We present a faithful axiomatization of von Mises' notion of a random sequence, using an abstract independence relation. A byproduct is a quantifier elimination theorem for Friedman's "almost all" quantifier in terms of this independence relation.

In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s theory of rest (...) and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017). The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises. (shrink)

We present a critical discussion of the claim (most forcefully propounded by Chaitin) that algorithmic information theory sheds new light on Godel's first incompleteness theorem.

We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.

Van Lambalgen's Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen's Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi'}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$ . We also prove that there exists A such (...) that, for each n , the real $\Omega^A_M$ is $\textrm{high}_n$ for some universal Turing machine M by using the extended van Lambalgen's Theorem. (shrink)

Although Kant (1998) envisaged a prominent role for logic in the argumentative structure of his Critique of Pure Reason, logicians and philosophers have generally judged Kantgeneralformaltranscendental logics is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first-order logic. The main technical application of the formalism developed here is a formal proof that Kants logic is after all a distinguished subsystem of first-order logic, namely what (...) is known as geometric logic. (shrink)

We sketch four applications of Marr's levels‐of‐analysis methodology to the relations between logic and experimental data in the cognitive neuroscience of language and reasoning. The first part of the paper illustrates the explanatory power of computational level theories based on logic. We show that a Bayesian treatment of the suppression task in reasoning with conditionals is ruled out by EEG data, supporting instead an analysis based on defeasible logic. Further, we describe how results from an EEG study on temporal prepositions (...) can be reanalyzed using formal semantics, addressing a potential confound. The second part of the article demonstrates the predictive power of logical theories drawing on EEG data on processing progressive constructions and on behavioral data on conditional reasoning in people with autism. Logical theories can constrain processing hypotheses all the way down to neurophysiology, and conversely neuroscience data can guide the selection of alternative computational level models of cognition. (shrink)

We review briefly the attempts to define random sequences. These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence; the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests.

Gaisi Takeuti has recently proposed a new operation on orthomodular lattices L, ⫫: $\scr{P}\rightarrow L$ . The properties of ⫫ suggest that the value of ⫫ $$ corresponds to the degree in which the elements of A behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular lattices L and the existence of two-valued homomorphisms on L.

In this paper we provide a mathematical model of Kant’s temporal continuum that yields formal correlates for Kant’s informal treatment of this concept in theCritique of Pure Reasonand in other works of his critical period. We show that the formal model satisfies Kant’s synthetic a priori principles for time and that it even illuminates what “faculties and functions” must be in place, as “conditions for the possibility of experience”, for time to satisfy such principles. We then present a mathematically precise (...) account of Kant’s transcendental theory of time—the most precise account to date.Moreover, we show that the Kantian continuum which we obtain has some affinities with the Brouwerian continuum but that it also has “infinitesimal intervals” consisting of nilpotent infinitesimals; these allow us to capture Kant’s theory of rest and motion in theMetaphysical Foundations of Natural Science.While our focus is on Kant’s theory of time the material in this paper is more generally relevant for the problem of developing a rigorous theory of the phenomenological continuum, in the tradition of Whitehead, Russell, and Weyl among others. (shrink)

We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner (...) sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers. (shrink)

Interpretation is the process whereby a hearer reasons to an interpretation of a speaker's discourse. The hearer normally adopts a credulous attitude to the discourse, at least for the purposes of interpreting it. That is to say the hearer tries to accommodate the truth of all the speaker's utterances in deriving an intended model. We present a nonmonotonic logical model of this process which defines unique minimal preferred models and efficiently simulates a kind of closed-world reasoning of particular interest for (...) human cognition. Byrne's “suppression” data (Byrne, 1989) are used to illustrate how variants on this logic can capture and motivate subtly different interpretative stances which different subjects adopt, thus indicating where more fine-grained empirical data are required to understand what subjects are doing in this task. We then show that this logical competence model can be implemented in spreading activation network models. A one pass process interprets the textual input by constructing a network which then computes minimal preferred models for (3-valued) valuations of the set of propositions of the text. The neural implementation distinguishes easy forward reasoning from more complex backward reasoning in a way that may be useful in explaining directionality in human reasoning. (shrink)

