My purpose here is purely historical. It is not an attempt to resolve the question as to whether Russell did or did not countenance nonclassical logics, and if so, which nonclassical logics, and still less to demonstrate whether he himself contributed, in any manner, to the development of nonclassical logic. Rather, I want merely to explore and insofar as possible document, whether, and to what extent, if any, Russell interacted with the various, either the various candidates or their, ideas that (...) Dejnožka and others have proposed as potentially influential in Russell’s intellectual reactions to nonclassical logic or to the philosophical concepts that might contribute to his reactions to nonclassical logics. (shrink)

This volume seeks to further the use of formal methods in clarifying one of the central problems of philosophy: that of our free human agency and its place in our indeterministic world. It celebrates the important contributions made in this area by Nuel Belnap, American logician and philosopher. Philosophically, indeterminism and free action can seem far apart, but in Belnap’s work, they are intimately linked. This book explores their philosophical interconnectedness through a selection of original research papers that build forth (...) on Belnap’s logical and philosophical work. Some contributions take the form of critical discussions of Belnap's published work, some develop points made in his publications in new directions, and others provide additional insights on the topics of indeterminism and free action. In Nuel Belnap’s work on indeterminism and free action, three formal frameworks figure prominently: the simple branching histories framework known as "branching time;" its relativistic spatio-temporal extension, branching space-times; and the “seeing to it that” logic of agency. As those frameworks provide the formal background for the contributed papers, the volume introduction gives an overview of the current state of their development. It also introduces case-intensional first order logic, a general intensional logic offering resources for a first-order extension of the mentioned frameworks and a recent research focus of Belnap’s. The volume also contains an extended biographical interview with Nuel Belnap. (shrink)

It is demonstrated that Hamblin's 15 tense theorem does not only hold for temporal logic with linear time but also for branching time. Furthermore three other theorems with finitely many tenses are shown.

The ‘Dialectical school’ denotes a group of early Hellenistic philosophers that were loosely connected by philosophizing in the — Socratic — tradition of Eubulides of Megara and by their interest in logical paradoxes, propositional logic and dialectical expertise. . Its two best known members, Diodorus Cronus and Philo the Logician, made groundbreaking contributions to the development of theories of conditionals and modal logic. Philo introduced a version of material implication; Diodorus devised a forerunner of strict implication. Each developed a system (...) of modal notions that satisfies the basic logical requirements laid down by modern standard modal theories. In antiquity, Diodorus Cronus was famous for his so called Master Argument, which aims to prove that only the actual is possible. (shrink)

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Content: 1. Pre-Aristotelian Logic 1.1 Syntax and Semantics 1.2 Argument Patterns and Valid Inference 2. Aristotle 2.1 Dialectics 2.2 Sub-sentential Classifications 2.3 Syntax and Semantics of Sentences 2.4 Non-modal Syllogistic 2.5 Modal Logic 3. The early Peripatetics: Theophrastus and Eudemus 3.1 Improvements and Modifications of Aristotle's Logic 3.2 Prosleptic Syllogisms 3.3 Forerunners (...) of Modus Ponens and Modus Tollens 3.4 Wholly Hypothetical Syllogisms 4. Diodorus Cronus and Philo the Logician 5. The Stoics 5.1 Logical Achievements Besides Propositional Logic 5.2 Syntax and Semantics of Complex Propositions 5.3 Arguments 5.4 Stoic Syllogistic 5.5 Logical Paradoxes 6. Epicurus and the Epicureans 7. Later Antiquity. (shrink)

We care not only about what experiences we have, but when we have them too. However, on the B-theory of time, something’s timing isn’t an intrinsic way for that thing to be or become. Given B-theory, should we be rationally indifferent about the timing per se of an experience? In this paper, I argue that B-theorists can justify time-biased preferences for pains to be past rather than present and for pleasures to be present rather than past. In support of this (...) argument, I appeal to the doctrine of temporal parts or “four-dimensionalism” for short. When held in conjunction with a certain evaluative principle about whose experiences matter, four-dimensionalism reconciles B-theory with some time-biased preferences. (shrink)

