Timothy Williamson has maintained that the applicability of classical mathematics in science raises a problem for the endorsement, in non-mathematical domains, of a wide range of non-classical logics. We show that this is false.

Validity : what follows from what? -- Truth functions,or not -- Names and quantifiers : is nothing something? -- Descriptions and existence : did the greeks worship Zeus? -- Self-reference : what is this chapter about? -- Necessity and possibility : what will be must be? -- Conditionals: what's in an if? -- The future and the past : is time real?? -- Identity and change : is anything ever the same? -- Vaguenes : how do you stop sliding (...) down a slippery slope? -- Probability : the strange case of the missing reference class? -- Inverse probability : you can't be indifferent about it! -- Decision theory : great expectations? -- A little history and some further reading? (shrink)

In “Truth and Probability” Ramsey claimed that the logic of consistency for probability is not a logic of truth. After supporting this claim, he proceeded to explore the possibilities for a logic of truth for probability. An examination of Ramsey's intent reveals that Ramsey was far from being an orthodox Bayesian when it comes to statistical reasoning. The relations between Ramsey's thought and the ideas of Keynes and Peirce are discussed.

Traditional logical empiricist and more recent historicist positions on the logic of discovery are briefly reviewed and both are found wanting. None have examined the historical detail now available from recent research on Darwin, from which there is evidence for gradual transition in descriptive and explanatory concepts. This episode also shows that revolutionary research can be directed by borrowed metascientific objectives and heuristics from other disciplines. Darwin's own revolutionary research took place within an ontological context borrowed from non evolutionary (...) predecessors with methodological objectives borrowed from and justified by their success in Newton's physics. The logic of discovery is not a special form of inference from observation to theory, but rather a theory of the rationality of research, including principles bearing upon the rational choice of problems, or epistemic objectives, and heuristic, or means to solving the problems. Such choices can be justified only locally in the context of a relatively stable background ontology and substantive epistemology, not globally for all science. (shrink)

Labeled transition systems are key structures for modeling computation. In this paper, we show how they lend themselves to ordinary logical analysis (without any special new formalisms), by introducing their standard first-order theory. This perspective enables us to raise several basic model-theoretic questions of definability, axiomatization and preservation for various notions of process equivalence found in the computational literature, and answer them using well-known logical techniques (including the Compactness theorem, Saturation and Ehrenfeucht games). Moreover, we consider what happens to this (...) general theory when one restricts attention to special classes of transition systems (in particular, finite ones), as well as extended logical languages (in particular, infinitary first-order logic). We hope that this puts standard logical formalisms on the map as a serious option for a theory of computational processes. As a side benefit, our approach increases comparability with several other existing formalisms over labeled transition systems (such as Process Algebra or Modal Logic). We provide some pointers to this effect, too. (shrink)

We continue our work [5] on the logic of multisets (or on the multiset semantics of linear logic), by interpreting further the additive disjunction . To this purpose we employ a more general class of processes, called free, the axiomatization of which requires a new rule (not compatible with the full LL), the cancellation rule. Disjunctive multisets are modeled as finite sets of multisets. The -Horn fragment of linear logic, with the cut rule slightly restricted, is sound (...) with respect to this semantics. Another rule, which is a slight modification of cancellation, added to HF makes the system sound and complete. (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky (...) describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

The approach to decision theory floated in my 1965 book is reviewed (I), challenged in various related ways (II–V) and defended, firstad hoc (II–IV) and then by a general argument of Ellery Ells's (VI). Finally, causal decision theory (in a version sketched in VII) is exhibited as a special case of my 1965 theory, according to the Eellsian argument.

