The papers that follow were read and discussed at the first Symposium on Exact Philosophy. This conference was held at Montreal on November 4th and 5th, 1971, to celebrate the sesquicentennial of McGill University and establish the Society for Exact Philosophy. The expression 'exact philosophy' is taken to signify mathematical phi losophy, i.e., philosophy done with the explicit help of mathematical logic and mathematics. So far the expression denotes an attitude rather than a fully blown discipline: it (...) intends to convey the intention to try and pro ceed in as exact a manner as we can in formulating and discussing phi losophical problems and theories. The kind of philosophy we wish to practice and promote is disciplined rather than wild, systematic rather than disconnected, and capable of being argued over rather than oracular. We believe that even metaphysics, notoriously riotous, can be subjected to the control of logic and mathematics. Even the history of philosophy, notoriously unsystematic, can benefit from an exact reconstruction of some classical ideas. (shrink)

There has recently been a revival of interest in panpsychism as a theory of consciousness. The hope of the contemporary proponents of panpsychism is that the view enables us to integrate consciousness into our overall theory of reality in a way that avoids the deep difficulties that plague the more conventional options of physicalism on the one hand and dualism on the other. However, panpsychism comes in two forms — strong and weak emergentist — and there are arguments that seem (...) to show that weak emergentist panpsychism faces problems analogous to those of physicalism whilst strong emergentist panpsychism faces problems analogous to those of dualism. In this paper, I will develop a new hybrid of the strong and weak emergentist forms of panpsychism, a view according to which subjects of experience are strongly emergent but their phenomenal properties are weakly emergent. I will argue that this hybrid view manages to avoid the challenges facing both physicalism and dualism, and the analogues of those challenges that seem to undermine standard forms of panpsychism. (shrink)

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than (...) 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (shrink)

We develop an exact truthmaker semantics for permission and obligation. The idea is that with every singular act, we associate a sphere of permissions and a sphere of requirements: the acts that are rendered permissible and the acts that are rendered required by the act. We propose the following clauses for permissions and obligations: -/- - a singular act is an exact truthmaker of Pφ iff every exact truthmaker of φ is in the sphere of permissibility of (...) the act, and -/- - a singular act is an exact truthmaker of Oφ iff some exact truthmaker of φ is in the sphere of requirements of the act. -/- We show that this semantics is hyperintensional, and that it can deal with some of the so-called paradoxes of deontic logic in a natural way. Finally, we give a sound and complete axiomatization of the semantics. (shrink)

The papers that follow were read and discussed at the first Symposium on Exact Philosophy. This conference was held at Montreal on November 4th and 5th, 1971, to celebrate the sesquicentennial of McGill University and establish the Society for Exact Philosophy. The expression 'exact philosophy' is taken to signify mathematical phi losophy, i.e., philosophy done with the explicit help of mathematical logic and mathematics. So far the expression denotes an attitude rather than a fully blown discipline: it (...) intends to convey the intention to try and pro ceed in as exact a manner as we can in formulating and discussing phi losophical problems and theories. The kind of philosophy we wish to practice and promote is disciplined rather than wild, systematic rather than disconnected, and capable of being argued over rather than oracular. We believe that even metaphysics, notoriously riotous, can be subjected to the control of logic and mathematics. Even the history of philosophy, notoriously unsystematic, can benefit from an exact reconstruction of some classical ideas. (shrink)

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers (...) can be generated by a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘non-falsity’ versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case (...) of infectious logics, namely to the case of Deutsch's logic Sfde introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are Setl and Snfl. We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between Setl, Snfl and two well-known systems, strong Kleene three-valued logic K3 and Priest's Logic of Paradox LP. This connection allows us to investigate the characterisation of the entailment relations associated with Setl and Snfl as well as to introduce the notion of ‘infectious analogue’ of a certain logic. We also study implicative extensions of Setl and Snfl and prove soundness and completeness theorems for them as well. (shrink)

