(with Lloyd Humberstone) Modal Logics That Are Both Monotone and Antitone: Makinson's Extension Results and Affinities Between Logics (Notre Dame Jouranal of Formal Logic forthcoming)
A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI, LV or LF, and every antitone logic, to LN, LV or LF, where LI, LN, LV and LF are logics axiomatized, respectively,by the schemas □α ↔ α, □α ↔ ¬α, □α ↔ ⊤ and □α ↔ ⊥. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV, LF and the minimum amphitone logic AM axiomatized by the schema □α → □β. These logics, along with LI, LN and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the non-modal connectives. We close by discussing these divergences and conditions under
which our results do carry over.
Draft
A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI, LV or LF, and every antitone logic, to LN, LV or LF, where LI, LN, LV and LF are logics axiomatized, respectively,by the schemas □α ↔ α, □α ↔ ¬α, □α ↔ ⊤ and □α ↔ ⊥. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV, LF and the minimum amphitone logic AM axiomatized by the schema □α → □β. These logics, along with LI, LN and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the non-modal connectives. We close by discussing these divergences and conditions under
which our results do carry over.
Draft
Necessary, Transcendental and Universal Truth (Themes from Kit Fine 2020)
A simple puzzle leads Fine to conclude that we should distinguish between worldly sentences like ‘Socrates exists,’ whose truth values depend on circumstances and unworldly ones like ‘Socrates is human,’ which are true or false independently of circumstances. The former, if true in every circumstance, express necessary propositions. The latter, if true, express transcendental propositions, which, for theoretical convenience, we regard as necessary in an extended sense. Here it is argued that this understanding is backwards. Transcendental truths and sentences true in every circumstances (here labeled universal truths) are both species of necessary truth. The revised understanding is clarified by a simple formal system with distinct operators for necessary, transcendental and universal truth. The system is axiomatized. Its universal-truth fragment coincides with something that Arthur Prior once proposed as System A. The ideas of necessary, transcendental truth are further clarified by considering their interaction with actual truth. Adding an operator for actually true to the formal system produces a system closely related to one of Crossley and Humberstone.
Draft Published Version (Oxford University Press)
A simple puzzle leads Fine to conclude that we should distinguish between worldly sentences like ‘Socrates exists,’ whose truth values depend on circumstances and unworldly ones like ‘Socrates is human,’ which are true or false independently of circumstances. The former, if true in every circumstance, express necessary propositions. The latter, if true, express transcendental propositions, which, for theoretical convenience, we regard as necessary in an extended sense. Here it is argued that this understanding is backwards. Transcendental truths and sentences true in every circumstances (here labeled universal truths) are both species of necessary truth. The revised understanding is clarified by a simple formal system with distinct operators for necessary, transcendental and universal truth. The system is axiomatized. Its universal-truth fragment coincides with something that Arthur Prior once proposed as System A. The ideas of necessary, transcendental truth are further clarified by considering their interaction with actual truth. Adding an operator for actually true to the formal system produces a system closely related to one of Crossley and Humberstone.
Draft Published Version (Oxford University Press)
(with Brian Weatherson) Notes on Some Ideas in Lloyd Humberstone's Philosophical Applications of Modal Logic (AAJL 2018)
We answer some questions raised in the book and extend some ideas developed there. In particular we show: 1) a logic is fully modalized in Humberstone's sense iff every theorem is a tautological consequence of a fully modalized formula, 2) Humberstone's observation that any reasonably well-behaved modal logic admits logical intermediaries between a formula and its consequences implies that all such logics are translationally equivalent, 3) the schemas 5' and F, which have received attention in the literature on epistemic logic, are interderivable in S4, and 4) Humberstone's proposed logic of "coming about" is tantamount to previously proposed logics of "essence and accident" and his infinite axiomatization can be replaced by a finite one.
Draft Published Version (AAJL)
We answer some questions raised in the book and extend some ideas developed there. In particular we show: 1) a logic is fully modalized in Humberstone's sense iff every theorem is a tautological consequence of a fully modalized formula, 2) Humberstone's observation that any reasonably well-behaved modal logic admits logical intermediaries between a formula and its consequences implies that all such logics are translationally equivalent, 3) the schemas 5' and F, which have received attention in the literature on epistemic logic, are interderivable in S4, and 4) Humberstone's proposed logic of "coming about" is tantamount to previously proposed logics of "essence and accident" and his infinite axiomatization can be replaced by a finite one.
Draft Published Version (AAJL)
On Connections between Classical and Modern Modal Logic (manuscript 2018)
It is noted that the "modern" notion of bisimulation can be characterized in terms of the "classical" notions of filtration and p-morphism.
