Doing and refraining from refraining

The main purpose of this paper is to prove that in everystit semantic structure that contains a busy choice sequence, neither does doing imply refraining from refraining from doing, nor does refraining from refraining from doing imply doing.     

Stit and the language of agency

Stit, a sentence form first introduced in Belnap and Perloff (1988), encourages a modal approach to agency. Von Wright, Chisholm, Kenny, and Castañeda have all attempted modal treatments of agency, while Davidson has rejected such treatments. After a brief explanation of the syntax and semantics of stit and a restatement of several of the important claims of the earlier paper, I discuss the virtues of stit against the background of proposals made by these philososphers.     

Decidability ofstit theory with a single agent andRefref Equivalence

The purpose of this paper is to prove the decidability ofstit theory (a logic of “seeing to it that”) with a single agent andRefref Equivalence. This result is obtained through an axiomatization of the theory and a proof that it has thefinite model property. A notion ofcompanions to stit formulas is introduced and extensively used in the proof.     

Propositional Q-Logic

Topic of the paper is Q-logic – a logic of agency in its temporal and modal context. Q-logic may be considered as a basal logic of agency since the most important stit-operators discussed in the literature can be defined or axiomatized easily within its semantical and syntactical framework. Its basic agent dependent operator, the Q-operator (also known as - or cstit-operator), which has been discussed independently by F. v. Kutschera and B. F. Chellas, is investigated here in respect of its relation to other temporal and modal operators. The main result of the paper, then, is a completeness result for a calculus of Q-logic with respect to a semantics defined on the tree-approach to agency as introduced and developed by, among others, F. v. Kutschera and N. D. Belnap.     

Alternative semantics for quantified first degree relevant logic

A system FDQ of first degree entailment with quantification, extending classical quantification logic Q by an entailment connective, is axiomatised, and the choice of axioms defended and also, from another viewpoint, criticised. The system proves to be the equivalent to the first degree part of the quantified entailmental system EQ studied by Anderson and Belnap; accordingly the semantics furnished are alternative to those provided for the first degree of EQ by Belnap. A worlds semantics for FDQ is presented, and the soundness and completeness of FDQ proved, the main work of the paper going into the proof of completeness. The adequacy result is applied to yield, as well as the usual corollaries, weak relevance of FDQ and the fact that FDQ is the common first degree of a wide variety of (constant domain) quantified relevant logics. Finally much unfinished business at the first degree is discussed.     

Truth in the Tractatus

My paper takes issue both with the standard view that the Tractatus contains a correspondence theory and with recent suggestions that it features a deflationary or semantic theory. Standard correspondence interpretations are mistaken, because they treat the isomorphism between a sentence and what it depicts as a sufficient condition of truth rather than of sense. The semantic/deflationary interpretation ignores passages that suggest some kind of correspondence theory. The official theory of truth in the Tractatus is an obtainment theory – a sentence is true iff the state of affairs it depicts obtains. This theory differs from deflationary theories in that it involves an ontology of states of affairs/facts; and it can be transformed into a type of correspondence theory: a sentence is true iff it corresponds to, i.e. depicts an obtaining state of affairs (fact). Admittedly, unlike correspondence theories as commonly portrayed, this account does not involve a genuinely truth-making relation. It features a relation of correspondence, yet it is that of depicting, between a meaningful sentence and its sense – a possible state of affairs. What makes for truth is not that relation, but the obtaining of the depicted state of affairs. This does not disqualify the Tractatus from holding a correspondence theory, however, since the correspondence theories of Moore and Russell are committed to a similar position. Alternatively, the obtainment theory can be seen as a synthesis of correspondence, semantic and deflationary approaches. It does justice to the idea that what is true depends solely on what is the case, and it combines a semantic explanation of the relation between a sentence and what it says with a deflationary account of the agreement between what the sentence says and what obtains or is the case if it is true     

