## Fine’s Theorem on First-Order Complete Modal Logics (2011)

### BibTeX

@MISC{Goldblatt11fine’stheorem,

author = {Robert Goldblatt},

title = {Fine’s Theorem on First-Order Complete Modal Logics},

year = {2011}

}

### OpenURL

### Abstract

Fine’s Canonicity Theorem states that if a modal logic is determined by a first-order definable class of Kripke frames, then it is valid in its canonical frames. This article reviews the background and context of this result, and the history of its influence on further research. It then develops a new characterisation of when a logic is canonically valid, providing a precise point of distinction with the property of firstorder completeness. 1 The Canonicity Theorem and Its Impact In his PhD research, completed in 1969, and over the next half-dozen years, Kit Fine made a series of fundamental contributions to the semantic analysis and metatheory of propositional modal logic, proving general theorems about notable classes of logics and providing examples of failure of some significant properties. This work included the following (in order of publication): • A study [6] of logics that have propositional quantifiers and are defined

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Citation Context ... similar relational structures, having any kind and number of finitary relations. In general a member of VC is then a Boolean algebra with operators, or BAO, a notion introduced by Jónsson and Tarski =-=[38, 39]-=-, who first showed how n+1-ary relations on a set X correspond to n-ary operations on the powerset algebra of X (a Kripke frame falls under the special case n = 1 of this). BAO’s include many kinds of... |

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Citation Context ...m which arbitrarily large canonical frames can be built for any given logic. The above body of work by Fine can be seen as part of a second wave of research that flowed from the publication by Kripke =-=[41]-=- of his seminal work on the relational semantics of normal propositional modal logics. As is well known, one reason for the great success of Kripke’s theory was that it provided models that were much ... |

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Citation Context ...a simple proof of the converse of the Canonicity Theorem for subframe logics. 2 Background This section briefly reviews what we need from the relational model theory of propositional modal logic (see =-=[3]-=- for more details). From a given class of variables pξ, one for each ordinal ξ, formulas are generated using Boolean connectives (⊥ and →) and the modality □. The connectives ¬, ∧, ∨, ↔, ♦ are defined... |

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Citation Context ... axioms of S4 ensure that the frame FS4 is a preorder, and that suffices to imply that S4 is determined by validity in all preorders. The adjective canonical was attached to the model ML by Segerberg =-=[49, 50]-=-. Its underlying frame FL is the canonical frame of L, and L itself may be called a canonical logic if it is valid in FL. 1 By the end of the 1960’s, 1 This use of ‘canonical’ will be refined later. S... |

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Citation Context ... [18] of my thesis. It has now become known as the Goldblatt–Thomason Theorem (see [3, p. 186] for background), and versions of it have been developed for other formalisms, including hybrid languages =-=[51]-=-, graded modal languages [47], and the logic of coalgebras [42]. A suitably saturated elementary extension of F can always be obtained as an ultrapower of F [4, §6.1], and so the hypothesis on C can b... |

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Citation Context ...xiom □♦p → ♦□p, for which no such results had been available. My interest here is in what was arguably the most influential contribution: the paper Some connections between elementary and modal logic =-=[14]-=-, and 2in particular its Theorem 3, which will be referred to as Fine’s Canonicity Theorem. It states that any logic that is complete with respect to a first-order definable class of Kripke frames mu... |

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Citation Context ...tent sets of formulas as their members (‘possible worlds’), and their use is an extension of the famous method of completeness proof introduced by Henkin for first-order logic and the theory of types =-=[33, 34, 35]-=-. Prior to [14], propositional modal logics were typically taken to be based on a denumerably infinite set of variables, but Fine took the step of allowing languages to have arbitrarily large sets of ... |

35 |
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Citation Context ...logic, proving general theorems about notable classes of logics and providing examples of failure of some significant properties. This work included the following (in order of publication): • A study =-=[6]-=- of logics that have propositional quantifiers and are defined semantically by constraints on the range of interpretation of the quantifiable variables as subsets of a Kripke model. Axiomatisations we... |

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Citation Context ... Makinson [46] had constructed a sublogic of S4 with these characteristics. • Axiomatisations of logics with ‘numerical’ modalities Mk, for positive integer k, meaning ‘in at least k possible worlds’ =-=[8]-=-. This topic later became known as graded modal logic. • Exhibition of a logic extending S4 that is incomplete for validity in its Kripke frames [11]. This paper and one of S. K. Thomason [53] indepen... |