Sieg has proposed axioms for computability whose models can be reduced to Turing machines. This lecture will investigate to what extent these axioms hold for reasoning. In particular we focus on the requirement that the configurations that a computing agent (whether human or machine) operates on must be ’immediately recognisable’. If one thinks of reasoning as derivation in a calculus, this requirement is satisfied; but even in contexts which are only slightly less formal, the requirement cannot be met. Our main (...) example will be the Wason selection task, a propositional reasoning task in which in a typical (undergraduate) subject group only around 5% arrive at the answer dictated by classical logic. The instructions for this task (as well as other standard tasks in the psychology of reasoning, such as syllogisms) do not contain any ’immediately recognisable’ configurations. The subject must try to find an interpretation of the task by making the various elements in the instructions cohere, in effect solving a difficult constraint satisfaction problem, which has no unique solution. The subject has given a complete interpretation of the task if she can formulate the problem posed in the task as a theorem to be proved. The complexity of such theorems can be quite high; e.g. for the propositional Wason selection task the theorem can be in Σ1 3 . This sounds implausible, but we’ll present experimental data confirming this point. (shrink)

This reply to Oaksford and Chater’s ’s critical discussion of our use of logic programming to model and predict patterns of conditional reasoning will frame the dispute in terms of the semantics of the conditional. We begin by outlining some common features of LP and probabilistic conditionals in knowledge-rich reasoning over long-term memory knowledge bases. For both, context determines causal strength; there are inferences from the absence of certain evidence; and both have analogues of the Ramsey test. Some current work (...) shows how a combination of counting defeaters and statistics from network monitoring can provide the information for graded responses from LP reasoning. With this much introduction, we then respond to O&C’s specific criticisms and misunderstandings. (shrink)

We review the various explanations that have been offered toaccount for subjects'' behaviour in Wason ''s famous selection task. Weargue that one element that is lacking is a good understanding ofsubjects'' semantics for the key expressions involved, and anunderstanding of how this semantics is affected by the demands the taskputs upon the subject''s cognitive system. We make novel proposals inthese terms for explaining the major content effects of deonticmaterials. Throughout we illustrate with excerpts from tutorialdialogues which motivate the kinds of (...) analysis proposed. Our long termgoal is an integration of the various insights about conditionalreasoning on offer from different cognitive science methodologies. Thepurpose of this paper is to try to draw the attention of logicians andsemanticists to this area, since we believe that empirical investigationof the cognitive processes involved could benefit from semanticanalyses. (shrink)

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)

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 • to develop logical means to let description vary with such degrees of refinement. The first task is formalised by means of an inverse systems of first order models, with reality appearing as its inverse limit. The second task is formalised by means of so-called conditional quantifiers, a new form of generalised quantification which can best be described as resource bounded quantification. We show that the logic provides for a semantics and pragmatics of direct perception reports. In particular, direct perception reports have a possibly nonveridical, approximative semantics, which becomes veridical only by virtue of our pragmatic expectation that what is perceived would continue to be the case, were we to perceive more accurately. It is a general feature of resource bounded logics that the underlying logics are weak, but that stronger principles can be obtained pragmatically, by strengthening the resource. For the logic of vision this feature is clarified by showing how changes in the resource capture different notions of partiality, and by studying how the perception verb interacts with connectives and quantifiers in different visual contexts. The inference Veridicality, which is now viewed rather as a nonmonotonic inference, is also studied in depth. We end with an attempt to buttress the proposed model by comparing it with suggestions put forward in Cognitive or Conceptual Semantics, in the literature on evidentials, and in Husserl's philosophy of perception. (shrink)