In this paper I argue that (at least many) philosophical thought experiments are unreliable. But I argue that this notion of unreliability has to be understood relative to the goal of thought experiments as knowledge producing. And relative to that goal many thought experiments in science are just as unreliable. But in fact thought experiments in science play a varied role and I will suggest that knowledge production is a goal only under quite limited circumstances. I defend the view that (...) these circumstances can (sometimes) arise in philosophy as well. (shrink)

In this paper we sketch a logic of message and reply. The logic is intended for application in a wide variety of situations, not restricted to the two-person, turn-taking situation. Each message has a body and a vector; the vector specifies the from, to, and the like. To reply to a message, it suffices to give either (1) a complete reply to the body or (2) a corrective reply to at least one presumption derivable from the vector. We (...) discuss the problems of achieving effectiveness and completeness with respect to certain aspects of communication. The results are mixed. In section 9 we argue semi-formally that, in a certain sense, dialogue is necessary. Finally we note that this logic is not a rival of other approaches but may be combinable with them. (shrink)

In this paper we sketch a logic of message and reply. The logic is intended for application in a wide variety of situations, not restricted to the two-person, turn-taking situation. Each message has a body and a vector; the vector specifies the from, to, and the like. To reply to a message, it suffices to give either (1) a complete reply to the body or (2) a corrective reply to at least one presumption derivable from the vector. We (...) discuss the problems of achieving effectiveness and completeness with respect to certain aspects of communication. The results are mixed. In section 9 we argue semi-formally that, in a certain sense, dialogue is necessary. Finally we note that this logic is not a rival of other approaches but may be combinable with them. (shrink)

As do all forms of science, medical theories have a factual as well as a logical basis. New information is presented in medical research articles. These papers have three separate arguments: the argument of the hypothesis, the argument of the experimental protocol, and the argument of the hypothesis's judgment. These arguments may be examples of the hypothetico-deductive or confirmational model of scientific inference. The logical form of these arguments are informal and inductive rather than formal and deductive. Understanding the nature (...) of the logic of the medical research article may help avoid erroneous conclusions. (shrink)

We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t: F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the well-known Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal (...) logic, there is a robust system of justifications. This renders a new, evidence-based foundation for epistemic logic. As a case study, we offer a resolution of the GoldmanRed Barns in Justification Logic. Furthermore, we formalize the well-known Gettier example and reveal hidden assumptions and redundancies in Gettier’s reasoning. (shrink)

Pit is a multi-player card game that simulates the commodities trading market, and where actions consist of bidding and of swapping cards. We present a formal description of the knowledge and change of knowledge in that game. The description is in a standard language for dynamic epistemics expanded with assignment. Assignment is necessary to describe that cards change hands. The formal description is a prerequisite to model Pit in game theory. The main contribution of this paper should be seen as (...) the rigorous formalization of all knowledge in Pit. (shrink)

I argue that some utterances of sentences containing occurrences of indexical expressions should not be evaluated with respect to the context of utterance. I suggest that we distinguish between context of utterance and context of interpret‐ation, and I employ this distinction in the analysis of recorded messages and other interesting linguistic phenomena. I then discuss the implications of my views on contexts with respect to the logic of indexicals. Against the traditional view, I argue that sentences such as ‘I (...) am here now’ or ‘something exists’ should not be treated as logical truths. (shrink)

Information-based epistemology maintains that ‘being informed’ is an independent cognitive state that cannot be reduced to knowledge or to belief, and the modal logic KTB has been proposed as a model. But what distinguishes the KTB analysis of ‘being informed’, the Brouwersche schema (B), is precisely its downfall, for no logic of information should include (B) and, more generally, no epistemic logic should include (B), either.

This book, one of the handful of truly pathbreaking works in twentieth-century philosophical theology, presents Hartshorne's persuasive rehabilitation of Anselm's Ontological Argument, recast in neoclassical form as "the Modal Proof.".

Proof Theory of Modal Logic is devoted to a thorough study of proof systems for modal logics, that is, logics of necessity, possibility, knowledge, belief, time, computations etc. It contains many new technical results and presentations of novel proof procedures. The volume is of immense importance for the interdisciplinary fields of logic, knowledge representation, and automated deduction.

This essay attempts to develop a psychologically informed semantics of perception reports, whose predictions match with the linguistic data. As suggested by the quotation from Miller and Johnson-Laird, we take a hallmark of perception to be its fallible nature; the resulting semantics thus necessarily differs from situation semantics. On the psychological side, our main inspiration is Marr's (1982) theory of vision, which can easily accomodate fallible perception. In Marr's theory, vision is a multi-layered process. The different layers have filters of (...) different gradation, which makes vision at each of them approximate. On the logical side, our task is therefore twofold - to formalise the layers and the ways in which they may refine each other, and. (shrink)

Two kinds of causal inference rules which are widely used by social scientists are investigated. Two conceptions of causation also widely used are explicated — the INUS and probabilistic conceptions of causation. It is shown that the causal inference rules which link correlation, a kind of partial correlation, and a conception of causation areinvalid. It is concluded anew methodology is required for causal inference.