'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...) Moreover, it provides possible gateways from modern deadlocks of theory either through approaches to consciousness or through historical critique of intellectual authorities. This work will be of interest to those either researching or studying in colleges and universities, especially in the departments of philosophy, history of science, philosophy of science, philosophy of physics and quantum mechanics, history of ideas and culture. Greek and Latin Literature students and instructors may also find this book to be both a fascinating and valuable point of reference. (shrink)

Essays on Aristotle's Sea-Battle, Lazy Argument, Argument Reaper, Diodorus' Master Argument -/- The book is devoted to the ancient logical theories, reconstruction of their semantic proprieties and possibilities of their interpretation by modern logical tools. The Ancient arguments are frequently misunderstood in modern interpretations since authors usually have tendency to ignore their historical proprieties and theoretical background what usually leads to a quite inappropriate picture of the argument’s original form and mission. Author’s primary intention was to draw attention to the (...) complexity of some historical arguments and to the theoretical context in which arguments were created, circulated, developed, and finally tuned. Four well-known ancient arguments – with a common central subject related to the future contingencies problem – are reconstructed from available historical sources: “The Sea Battle”, which is drawn from Aristotle’s treatise De Interpretatione; two arguments, usually ascribed to the Stoics, “The Lazy Argument” and “The Reaper”; “The Master Argument” of the Megarian philosopher Diodorus. Arguments are linguistically and semantically detaily analyzed, formally presented by reflecting some relevant corresponding hypotheses based on physical or logical theories of their ancient authors, and finally covered by appropriate logical tools familiar to a modern reader. Two appendices are added at the closing part of the book. One covering some assumptions relevant for understanding of rival streams in ancient theories of meaning related to the nature of names and naming; the other is devoted to the ancient understanding of logical proposition and attempts to find an adequate Latin translation of the Greek delicate philosophical term “ἀξίωμα”. (shrink)

Some apparently valid arguments crucially rely on context change. To take a kind of example first discussed by Frege, ‘Tomorrow, it’ll be sunny’ taken on a day seems to entail ‘Today, it’s sunny’ taken on the next day, but the first sentence taken on a day sadly does not seem to entail the second sentence taken on the second next day. Mid-argument context change has not been accounted for by the tradition that has extensively studied the distinctive logical properties of (...) context-dependent languages, for that tradition has focussed on arguments whose premises and conclusions are taken at the same context. I first argue for the desiderability of having a logic that accounts for mid-argument context change and I explain how one can informally understand such context change in a standard framework in which the relation of logical consequence holds among sentences. I then propose a family of simple temporal “intercontextual” logics that adequately model the validity of certain arguments in which the context changes. In particular, such logics validate the apparently valid argument in the Fregean example. The logics lack many traditional structural properties (reflexivity, contraction, commutativity etc.) as a consequence of the logical significance acquired by the sequence structure of premises and conclusions. The logics are however strong enough to capture in the form of logical truths all the valid arguments of both classical logic and Kaplan-style “intracontextual” logic. Finally, I extend the framework by introducing new operations into the object language, such as intercontextual conjunction, disjunction and implication, which, contrary to intracontextual conjunction, disjunction and implication, perfectly match the metalinguistic, intercontextual notions of premise combination, conclusion combination and logical consequence by representing their respective two operands as taken at different contexts. (shrink)

The Megarians, as well as their Stoic heirs, are known to have been fatalists or logical determinists in the following, very broad sense of these terms: with respect to at least certain classes or kinds of nontautologous propositions, they held that the mere truth of a proposition entails its necessity. This paper explores, in a very tentative fashion, the relation between several versions of logical determinism and two passages in the Aristotelian corpus, one of which is specifically directed against Megarian (...) doctrine and the other of which is an argument purporting to establish a thesis concerning the necessity of future events which Aristotle wished to reject but which the Megarian logician Diodorus Cronus apparently accepted. (shrink)