I defend, to a certain extent, the traditional view that perceptual indiscriminability is non-transitive. The argument proceeds by considering important recent work by Benj Hellie: Hellie argues that colour perception represents ‘inexactly’, and that this results in violations of the transitivity of colour indiscriminability. I show that Hellie’s argument remains inconclusive, since he does not demonstrate conclusively that colour perception really does represent inexactly. My own argument for the non-transitivity of perceptual indiscriminability uses inexactness instead as one horn of a (...) dilemma: the key idea is that there is a class of perceptual experiences which might plausibly be supposed either to represent inexactly or to represent exactly—but which demonstrate the non-transitivity of perceptual indiscriminability either way. (shrink)

In this paper we propose that the term skill acquisition, as commonly used in traditional psychology, and the philosophy, education, movement science and performance development literatures, has been biased by an organismic asymmetry. In cognitive and experimental psychology, for example, it refers to the establishment of an internal state or representation of an act which is believed to be acquired as a result of learning and task experience. Here we elucidate an ecological perspective which suggests that the term skill acquisition (...) may not refer to an entity but rather to the emergence of an adaptive, functional relationship between an organism and its environment, thus avoiding an inherent organismic asymmetry in theorizing. In this respect, the terms 'skill adaptation' or 'skill attunement' might be more suitable to describe this process. (shrink)

We determine the exact bounds for the length of an arbitrary reduction sequence of a term in the typed λ-calculus with β-, ξ- and η-conversion. There will be two essentially different classifications, one depending on the height and the degree of the term and the other depending on the length and the degree of the term.

We aim to construct exact and explicit solutions to a generalized Bogoyavlensky-Konopelchenko equation through the Maple computer algebra system. The considered nonlinear equation is transformed into a Hirota bilinear form, and symbolic computations are made for solving both the nonlinear equation and the corresponding bilinear equation. A few classes of exact and explicit solutions are generated from different ansätze on solution forms, including traveling wave solutions, two-wave solutions, and polynomial solutions.

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and (...) unify Theorems 1 and 2 of [5], due to the first author. (shrink)

TheK-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists aK-trivial degreedsuch that (...) for all degreesa, bwhich are notK-trivial anda > d, b > dthere exists a degreevwhich is notK-trivial anda > v, b > v. This work sheds light to the question of the definability of theK-trivial degrees in the c.e. degrees. (shrink)

In this new book, Michael Friedman argues that Kant's continuing efforts to find a metaphysics that could provide a foundation for the sciences is of the utmost ...

Physicien de formation, Francois Bastien a enseigne dans de nombreux domaines: mathematiques, electricite, optique, thermodynamique, physique des vibrations, physique des capteurs, acoustique des solides, informatique, electronique numerique et electrotechnique. Il a tenu un blog ou se sont regroupees certaines interrogations sur les sciences exactes. Son livre presente donc des reflexions sur la recherche scientifique. Un changement d'echelle de la communaute scientifique entraine necessairement un bouleversement. L'auteur tend a sortir de la doctrine hors de l'ecole point de salut qui freine la (...) promotion sociale, en proposant par exemple un Musee de la Science inconnue. Il invite egalement a lutter contre les effets de la multiplication des publications scientifiques, en faisant naitre des journaux de critiques scientifiques comme il en existe pour la litterature. Par ailleurs, la prise en compte des limites de la science, d'un point de vue philosophique et pratique, est discutee. (shrink)

Gary Becker and others have done important work to broaden the content of self interest, but have not departed from seeing rationality in terms of the exclusive pursuit of self-interest. One reason why committed behavior is important is that a person can have good reason to pursue objectives other than self interest maximization (no matter how broadly it is construed). Indeed, one can also follow rules of behavior that go beyond the pursuit of one's own goals, even if the goals (...) include non-self-interested concerns. By living in a society, one develops possible reasons for considering other people's goals as well, which takes one beyond an exclusive concentration on one's own goals, not to mention the single-minded pursuit of one's own self interest. The recognition of other people's goals may be a part of rational thought. If rational behavior may depart from the relentless pursuit of one's own goals, commitment has to be important in a theory of rationality. Furthermore, seeing the role of commitment in human behavior can have explanatory importance in allowing us to understand behavior patterns that are hard to fit into the narrow format of contemporary rational choice theory. Commitment is, thus, important both for practical reason and for causal explanation. (shrink)