Internal approach to external sets and universes

In this article we show how the universe ofBST,bounded set theory (a modification ofIST which is, briefly, a theory for the family of those sets inIST which are members of standard sets) can be enlarged by definable subclasses of sets (which are not necessarily sest in internal theories likeBST orIST) so that Separation and Replacement are true in the enlargement for all formulas, including those in which the standardness predicate may occur.ThusBST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop topics in nonstandard analysis inaccessible in the framework of a purely internal approach, such as Loeb measures.Presented byRobert Goldblatt     

The Rationale Behind Revision-Rule Semantics

According to Gupta and Belnap, the “extensional behavior” of ‘true’ matches that of a circularly defined predicate. Besides promising to explain semantic paradoxicality, their general theory of circular predicates significantly liberalizes the framework of truth-conditional semantics. The authors’ discussions of the rationale behind that liberalization invoke two distinct senses in which a circular predicate’s semantic behavior is explained by a “revision rule” carrying hypothetical information about its extension. Neither attempted explanation succeeds. Their theory may however be modified to employ a relativized notion of extension. The resulting contextualist semantics for ‘true’ construes circularity as a pragmatic phenomenon.     

Actions as Events

We present a theory of actions based on a theory of events in branching time, in which “particular” or “token” actions are taken to be sets of transitions from their initial states to the outcomes. We also present a simple theory of composition of events by which composite events can be formed out of other events. Various kinds of actions, including instantaneous group actions and sequential group actions, are introduced by way of composition, and an extended stit theory of agency is proposed, in which the stit operators are combined or equipped with reified group actions.     

Internal approach to external sets and universes

A problem which enthusiasts ofIST, Nelson's internal set theory, usually face is how to treat external sets in the internal universe which does not contain them directly. To solve this problem, we considerBST,bounded set theory, a modification ofIST which is, briefly, a theory for the family of thoseIST sets which are members of standard sets.We show thatBST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop the most advanced applications of nonstandard methods. In particular, we define inBST an enlargement of theBST universe which satisfies the axioms ofHST, an external theory close to a theory introduced by Hrbaek. HST includes Replacement and Saturation for all formulas but contradicts the Power Set and Choice axioms (either of them is incompatible with Replacement plus Saturation), therefore to get an external universe which satisfies all ofZFC minus Regularity one has to pay by a restriction of Saturation. We prove thatHST admits a system of subuniverses which modelZFC (minus Regularity but with Power Set and Choice) and Saturation in a form restricted by a fixed but arbitrary standard cardinal.Thus the proposed system of set theoretic foundations for nonstandard mathematics, based on the simple and natural axioms of the internal theoryBST, provides the treatment of external sets sufficient to carry out elaborate external constructions.Partially supported by AMS grants in 1993 and 1994 and DFG grant in 1994.Presented byRobert Goldblatt     

Truth,Predication and a Family of Contingent Paradoxes

In truth theory one aims at general formal laws governing the attribution of truth to statements. Gupta’s and Belnap’s revision-theoretic approach provides various well-motivated theories of truth, in particular T* and T#, which tame the Liar and related paradoxes without a Tarskian hierarchy of languages. In property theory, one similarly aims at general formal laws governing the predication of properties. To avoid Russell’s paradox in this area a recourse to type theory is still popular, as testified by recent work in formal metaphysics by Williamson and Hale. There is a contingent Liar that has been taken to be a problem for type theory. But this is because this Liar has been presented without an explicit recourse to a truth predicate. Thus, type theory could avoid this paradox by incorporating such a predicate and accepting an appropriate theory of truth. There is however a contingent paradox of predication that more clearly undermines the viability of type theory. It is then suggested that a type-free property theory is a better option. One can pursue it, by generalizing the revision-theoretic approach to predication, as it has been done by Orilia with his system P*, based on T*. Although Gupta and Belnap do not explicitly declare a preference for T# over T*, they show that the latter has some advantages, such as the recovery of intuitively acceptable principles concerning truth and a better reconstruction of informal arguments involving this notion. A type-free system based on T# rather than T* extends these advantages to predication and thus fares better than P* in the intended applications of property theory.

