We show, under the assumption that factoring is hard, that a model of PV exists in which the polynomial hierarchy does not collapse to the linear hierarchy; that a model of S21 exists in which NP is not in the second level of the linear hierarchy; and that a model of S21 exists in which the polynomial hierarchy collapses to the linear hierarchy. Our methods are model-theoretic. We use the assumption about factoring to get a model in which the weak (...) pigeonhole principle fails in a certain way, and then work with this failure to obtain our results. As a corollary of one of the proofs, we also show that in S21 the failure of WPHP (for Σb1 definable relations) implies that the strict version of PH does not collapse to a finite level. (shrink)

Authenticity and theatricality: Andrzej Falkiewicz’s model of homo theatralis in the context of Helmuth Plessner’s criticism of radicalismThe aim of this work is to confront Andrzej Falkiewicz’s model of “homo theatralis” and—to a lesser degree—“being-as-metaphor” with Helmuth Plessner’s criticism of radicalism. In the first step the author shows generally the problem of “self” and moral sources of identity. In the second step the author analyzes the concept of “homo theatralis” “theater man” and “being-as-metaphor” by the Polish writer, critic, and philosopher (...) Andrzej Falkiewicz—Leszek Nowak’s cooperator. In the third step the author is trying to connect Falkiewicz’s ideas with Helmuth Plessner’s model of radicalism connected with his concept of commonwealth and society and difference between them. Then the author answers the question of the possibility of freedom in building our individual moral horizon. (shrink)

Modifying the methods of Z. Adamowicz's paper Herbrand consistency and bounded arithmetic [3] we show that there exists a number n such that ⋃m Sm (the union of the bounded arithmetic theories Sm) does not prove the Herbrand consistency of the finitely axiomatizable theory $S_{3}^{n}$.

THE CONCEPT OF MOTIVATION AND THE UNITY OF A HUMAN SUBJECT IN SCHOPENHAUER’S “ON THE FREEDOM OF THE WILL” In the following article I am going to present the detailed analysis of the concept of motivation in philosophy of Arthur Schopenhauer. It will be particularly important for me to present this concept in relation to the problem of identity of two aspects of human subject: cognitive and volitional. Moreover I will show that there are at least several important problems and (...) obscurities connected with Shopenhauer’s standpoint even though his concept of action is unquestionably close to our “natural” and “common sense” beliefs. (shrink)

We show that the arithmetical theory -INDx5, formalized in the language of Buss, i.e. with x/2 but without the MSP function x/2y, does not prove that every nontrivial divisor of a power of 2 is even. It follows that this theory proves neither NP=coNP nor.

We study the long-standing open problem of giving$\forall {\rm{\Sigma }}_1^b$separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the$\forall {\rm{\Sigma }}_1^b$Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole (...) principle for FPNPfunctions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of$T_2^1$augmented with the surjective weak pigeonhole principle for polynomial time functions. (shrink)

D’Aquino et al. (J Symb Log 75(1):1–11, 2010) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by IΔ0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss’ bounded arithmetic: PV or $${\Sigma^b_1-IND^{|x|_k}}$$. It also holds for IΔ0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it (...) fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality. (shrink)

We prove that: • if there is a model of I∆₀ + ¬ exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then I∆₀ + ¬ exp +¬BΣ₁ is consistent, • there is a model of I∆₀ Ω₁ + ¬ exp with cofinal Σ₁-definable elements, both a Σ₂ and a ∏₂ truth definition for Σ₁ sentences, and for each n > 2, a Σ n truth definition for Σ n sentences. The latter result is obtained by (...) constructing a model with a recursive truth-preserving translation of Σ₁ sentences into boolean combinations of $\exists \sum {\begin{array}{*{20}{c}} h \\ 0 \\ \end{array} } $ sentences. We also present an old but previously unpublished proof of the consistency of I∆₀ + ¬ exp + ¬BΣ₁ under the assumption that the size parameter in Lessan's ∆₀ universal formula is optimal. We then discuss a possible reason why proving the consistency of I∆₀ + ¬ exp + ¬BΣ₁ unconditionally has turned out to be so difficult. (shrink)