Manuscript
It is noted that the "modern" notion of bisimulation can be characterized in terms of the "classical" notions of filtration and p-morphism.
Manuscript
Two-dimensional Logic and Two-dimensionalism in Philosophy (Routledge Companion to Philosophy of Language 2012)
Survey article, containing a new normal form theorem for 2D logic with full complement of connectives.
Draft Published Version (Routledge)
Survey article, containing a new normal form theorem for 2D logic with full complement of connectives.
Draft Published Version (Routledge)
A Simple Embedding of T into Double S5 (Notre Dame Journal of Formal Logic 2004).
Double S5 is the modal system with operators [1] and [2] interpreted by equivalence relations and the usual Kripke semantics. It is shown that the translation mapping □ to [1][2] embeds the system T into Double S5, and that this remains so in the presence of full propositional quantification. This provides a simple proof that Double S5 with full propositional quantification (unlike the corresponding one-operator system) is undecidable.
Draft Published Version (Project Euclid)
Double S5 is the modal system with operators [1] and [2] interpreted by equivalence relations and the usual Kripke semantics. It is shown that the translation mapping □ to [1][2] embeds the system T into Double S5, and that this remains so in the presence of full propositional quantification. This provides a simple proof that Double S5 with full propositional quantification (unlike the corresponding one-operator system) is undecidable.
Draft Published Version (Project Euclid)
(with Paul Portner) Tense and Time (Handbook of Philosophical Logic 2002)
Extensive survey for handbook.
Draft Published Version (Springer)
Extensive survey for handbook.
Draft Published Version (Springer)
Embedded Definite Descriptions: Russelian Analysis and Semantic Puzzles (Mind 2000).
A sentence containing a number of definite descriptions, each lying within the scope of its predecessor, is naturally read as asserting the uniqueness of a sequence of objects satisfying the descriptions. The project of providing a general uniform procedure for eliminating embedded definite descriptions that gets this and other logical forms right is impeded by several puzzles.
Draft Published Version (Oxford)
A sentence containing a number of definite descriptions, each lying within the scope of its predecessor, is naturally read as asserting the uniqueness of a sequence of objects satisfying the descriptions. The project of providing a general uniform procedure for eliminating embedded definite descriptions that gets this and other logical forms right is impeded by several puzzles.
Draft Published Version (Oxford)
Modal Logic (Routledge 1998).
Entry for Routledge Encyclopedia of Philosophy.
Draft Published Version (Routledge)
Entry for Routledge Encyclopedia of Philosophy.
Draft Published Version (Routledge)
Minimal Non-Contingency Logic (Notre Dame Journal of Formal Logic 1995).
Simple finite axiomatizations are given for versions of the modal logics K and K4 with non-contingency (or contingency) as the sole modal primitive. This answers two questions of I. L. Humberstone.
Draft Published Version (Project Euclid)
Simple finite axiomatizations are given for versions of the modal logics K and K4 with non-contingency (or contingency) as the sole modal primitive. This answers two questions of I. L. Humberstone.
Draft Published Version (Project Euclid)
(with Brendan Henry and Sharon Pizzo) Logic and Computers at Georgetown (Abstract 1990)
Abstract describing derivation checker for the system in Lemmon's Beginning Logic. Program available on request.
Manuscript
Abstract describing derivation checker for the system in Lemmon's Beginning Logic. Program available on request.
Manuscript
The Domino Relation, (Journal of Philosophical Logic 1989).
The domino relation is the relation that holds between ordered pairs (x,y) and (z,w) iff y=z. Modal connectives interpreted by this relation illuminate some disparate recent discussions: absolute and relative necessity, modal logic as description of directed graphs, and quantification in Quine’s predicate functor logic. We provide necessary and sufficient conditions for R to be isomorphic to a domino relation and axiomatize the modal logic determined by relations satisfying these conditions.
Draft Published Version (Springer)
The domino relation is the relation that holds between ordered pairs (x,y) and (z,w) iff y=z. Modal connectives interpreted by this relation illuminate some disparate recent discussions: absolute and relative necessity, modal logic as description of directed graphs, and quantification in Quine’s predicate functor logic. We provide necessary and sufficient conditions for R to be isomorphic to a domino relation and axiomatize the modal logic determined by relations satisfying these conditions.
Draft Published Version (Springer)
Notes On a Problem of W.V. Quine (Manuscript 1984)
This is a working paper on the problem of simplifying the axiomatization of predicate functor logic given in my NDJFL paper below.
