Theories and Theories of Truth

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.     

Realisability for infinitary intuitionistic set theory

We introduce a realisability semantics for infinitary intuitionistic set theory that is based on Ordinal Turing Machines (OTMs). We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. Finally, we use a variant of our notion of realisability to show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic.     

Discounting axioms imply risk neutrality

Although most applications of discounting occur in risky settings, the best-known axiomatic justifications are deterministic. This paper provides an axiomatic rationale for discounting in a stochastic framework. Consider a representation of time and risk preferences with a binary relation on a real vector space of vector-valued discrete-time stochastic processes on a probability space. Four axioms imply that there are unique discount factors such that preferences among stochastic processes correspond to preferences among present value random vectors. The familiar axioms are weak ordering, continuity and nontriviality. The fourth axiom, decomposition, is non-standard and key. These axioms and the converse of decomposition are assumed in previous axiomatic justifications for discounting with nonlinear intraperiod utility functions in deterministic frameworks. Thus, the results here provide the weakest known sufficient conditions for discounting in deterministic or stochastic settings. In addition to the four axioms, if there exists a von Neumann-Morgenstern utility function corresponding to the binary relation, then that function is risk neutral (i.e., affine). In this sense, discounting axioms imply risk neutrality.     

Torsten Brodén's work on the foundations of Euclidean geometry

The Swedish mathematician Torsten Brodén (1857–1931) wrote two articles on the foundations of Euclidean geometry. The first was published in 1890, almost a decade before Hilbert's first attempt, and the second was published in 1912. Brodén's philosophical view of the nature of geometry is discussed and his thoughts on axiomatic systems are described. His axiomatic system for Euclidean geometry from 1890 is considered in detail and compared with his later work on the foundations of geometry. The two continuity axioms given are compared to and proved to imply Hilbert's two continuity axioms of 1903.     

Admissible rules in the implication-negation fragment of intuitionistic logic

Uniform infinite bases are defined for the single-conclusion and multiple-conclusion admissible rules of the implication-negation fragments of intuitionistic logic and its consistent axiomatic extensions (intermediate logics). A Kripke semantics characterization is given for the (hereditarily) structurally complete implication-negation fragments of intermediate logics, and it is shown that the admissible rules of this fragment of form a PSPACE-complete set and have no finite basis.     

Largest fixed points of set continuous operators and Boffa's Anti‐Foundation

In Aczel [1], the existence of largest (written “greatest” in Barwise and Moss [2]) fixed points of set continuous operators is proved assuming the schema version of dependent choices in Zermelo‐Fraenkel set theory without the axiom of Foundation. In the present paper, we study whether the existence of largest fixed points of set continuous operators is provable without the schema version of dependent choices, using Boffa's weak antifoundation axioms. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on Fung-Fu's theorem

The axiomatic approach to rational decision malting in a fuzzy environment proposed by Fung and Fu (1975) lays down a too restrictive family of aggregation rules. Because of such axioms, the independence of the size of the groups under aggregation is assumed; moveover, in a general problem, these groups are fuzzy sets.     

Finite axiomatizability of local set theory

The need for modifying axiomatic set theories was caused, in particular, by the development of category theory. The ZF and NBG axiomatic theories turned out to be unsuitable for defining the notion of a model of category theory. The point is that there are constructions such as the category of categories in naïve category theory, while constructions like the set of sets are strongly restricted in the ZF and NBG axiomatic theories. Thus, it was required, on the one hand, to restrict constructions similar to the category of categories and, on the other hand, adapt axiomatic set theory in order to give a definition of a category which survives restricted construction similar to the category of categories. This task was accomplished by promptly inventing the axiom of universality (AU) asserting that each set is an element of a universal set closed under all NBG constructions. Unfortunately, in the theories ZF + AU and NBG + AU, there are toomany universal sets (as many as the number of all ordinals), whereas to solve the problem stated above, a countable collection of universal sets would suffice. For this reason, in 2005, the first-named author introduced local-minimal set theory, which preserves the axiom AU of universality and has an at most countable collection of universal sets. This was achieved at the expense of rejecting the global replacement axiom and using the local replacement axiom for each universal class instead. Local-minimal set theory has 14 axioms and one axiom scheme (of comprehension). It is shown that this axiom scheme can be replaced by finitely many axioms that are special cases of the comprehension scheme. The proof follows Bernays’ scheme with significant modifications required by the presence of the restricted predicativity condition on the formula in the comprehension axiom scheme.     

Locally averaged risk

Summary A heuristic method of reducing a class of admissible or Bayes decision rules is given. A new risk function is defined which is called the locally averaged risk. Bayes and admissible rules with respect to the new risk function are calledG-Bayes andG-admissible, respectively. It is shown under general assumptions that the class ofG-Bayes decision rules is a subset of the class of Bayes decision rules and the class ofG-admissible decision rules is a subset of the class of admissible decision rules. Some examples are considered, showing that the usual estimates of the parameter of a distribution with squared error as loss function, which are known to be admissible, are alsoG-admissible. This work was supported in part by NASA Grant-NGR 15-003-064 and NSF Grant-GP 7496 at Indiana University.     