30 |
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Citation Context ... similar relational structures, having any kind and number of finitary relations. In general a member of VC is then a Boolean algebra with operators, or BAO, a notion introduced by Jónsson and Tarski =-=[38, 39]-=-, who first showed how n+1-ary relations on a set X correspond to n-ary operations on the powerset algebra of X (a Kripke frame falls under the special case n = 1 of this). BAO’s include many kinds of... |

29 | Mathematical Modal Logic: a View of its Evolution
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(Show Context)
Citation Context ... reduction of the full monadic second-order theory of a binary relation to the propositional logic of a single modality. 3 1 This use of ‘canonical’ will be refined later. See footnote 7. 2 See §6 of =-=[29]-=- for a survey of this metatheory from the 1970’s. 3 See [29, §6.4] for a summary. 3Every logic that had been shown to be canonical had been done so in a way, explained two paragraphs ago, that also s... |

27 |
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Citation Context ...aper and one of S. K. Thomason [53] independently provided the first examples of incomplete modal logics. Thomason’s was a sublogic of S4, following his earlier discovery of an incomplete tense logic =-=[52]-=-. • A proof [10] that the lattice of logics extending S4 is uncountable and includes an isomorphic copy of the powerset (P(ω), ⊆) of the natural numbers, ordered by inclusion. • An extensive study [12... |

22 |
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Citation Context ...way in which the canonical structure CA of each A in VC could be constructed from members of C by forming images of bounded morphisms, inner subframes and disjoint unions. This was studied further in =-=[25]-=-, leading to strengthenings of (1.2) that gave information about other first-order definable classes that generate VC. In particular, if C is first-order definable, then: • VC is also generated by som... |

18 |
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Citation Context ...etc. The first wave of research focused on exploring this phenomenon for various logics. The Henkin method was applied to relational semantics by a number of people, including Cresswell [5], Makinson =-=[45]-=- and Lemmon and Scott [44]. The latter defined a particular model ML for any logic L, based on the frame FL of all maximally L-consistent sets. This model determines L: the formulas true in ML are pre... |

18 |
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Citation Context ... worlds’ [8]. This topic later became known as graded modal logic. • Exhibition of a logic extending S4 that is incomplete for validity in its Kripke frames [11]. This paper and one of S. K. Thomason =-=[55]-=- independently provided the first examples of incomplete modal logics. Thomason’s was a sublogic of S4, following his earlier discovery of an incomplete tense logic [54]. • A proof [10] that the latti... |

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14 |
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Citation Context ... integer k, meaning ‘in at least k possible worlds’ [8]. This topic later became known as graded modal logic. • Exhibition of a logic extending S4 that is incomplete for validity in its Kripke frames =-=[11]-=-. This paper and one of S. K. Thomason [53] independently provided the first examples of incomplete modal logics. Thomason’s was a sublogic of S4, following his earlier discovery of an incomplete tens... |

13 |
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Citation Context ...intrinsically second-order, since it refers to truth in all models on a frame, hence has the effect of allowing propositional variables to range in value over arbitrary subsets of the frame. Thomason =-=[54]-=- gave a reduction of the full monadic second-order theory of a binary relation to the propositional logic of a single modality. 3 Every logic that had been shown to be canonical had been done so in a ... |

12 | Erdös graphs resolve Fine’s canonicity problem
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Citation Context ...ving the transitivity restriction. But eventually, after three decades, it was found that the answer to Fine’s converse question is negative in general. Uncountably many counterexamples were given in =-=[31]-=- and [30]. So canonicity is not equivalent to first-order completeness. The other question was whether a logic that is validated by its canonical frame built from a countable language must be validate... |

11 |
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Citation Context ...52]. • A proof [10] that the lattice of logics extending S4 is uncountable and includes an isomorphic copy of the powerset (P(ω), ⊆) of the natural numbers, ordered by inclusion. • An extensive study =-=[12]-=- of the model theory of logics that extend the system K4, i.e. their Kripke frames are transitive. This included a completeness proof for any such logic whose frames have a fixed bound on their ‘width... |

11 |
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Citation Context ...ass of frames that is closed under subframes. There are uncountably many of them, and all were shown to all have the finite model property, and other features that will be mentioned later. • A theory =-=[13]-=- of normal forms in modal logic, leading to a proof that all members of a certain class of ‘uniform’ logics have the finite model property, hence are Kripke-complete, and are decidable if finitely axi... |