Though Arthur Prior is now best known for his founding of modern temporal logic and hybrid logic, much of his early philosophical career was devoted to history of logic and historical logic. This interest laid the foundations for both of his ground-breaking innovations in the 1950s and 1960s. Because of the important rôle played by Prior's research in ancient and medieval logic in his development of temporal and hybrid logic, any student of Prior, temporal logic, or hybrid logic should be (...) familiar with the medieval logicians and their work. In this article we give an overview of Prior's work in ancient and medieval logic. (shrink)

In the 4th century BC, the Greek philosopher Diodoros Chronos gave a temporal definition of necessity. Because it connects modality and temporality, this definition is of interest to philosophers working within branching time or branching space-time models. This definition of necessity can be formalized and treated within a logical framework. We give a survey of the several known modal and temporal logics of abstract space-time structures based on the real numbers and the integers, considering three different accessibility relations between spatio-temporal (...) points. (shrink)

Arthur Prior was one of the logicians who participated in the invention of the possible worlds’ semantics. The ontology, which is connected with his systems of modal logic, is unique. Prior tried to reduce the number of abstract entities as much as possible. Hence he did not elect to introduce possible worlds and possibilia into his ontology. In addition, he held a reductionist view, which is called modal actualism by Fine or modalism by Melia. Prior was inspired by various authors (...) but this paper mainly discusses the influence of two prominent sources, Wittgenstein and logicians from Warsaw School. Prior acknowledged Wittgenstein’s influence, when he discuss the view that possible worlds consist of propositions. The second important feature of Prior’s concept is the denial of the real existence of possible worlds and the denial of the existence of entities which are in them. It was probably enabled by Prior’s adoption of some systems of logic invented by logicians from the Warsaw school, especially Lesniewski’s and Łukasiewicz’s systems. (shrink)

This paper presents the link between Arthur N. Prior and logicians that belonged to the Lvov-Warsaw School. Although certain members of the Lvov-Warsaw School influenced Prior’s views, the amount and the form of the impact are still under discussion. Prior also cooperated with some of them in the development of his systems of logic. This paper focuses on four main areas in which Prior admitted adopting ideas from the Lvov-Warsaw School: systems of propositional logic, the history of logic, modal and (...) temporal logic and ontology. Prior’s published works as well as his correspondence, which is stored in Prior’s Nachlass, are used to describe the influence. (shrink)

We introduce a new way of defining metric temporal logic on a real-numbers flow of time. The idea is based on having semantics which allow us to refer to a single universal clock of arbitrary precision in order to impose metric constraints. This gives us a new metric temporal logic which is very expressive, is natural to use, can be applied in very general situations, affords a wide range of useful abbreviations and operators, has a PSPACE decision procedure, and has (...) the promise of being amenable to standard reasoning techniques. Thus it has many advantages over the existing metric temporal logics. We provide a decision procedure via conversion to a non-metric temporal logic, hence doing metric reasoning with no clocks. (shrink)

This paper deals with the historical and philosophical background of the introduction of the notion of branching time in philosophical logic as it is revealed in the hitherto unpublished mail-correspondence between Saul Kripke and A.N. Prior in the late 1950s. The paper reveals that the idea was first suggested by Saul Kripke in a letter to A.N. Prior, dated September 3, 1958, and it is shown how the elaboration of the idea in the course of the correspondence was intimately intervowen (...) with considerations of how to represent indeterminism and of the adequacy of tensed logic in light of special relativity. The correspondence underpins the point that Prior's later development of branching time may be understood as a crucial part of his attempt at the formulating a conceptual framework integrating basic human notions of time and free choice. (shrink)

This paper presents structural similarities and historical connections between Prior’s rejection of the Barcan formula and his critique of Berkeley’s master argument for idealism in his 1955 paper “Berkeley in Logical Form”. Making use of Mackie’s paper “Self-Refutation—A Formal Analysis”, it concludes with some suggestions concerning what is at stake in the debate between Prior and Berkeley and in structurally similar debates such as whether to accept the Barcan formula.