     

Relevant Robinson's arithmetic

In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in relevant arithmetic.This paper has been greatly influenced by the (largely unpublished) work of E. K. Meyer (cf. [7]) on relevant arithmetic, and I wish to thank him, and also N. D. Belnap, Jr. and D. Cohen for helpful advice. In fairness to Meyer it must be said that he finds my axioms 13 and 13(1) too strong (they are not theorems of his system R# - cf. §4 below). Meyer tells me be finds vindication for his view in my chief theorem of §2. For myself, I find the insights behind Meyer's work on R# to be both stable and fruitful, and if I now had to make a choice, I would follow Meyer in his rejection of my axioms. However, the systems I explore in this paper themselves have a surprising amount of internal consistency of motivation (cf. §5). Let a hundred formal systems bloom.     

Semantic Holism is seriously false

Semantic Holism is the claim that any semantic path from inferential semantics (the indeterminate semantics forced by the classical inference rules of PC) reaches all the way to classical semantics if it is even one step long. In our joint paper Semantic Holism, Belnap and I showed that some such semantic paths are two steps long, but we left open a number of questions about the lengths of semantic paths. Here I answer the most important of these questions by showing that there are infinitely long semantic paths that begin at inferential semantics but that do not even reach classical semantics. I do this by showing how to construct such an infinite semantic path from the members of the family of (n–1)-out-of-n-disjunction connectives.     

A pragmatic interpretation of intuitionistic propositional logic

We construct an extension ^P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of ^P;rfs preceded by theassertion sign constituteelementary assertive formulas of ^P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of ^P is endowed with atruth value defined classically, and everyaf is endowed with ajustification value, defined in terms of the intuitive notion of proof and depending on the truth values of its radical subformulas. In this framework, we define the notion ofpragmatic validity in ^P and yield a list of criteria of pragmatic validity which hold under the assumption that only classical metalinguistic procedures of proof be accepted. We translate the classical propositional calculus (CPC) and the intuitionistic propositional calculus (IPC) into the assertive part of ^P and show that this translation allows us to interpret Intuitionistic Logic as an axiomatic theory of the constructive proof concept rather than an alternative to Classical Logic. Finally, we show that our framework provides a suitable background for discussing classical problems in the philosophy of logic.This paper is an enlarged and entirely revised version of the paper by Dalla Pozza (1991) worked out in the framework of C.N.R. project n. 89.02281.08, and published in Italian. The basic ideas in it have been propounded since 1986 by Dalla Pozza in a series of seminars given at the University of Lecce and in other Italian Universities. C. Garola collected the scattered parts of the work, helped in solving some conceptual difficulties and refining the formalism, yielded the proofs of some propositions (in particular, in Section 3) and provided physical examples (see in particular Remark 2.3.1).     

Nested Deontic Modalities: Another View of Parking on Highways

A suggestion is made for representing iterated deontic modalities in stit theory, the “seeing-to-it-that” theory of agency. The formalization is such that normative sentences are represented as agentive sentences and therefore have history dependent truth conditions. In contrast to investigations in alethic modal logic, in the construction of systems of deontic logic little attention has been paid to the iteration... of the deontic modalities. L. Goble (1966, p. 197) N. Belnap and P. Bartha (1995) present a formalization of iterated deontic modalities in stit theory, the “seeing-to-it-that” theory of agency, due to Belnap, Perloff, and Xu (1996). In the present paper a simplification of Belnap and Bartha's approach is suggested. In order to support easy comparison with (1995), I shall take up Belnap and Bartha's discussion of R. Barcan Marcus' (1966) example Parking on highways ought to be forbidden. The simplified account underlines the power of stit theory and represents obligations, prohibitions and permissions as agentive sentences, as required by iteration. Therefore, in particular, the truth of obligations, prohibitions and permissions is it history dependent. This revised version was published online in July 2006 with corrections to the Cover Date.     