11 |
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Citation Context ...[31, pages 189–190]. It includes the subframe logics, introduced by Fine in [15] where he showed that every canonical subframe logic having transitive frames must be first-order complete, with Wolter =-=[58]-=- later removing the transitivity restriction. But eventually, after three decades, it was found that the answer to Fine’s converse question is negative in general. Uncountably many counterexamples wer... |

11 |
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Citation Context ...omason Theorem (see [3, p. 186] for background), and versions of it have been developed for other formalisms, including hybrid languages [53], graded modal languages [49], and the logic of coalgebras =-=[43]-=-. A suitably saturated elementary extension of F can always be obtained as an ultrapower of F [4, §6.1], and so the hypothesis on C can be stated as closure under ultrapowers. Ultimately it can be wea... |

10 |
Relation Algebras by Games, volume 147
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Citation Context ... under the special case n = 1 of this). BAO’s include many kinds of algebra that have been studied by algebraic logicians, including cylindric algebras [36], polyadic algebras [32], relation algebras =-=[40, 37]-=-, and algebras for temporal logic, dynamic logic and other kinds of multi-modal logic. In [22] I gave a quite different second proof of (1.2) that analysed the way in which the canonical structure CA ... |

10 |
Boolean algebras with operators, part II
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(Show Context)
Citation Context ... under the special case n = 1 of this). BAO’s include many kinds of algebra that have been studied by algebraic logicians, including cylindric algebras [36], polyadic algebras [32], relation algebras =-=[40, 37]-=-, and algebras for temporal logic, dynamic logic and other kinds of multi-modal logic. In [22] I gave a quite different second proof of (1.2) that analysed the way in which the canonical structure CA ... |

8 |
On closure under canonical embedding algebras
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(Show Context)
Citation Context ... by algebraic logicians, including cylindric algebras [36], polyadic algebras [32], relation algebras [40, 37], and algebras for temporal logic, dynamic logic and other kinds of multi-modal logic. In =-=[22]-=- I gave a quite different second proof of (1.2) that analysed the way in which the canonical structure CA of each A in VC could be constructed from members of C by forming images of bounded morphisms,... |

7 |
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Citation Context ...inear orders; etc. The first wave of research focused on exploring this phenomenon for various logics. The Henkin method was applied to relational semantics by a number of people, including Cresswell =-=[5]-=-, Makinson [45] and Lemmon and Scott [44]. The latter defined a particular model ML for any logic L, based on the frame FL of all maximally L-consistent sets. This model determines L: the formulas tru... |

7 |
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(Show Context)
Citation Context ...or first-order logic (̷Lo´s’s Theorem), and the relationship (2.1). From this it can be shown that ∏ U Mi |= A iff {i ∈ I : Mi |= A} ∈ U, and that ∏ U Fi |= A implies {i ∈ I : Fi |= A} ∈ U (2.3) (cf. =-=[20]-=-). Now when the frames Fi for all i ∈ I are equal to a single frame F, then the ultraproduct is called the ultrapower of F modulo U, denoted F U , and F is the ultraroot of F U . Then from (2.3), F U ... |

7 |
Algebraic Polymodal Logic: a survey. Logic
- Goldblatt
(Show Context)
Citation Context ...ed the name ‘pseudo-equational’ rather than ‘quasi-modal’, but the latter seems more apposite in the modal context. 154 Canonical Varieties We now review the algebraic semantics of modal logics (see =-=[27]-=- or [3] for more details), and then derive a new characterisation of canonicity of a logic in Theorem 4.3. A modal algebra A = (B, l) consists of a Boolean algebra B with a function l : B → B that pre... |

7 |
Intensional logic. Preliminary draft of initial chapters by E.J.Lemmon
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(Show Context)
Citation Context ...earch focused on exploring this phenomenon for various logics. The Henkin method was applied to relational semantics by a number of people, including Cresswell [5], Makinson [45] and Lemmon and Scott =-=[44]-=-. The latter defined a particular model ML for any logic L, based on the frame FL of all maximally L-consistent sets. This model determines L: the formulas true in ML are precisely the L-theorems. The... |