Cognitive developmentalists have had a long-standing interest in neurodevelopmental conditions, such as autism. This is not only out of a desire to understand the causes of such atypical development, in order to advance medical science and develop interventions. It is also because studying the processes that cause atypicality can sometimes throw light on typical development. It is this two-way influence that characterises the field of developmental psychopathology. In this chapter, we focus on autism. We bring out this interaction between what (...) we now understand about autistic cognition, and how this has helped us understand ‘normality’. (shrink)

It is widely agreed among contemporary philosophers of mind that science leaves us with an ‘explanatory gap’—that even after we know everything that science can tell us about the conscious mind and the brain, their relationship still remains mysterious. I argue that this agreed view is quite mistaken. The feeling of a ‘explanatory gap’ arises only because we cannot stop ourselves thinking about the mind–brain relation in a dualist way.

The present paper attempts to provide an exact truthmaker semantical analysis of modalized propositions. According to the present proposal, an exact truthmaker for “Necessarily _P_” is a state that bans every exact truthmaker for “Not _P_”, and an exact truthmaker for “Possibly _P_” is a state that allows an exact truthmaker for _P_. Based on this proposal, a formal semantics will be developed; and the soundness and completeness results for a well-known family of the systems (...) of normal modal propositional logic will be established. It shall be seen that the present analysis offers an exactification of the standard Kripke semantics in the sense that it analyzes the accessibility relation between possible worlds in terms of the banning and allowing relations between the constituent states, and thereby gives an account of “truth at a possible world” in terms of exact truthmaking. (shrink)

We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This (...) had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analog of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories. (shrink)

Our lives are full of inexactitude. We say a person is tall or an action is just without the precision of measurement on a dial. In this engaging account, Kees van Deemter explores vagueness, cutting across areas such as language, mathematical logic, and computing. He considers why vagueness is inherent, and why it is important in how we function.

Proof uses forcing on perfect trees for 2-quantifier sentences in the language of arithmetic. The result extends to exact pairs for the hyperarithmetic degrees.

The last two decades have seen a rising interest in (a) the notion of a scientific phenomenon as distinct from theories and data, and (b) the intricacies of experimentally producing and stabilizing phenomena. This paper develops an analysis of the stabilization of phenomena that integrates two aspects that have largely been treated separately in the literature: one concerns the skills required for empirical work; the other concerns the strategies by which claims about phenomena are validated. I argue that in order (...) to make sense of the process of stabilization, we need to distinguish between two types of phenomena: phenomena as patterns in the data ( surface regularities ) and phenomena as underlying (or hidden ) regularities. I show that the epistemic relationships that data bear to each of these types of phenomena are different: Data patterns are instantiated by individual data, whereas underlying regularities are indicated by individual data, insofar as they instantiate a data pattern. Drawing on an example from memory research, I argue that neither of these two kinds of phenomenon can be stabilized in isolation. I conclude that what is stabilized when phenomena are stabilized is the fit between surface regularities and hidden regularities. (shrink)

We take seriously the idea that paradoxes come in quantifiable degree by offering an exact measure of paradox. We consider three factors relevant to the degree of paradox, which are a function of the degree of belief in each of the individual propositions in the paradox set and the degree of belief in the set as a whole. We illustrate the proposal with a particular measure, and conclude the discussion with some critical remarks.

A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: see text]-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.