构造自然数算术系统的一种新方法

用公理化方法来定义非空集上的二元关系〈，使得〈与该集合构成全序集，在全序集中给出最小元素原理的定义，再构造一个含有最小元素原理的适当公理系统来重新给出自然数的公理化定义，然后从构造的自然数公理系统中严格推导出一些基本命题，最后根据这些基本命题来完成对自然数算术系统的精确刻画，从而得到一种具体构造自然算术系统的新方法。     

Some axioms for constructive analysis

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CF_d asserting that every decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC₀₀ of countable choice can be decomposed into a monotone choice schema AC ₀₀ ^m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC₀₀. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC_1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms ${AB_{1/2,0}^{2^{\mathbb{N}}}}$ . We also introduce a version WKL!! of Weak K?nig’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FT_d.     

Permutability of rules in lattice theory

Lattice theory with the meet and join operations is formulated as a system of rules of inference. The order of application of these rules can be permuted so that a subterm property follows: If an atomic formula is derivable from given atomic formulas by the rules, it has a derivation all terms of which are terms in the given formulas or the conclusion. A direct decision method for universal formulas in lattice theory with the meet and join operations follows.     

Table admissible inference rules

A recursive basis of inference rules is described which are instantaneously admissible in all table (residually finite) logics extending one of the logics Int and Grz. A rather simple semantic criterion is derived to determine whether a given inference rule is admissible in all table superintuitionistic logics, and the relationship is established between admissibility of a rule in all table (residually finite) superintuitionistic logics and its truth values in Int. Translated from Algebra i Logika, Vol. 48, No. 3, pp. 400–414, May–June, 2009.     

Toward an Elementary Axiomatic Theory of the Category of LP-Matroids

The purpose of this paper is to provide the beginnings of an elementary theory for the category of loopless pointed matroids and strong maps. We propose a finite set of elementary axioms that is the beginning of an elementary axiomatic theory for this category.     

Characterizing properties of the value function of stochastic games

In the present note, the axiomatic characterization of the value function of two-person, zero-sum games in normal form by Vilkas and Tijs is extended to the value function of discounted, two-person, zero-sum stochastic games. The characterizing axioms can be indicated by the following terms: objectivity, monotony, and sufficiency for both players; or sufficiency for one of the players and symmetry. Also, a characterization without using the monotony axiom is given.     

Induction and foundation in the theory of hereditarily finite sets

The paper contains an axiomatic treatment of the intuitionistic theory of hereditarily finite sets, based on an induction axiom-schema and a finite set of single axioms. The main feature of the principle of induction used (due to Givant and Tarski) is that it incorporates Foundation. On the analogy of what is done in Arithmetic, in the axiomatic system selected the transitive closure of the membership relation is taken as a primitive notion, so as to permit an immediate adaptation of the theory to cases of restricted induction (in particular primitive recursive induction). At the end of the paper several different forms of induction, which play an important role in the development of the theory, are compared. An alternative axiomatization of the theory, which is of intrinsic interest, is also discussed.     

A shapley value for games with restricted coalitions

A restriction is a monotonic projection assigning to each coalition of a finite player setN a subcoalition. On the class of transferable utility games with player setN, a Shapley value is associated with each restriction by replacing, in the familiar probabilistic formula, each coalition by the subcoalition assigned to it. Alternatively, such a Shapley value can be characterized by restricted dividends. This method generalizes several other approaches known in literature. The main result is an axiomatic characterization with the property that the restriction is determined endogenously by the axioms.     

An ordinal analysis for theories of self-referential truth

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of ${\Sigma^1_1}The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of S¹₁{\Sigma^1_1} dependent choice.     

Two axiomatic approaches to decision making using possibility theory

This paper proposes a utility theory for decision making under uncertainty that is described by possibility theory. We show that our approach is a natural generalization of the two axiomatic systems that correspond to pessimistic and optimistic decision criteria proposed by Dubois et al. The generalization is achieved by removing axioms that are supposed to reflect attitudes toward uncertainty, namely, pessimism and optimism. In their place we adopt an axiom that imposes an order on a class of canonical lotteries that realize either in the best or in the worst prize. We prove an expected utility theorem for the generalized axiomatic system based on the newly introduced concept of binary utility.     

Model Approach to Nonstandard Analysis in the Context of Axiomatic Set Theory

In this paper, the model approach to nonstandard analysis is developed on the basis of Zermelo—Fraenkel axiomatic set theory with atoms. The traditional consideration of the standard superstructure V as the primary object of nonstandard analysis is justified. Set-theoretic axioms for the nonstandard system *V are obtained.