Manuscript
This is a working paper on the problem of simplifying the axiomatization of predicate functor logic given in my NDJFL paper below.
Manuscript
An Axiomatization of Predicate Functor Logic (Notre Dame Journal of Formal Logic 1983).
W.V. Quine asks for an effective proof procedure for his logical system in which the work of quantifiers and variables is done by "functors" that transform predicates into predicates. This paper provides such a proof procedure. It is less simple than might be desired, in that non-primitive symbols are needed to display the axioms conveniently. If one is content with a version in which functors replace only variable-binding (while free variables remain), a simple and perspicuous axiomatization is possible.
Draft Published Version (Project Euclid)
W.V. Quine asks for an effective proof procedure for his logical system in which the work of quantifiers and variables is done by "functors" that transform predicates into predicates. This paper provides such a proof procedure. It is less simple than might be desired, in that non-primitive symbols are needed to display the axioms conveniently. If one is content with a version in which functors replace only variable-binding (while free variables remain), a simple and perspicuous axiomatization is possible.
Draft Published Version (Project Euclid)
Logical Expressions, Constants and Operator Logic (Journal of Philosophy 1981)
Renewed interest in modal logic has given urgency to foundational questions. Common accounts of the nature of logic ignore or insufficiently emphasize several simple facts. One result is that logical expressions are mistakenly identified with constant expressions. A second is that a wide variety of modal systems have been construed as rival alternative logics. Many of these are better construed as distinct theories of a more general operator logic, in which the interpretation of one or more purely schematic sentential connectives is narrowed as appropriate for a particular subject matter.
Draft Published Version (JStor)
Renewed interest in modal logic has given urgency to foundational questions. Common accounts of the nature of logic ignore or insufficiently emphasize several simple facts. One result is that logical expressions are mistakenly identified with constant expressions. A second is that a wide variety of modal systems have been construed as rival alternative logics. Many of these are better construed as distinct theories of a more general operator logic, in which the interpretation of one or more purely schematic sentential connectives is narrowed as appropriate for a particular subject matter.
Draft Published Version (JStor)
Quantifiers as Modal Operators, (Studia Logica 1980).
Montague, Prior, von Wright and others drew attention to resemblances between modal operators and quantifiers. In this paper we show that classical quantifiers can in fact be regarded as "S5-like" operators in a purely propositional modal logic. This logic is axiomatized and some interesting fragments of it are investigated.
Draft Published Version (Springer)
Montague, Prior, von Wright and others drew attention to resemblances between modal operators and quantifiers. In this paper we show that classical quantifiers can in fact be regarded as "S5-like" operators in a purely propositional modal logic. This logic is axiomatized and some interesting fragments of it are investigated.
Draft Published Version (Springer)
(with A.K. Joshi) Centered Logic: The Role of Entity-Centered Sentence Representation in Natural Language Inferencing" (in IJCAI Proceedings 1979)
A salient difference between sentences of natural language and formulas of predicate logic is that in the former a term is “centered,” usually, but not always, by being put in subject position. Such a sentence may be viewed as attributing a property to the object denoted by its center. Other terms may be viewed as “masked” within the predicate. We begin investigation of a formal logical system sharing this feature, with the idea of illuminating not just whether an inference is valid, but how difficult it is. Difficulty is raised when a centered and masked terms change roles (change of topic) and when a temporary assumption with a new center is introduced. Questions about the scope of limited-difficulty fragments of the system remain open.
Draft IJCAI Russian translation
A salient difference between sentences of natural language and formulas of predicate logic is that in the former a term is “centered,” usually, but not always, by being put in subject position. Such a sentence may be viewed as attributing a property to the object denoted by its center. Other terms may be viewed as “masked” within the predicate. We begin investigation of a formal logical system sharing this feature, with the idea of illuminating not just whether an inference is valid, but how difficult it is. Difficulty is raised when a centered and masked terms change roles (change of topic) and when a temporary assumption with a new center is introduced. Questions about the scope of limited-difficulty fragments of the system remain open.
Draft IJCAI Russian translation
What is Segerberg's Two Dimensional Modal Logic? (in Box and Diamond: Mini Essays in Honor of Krister Segerberg, 1977)
It is the the two-dimensional fragment of classical dyadic predicate logic.
Draft Libris
It is the the two-dimensional fragment of classical dyadic predicate logic.
Draft Libris
Many-sorted Modal Logics (1976)
Stanford University PhD thesis, which was reprinted in the Uppsala University Philosophical Studies series.
University Microfilms Uppsala
Stanford University PhD thesis, which was reprinted in the Uppsala University Philosophical Studies series.
University Microfilms Uppsala