How to eliminate self-reference: a précis

We provide a systematic recipe for eliminating self-reference from a simple language in which semantic paradoxes (whether purely logical or empirical) can be expressed. We start from a non-quantificational language L which contains a truth predicate and sentence names, and we associate to each sentence F of L an infinite series of translations h ₀(F), h ₁(F), ..., stated in a quantificational language L ^*. Under certain conditions, we show that none of the translations is self-referential, but that any one of them perfectly mirrors the semantic behavior of the original. The result, which can be seen as a generalization of recent work by Yablo (1993, Analysis, 53, 251–252; 2004, Self-reference, CSLI) and Cook (2004, Journal of Symbolic Logic, 69(3), 767–774), shows that under certain conditions self-reference is not essential to any of the semantic phenomena that can be obtained in a simple language.     

The non-definability notion and first order logic

The theorem to the effect that the languageL introduced in [2] is mutually interpretable with the first order language is proved. This yields several model-theoretical results concerningL .     

Quine’s Substitutional Definition of Logical Truth and the Philosophical Significance of the Löwenheim-Hilbert-Bernays Theorem

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S results in true sentences of L. For two reasons, this theorem is relevant to issues relative to Quine’s substitutional definition of logical truth. First, it makes it possible for Quine to reply to widespread objections raised against his account (the lexicon-dependence problem and the cardinality-dependence problem). These objections purport to show that Quine’s account overgenerates: it would count as logically true sentences which intuitively or model-theoretically are not so. Second, since this theorem is a crucial premise in Quine’s proof of the equivalence between his substitutional account and the model-theoretic one, it enables him to show that, from a metamathematical point of view, there is no need to favour the model-theoretic account over one in terms of substitutions. The purpose of that essay is thus to explore the philosophical bearings of the Löwenheim-Hilbert-Bernays theorem on Quine’s definition of logical truth. This neglected aspect of Quine’s argumentation in favour of a substitutional definition is shown to be part of a struggle against the model-theoretic prejudice in logic. Such an exploration leads to reassess Quine’s peculiar position in the history of logic.     

Written Argumentation by a 10-Year-Old Pupil in Sweden

Most pupils become confident with narrative texts. However, studies show that pupils do not learn to master discursive genres in a satisfactory way. Therefore it is important to study pupils’ written argumentation and to develop knowledge about text production in an education that also highlights linguistic structures. The present article investigates written argumentations produced by 10–12 year-old pupils. The aim is to investigate perspectives in the texts, and thereby catch the entire texts—their content, function and form—and to relate text analysis to interaction in the classroom. The theoretical framework emanates from the dialogical and triadic conception of language and text where ideational, relational and textual aspects play a central role. In this article the focus is on the three perspective dimensions—relief, hierarchy and sequences—in one argumentative text written by a 10-year-old girl and the discursive practice within which her text is produced.     

Same-Sex Marriage and Equality

Some argue that same-sex marriage is not an equal rights issue because, where same-sex marriage is illegal, heterosexuals and homosexuals have the exact same right to marry—i.e., the right to marry one adult of the opposite sex. I dispute this argument by pointing out that while societies that prohibit same-sex marriage equally permit individual heterosexuals and homosexuals to marry one adult of the opposite sex, same-sex couples in such societies are denied an important right that opposite-sex couples enjoy—i.e., the right to marry. I argue that the right to marry is fundamentally, not an individual right, but a couple’s collective right, analogous to assembly rights.     

The impossibility of a bivalent truth-functional semantics for the non-Boolean propositional structures of quantum mechanics