On the basis of the exact solutions to the non-stationary spherically symmetric Einstein and Maxwell equations for dust matter and radial electromagnetic field, a model of a wormhole with the pulsating in time inner world and two static throats has been developed. It has been shown that such a wormhole with an arbitrary radius of the Gaussian curvature can connect both two different asymptotically flat space-times and two regions of the selfsame space-time. The problem of the fulfilment of the (...) energy conditions in this wormhole has been investigated, as well as the problem of its traversability investigation has been set. (shrink)

There have been two parallel views regarding the role of voting in deliberation. The first is that deliberation before the fabrication of balloting was completely devoid of voting. The second is that voting is, not just part of deliberation, but is standard to deliberation. I argue in this article that neither of these views is correct. Implicit voting has always existed across time and space but only as a last resort in the event of a failure of natural unanimity. What (...) is relatively modern is the establishment of what I call explicit voting, namely, balloting, outside deliberation and often without deliberation. I also distinguish between natural and artificial unanimities, and clarify that artificial unanimities are products of implicit voting. I demonstrate these clarifications with some examples of deliberation. I deploy these clarifications to rid a certain debate of confusion regarding the precise role of voting in consensual deliberation. (shrink)

In an attempt to give a unified account of common properties of various resource bounded reducibilities, we introduce conditions on a binary relation ≤r between subsets of the natural numbers, where ≤r is meant as a resource bounded reducibility. The conditions are a formalization of basic features shared by most resource bounded reducibilities which can be found in the literature. As our main technical result, we show that these conditions imply a result about exact pairs which has been previously (...) shown by Ambos-Spies [2] in a setting of polynomial time bounds: given some recursively presentable ≤r-ideal I and some recursive ≤r-hard set B for I which is not contained in I, there is some recursive set C, where B and C are an exact pair for I, that is, I is equal to the intersection of the lower ≤r-cones of B and C, where C is not in I. In particular, if the relation ≤r is in addition transitive and there are least sets, then every recursive set which is not in the least degree is half of a minimal pair of recursive sets. (shrink)

We represent continuous functions on compact intervals by sequences of functions defined on finite sets of rational numbers. We call this an exact representation. This enables us to calculate the values of the function arbitrarily exactly, without roundoff errors. As an application we develop a procedure to transfer an exact representation of an increasing function into an exact representation of the corresponding inverse function.

In his paper ‘Pretending’ J. L. Austin says that philosophers have exaggerated the scope and distorted the meaning of pretending, and the clarification of this notion has a place in the ‘long-term project of classifying and clarifying all possible ways and varieties of not exactly doing things , which has to be carried through if we are ever to understand properly what doing things is.’.

This chapter analyzes the explanation of resemblance provided by the twelfth-century Latin philosopher Gilbert of Poitiers. As Gilbert holds that everything that exists is particular and rejects immanent universals, he is in need of an explanation for the resemblance of co-specific properties. His solution stems from a complex consideration of exact resemblance, which he calls ‘conformitas,’ based upon a thorough reflection about properties, as for him, only properties and not individuals can be exactly similar. In order to discuss his (...) theory of resemblance, the chapter first offers a reconstruction of the main tenets of Gilbert’s particularist ontology; it argues that Gilbert can adequately be categorized as a trope theorist. Gilbert analyzes the metaphysical constitution of an individual as a set of hierarchically organized co-present tropes, starting from an essential nucleus. For Gilbert, a universal is a collection of properties gathered on the basis of resemblance. At the end of the chapter, the possible role of divine ideas in his explanation of resemblance is discussed. (shrink)

(No abstract is available for this citation).

The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for the (...) infinitary languages $\mathcal {L}_{\text {Ord},\omega }$, allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega,\omega }$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$. Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$. (shrink)

Given a locale L and any set-indexed family of continuous mappings , fi:L→Li with compact and completely regular co-domain, a compactification η:L→Lγ of L is constructed enjoying the following extension property: for every a unique continuous mapping exists such that . Considered in ordinary set theory, this compactification also enjoys certain convenient weight limitations.Stone–Čech compactification is obtained as a particular case of this construction in those settings in which the class of [0,1]-valued continuous mappings is a set for all L. (...) This will follow by the proof that–also in the point-free context–a compactification that allows for the extension of [0,1]-valued mappings suffices for deriving the full reflection.A constructive proof that the class is a set whenever L is locally compact and L′ is set-presented and regular is also obtained; together with the described compactification, this makes it possible to characterize the class of locales for which Stone–Čech compactification can be defined constructively. (shrink)

We present a novel generative model for natural language tree structures in which semantic (lexical dependency) and syntactic (PCFG) structures are scored with separate models. This factorization provides conceptual simplicity, straightforward opportunities for separately improving the component models, and a level of performance comparable to similar, non-factored models. Most importantly, unlike other modern parsing models, the factored model admits an extremely effective A* parsing algorithm, which enables efficient, exact inference.

