43: 42 Robert GoldblattDiodorean interp
- Page 44 and 45: 44 Robert Goldblattthat the formula
- Page 46 and 47: 46 Robert Goldblatt6.1 Incompletene
- Page 48 and 49: 48 Robert Goldblatttions: every nor
- Page 50 and 51: 50 Robert Goldblatttrue at some poi
- Page 52 and 53: 52 Robert Goldblattof the monadic s
- Page 54 and 55: 54 Robert Goldblattversion [van Ben
- Page 56 and 57: 56 Robert Goldblatt6.5 Duality and
- Page 58 and 59: 58 Robert Goldblattfrom a suitably
- Page 60 and 61: 60 Robert GoldblattAnother way to d
- Page 62 and 63: 62 Robert Goldblattwhether a variet
- Page 64 and 65: 64 Robert Goldblattatomic commands
- Page 66 and 67: 66 Robert Goldblattmodalities 〈 i
- Page 68 and 69: 68 Robert Goldblatt[Hennessy and Li
- Page 70 and 71: 70 Robert GoldblattThe logic CTL* w
- Page 72 and 73: 72 Robert GoldblattThe meaning of
- Page 74 and 75: 74 Robert Goldblattwhich shows that
- Page 76 and 77: 76 Robert Goldblattmodal formulas s
- Page 80 and 81: 80 Robert GoldblattNow if Y and Z a
- Page 82 and 83: 82 Robert Goldblatt7.7 Modal Logic
- Page 84 and 85: 84 Robert GoldblattThis abstracts t
- Page 86 and 87: 86 Robert Goldblattextensions [Gold
- Page 88 and 89: 88 Robert Goldblatt[Clarke and Emer
- Page 90 and 91: 90 Robert Goldblatt[Gerson, 1976] M
- Page 92 and 93: 92 Robert Goldblatt[Hoare, 1969] C.
- Page 94 and 95: 94 Robert Goldblatt[̷Lukasiewicz a
- Page 96 and 97: 96 Robert Goldblatt[Prior, 1967] Ar
- Page 98: 98 Robert Goldblatt[Tarski, 1956] A
78 Robert GoldblattGrothendieck generalised the notion of a topology on a set to that of a topologyon a category, by generalising the notion of an open covering of a set. He usedthis as a basis on which to formulate sheaf theory. F. William Lawvere and MilesTierney showed that the theory could be developed axiomatically by starting witha topos E having a morphism j : Ω → Ω, called a topology on E, satisfyingproperties that allow the construction of a certain sub-topos of “j-sheaves”. Thepair (E, j) will be called a site. The axioms for j are categorical versions of therequirement that an operation on a lattice bemultiplicative : j(x · y) = jx · jy,idempotent : j(jx) = jx, andinflationary : x ≤ jx.In the address at which he first announced this new theory Lawvere [1970] statedthatA Grothendieck “topology” appears most naturally as a modal operator ofthe nature “it is locally the case that”.Intuitively, a property holds locally at a point x of a topological space if it holdsat all points “near” to x, or throughout some neighbourhood of x. Alternatively,a property holds locally of an object if it is covered by open sets for each of whichthe property holds. For example a locally constant function is one whose domainis covered by open sets on each of which the function is constant.Define a local operator 67 on a Heyting algebra H to be any operation j that ismultiplicative, idempotent and inflationary, and call the pair A = (H, j) a localalgebra. The general theory of these algebras has been studied by Donald Macnab[1976; 1981], who showed that local operators can be alternatively defined by thesingle equation(x ⇒ jy) = (jx ⇒ jy).Any local algebra is a candidate for modelling a modal logic based on the intuitionisticcalculus IPC. Since j is multiplicative and has j1 = 1, this will be anormal logic when ✷ is interpreted as j, but there has been some uncertaintyas to whether a modality modelled by j is of universal or existential character.Note that a local operator has a mixture of the properties of topological interiorand closure operators. It fulfills all of the axioms of an interior operator exceptIx ≤ x, satisfying instead the inflationary condition which is possessed by closureoperators. But topological closure operators are additive (C(x + y) = Cx + Cy), aproperty not required of j.Let J be the set of all modal propositional formulas satisfied by all local algebraswith 1 designated. The proof theory and semantics (algebraic, relational,neighbourhood, topos-theoretic) of this logic was investigated in [Goldblatt, 1981]where the symbol ∇ was used in place of ✷ and interpreted as a “geometric”67 Also known in the literature as a “nucleus”.
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