In his Formal Logic, Arthur N. Prior declared that Jan Łukasiewicz's logical notation is ‘unquestionably the best logical symbolism for most purposes’. Whether he had a substantive, and...

In this paper we examine Prior’s reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic K t 4 plus a new axiom ( P ): ‘ p Λ G p ⊃ P G p ’. This formula was used by Prior in his original analysis of Master Argument. ( P ) (...) is usually added as an extra axiom to an axiomatization of the logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by ( P ) in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. (shrink)

In this paper we examine Prior's reconstruction of Master Argument [4] in some modal-tense logic. This logic consists of a purely tense part and Diodorean definitions of modal alethic operators. Next we study this tense logic in the pure tense language. It is the logic $K_t 4$ plus a new axiom $:p \wedge Gp \supset PGp'$. This formula was used by Prior in his original analysis of Master Argument. is usually added as an extra axiom to an axiomatization of the (...) logic of linear time. In that case the set of moments is a total order and must be left-discrete without the least moment. However, the logic of Master Argument does not require linear time. We show what properties of the set of moments are exactly forced by in the reconstruction of Prior. We make also some philosophical remarks on the analyzed reconstruction. (shrink)

Although it receives less attention, (Lewis in Noûs 13:455–476, 1979. https://doi.org/10.2307/2215339) admitted that the branching-time(-like) model fits a wide range of counterfactuals, including (Nix) ‘If Nixon had pressed the button, there would have been a nuclear war’, which was raised by (Fine in Mind 84:451–458, 1975). However, Lewis then claimed that similarity analysis is more general than temporality analysis. In this paper, we do not scrutinise his claim. Instead, we re-analyse (Nix) not only model-theoretically but also proof-theoretically from the ‘meaning-as-use’ (...) and ‘inferentialist’ points of view. Then, we re-formalise (Nix) in a natural extension of hybrid tense logic, which we refer to as hybrid tense logic for temporal conditionals (HTL$$_{TC}$$ TC ). Consequently, we find that not only among counterfactuals, but also among indicatives, there is a wide range of conditionals whose formalisation in HTL$$_{TC}$$ TC is appropriate. We refer to these conditionals as temporal conditionals. This suggests a new logical generality that temporality analysis has but similarity analysis does not, from which emerges a new logical perspective on conditionals in general: temporal ones and others. (shrink)

Controversy has surrounded the interpretation of the so-called 'Diodorean' and 'Chrysippean' conditionals of the Stoics. I critically evaluate and reject two interpretations of each of them: as expressing natural laws, and as strict conditionals. In doing so I engage with the work of authors such as Frede, Gould, Hurst, the Kneales, Mates, and Prior. I conclude by offering my own proposal for where these Stoic conditionals should be located on a 'ladder' of logical strength.

Arthur Norman Prior's early theological writings have been relatively neglected for many years. Moreover, to the extent that they have been discussed at all they have been treated mainly as youthful work quite separate from Prior's later work as a philosopher and logician. However, as interest in Prior's achievements has been growing significantly in recent years it has become more important to investigate the development with his overall work. In fact, Prior's putatively "youthful" theological work overlapped his work as a (...) philosopher and logician for many years, as is richly documented by examples discussed in this paper. A particularly important theme is the problem of predestination. This paper presents comprehensive evidence that this theme, which was Prior's most important single preoccupation as a theological writer, was a most important source of inspiration for his development of tense logic. Via questions regarding divine foreknowledge and human free will, predestination was to motivate Prior as a logician to focus on time and tense. Whilst investigating this development, the paper also traces Prior's parallel development from Calvinist Christian believer to a more agnostic position. (shrink)