Gary Becker and others have done important work to broaden the content of self interest, but have not departed from seeing rationality in terms of the exclusive pursuit of self-interest. One reason why committed behavior is important is that a person can have good reason to pursue objectives other than self interest maximization (no matter how broadly it is construed). Indeed, one can also follow rules of behavior that go beyond the pursuit of one's own goals, even if the goals (...) include non-self-interested concerns. By living in a society, one develops possible reasons for considering other people's goals as well, which takes one beyond an exclusive concentration on one's own goals, not to mention the single-minded pursuit of one's own self interest. The recognition of other people's goals may be a part of rational thought. If rational behavior may depart from the relentless pursuit of one's own goals, commitment has to be important in a theory of rationality. Furthermore, seeing the role of commitment in human behavior can have explanatory importance in allowing us to understand behavior patterns that are hard to fit into the narrow format of contemporary rational choice theory. Commitment is, thus, important both for practical reason and for causal explanation. Footnotes1 Paper presented at a Workshop on Rationality and Commitment at the University of St. Gallen, May 13–15, 2004. (shrink)

In this paper we prove general exact unprovability results that show how a threshold between provability and unprovability of a finite well-quasi-orderedness assertion of a combinatorial class is transformed by the sequence-construction, multiset-construction, cycle-construction and labeled-tree-construction. Provability proofs use the asymptotic pigeonhole principle, unprovability proofs use Weiermann-style compression techniques and results from analytic combinatorics.

The paper seeks to contribute to clarifying agnotology as an ‘epistemic strategy’, conceived as ‘epistemically damaging and hurt[ing] the production of knowledge’. My general claim is that the grammar of intentions ‘embedded’ in agnotological arguments is often not considered accurately. I use considerations from the philosophy of action as a theoretical framework to make more explicit what is implied in agnogenetic manoeuvres. Agnotology, as a ‘theory’ about epistemic states, in particular knowledge and ignorance, would be seriously incomplete without that component. (...) The following can thus be read as a contribution to an analysis of the presuppositions of the strategic variant of Agnotology. My first claim is that the more common objections to the introduction of intentions are in no way definitive. My second, more specific, claim is that we need a room, in our conceptual toolbox, for ‘anti-epistemic intentions’, which play a key role in agnotological arguments. (shrink)

We present a novel generative model for natural language tree structures in which semantic (lexical dependency) and syntactic (PCFG) structures are scored with separate models. This factorization provides conceptual simplicity, straightforward opportunities for separately improving the component models, and a level of performance comparable to similar, non-factored models. Most importantly, unlike other modern parsing models, the factored model admits an extremely effective A* parsing algorithm, which enables efficient, exact inference.

Confucianism’s emphasis on filial piety is both a hallmark of its approach to ethics and a source of concern. Critics charge that filial piety’s extreme partialism corrupts Chinese society and should therefore be expunged from the tradition. Are the critics correct? In this article, we outline the criticism and note its persistence over the last century. We then evaluate data from the empirical study of corruption to see whether they support the claim that partialism corrupts. Finally, we report some recent (...) experimental work done with colleagues testing the claim that filial piety is associated with tolerance of corruption in Chinese societies. The results suggest that the critics are on to something. However, partialism (or kin affection) is not a cause of concern. Instead, authoritarianism (another aspect of filial piety) is associated with tolerance of corruption. We conclude that critics should reformulate their criticisms if they seek to combat corruption effectively. (shrink)