In this paper individual proofs are integrated into provability logic. Systems of axioms for a logic with operators “A is provable” and “p is a proof of A” are introduced, provided with Kripke semantics and decision procedure. Completeness theorems with respect to the arithmetical interpretation are proved.

A logic of questions and answers exists within the logic of statements, if we make the following identifications (roughly): "Whether" questions are identified with true exclusive disjunctions, and "which" questions are identified with true existential quantifications. The question-and-answer process is interpreted as an information-matching game. The question mark is not needed except as a device of abbreviation. Complete and partial answers can be distinguished and various relations of relevance, independence, and resolution defined.

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

Charles Peirce’s Illustrations of the Logic of Science is an early work in the philosophy of science and the official birthplace of pragmatism. It contains Peirce’s two most influential papers: “The Fixation of Belief” and “How to Make Our Ideas Clear,” as well as discussions on the theory of probability, the ground of induction, the relation between science and religion, and the logic of abduction. Unsatisfied with the result and driven by a constant, almost feverish urge to improve (...) his work, Peirce spent considerable time and effort revising these papers. After the turn of the century these efforts gained significant momentum when Peirce sought to establish his role in the development of pragmatism while distancing himself from the more popular versions that had become current. The present edition brings together the original series as it appeared in Popular Science Monthly and a selection of Peirce’s later revisions, many of which remained hidden in the mass of messy manuscripts that were left behind after his death in 1914. (shrink)

This paper is a presentation of astatus quæstionis, to wit of the problemof the interpretability logic of all reasonablearithmetical theories.We present both the arithmetical side and themodal side of the question.Dedicated to Dick de Jongh on the occasion of his 60th birthday.

The argument is that on moore's analysis of normative language, which is both nonnaturalistic and cognitivistic, one must adopt two distinct criteria of truth. and that any theory which fundamentally assumes two distinct and independent types of truth need not be committed to a logical dualism, that there are some sets of deductive rules homomorphic to the rules of propositional logic which validate certain arguments the premises of which may be true in different ways, and that a system having (...) these properties is a normal alethic modal logic. (shrink)

In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result (...) where negation is included is also discussed). It is proved that BG satisfies the variable sharing property and is Halldén complete, while failing to satisfy the disjunction property. Sound and complete subscripted tableaux are presented for BG. An axiomatization (in the positive language extended with fusion) is presented, which is sound with respect to BG and, with respect to a related ternary relational semantics, complete; the problem of identifying a complete axiomatization for BG itself is left open. Connections between BG and the quasi-relevance logic KR are also discussed. (shrink)

In the course of his researches in electromagnetism and the kinetic theory of gases, James Clerk Maxwell gave some thought to the nature of science itself. His observations in this field are of interest today not only because they are his, but because they are still instructive. Maxwell's views are to be found in the many asides with which he enlivened his scientific papers and treatises and in the various articles and reviews which he prepared for more popular consumption. The (...) topics discussed bear on his own contributions to physics; they include the logic of dynamical explanation, the method of physical analogy, and the perennial question of action at a distance versus contiguity. In the present essay, I wish to discuss Maxwell's views on dynamical explanation. (shrink)

: A continuation of the debate over the intelligibility, and plausibility, of Yolton's reading of Locke's account of perception. Here, the issue turns on the de‐reification of ideas and its implications for the putative axioms of symmetry and transitivity governing the identity of ideas. The issue is illustrated by what Locke says about confused ideas.

We present in this paper an axiomatization of Belnap and Perloff's stit theory (a logic of "seeing to it that") with a single agent. The idea of the proof is to apply the notion of companion sets--the same notion as used in another paper by the author that showed the decidability of stit theory with a single agent and Refref equivalence.

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

J P White; Intelligence and the Logic of the Nature-Nurture Issue, Journal of Philosophy of Education, Volume 8, Issue 1, 30 May 2006, Pages 30–51, https://doi.