Resumen: El presente artículo tiene por objetivo la reconstrucción del argumento dominador atribuido a Diodoro Crono y su revisión filosófica a la luz de la crítica aristotélica al actualismo extremo de la Escuela de Mégara en Metafísica IX, 3. Desde esta perspectiva, el trabajo aspira a situar el proyecto teórico de Diodoro en el marco de su disputa filosófica con la metafísica aristotélica y su concepto de δύναµις, con el fin de traducir esta querella dialéctica en lo que, a nuestro (...) juicio, constituye una disputa teórica de importantes consecuencias prácticas: la disyunción excluyente entre el azar y el destino como paradigmas de interpretación conceptual y de acción moral.: This paper aims to reconstruct and analyze the Master Argument attributed to Diodorus Chronus under the perspective of Aristotle's criticism of extreme actualism in Metaphysics IX. 3. From this point of view, the paper locates Diodorus' theoretical project within the framework of his philosophical dispute with Aristotle's metaphysics and the Aristotelian concept of δύναµις, trying to show that what really lies behind this dialectical confrontation is a theoretical problem of important practical consequences: the excluding disjunction between chance and destiny as paradigms of conceptual interpretation and moral action. (shrink)

We defend the thesis that every necessarily true proposition is always true. Since not every proposition that is always true is necessarily true, our thesis is at odds with theories of modality and time, such as those of Kit Fine and David Kaplan, which posit a fundamental symmetry between modal and tense operators. According to such theories, just as it is a contingent matter what is true at a given time, it is likewise a temporary matter what is true at (...) a given possible world; so a proposition that is now true at all worlds, and thus necessarily true, may yet at some past or future time be false in the actual world, and thus not always true. We reconstruct and criticize several lines of argument in favor of this picture, and then argue against the picture on the grounds that it is inconsistent with certain sorts of contingency in the structure of time. (shrink)

Developed by Arthur Prior, Temporal Logic allows to represent temporal information on a logical system using modal operators such as P, F, H or G, whose intuitive meaning is “it was sometime in the Past...”, “it will be sometime in the Future...”, “it Has always been in the past...” and “it will always Going to be in the future...” respectively. Valuation of formulae built from these operators are carried out on Kripke semantics, so Modal Logic and Temporal Logic are consequently (...) related. In fact, Temporal Logic is an extension of Modal one. Even when both logics mechanisms are able to formalize modal-temporal information with some accuracy, they suffer from a lack of expressiveness which Hybrid Logic can solve. Indeed, one of the problems of Modal Logic consists in its incapacity of naming specific points inside a model. As Temporal Logic is based on it, it cannot make such a thing neither. But First-Order Logic does can by means of constants and equality relation. Hybrid Logic, which results from combining Modal Logic and First-Order Logic, may solve this shortcoming. The main aim of this paper is to explain how Hybrid Logic emanates from Modal and Temporal ones in order to show what it adds to both logics with regard to information representation, why it is more expressive than them and what relation it maintains with the First-Order Correspondence Language. (shrink)

ABSTRACT A spatial logic is a modal logic of which the models are the mathematical models of space. Successively considering the mathematical models of space that are the incidence geometry and the projective geometry, we will successively establish the language, the semantical basis, the axiomatical presentation, the proof of the decidability and the proof of the completeness of INC, the modal multilogic of incidence geometry, and PRO, the modal multilogic of projective geometry.