On the intended interpretation of intuitionistic logic, Heyting's Proof Interpretation, a proof of a proposition of the form p -> q consists in a construction method that transforms any possible proof of p into a proof of q. This involves the notion of the totality of all proofs in an essential way, and this interpretation has therefore been objected to on grounds of impredicativity (e.g. Gödel 1933). In fact this hardly ever leads to problems as in proofs of implications usually (...) nothing more is assumed about a proof of the antecedent than that it indeed is one, and this assumption does not require a further grasp of the totality of proofs. The prime example of an intuitionistic theorem that goes beyond that assumption is Brouwer's proof of the 'bar theorem': For every tree x, if x contains a decidable subset of nodes such that every path through the tree meets it (a 'bar'), then there is a well-ordered subtree of x that contains a bar for the whole of x. Instantiated with an arbitrary tree t, this proposition takes the form P(t) -> Q(t). Brouwer's proof of the bar theorem mainly consists in an analysis of the inner structure that a proof of P(t) must have, where proofs are taken to be primarily mental objects. So here Brouwer engages in phenomenological reflection by considering the acts in which we think about bars. From that analysis he obtains the information from which to construct a proof of Q(t). In this talk I will argue that Brouwer circumvents the problem of impredicativity by resorting to a transcendental argument based on phenomenological description, and defend this application by showing how common objections to transcendental arguments do not apply here. Finally, I will indulge in some historical speculation by relating the foregoing considerations to the remarkable change that Gödel's view on the Proof Interpretation underwent between his Yale Lecture (1941) and the Dialectica paper (1958). (shrink)

This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a (...) space-saving lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted. (shrink)

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

This paper presents a new modal logic for ceteris paribus preferences understood in the sense of "all other things being equal". This reading goes back to the seminal work of Von Wright in the early 1960's and has returned in computer science in the 1990' s and in more abstract "dependency logics" today. We show how it differs from ceteris paribus as "all other things being normal", which is used in contexts with preference defeaters. We provide a semantic analysis and (...) several completeness theorems. We show how our system links up with Von Wright's work, and how it applies to game-theoretic solution concepts, to agenda setting in investigation, and to preference change. We finally consider its relation with infinitary modal logics. (shrink)

Compositionality remains effective as an explanation of cases in which processing complexity increases due to syntactic factors only. It falls short of accounting for situations in which complexity arises from interactions with the sentence or discourse context, perceptual cues, and stored knowledge. The idea of compositionality as a methodological principle is appealing, but imputing the complexity to one component of the grammar or another, instead of enriching the notion of composition, is not always an innocuous move, leading to fully equivalent (...) theories. Compositionality sets an upper bound on the degree of informational encapsulation that can be posited by modular or component-based theories of language: simple composition ties in with a strongly modular take on meaning assembly, which is seen as sealed off from information streams other than the lexicon and the syntax. (shrink)

I started out as a student of physics, hard-working, interested, but alas, not ‘in love’ with my subject. Then logic struck, and having become interested in this subject for various reasons – including the fascinating personality of my first teacher –, I switched after my candidate’s program, to take two master’s degrees, in mathematics and in philosophy. The beauty of mathematics was clear to me at once, with the amazing power, surprising twists, and indeed the music, of abstract arguments. As (...) our professor of Analysis wrote at the time in our study guide “Mathematics is about the delight in the purity of trains of thought”, and oldfashioned though this phrasing sounded in the revolutionary 1960s, it did resonate with me. Then I had the privilege of being taught set-theoretic topology by a group of brilliant students around De Groot, our leading expert around the time, who worked with Moore’s method of discovering a subject for oneself. Topology unfolded from a few definitions and examples to real theorems that we had to prove ourselves – and the take-home exam took sleepless nights, as it included proving some results from scratch which came from a recent dissertation (as it turned out later). Only at the very end did De Groot appear, to give one lecture on Tychonoff’s Theorem where an application was made of the Axiom of Choice, a sacral act only to be performed by tenured full professors. (shrink)