The bookModeling Reality covers a wide range of fascinating subjects, accessible to anyone who wants to learn about the use of computer modeling to solve a diverse range of problems, but who does not possess a specialized training in mathematics or computer science. The material presented is pitched at the level of high-school graduates, even though it covers some advanced topics (cellular automata, Shannon's measure of information, deterministic chaos, fractals, game theory, neural networks, genetic algorithms, and Turing machines). These advanced (...) topics are explained in terms of well known simple concepts: Cellular automata - Game of Life, Shannon's formula - Game of twenty questions, Game theory - Television quiz, etc. The book is unique in explaining in a straightforward, yet complete, fashion many important ideas, related to various models of reality and their applications. Twenty-five programs, written especially for this book, are provided on an accompanying CD. They greatly enhance its pedagogical value and make learning of even the more complex topics an enjoyable pleasure. (shrink)

The goal of the paper is to expound a notion of conceptual apparatus, by revealing structure and functions of objects it designates. The notion has been developed by Kazimierz Ajdukiewicz in the mid of 30ties, within a standpoint of radical conventionalism. According to its proponents, a picture of world one bears is not stated directly by the data of experience, but depends on conceptual apparatus chosen. Ajdukiewicz defines the notion consequently, referring it to a set of all meanings attributed to (...) expressions in a closed and compact language. Nevertheless, it turns out to be vague, since it inherits the vagueness from presupposed categories, especially from ambiguously used category of meaning. (shrink)

Michael Polanyi’s philosophical ideas are interpret in various ways worldwide. In Poland the name remains (barely) listed among such philosophers of science as Thomas Kuhn and Paul Feyerabend, whereas English or German authors regard him rather as a theorist of knowledge and place aside Gilbert Ryle, Charles Sanders Peirce, Hans-Georg Gadamer or Maurice Merleau-Ponty. The aim of the paper is to describe typical ways of how Polanyi’s ideas are being currently received and to report his main statements. It is proceeded (...) in four steps. After short biography of the author briefed in first point, second one sums up the content of all his main works, depicting thereby the evolution of his views. In third point receptions published in Polish, English and German literature are screened and discussed, focusing on issues of rationality, personal knowledge, tacit knowledge and tacit knowing. Fourth point states a problem of alleged holism and eclecticism as interpretative characteristics of Polanyi’s philosophical views. (shrink)

How the idea of tacit knowledge is being understood typically? The article reconstructs interpretations in context of three different disciplines: (1) linguistics, (2) cognitive psychology and (3) sociology of knowledge. Furthermore, it proposes (4) definitional criteria for a general notion of tacit knowledge.

Tacit premises of science constitute researcher’s cognitive scheme, i.e. a set of a priori conditions of knowledge acquisition and application. Couple of assumptions make Polanyi’s idea considerably different than Kantian or behavioural or structural interpretations of cognitive scheme. He sees it more in hermeneutical or habitual terms — as system of (a) skills (dispositions to act), which (b) defines the level of competence; (c) cannot be verbally articulated; (d) is embodied (and hence unaware); (e) innate or acquired through practice — (...) in master-pupil relations; (f) undergoes constant modifications when applied; (g) conditions both theoretical actions (eg. categorisation, acts of assertion), as well as practical (manual skills, acts of perception). (shrink)