At the heart of the Stoic theory of modality is a strict commitment to bivalence, even for future contingents. A commitment to both future truth and contingency has often been thought paradoxical. This paper argues that the Stoic retreat from necessity is successful. it maintains that the Stoics recognized three distinct senses of necessity and possibility: logical, metaphysical and providential. Logical necessity consists of truths that are knowable a priori. Metaphysical necessity consists of truths that are knowable a posteriori, a (...) world order according to certain metaphysical principles and natures that god crafts within the constraints of matter. Finally, what is providentially necessary is what occurs according to the chain of fate, but only once it is in process or past. -/- The method of the paper is a close reading of Diogenes Laertius 7.75, adducing broad textual evidence along the way, to show that the Stoic theory of modality embraces Philonian possibility, both that which is capable of being true as a matter of logical consistency, and that which is possible according to the bare fitness of the entity. What differentiates the Stoics from Philo is their additional commitment to possibility as opportunity, resisting the collapse of determinism into necessity. (shrink)

Logic begins but does not end with the study of truth and falsity. Within truth there are the modes of truth, ways of being true: necessary truth and contingent truth. When a proposition is true, we may ask whether it could have been false. If so, then it is contingently true. If not, then it is necessarily true; it must be true; it could not have been false. Falsity has modes as well: a false proposition that could not have been (...) true is impossible or necessarily false; one that could have been true is merely contingently false. The proposition that some humans are over seven feet tall is contingently true; the proposition that all humans over seven feet tall are over six feet tall is necessarily true; the proposition that some humans are over seven feet tall and under six feet tall is impossible, and the proposition that some humans are over nine feet tall is contingently false. Of these four modes of truth, let us focus on necessity, plus a fifth: possibility. A proposition is possible if it is or could have been true; hence propositions that are either necessarily true, contingently true, or contingently false are possible. Notions that are similar to the modes of truth in being concerned with what might have been are called modal. Dispositions are modal notions, for example the disposition of fragility. Relatedly, there are counterfactual conditionals, for example “if this glass were dropped, it would break.” And the notion of supervenience is modal.1 But let us focus here on necessity and possibility. Modal words are notoriously ambiguous (or at least context-sensitive2). I may reply to an invitation to give a talk in England by saying “I can’t come; I have to give a talk in California the day before”. This use of “can’t” is perfectly appropriate. But it would be equally appropriate for me to say that I could cancel my talk in California (although that would be rude) and give the talk in England instead. What I cannot do is give both talks.. (shrink)

This chapter argues for a notion of time that allows time travel. In order to time traveling to happen, in contrast to Presentism, the chapter demonstrates that we can change the past and we have some place where to travel. It shows the advantages of a non-presentist ontology that advocates for indeterminacy of future facts based not on its absence of truth-value, but on the overdetermination of future facts. The conclusion is that to break the causal chain is impossible in (...) we are placed in the same causal line. But if we rethink the time traveling as a trans-world traveling, it is possible to open a new causal line anytime that someone travel in time, to the past as well as to the future. (shrink)

There is an ancient puzzle about divine foreknowledge and human freedom. If God has already known that you will do a certain thing tomorrow, then it must already be a settled fact that God has known this. Since knowledge entails truth, it must also be a settled fact that you will do it. In that case, you really cannot avoid doing it. If so, then when you do it tomorrow, you won't do it freely. ;This dissertation consists of a careful (...) statement of the puzzle and an examination of its principal solutions. These are fatalism, the view that nothing about the past, present, or future is open, eternalism, the view that God exists outside of time, semantic indeterminism, the view that some propositions are neither true nor false, and Ockhamism, the view that some things about the past are open. ;In Chapter I the puzzle is stated carefully and its solutions are sketched. Chapters II through V each discuss one of the principal solutions. It is argued that fatalism and semantic indeterminism are not adequate solutions to the puzzle. These positions cannot accommodate or adequately explain away our intuition that there is something someone can but will not do. Eternalism is seen to be a formally coherent view, but it also does not provide a satisfying solution to the puzzle. It is argued that Ockham's solution succeeds where the others fail. Finally, it is argued that although Ockham denies the necessity of the past, endorsing his solution does not force one to hold that we can literally change the past. It is concluded that Ockham's solution accommodates our intuitions about freedom and the future without charging a price. (shrink)