A critical review of theS/L reliability index

The meaning and properties of a commonly used index of reliability, S/L,were examined critically. It was found that the index does not reflect any conventional concept of reliability. When used for an identical behavioral observation session, it is not statistically correlated with other reliability indices. Within an observation session, the standardizing measure of Lis beyond the control of the investigator. Furthermore, the reason for the choice of Las the standard is unclear. The role of chance agreement in S/Lis not known. The exact interpretation of the index depends on which observer reports L.Overall the conceptual and mathematical meaning of S/Lis dubious. It is suggested that the S/Lindex should not be used until its nature is shown to be a measure of reliability. Other approaches such as the intraclass correlations and generalizability coefficients should be used instead.The authors are indebted to Johnny Matson for his critique of an earlier version of this paper.     

Effects of Weight and Syntactic Priming on the Production of Cantonese Verb-Doubling

Verb-doubling, where a copy of the main verb occurs both before and after the direct object, is a structure commonly used in Chinese in sentences containing a frequency or duration phrase. In Cantonese, verb-doubling is highly optional and therefore problematic for existing syntactic, semantic, and pragmatic accounts of its distribution in Mandarin. The current study investigates the role of grammatical weight and syntactic priming in the choice of verb-doubling in Cantonese. Following (Hawkins in Efficiency and complexity in grammars, Oxford University Press, Oxford, 2004) theory of efficiency and complexity in grammars, we hypothesized that speakers would choose verb-doubling over the canonical structure more often when the object NP was heavy, in order to minimize processing domains. In addition, we expected an effect of syntactic priming whereby the choice of structure is influenced by a previously encountered structure. The results of two elicited production experiments revealed no weight-based preference for verb-doubling, and only minor effects of grammatical weight, but clear effects of syntactic priming: (1) for both canonical primes and verb-doubling primes, speakers tended to repeat previously heard structures; (2) the priming effect was just as strong in the heavy NP condition, where speakers made more errors recalling the semantic content of the sentence, suggesting that semantic information is represented separately from syntactic information; (3) the priming effect was stronger for informationally accurate responses, suggesting that recently activated structures are easier to produce than other structures under conditions of increased cognitive load. We conclude that the choice between verb-doubling and canonical structure in Cantonese is easily influenced by the structure of recently encountered sentences, with weight effects, if any, being more difficult to detect.     

The theory of boolean algebras with an additional binary operation

This paper deals with Boolean algebras supplied with an additional binary operation, calledB-algebras for short.The aim of the paper is to generalize some theorems concerning topological Boolean algebras to more comprehensive classes ofB-algebras, to formulate fundamental properties ofB-algebras, and to find more important relationships of these algebras to other known algebras.The paper consists of two parts. At the beginning of the first one, several subclasses ofB-algebras are distinguished, and then, their basic properties, connections between them as well as certain relationships with other algebras, are investigated. In particular, it is shown that the class of Boolean algebras together with an arbitrary unary operation is polynomially equivalent to the class ofB ₁-algebras.The second part of the paper is concerned with the theory of filters and congruences inB-algebras.     

How Truth Behaves When There’s No Vicious Reference

In The Revision Theory of Truth (MIT Press), Gupta and Belnap (1993) claim as an advantage of their approach to truth “its consequence that truth behaves like an ordinary classical concept under certain conditions—conditions that can roughly be characterized as those in which there is no vicious reference in the language.” To clarify this remark, they define Thomason models, nonpathological models in which truth behaves like a classical concept, and investigate conditions under which a model is Thomason: they argue that a model is Thomason when there is no vicious reference in it. We extend their investigation, considering notions of nonpathologicality and senses of “no vicious reference” generated both by revision theories of truth and by fixed-point theories of truth. We show that some of the fixed-point theories have an advantage analogous to that which Gupta and Belnap claim for their approach, and that at least one revision theory does not. This calls into question the claim that the revision theories have a distinctive advantage in this regard.