6 |
The logics containing S4.3. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
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(Show Context)
Citation Context ...shown for some cases where the range includes all subsets and the underlying propositional logic is weaker than S5. Decidability and undecidability results were also proved. • A model-theoretic proof =-=[7]-=- of Bull’s theorem (originally proved algebraically) that all normal extensions of S4.3 have the finite model 1property. It was also shown that these logics are all finitely axiomatisable and decidab... |

6 |
An ascending chain of S4 logics
- Fine
- 1974
(Show Context)
Citation Context ...S. K. Thomason [53] independently provided the first examples of incomplete modal logics. Thomason’s was a sublogic of S4, following his earlier discovery of an incomplete tense logic [52]. • A proof =-=[10]-=- that the lattice of logics extending S4 is uncountable and includes an isomorphic copy of the powerset (P(ω), ⊆) of the natural numbers, ordered by inclusion. • An extensive study [12] of the model t... |

6 | On canonical modal logics that are not elementarily determined. Logique et Analyse
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- 2004
(Show Context)
Citation Context ...transitivity restriction. But eventually, after three decades, it was found that the answer to Fine’s converse question is negative in general. Uncountably many counterexamples were given in [31] and =-=[30]-=-. So canonicity is not equivalent to first-order completeness. The other question was whether a logic that is validated by its canonical frame built from a countable language must be validated by its ... |

6 | Quasi-modal equivalence of canonical structures
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(Show Context)
Citation Context ...s: if L is determined by some first-order conditions, then it is determined by the first-order conditions that are satisfied by the canonical frame FL. This conclusion was further refined in [23] and =-=[28]-=- by studying certain ‘quasi-modal’ sentences which are defined syntactically, and are the first-order sentences whose truth is preserved by the key modal-validity preserving constructions of bounded m... |

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5 | The discovery of my completeness proofs
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(Show Context)
Citation Context ...tent sets of formulas as their members (‘possible worlds’), and their use is an extension of the famous method of completeness proof introduced by Henkin for first-order logic and the theory of types =-=[33, 34, 35]-=-. Prior to [14], propositional modal logics were typically taken to be based on a denumerably infinite set of variables, but Fine took the step of allowing languages to have arbitrarily large sets of ... |

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5 |
Decidability of S4.1
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(Show Context)
Citation Context ... axioms of S4 ensure that the frame FS4 is a preorder, and that suffices to imply that S4 is determined by validity in all preorders. The adjective canonical was attached to the model ML by Segerberg =-=[49, 50]-=-. Its underlying frame FL is the canonical frame of L, and L itself may be called a canonical logic if it is valid in FL. 1 By the end of the 1960’s, 1 This use of ‘canonical’ will be refined later. S... |

4 |
Elementary logics are canonical and pseudo-equational
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(Show Context)
Citation Context ...In other words: if L is determined by some first-order conditions, then it is determined by first-order conditions that are satisfied by the canonical frame FL. This conclusion was further refined in =-=[23]-=- and [28] by studying certain ‘quasi-modal’ sentences which are defined syntactically, and are the firstorder sentences whose truth is preserved by the key modal-validity preserving constructions of b... |

4 | Reflections on a proof of elementarity
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(Show Context)
Citation Context ...This led to 5 van Benthem’s original proof involved the use of first-order compactness, but in the published version [56] he chose to give my structural argument about closure under ultraproducts. In =-=[26]-=- I was able to return the favour by publishing an account of his original proof. 5the question of which first-order definable properties of a binary relation are expressible by modal formulas, given ... |

3 |
Metamathematics of modal logic, parts I
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(Show Context)
Citation Context ...′ here satisfies the same first-order sentences as F, so the assumption on C can be weakened to closure under first-order equivalence. In this form (1.1) appeared in [17] and in the published version =-=[18]-=- of my thesis. It has now become known as the Goldblatt–Thomason Theorem (see [3, p. 186] for background), and versions of it have been developed for other formalisms, including hybrid languages [51],... |

3 |
A normal modal calculus between T and S4 without the finite model property
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(Show Context)
Citation Context ...ably infinite. • Construction [9] of a modal logic extending S4, and a superintuitionistic propositional logic, that are finitely axiomatisable and lack the finite model property. Previously Makinson =-=[46]-=- had constructed a sublogic of S4 with these characteristics. • Axiomatisations of logics with ‘numerical’ modalities Mk, for positive integer k, meaning ‘in at least k possible worlds’ [8]. This topi... |