We study the first-order consequences of Ramsey’s Theorem fork-colourings ofn-tuples, for fixed$n, k \ge 2$, over the relatively weak second-order arithmetic theory$\mathrm {RCA}^*_0$. Using the Chong–Mourad coding lemma, we show that in a model of$\mathrm {RCA}^*_0$that does not satisfy$\Sigma ^0_1$induction,$\mathrm {RT}^n_k$is equivalent to its relativization to any proper$\Sigma ^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe.We give a complete axiomatization of the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^n_k$for$n \ge (...) 3$. We show that they form a non-finitely axiomatizable subtheory of$\mathrm {PA}$whose$\Pi _3$fragment coincides with$\mathrm {B} \Sigma _1 + \exp $and whose$\Pi _{\ell +3}$fragment for$\ell \ge 1$lies between$\mathrm {I} \Sigma _\ell \Rightarrow \mathrm {B} \Sigma _{\ell +1}$and$\mathrm {B} \Sigma _{\ell +1}$. We also give a complete axiomatization of the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^2_k + \neg \mathrm {I} \Sigma _1$. In general, we show that the first-order consequences of$\mathrm {RCA}^*_0 + \mathrm {RT}^2_k$form a subtheory of$\mathrm {I} \Sigma _2$whose$\Pi _3$fragment coincides with$\mathrm {B} \Sigma _1 + \exp $and whose$\Pi _4$fragment is strictly weaker than$\mathrm {B} \Sigma _2$but not contained in$\mathrm {I} \Sigma _1$.Additionally, we consider a principle$\Delta ^0_2$-$\mathrm {RT}^2_2$which is defined like$\mathrm {RT}^2_2$but with both the$2$-colourings and the solutions allowed to be$\Delta ^0_2$-sets rather than just sets. We show that the behaviour of$\Delta ^0_2$-$\mathrm {RT}^2_2$over$\mathrm {RCA}_0 + \mathrm {B}\Sigma ^0_2$is in many ways analogous to that of$\mathrm {RT}^2_2$over$\mathrm {RCA}^*_0$, and that$\mathrm {RCA}_0 + \mathrm {B} \Sigma ^0_2 + \Delta ^0_2$-$\mathrm {RT}^2_2$is$\Pi _4$- but not$\Pi _5$-conservative over$\mathrm {B} \Sigma _2$. However, the statement we use to witness failure of$\Pi _5$-conservativity is not provable in$\mathrm {RCA}_0 +\mathrm {RT}^2_2$. (shrink)

We define a new NP search problem, the “local improvement” principle, about labellings of an acyclic, bounded-degree graph. We show that, provably in , it characterizes the consequences of and that natural restrictions of it characterize the consequences of and of the bounded arithmetic hierarchy. We also show that over V0 it characterizes the consequences of V1 and hence that, in some sense, a miniaturized version of the principle gives a new characterization of the consequences of . Throughout our search (...) problems are “type-2” NP search problems, which take second-order objects as parameters. (shrink)

We show that for each n ≥ 1, if T2n does not prove the weak pigeonhole principle for Σbn functions, then the collection scheme B Σ1 is not finitely axiomatizable over T2n. The same result holds with Sn2 in place of T 2n.

For any n, we construct a model of in which each formula is equivalent to an formula.

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic , has recently been unconditionally separated from full bounded arithmetic S2. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer (...) many results and methods from open to sharply bounded induction with relative ease.We provide two applications: the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes. (shrink)

Author: Kołodziejczyk Sebastian Tomasz Title: ON THE MUTUAL RELATIONSHIP BETWEEN THE CONCEPT OF BEING AND THE PLURALITY OF METAPHYSICAL DISCOURSES (O związkach między wielością dyskursów metafizycznych a pojęciem bytu) Source: Filo-Sofija year: 2011, vol:.15, number: 2011/4, pages: 865-880 Keywords: METAPHYSICS, ARISTOTLE, BEING, CONCEPT OF BEING, CATEGORIES, UNIVERSALS, PARTICULARS, PARTICULARITY Discipline: PHILOSOPHY Language: POLISH Document type: ARTICLE Publication order reference (Primary author’s office address): E-mail: www:The aim of the paper is to argue that there is a mutual relationship between the (...) concept of being as it is conceived of in the classical Aristotelian metaphysics and the plurality of possible metaphysical discourses. That thesis is rooted in the inherent ambiguity of the concept of being and some consequences connected therewith. Three forms of metaphysical thinking are taken into account: the metaphysics of formal interpretation of the concept of being, metaphysics of existential determination of being, and metaphysics of the material content of every conceptual representation of being. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory [math] of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational (...) characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory [math], this shows that [math] does not prove every [math] sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496–525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the “fixing lemma” from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185–239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a [math] oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic. (shrink)