In the late summer of 1998, the authors, a cognitive scientist and a logician, started talking about the relevance of modern mathematical logic to the study of human reasoning, and we have been talking ever since. This book is an interim report of that conversation. It argues that results such as those on the Wason selection task, purportedly showing the irrelevance of formal logic to actual human reasoning, have been widely misinterpreted, mainly because the picture of logic current in psychology (...) and cognitive science is completely mistaken. We aim to give the reader a more accurate picture of mathematical logic and, in doing so, hope to show that logic, properly conceived, is still a very helpful tool in cognitive science. The main thrust of the book is therefore constructive. We give a number of examples in which logical theorizing helps in understanding and modeling observed behavior in reasoning tasks, deviations of that behavior in a psychiatric disorder (autism), and even the roots of that behavior in the evolution of the brain. (shrink)

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of (...) structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation. (shrink)

In a brief but remarkable section of the Inquiry into the Human Mind, Thomas Reid argued that the visual field is governed by principles other than the familiar theorems of Euclid—theorems we would nowadays classify as Riemannian. On the strength of this section, he has been credited by Norman Daniels, R. B. Angell, and others with discovering non-Euclidean geometry over half a century before the mathematicians—sixty years before Lobachevsky and ninety years before Riemann. I believe that Reid does indeed have (...) a fair claim to have discovered a non-Euclidean geometry. However, the real basis of Reid’s geometry of visibles is subtle and easy to misidentify. My main aim in what follows is to make clear what the real basis is, separating it from several fallacious or irrelevant considerations on which Reid may seem to be relying. A secondary aim is to air the worry that Reid’s case for his geometry can succeed only at the cost of compromising the achievement for which Reid is best known—his direct realist theory of perception. (shrink)

In this paper we present a semantic analysis of the imperfective paradox based on the Event Calculus, a planning formalism characterizing a class of models which can be computed by connectionist networks. We report the results of a questionnaire that support the semantic theory and suggest that different aspectual classes of VPs in the progressive give rise to different entailment patterns. Further, a processing model is outlined, combining the semantic analysis with the psycholinguistic principle of immediacy in the framework of (...) recurrent networks. The model is used to derive predictions concerning the electrophysiological correlates of the computations described by the Event Calculus. (shrink)

It is well known that, in classical theory, Liouville's theorem shows that if an ensemble of systems is distributed over a small element of volume in phase space, the ensemble fills a region of equal volume at all later instants of time. In quantum mechanics, the uncertainty principle is associated with the products of the errors in conjugate coordinates and momenta, and such products can be interpreted in terms of volume elements in phase space. Comparison of these two facts (...) leads to the assertion in the interesting volume on “Statistical Mechanics” recently published by J. E. and M. G. Mayer that “The Liouville theorem is essential for the complete understanding of the uncertainty principle. (shrink)

In this paper we present a semantic analysis of the imperfective paradox based on the Event Calculus, a planning formalism characterizing a class of models which can be computed by connectionist networks. We report the results of a questionnaire that support the semantic theory and suggest that different aspectual classes of VPs in the progressive give rise to different entailment patterns. Further, a processing model is outlined, combining the semantic analysis with the psycholinguistic principle of immediacy in the framework of (...) recurrent networks. The model is used to derive predictions concerning the electrophysiological correlates of the computations described by the Event Calculus. (shrink)

In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all trueΠ 2 (...) 0 -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA. (shrink)

In a brief but remarkable section of the Inquiry into the Human Mind, Thomas Reid argued that the visual field is governed by principles other than the familiar theorems of Euclid—theorems we would nowadays classify as Riemannian. On the strength of this section, he has been credited by Norman Daniels, R. B. Angell, and others with discovering non-Euclidean geometry over half a century before the mathematicians—sixty years before Lobachevsky and ninety years before Riemann. I believe that Reid does indeed have (...) a fair claim to have discovered a non-Euclidean geometry. However, the real basis of Reid’s geometry of visibles is subtle and easy to misidentify. My main aim in what follows is to make clear what the real basis is, separating it from several fallacious or irrelevant considerations on which Reid may seem to be relying. A secondary aim is to air the worry that Reid’s case for his geometry can succeed only at the cost of compromising the achievement for which Reid is best known—his direct realist theory of perception. (shrink)

A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles.