Let Con↾x denote the finite consistency statement “there are no proofs of contradiction in T with ≤x symbols.” For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con↾n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con)↾n. In contrast, we show that Peano arithmetic (...) has polynomial proofs of Con)↾n, where Con∗ is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen, and Weiermann. We also obtain a new proof of the result that the usual consistency statement Con is equivalent to ε0 iterations of slow consistency. Our argument is proof-theoretic, whereas previous investigations of slow consistency relied on nonstandard models of arithmetic. (shrink)

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)

In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as $\mathsf {P}\neq \mathsf {NP}$ [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results (...) of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies. (shrink)

Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can (...) be overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε0. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε0 are finitary objects despite being infinite. (shrink)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? Once the question has been properly formulated, the answer turns out to be about as elegant as one could want: Adding a theory of truth to a finitely axiomatized theory T is more or less equivalent to a (...) kind of abstract consistency statement. A large part of the interest of the paper lies in the way syntactic theories are 'disentangled' from object theories. (shrink)

A fast consistency prover is a consistent polytime axiomatized theory that has short proofs of the finite consistency statements of any other polytime axiomatized theory. Krajíček and Pudlák have proved that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover. It is an easy observation that NP = coNP implies the existence of a fast consistency prover. The reverse implication is an open question. In this (...) paper we define the notion of an unlikely fast consistency prover and prove that its existence is equivalent to NP = coNP. Next it is proved that fast consistency provers do not exist if one considers RE axiomatized theories rather than theories with an axiom set that is recognizable in polynomial time. (shrink)

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. (...) Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part. (shrink)

For a set x, let S(x) be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF: (1) There is an infinite set x such that |p(x)|<|S(x)|<|seq^1-1(x)|<|seq(x)|, where p(x) is the powerset of x, seq(x) is the set of all finite sequences of elements of x, and seq^1-1(x) is the set of all finite sequences of elements of x without repetition. (2) There is a (...) Dedekind infinite set x such that |S(x)|<|[x]^3| and such that there exists a surjection from x onto S(x). (3) There is an infinite set x such that there is a finite-to-one function from S(x) into x. (shrink)

In this paper we will state and prove some comparative theorems concerning PRA and IΣ1. We shall provide a characterization of IΣ1 in terms of PRA and iterations of a class of functions. In particular, we prove that for this class of functions the difference between IΣ1 and PRA is exactly that, where PRA is closed under iterations of these functions, IΣ1 is moreover provably closed under iteration. We will formulate a sufficient condition for a model of PRA to be (...) a model of IΣ1. This condition is used to give a model-theoretic proof of Parsons’ theorem, that is, IΣ1 is Π2-conservative over PRA. We shall also give a purely syntactical proof of Parsons’ theorem. Finally, we show that IΣ1 proves the consistency of PRA on a definable IΣ1-cut. This implies that proofs in IΣ1 can have non-elementary speed up over proofs in PRA. (shrink)

We study belief change in the branching-time structures introduced in Bonanno (Artif Intell 171:144–160, 2007 ). First, we identify a property of branching-time frames that is equivalent (when the set of states is finite) to AGM-consistency, which is defined as follows. A frame is AGM-consistent if the partial belief revision function associated with an arbitrary state-instant pair and an arbitrary model based on that frame can be extended to a full belief revision function that satisfies the AGM postulates. (...) Second, we provide a set of modal axioms that characterize the class of AGM-consistent frames within the modal logic introduced in Bonanno (Artif Intell 171:144–160, 2007 ). Third, we introduce a generalization of AGM belief revision functions that allows a clear statement of principles of iterated belief revision and discuss iterated revision both semantically and syntactically. (shrink)

Samuel R. Buss and Aleksandar Ignjatović. Unprovability of Consistency Statements in Fragments of Bounded Arithmetic.

We define and study a new restricted consistency notion RCon ∗ for bounded arithmetic theories T 2 j . It is the strongest ∀ Π 1 b -statement over S 2 1 provable in T 2 j , similar to Con in Krajíček and Pudlák, 29) or RCon in Krajı́ček and Takeuti 107). The advantage of our notion over the others is that RCon ∗ can directly be used to construct models of T 2 j . We apply this (...) by proving preservation theorems for theories of bounded arithmetic of the following well-known kind: The ∀ Π 1 b -separation of bounded arithmetic theories S 2 i from T 2 j is equivalent to the existence of a model of S 2 i which does not have a Δ 0 b -elementary extension to a model of T 2 j . More specific, let M ⊨ Ω 1 nst denote that there is a nonstandard element c in M such that the function n ↦ 2 log c is total in M . Let BLΣ 1 b be the bounded collection schema for Σ 1 b -formulas. We obtain the following preservation results: the ∀ Π 1 b -separation of S 2 i from T 2 j is equivalent to the existence of 1. a model of S 2 i +Ω 1 nst which is 1 b -closed w.r.t. T 2 j , 2. a countable model of S 2 i + BLΣ 1 b without weak end extensions to models of T 2 j. (shrink)

We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of IΔ0 whose Herbrand Consistency is not provable in IΔ0. We also show the existence of an IΔ0-derivable Π1-sentence such that IΔ0 cannot prove its Herbrand Consistency.

David Hilbert proposed his well-known Hilbert Program in the early 1920s for foundations of mathematics. The purpose of his program was to prove the consistency of mathematics by using the finitary methods and relying on axiomatic system. Thus, riddles and paradoxes related with the foundations of mathematics could be solved. Hilbert considers, formalizing whole mathematics in a consistent finite way depending on axioms, as an effort to develop a proof theory. So much so that any problems which may (...) occur in a mathematical system, including those related to infinity, will be solved. Hilbert begins with finite number of signs and rules and then proceeds to develop various proved consistent statements. From there, he continues to a higher-order mathematics that includes ideal objects. Hilbert's development of the program started in 1899 with the Foundations of Geometry, and completed by The Grounding of Elementary Number Theory in 1931. In this paper, we will exhibit the various stages and attempts of Hilbert to found the numbers. Our thesis states that Hilbert has always maintained an intuitive apprehension in these foundational attempts i.e. Hilbert advocates intuitional insight prior to any logical inference that makes it possible to found the numbers. Although Hilbert did not define himself as a formalist in any of his works, he was named as a formalist because of Brouwer's criticism of him. Hilbert’s emphasis on intuition reveals that a kind of formalism which is a mere play of signs can be associated with him. In this paper, we will also deal with the relations Hilbert established between signs, axioms, logic and intuition. (shrink)

A statement of hyperarithmetic analysis is a sentence of second order arithmetic S such that for every Y⊆ω, the minimum ω-model containing Y of RCA0 + S is HYP, the ω-model consisting of the sets hyperarithmetic in Y. We provide an example of a mathematical theorem which is a statement of hyperarithmetic analysis. This statement, that we call INDEC, is due to Jullien [13]. To the author's knowledge, no other already published, purely mathematical statement has been found with this property (...) until now. We also prove that, over RCA0, INDEC is implied by [Formula: see text] and implies ACA0, but of course, neither ACA0, nor ACA 0+ imply it. We introduce five other statements of hyperarithmetic analysis and study the relations among them. Four of them are related to finitely-terminating games. The fifth one, related to iterations of the Turing jump, is strictly weaker than all the other statements that we study in this paper, as we prove using Steel's method of forcing with tagged trees. (shrink)

This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. (...) E. J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis. (shrink)

A new technique for proving realisability results is presented, and is illustrated in detail for the simple case of arithmetic minus induction. CL is a Gentzen formulation of classical logic. CPQ is CL minus the Cut Rule. The basic proof theory and model theory of CPQ and CL is developed. For the semantics presented CPQ is a paraconsistent logic, i.e. there are non-trivial CPQ models in which some sentences are both true and false. Two systems of arithmetic minus induction are (...) introduced, CL-A and CPQ-A based on CL and CPQ, respectively. The realisability theorem for CPQ-A is proved: It is shown constructively that to each theorem A of CPQ-A there is a formula A *, a so-called “realised disjunctive form of A ”, such that variables bound by essentially existential quantifiers in A * can be written as recursive functions of free variables and variables bound by essentially universal quantifiers. Realisability is then applied to prove the consistency of CL-A, making use of certain finite non-trivial inconsistent models of CPQ-A. (shrink)

In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is (...) no paradox, and that what it shows is that conditionalisation, often claimed to be integral to the Bayesian canon, has to be rejected as a general rule in a finitely additive environment. (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 logic Kf of the modalities of finite, devised to capture the notion of 'there exists a finite number of accessible worlds such that . . . is true', was introduced and axiomatized by Fattorosi. In this paper we enrich the logical framework of Kf: we give consistency properties and a tableau system (which yields the decidability) explicitly designed for Kf, and we introduce a shorter and more natural axiomatization. Moreover, we show the strong and suggestive relationship (...) between Kf and the much older logic of the physical modalities of Burks. (shrink)

Partial consistency statements can be expressed as polynomial-size propositional formulas. Frege proof systems have polynomial-size partial self-consistency proofs. Frege proof systems have polynomial-size proofs of partial consistency of extended Frege proof systems if and only if Frege proof systems polynomially simulate extended Frege proof systems. We give a new proof of Reckhow's theorem that any two Frege proof systems p-simulate each other. The proofs depend on polynomial size propositional formulas defining the truth of propositional formulas. These (...) are already known to exist since the Boolean formula value problem is in alternating logarithmic time; this paper presents a proof of this fact based on a construction which is somewhat simpler than the prior proofs of Buss and of Buss-Cook- Gupta-Ramachandran. (shrink)

We write and for the cardinalities of the set of finite subsets and the set of permutations with finitely many non‐fixed points, respectively, of a set which is of cardinality. In this paper, we investigate relationships between and for an infinite cardinal in the absence of the Axiom of Choice. We give conditions that make and comparable as well as give related consistency results.

We say that a first order theoryTislocally finiteif every finite part ofThas a finite model. It is the purpose of this paper to construct in a uniform way for any consistent theoryTa locally finite theory FIN which is syntactically isomorphic toT.Our construction draws upon the main idea of Paris and Harrington [6] and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories. The first mathematically strong locally finite theory, called FIN, was defined in (...) [1]. Now we get much stronger ones, e.g. FIN.From a physicalistic point of view the theorems of ZF and their FIN-counterparts may have the same meaning. Therefore FIN is a solution of Hilbert's second problem. It eliminates ideal objects from the proofs of properties of concrete objects.In [4] we will demonstrate that one can develop a direct finitistic intuition that FIN is locally finite. We will also prove a variant of Gödel's second incompleteness theorem for the theory FIN and for all its primitively recursively axiomatizable consistent extensions.The results of this paper were announced in [3]. (shrink)

David Furley's work on the cosmologies of classical antiquity is structured around what he calls "two pictures of the world." The first picture, defended by both Plato and Aristotle, portrays the universe, or all that there is (to pan), as identical with our particular ordered world-system. Thus, the adherents of this view claim that the universe is finite and unique. The second system, defended by Leucippus and Democritus, portrays an infinite universe within which our particular kosmos is only one (...) of countless kosmoi. Aristotle's argument in De caelo I.9 that the world is necessarily unique is an important contribution to this debate. This argument holds interest because it shows Aristotle wrestling with an apparent inconsistency in his own philosophy, as deeply-held convictions within his cosmology collide with an equally deeply-held conviction within his metaphysics. The following three principles, each of which Aristotle appears committed to, are inconsistent: -/- The cosmic uniqueness principle. The world is necessarily unique. The cosmic form principle. The world is an ordered, structured unity. As such, the world has a form. The possibility of multiple instantiation principle. For all F, if F is a form, it is possible that there exist multiple Fs. In De caelo I.9, Aristotle argues that we can establish the uniqueness of the universe, reject the multiple instantiation principle, yet still retain the distinction between 'this world' and 'world in general,' if the following is true (as it is): the world takes up all the matter that exists. Aristotle illustrates this argument with one of the stranger analogies in his corpus: imagine an aquiline nose that takes up all the flesh in the universe. If this were so, then there could not exist any other aquiline objects whatsoever. (For this reason, we dub the De caelo I.9 argument the 'Cosmic Nose argument.') This paper is an interpretation of how this argument is supposed to proceed and an assessment of its success. The first section states the problem Aristotle is confronted with, sorts through Aristotle's various statements of the Cosmic Nose argument, which exhibit some sloppiness, and reconstructs charitably a single argument. We also spend some time examining the significance of Aristotle's example of a gigantic aquiline nose. We argue that, even charitably reconstructed, the argument appears to commit a serious modal fallacy. The remainder of the paper explores whether this modal fallacy can be overcome. We conclude that, although not a cogent argument for the uniqueness of the world (as this would require a significant revision of our current astronomy), the Cosmic Nose argument does succeed on its own terms. However, it should not be regarded as a free-standing argument for the uniqueness of the world. Instead, it depends crucially on the earlier argument in De caelo I.8 for the universe's uniqueness; De caelo I.9 should be viewed as an attempt to extend the conclusion of De caelo I.8 and to show how this conclusion can be made consistent with Aristotle's metaphysical principles about the nature of form. (shrink)

Informal statements of Gödel's Second Incompleteness Theorem, referred to here as Informal Second Incompleteness, are simple and dramatic. However, current versions of Formal Second Incompleteness are complicated and awkward. We present new versions of Formal Second Incompleteness that are simple, and informally imply Informal Second Incompleteness. These results rest on the isolation of simple formal properties shared by consistency statements. Here we do not address any issues concerning proofs of Second Incompleteness.

3.0. Well-being as a Challenge to Naturalism. In Chapter Three Adams discusses and criticizes those accounts of a person’s well being which characterize it in terms of counterfactuals regarding her actual desires and preferences. These criticisms are important for the question of ethical naturalism because any plausible naturalist position will have to portray a person’s well-being as somehow or other supervening on features of her psychology and her environment. The sorts of analyses Adams criticizes are the most prominent analyses consistent (...) with this constraint, so it is important to see whether or not Adams’ criticisms of them undermine the prospects for ethical naturalism generally. (shrink)

We study properties of certain subclasses of the Dedekind finite sets in set theory without the axiom of choice with respect to the comparability of their elements and to the boundedness of such classes, and we answer related open problems from Herrlich’s “The Finite and the Infinite.” The main results are as follows: 1. It is relatively consistent with ZF that the class of all finite sets is not the only finiteness class such that any two of (...) its elements are comparable. 2. The principle “Small Violations of Choice” —introduced by A. Blass—implies that the class of all Dedekind finite sets is bounded above. 3. “The class of all Dedekind finite sets is bounded above” is true in every permutation model of ZFA in which the class of atoms is a set, and in every symmetric model of ZF. 4. There exists a model of ZFA set theory in which the class of all atoms is a proper class and in which the class of all infinite Dedekind finite sets is not bounded above. 5. There exists a model of ZF in which the class of all infinite Dedekind finite sets is not bounded above. (shrink)

3.0. Well-being as a Challenge to Naturalism. In Chapter Three Adams discusses and criticizes those accounts of a person’s well being which characterize it in terms of counterfactuals regarding her actual desires and preferences. These criticisms are important for the question of ethical naturalism because any plausible naturalist position will have to portray a person’s well-being as somehow or other supervening on features of her psychology and her environment. The sorts of analyses Adams criticizes are the most prominent analyses consistent (...) with this constraint, so it is important to see whether or not Adams’ criticisms of them undermine the prospects for ethical naturalism generally. (shrink)

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider (...) infinitary variants and extensions by so-called counting quantifiers.Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time or polynomial space can be characterized as collections of all classes of finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth. (shrink)

Given a finite relational language L is there an algorithm that, given two finite sets A and B of structures in the language, determines how many homogeneous L structures there are omitting every structure in B and embedding every structure in A? For directed graphs this question reduces to: Is there an algorithm that, given a finite set of tournaments Γ, determines whether QΓ, the class of finite tournaments omitting every tournament in Γ, is well-quasi-order? First, (...) we give a nonconstructive proof of the existence of an algorithm for the case in which Γ consists of one tournament. Then we determine explicitly the set of tournaments each of which does not have an antichain omitting it. Two antichains are exhibited and a summary is given of two structure theorems which allow the application of Kruskal's Tree Theorem. Detailed proofs of these structure theorems will be given elsewhere. The case in which Γ consists of two tournaments is also discussed. (shrink)

The grounding of semiotics in the finiteness of cognition is extended into constructs and methods for analysis by incorporating the assumption that cognition can be similar within and between agents. After examining and formalizing cognitive similarity as an ontological commitment, the recurrence of cognitive states is examined in terms of a “cognitive set.” In the individual, the cognitive set is seen as evolving under the bidirectional, cyclical determination of thought by the historical environment. At the population level, the distributed “global” (...) cognitive set is argued to be constrained to a manifold in which the cognition of individuals is determined only when their cognitive sets meet certain conditions in the world: a result seen as consistent with Lotman’s semiosphere. With these foundations in place, dimensional modelling of the semiosic field is inaugurated. Firstly, measures of cognitive similarity are formalized as cognitive “distance” and on this basis the concept of a semiotic vector is defined. Secondly, semiotic vectors are seen to shape a general pattern of oscillation in semiosis, and thus to imply zero points in semiosic potential. Thirdly, semiosic oscillation in individual agents is shown to be consistent with a novel diachronic or longitudinal interpretation of Greimas’ semiotic square expanded into a “semiotic pipe” in which cognition traverses an n-dimensional space structured by axes of oscillation. Finally, the expanded theory of finite semiotics is advanced as a useful basis for two new complementary disciplines: a computational, mathematical science of “natural semiotic processing” to trace and model semiotic vectors and oscillation; and an ethical, rhetorical art of “technological influencing” to guide its inputs and applications. (shrink)

