We give proofs of the effective monotone interpolation property for the system of modal logic K, and others, and the system IL of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 72–90]; here, we give considerably simplified (...) proofs, as well as some generalisations. (shrink)
We give an exponential lower bound on the number of proof-lines in intuitionistic propositional logic, IL, axiomatised in the usual Frege-style fashion; i.e., we give an example of IL-tautologies A1,A2,… s.t. every IL-proof of Ai must have a number of proof-lines exponential in terms of the size of Ai. We show that the results do not apply to the system of classical logic and we obtain an exponential speed-up between classical and intuitionistic logic.
We give an exponential lower bound on number of proof-lines in the proof system K of modal logic, i.e., we give an example of K-tautologies ψ₁, ψ₂,... s.t. every K-proof of ψi must have a number of proof-lines exponential in terms of the size of ψi. The result extends, for the same sequence of K-tautologies, to the systems K4, Gödel—Löb's logic, S and S4. We also determine some speed-up relations between different systems of modal logic on formulas of modal-depth one.
We give four examples of theories in which Kreisel's Conjecture is false: (1) the theory PA(-) obtained by adding a function symbol minus, '−', to the language of PA, and the axiom ∀x∀y∀z (x − y = z) ≡ (x = y + z ⋁ (x < y ⋀ z = 0)); (2) the theory T of integers; (3) the theory PA(q) obtained by adding a function symbol q (of arity ≥ 1) to PA, assuming nothing about q; (4) the (...) theory PA(N) containing a unary predicate N(x) meaning 'x is a natural number'. In Section 6 we suggest a counterexample to the so called Sharpened Kreisel's Conjecture. (shrink)
We prove that Kreisel's Conjecture is true, if Peano arithmetic is axiomatised using minimality principle and axioms of identity (theory $PA_M $ )-The result is independent on the choice of language of $PA_M $ . We also show that if infinitely many instances of A(x) are provable in a bounded number of steps in $PA_M $ then there existe k ∈ ω s. t. $PA_M $ ┤ ∀x > k̄ A(x). The results imply that $PA_M $ does not prove scheme (...) of induction or identity schemes in a bounded number of steps. (shrink)
Rédei and Gyenis recently displayed strong constraints of Bayesian learning. However, they also presented a positive result for Bayesianism. Despite the limited significance of this positive result, I find it useful to discuss its two possible strengthenings to present new results and open new questions about the limits of Bayesianism. First, I will show that one cannot strengthen the positive result by restricting the evidence to so-called “certain evidence”. Secondly, strengthening the result by restricting the partitions—as parts of one’s evidence—to (...) Jeffrey-independent partitions requires additional constraints on one’s evidence to preserve its commutativity. So, my results provide additional grounds for caution and support for the limitations of Bayesian learning. (shrink)
Traxler, Pickering, and Clifton found that ambiguous sentences are read faster than their unambiguous counterparts. This so-called ambiguity advantage has presented a major challenge to classical theories of human sentence comprehension because its most prominent explanation, in the form of the unrestricted race model, assumes that parsing is non-deterministic. Recently, Swets, Desmet, Clifton, and Ferreira have challenged the URM. They argue that readers strategically underspecify the representation of ambiguous sentences to save time, unless disambiguation is required by task demands. When (...) disambiguation is required, however, readers assign sentences full structure—and Swets et al. provide experimental evidence to this end. On the basis of their findings, they argue against the URM and in favor of a model of task-dependent sentence comprehension. We show through simulations that the Swets et al. data do not constitute evidence for task-dependent parsing because they can be explained by the URM. However, we provide decisive evidence from a German self-paced reading study consistent with Swets et al.'s general claim about task-dependent parsing. Specifically, we show that under certain conditions, ambiguous sentences can be read more slowly than their unambiguous counterparts, suggesting that the parser may create several parses, when required. Finally, we present the first quantitative model of task-driven disambiguation that subsumes the URM, and we show that it can explain both Swets et al.'s results and our findings. (shrink)
I. The framework. 1, Aristotle's project and methods. 2, The perceptual capacity of the soul. 3, The sensory apparatus. 4, The common sense and the related capacities -- II. The terminology. 1, Overlooked occurrences of the phrase 'common sense'. 2, De anima III.1 425a27. 3, De partibus animalium IV.10 686a31. 4, De memoria et reminiscentia 1 450a10. 5, De anima III.7 431b5. 6, Conclusions on the terminology -- III. Functions of the common sense. 1, Simultaneous perception and cross-modal binding. 2, (...) Perceptual discrimination. 3, Waking, sleep, and control of the senses. 4, Perceiving that what we see and hear, and monitoring of the senses. 5, Other roles of the common sense -- Conclusion. (shrink)
Roman Catholic and Eastern Orthodox theologians claim that the unique event of Christ’s sacrifice on Calvary is present in Eucharistic liturgies. A popular explanatory strategy for this miraculous presence suggests that due to its supernatural character the Eucharist “conquers time,” transcends its boundaries, and allows for temporal coincidence of two chronologically distant events. I discuss the four main approaches within this strategy that can be discovered in contemporary theological writings. The first approach implies a time travel of the Calvary event. (...) The second suggests the time travel of Eucharistic participants. The third eliminates the chronological distance by relocating one of the events into a timeless reality. The fourth assumes multilocation of the event across time. I argue that each of these approaches is untenable on philosophical or theological grounds. (shrink)
Pavel Florensky, a Russian theologian, philosopher, and mathematician, argued that the religious discourse is essentially contradictory and put forward the idea of the logical theory of antinomies. Recently his views raised interesting discussions among logicians who consider him a forerunner of many non-classical logics. In this paper I discuss four interpretations of Florensky’s views: paraconsistent, L-contradictory, non-monotonic and rhetorical. In conclusion I argue for the integral interpretation which unites these four approaches.
The article offers a rigorous explication of the intuitive notion of verisimilitude, I.E., Of the distance of a theory from the truth. The proposal is defended against charges of material inadequacy made by popper, Niniluoto, And miller.
In this paper we argue that Aristotle operates with a particular theoretical model in his explanation of animal locomotion, what we call the ‘centralized incoming and outgoing motions’ model. We show how the model accommodates more complex cases of animal motion and how it allows Aristotle to preserve the intuition that animals are self-movers, without jeopardizing his arguments for the eternity of motion and the necessary existence of one eternal unmoved mover in Physics VIII. The CIOM model helps to elucidate (...) Aristotle’s two central yet problematic claims, namely that the soul is the efficient cause of animal motion and that it is the internal supporting-point necessary for animal motion. Moreover, the CIOM model helps us to explain the difference between voluntary, involuntary and non-voluntary motions, and to square Aristotle’s cardiocentrism with his hylomorphism, but also, more generally, it provides an interesting way of thinking about the place of intentionality in the causal structure of the world. (shrink)
An investigation of the views on space and time of the Russian polymath Pavel Florensky. After a brief account of his life, I study Florensky’s conception of time in The Meaning of Idealism, where he first confronts Einstein’s theory of special relativity, comparing it to Plato’s metaphor of the Cave and Goethe’s myth of the Mothers. Later, in his Analysis of spatiality and time, Florensky speaks of a person’s biography as a four-dimensional unity, in which the temporal coordinate is (...) examined in sections. In On the Imaginaries in Geometry, Florensky argues that the speed of light is not, as in Relativity, an absolute speed limit in the universe. When bodies approach and then surpass the speed of light, they are transformed into unextended, eternal Platonic forms. Beyond this point, time runs in reverse, effects precede their causes, and efficient causality is transformed into final or teleological causality, a concept on which Florensky elaborates in his Iconostasis. Florensky thus transformed the findings of Einsteinian relativity in order to make room for Plato’s intelligible Ideas, the Aristotelian distinction between a changing realm of earth and the immutable realm of the heavens, and the notion of teleology or final causation. His notion that man can approximate God’s vision of past, present and future all at once, as if from above, is reminiscent of Boethius’ ideas. (shrink)
My book is about the human creativity being a source of progress, and cycling of evolution caused by platitude and triviality of once high-reaching idealism. In essence the book presents an original perception of human history, based on Christian values as vital coordinates system. I hope this book will revive the interest to the Russian school of thoughts and to humanism in general.
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond NP ≠ coNP. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to be (...) special cases of such general statements and we want to formalize and fully understand these statements. Roughly speaking, we are trying to connect syntactic complexity, by which we mean the complexity of sentences and strengths of the theories in which they are provable, with the semantic concept of complexity of the computational problems represented by these sentences. -/- We have introduced the most fundamental conjectures in our earlier works. Our aim in this article is to present them in a more systematic way, along with several new conjectures, and prove new connections between them and some other statements studied before. (shrink)
“Eco-labels” are an increasingly important form of private regulation for sustainability in areas such as carbon emissions, water consumption, ethical sourcing, or organic produce. The growing interest and popularity of eco-labels has also been coupled with growing concerns about their credibility, in part because the standard-setting and conformity assessment practices that eco-labels adopt exhibit striking differences. In this paper, we assess which assurance practices contribute to eco-labels being perceived as better governed, in the eyes of experts as well as the (...) media. Unlike previous studies, which are mostly conceptual, qualitative, or focused on one or few eco-labels, we study a large set of eco-labels, combining data from three different sources. Our findings suggest that experts and media are primarily concerned about “re-assurance” practices, looking for one or preferably multiple layers of “re-assurance” that independent parties are overseeing the eco-label and the firms certified under it. (shrink)
The success of the atheistic hiddenness argument depends on the “consciousness constraint” it imposes on the divine-human loving relationship: namely, that this relationship requires human conscious awareness of being in the relationship with God. I challenge the truth of this proposition by introducing the concept of a physical relationship with God that is not subject to this constraint. I argue, first, that a physical relationship with God is metaphysically possible; second, that its plausibility is supported by natural theology; and third, (...) that a perfectly loving God would prefer physical relationships with human beings over consciousness-constrained relationships, because a perfectly loving God would prefer to preserve the integrity of human freedom of participation and allow inclusion of all people regardless of their natural cognitive capabilities. I also offer an interpretation of apparent divine hiddenness in the light of the idea of God’s openness for physical relationships. (shrink)
We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.
Aristotle's notion of experience plays an important role in his epistemology as the link between perception and memory on the one side, and higher cognitive capacities on the other side. However, Aristotle does not say much about it, and what he does say seems inconsistent. Notably, some passages suggest that it is a non-rational capacity, others that it is a rational capacity and that it provides the principles of science. This paper presents a unitary account of experience. It explains how (...) experience grows from perception and memory into a rational capacity, and in what way it provides the principles. (shrink)
It is well known that the manner in which a definitely descriptive term contributes to the meaning of a sentence depends on the place the term occupies in the sentence. A distinction is accordingly drawn between ordinary contexts and contexts variously termed non-referential, intensional, oblique, or opaque. The aim of the present article is to offer a general account of the phenomenon, based on transparent intensional logic. It turns out that on this approach there is no need to say (as (...) Frege does) that descriptive terms are referentially ambiguous or to deny (as Russell does) that descriptive terms represent self-contained units of meaning. There is also no need to tolerate (as Montague does) exceptions to the Principle of Functionality. The notion of an ordinary (i.e., non-intensional) context is explicated exclusively in terms of logical structure and it is argued that two aspects of ordinariness (termed hospitality and exposure) must be distinguished. (shrink)
Two different types of functional dependencies are compared: dependencies that are functional due to the laws of nature and dependencies that are functional if all involved agents behave rationally. The first type of dependencies was axiomatized by Armstrong. This article gives a formal definition of the second type of functional dependencies in terms of strategic games and describes a sound and complete axiomatization of their properties. The axiomatization is significantly different from the Armstrong’s axioms.
Despite relative neglect in the literature, Kant’s published and unpublished writings in theoretical philosophy reveal a sustained and at times ambivalent effort to come to terms with the problem of miracles. Because they entail a form of supernatural causation that undermines the law-governedness of the order of nature, miracles pose a significant problem for Kant’s metaphysics. I explore in detail Kant’s account of miracles in conjunction with the relevant aspects of his metaphysics of nature in order to establish in what (...) sense miracles are possible, and how they fit into Kant’s architectonic more generally. (shrink)
The paper deals with the semantics of mathematical notation. In arithmetic, for example, the syntactic shape of a formula represents a particular way of specifying, arriving at, or constructing an arithmetical object (that is, a number, a function, or a truth value). A general definition of this sense of "construction" is proposed and compared with related notions, in particular with Frege's concept of "function" and Carnap's concept of "intensional isomorphism." It is argued that constructions constitute the proper subject matter of (...) both logic and mathematics, and that a coherent semantic account of mathematical formulas cannot be given without assuming that they serve as names of constructions. (shrink)
The article offers a rigorous truth condition for subjunctively conditional statements. The theory is framed in the system of transparent intensional logic and takes connections (especially the cause-Effect relation) as basic. Counterexamples are given to rival theories based on the notion of world similarity.