The paper discusses the notion of finite model truth definitions (or FM-truth definitions), introduced by M. Mostowski as a finite model analogue of Tarski's classical notion of truth definition. We compare FM-truth definitions with Vardi's concept of the combined complexity of logics, noting an important difference: the difficulty of defining FM-truth for a logic ᵍ does not depend on the syntax of L, as long as it is decidable. It follows that for a natural ᵍ there exist FM-truth definitions whose (...) evaluation is much easier than the combined complexiy of ᵍ would suggest. We apply the general theory to give a complexity-theoretical characterization of the logics for which the $\Sigma_{m}^{d}$ classes (prenex classes of higher order logics) define FM-truth. For any $d \geq 2.m \geq 1$ we construct a family $\{[\Sigma_{m}^{d}]^{\leg\kappa}\}\kappa\in\omega$ of syntactically defined fragments of $\Sigma_{m}^{d}$ which satisfy this characterization. We also use the $[\Sigma_{m}^{d}]\leg\kappa$ classes to give a refinement of known results on the complexity classes captured by $\Sigma_{m}^{d}$ . We close with a few simple corollaries, one of which gives a sufficient condition for the existence, given a vocabulary $\sigma$ , of a fixed number $\kappa$ such that model checking for all first order sentences over $\sigma$ can be done in deterministic time $n^{\kappa}$. (shrink)

We prove that any proof of a [Formula: see text] sentence in the theory [Formula: see text] can be translated into a proof in [Formula: see text] at the cost of a polynomial increase in size. In fact, the proof in [Formula: see text] can be obtained by a polynomial-time algorithm. On the other hand, [Formula: see text] has nonelementary speedup over the weaker base theory [Formula: see text] for proofs of [Formula: see text] sentences. We also show that for (...) [Formula: see text], proofs of [Formula: see text] sentences in [Formula: see text] can be translated into proofs in [Formula: see text] at a polynomial cost in size. Moreover, the [Formula: see text]-conservativity of [Formula: see text] over [Formula: see text] can be proved in [Formula: see text], a fragment of bounded arithmetic corresponding to polynomial-time computation. For [Formula: see text], this answers a question of Clote, Hájek and Paris. (shrink)

Throughout the centuries, the doctrine of divine simplicity laid at the heart of Christian theology. In Thomas Aquinas’ mature thought, simplicity occupies a privileged place among other attributes of God. A simple simple being is a being that does not consist of any parts or constituents. God’s being simple means that he does not exhibit any of the seven types of composition. The arguments that serve as the justification for this thesis can be divided into four groups which I labeled: (...) the argument from the lack of potentiality, the argument from the priority of parts to the whole, the argument from the anterior maker and the argument from perfection. All of them begin with identifying God with the first cause of all things and add other premises that are specific to them. The first argument — that the first cause does not have potentiality and every composite being does have it. The second — that everything that is composite depends on it’s parts while the first cause is not dependent upon anything. The third — that every composite being needs a cause or explanation and the first cause has neither. The fourth — that the first cause is a perfectly good being and such a being cannot be composite. The analyses that were conducted show that the first two reasonings seem to be unsound. The third of the arguments provides a fine justification for the thesis in question. The fourth may serve as an additional, but weaker justification for it. (shrink)

In this book, an international team of scholars from leading American, British and Continental European universities presents original ideas about religious epistemology, the philosophy of God's action in the world, including the problem of evil and Divine Providence, and the philosophical challenge of religious diversity.

The article attempts to prove whether some elements of New Deal make it a revolutionary phenomenon. When thinking of New Deal as a model of revolution it can be said that the revolutionary change was the liberation of the bureaucratic class. It was the government that was liberated from the power of citizens. Therefore, it can also be said that the change was made without no change in the political system. The United States of America remained a democratic country but (...) the power was transferred from the citizens to the bureaucratic apparatus. This, apart from the violent character of the changes, could have caused Garrett to define Neal Deal as a revolution that happened behind the back of the biggest opponents of revolution. (shrink)

Presented paper is my reply to Prof. Andrzej Biłat’s review of the book Przewodnik po metafizyce (A Companion to Metaphysics) edited by myself and published by WAM Press. It contains answers to both general and particular remarks and comments risen by Prof. Biłat.