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from (...) these classes have the finite model property with respect to the class of -formulae, i.e. each -formula has a -model iff it has a finite -model. Roughly speaking, a -formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators. (shrink)

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from (...) these classes have the finite model property with respect to the class of ♦-formulae, i.e. each ♦-formula has a ℒ-model iff it has a finite ℒ-model. Roughly speaking, a ♦-formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators. (shrink)

We present the notion of finite high-order Gowers games, and prove a statement parallel to Gowers's Combinatorial Lemma for these games. We derive ‘quantitative’ versions of the original Gowers Combinatorial Lemma and of Gowers's Dichotomy, which place them in the context of the recently introduced infinite dimensional asymptotic theory for Banach spaces.

Let φ be a monadic second order sentence about a finite structure from a class K which is closed under disjoint unions and has components. Compton has conjectured that if the number of n element structures has appropriate asymptotics, then unlabelled (labelled) asymptotic probabilities ν(φ) (μ(φ) respectively) for φ always exist. By applying generating series methods to count finite models, and a tailor made Tauberian lemma, this conjecture is proved under a mild additional condition on the asymptotics of (...) the number of single component K-structures. Prominent among examples covered, are structures consisting of a single unary function (or partial function) and a fixed number of unary predicates. (shrink)

The view that chances are relative frequencies of occurrence within actual, finite reference classes has long been written off. I argue that it ought to be reconsidered. Focusing on non-deterministic chance, I defend a version of finite frequentism in which reference classmates are required to have qualitatively identical pasts. While my analysis can evade or resist several standard objections, it has a counterintuitive consequence: non-trivial chances entail the existence of past light cones that are perfect intrinsic duplicates. In (...) mitigation, I argue that our scientific knowledge is consistent with the hypothesis that there are many such duplicates in the actual world. Moreover, my analysis has some striking advantages: it is simple, it is metaphysically undemanding, and it makes possible a satisfying explanation of the chance–credence connection. 1 Introduction2 Target3 Sketch and Skirmish4 Articulation5 Standard Worries5.1 Frequency tolerance5.2 Leibniz’s dictum5.3 Single case chance5.4 Missing values5.5 Explanation5.6 Counterfactual chances6 Chance and Credence6.1 An indefeasible indifference intuition6.2 Articulating the credence principle6.3 Appraising the credence principle7 Confirmation8 Laws9 Summing Up. (shrink)

One of the main themes in Spinoza’s Ethics is the issue of human freedom: What does it consist in and how may it be attained? Spinoza’s ethical views crucially depend on his metaphysical theory, and this close connection provides the answer to several central questions concerning Spinoza’s conception of human freedom. Firstly, how can we accommodate human freedom within Spinoza’s necessitarianism—in the context of which Spinoza rejects the notion of a free will? Secondly, how can humans, as merely finite (...) beings, genuinely attain freedom? Can Spinoza defend his claim that we may even attain blessedness? I will argue that these questions are answered by appeal to a twofold in human nature. According to Spinoza, we are finite in infinity. (shrink)

The traditional model theory of first-order logic assumes that the interpretation of a formula can be given without reference to its deductive context. This paper investigates an interpretation which depends on a formula's location within a derivation. The key step is to drop the assumption that all quantified variables must have the same range and to require only that the ranges of variables in a derivation must be related in such way as to preserve the soundness of the inference rules. (...) With each (consistent) derivation there is associated a Buridan-Volpin (orBV) structure [M, {r(x)}] which is simply a Tarski structureM for the language and a map giving the ranger(x) of each variablex in the derivation. IfLK* is (approximately) the classical sequent calculusLK of Gentzen from which the structural contraction rules have been dropped, then our main result reads: If a set of first-ordered formulas has a Tarski modelM, then from any normal derivationD inLK* of can be constructed aBV modelM D=[M, {r(x)}] of where each ranger(x) is finite. (shrink)

Suppose one accepts the argument that past infinity is not acceptable. This does not eliminate the possibility that the beginning of time is not equivalent across objects. Along with breakdown of absolute simultaneity of events in relativity, there may even be no agreement on whether an event existed. There may be no consistent way to totally order events. In such a case, despite every object, conscious or not, having finite lifetime, there may be no single point called “the beginning,” (...) and the universe stays as it is without requiring a cause of existence. (shrink)