Summary The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular latticesP _L or partial-Boolean algebrasP _A; according to whether the semantic mappings are required to be truth-functional or truth-functional ; and according to whether two-or-higher dimensional Hilbert spaceP structures or three-or-higher dimensional Hilbert spaceP structures are being considered. If the quantumP structures are taken to be orthomodular latticesP _L, then bivalent mappings which preserve the operations and relations of aP _L must be truth-functional . Then as suggested by von Neumann and Jauch-Piron and as proven in this paper, the mere presence of incompatible elements in aP _L is sufficient to rule out any semantical or hidden-variable proposal which imposes this strong condition, for anytwo-or-higher dimensional Hilbert spaceP _L structure. Thus from the orthomodular lattice perspective, the peculiarly non-classical feature of quantum mechanics and the peculiarly non-Boolean feature of the quantum propositional structures is the existence of incompatible magnitudes and propositions. However, the weaker truth-functionality condition can instead be imposed upon the semantic or hidden-variable mappings on theP _L structures, although such mappings ignore the lattice meets and joins of incompatibles and preserve only the partial-Boolean algebra structural features of theP _L structures. Or alternatively, the quantum propositional structures can be taken to be partial-Boolean algebrasP _A, where bivalent mappings which preserve the operations and relations of aP _A need only be truth-functional (c). In either case, the Gleason, Kochen-Specker proofs show that any semantical or hidden variable proposal which imposes this truth-functionality (c) condition is impossible for anythree-or-higher dimensional Hilbert spaceP _A orP _L structures. But such semantical or hidden-variable proposals are possible for any two dimensional Hilbert spaceP _A orP _L structures, in spite of the presence of incompatibles in these structures, in spite of the fact that Heisenberg's Uncertainty Principle applies to the incompatible elements in these structures, and in spite of the fact that these structures are non-Boolean in the Piron sense. The present paper is a sequel of the proceedings of theSociety of Exact Philosophy annual meeting, published in our issue 9:2, pp. 187–278. — Ed. I am indebted to my supervisor, Dr. Edwin Levy, for many hours of helpful discussion on the drafts leading to this paper.     

Proof Systems Combining Classical and Paraconsistent Negations

New propositional and first-order paraconsistent logics (called L _ω and FL _ω , respectively) are introduced as Gentzen-type sequent calculi with classical and paraconsistent negations. The embedding theorems of L _ω and FL _ω into propositional (first-order, respectively) classical logic are shown, and the completeness theorems with respect to simple semantics for L _ω and FL _ω are proved. The cut-elimination theorems for L _ω and FL _ω are shown using both syntactical ways via the embedding theorems and semantical ways via the completeness theorems. Presented by Yaroslav Shramko and Heinrich Wansing     

Don’t Ever Do That! Long-term Duties in <Emphasis Type="Italic">PD</Emphasis><Subscript><Emphasis Type="Italic">e</Emphasis></Subscript><Emphasis Type="Italic">L</Emphasis>

This paper studies long-term norms concerning actions. In Meyer’s Propositional Deontic Logic (PD _e L), only immediate duties can be expressed, however, often one has duties of longer durations such as: “Never do that”, or “Do this someday”. In this paper, we will investigate how to amend PD _e L so that such long-term duties can be expressed. This leads to the interesting and suprising consequence that the long-term prohibition and obligation are not interdefinable in our semantics, while there is a duality between these two notions. As a consequence, we have provided a new analysis of the long-term obligation by introducing a new atomic proposition I (indebtedness) to represent the condition that an agent has some unfulfilled obligation. Presented by Jacek Malinowski     

Satisfying Predicates: Kleene's Proof of the Hilbert–Bernays Theorem

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of the arithmetization is still regarded as the standard one. It is highly compressed, however, and very difficult to read. My goals in this paper are expository: to present the basics of Kleene's arithmetization in a less compressed, more easily readable form, in a setting that highlights its relevance to issues in the philosophy of logic, especially to Quine's substitutional definition of logical truth, and to formulate the Hilbert–Bernays Theorem in a way that incorporates Kreisel's, Mostowski's, and Putnam's refinements of Kleene's result.     

The inclosure scheme and the solution to the paradoxes of self-reference

Proponents of the explanatory gap claim that consciousness is a mystery. No one has ever given an account of how a physical thing could be identical to a phenomenal one. We fully understand the identity between water and H₂O but the identity between pain and the firing of C-fibers is inconceivable. Mark Johnston [Journal of philosophy (1997), 564–583] suggests that if water is constituted by H₂O, not identical to it, then the explanatory gap becomes a pseudo-problem. This is because all “manifest kinds”—those identified in experience—are on a par in not being identical to their physical bases, so that the special problem of the inconceivability of ‘pain = the firing of C-fibers’ vanishes. Moreover, the substitute relation, constitution, raises no explanatory difficulties: pain can be constituted by its physical base, as can water. The thesis of this paper is that the EG does not disappear when we substitute constitution for identity. I examine four arguments for the EG, and show that none of them is undermined by the move from constitution to identity.     

On logic of complex algorithms

An algebraic approach to programs called recursive coroutines — due to Janicki [3] — is based on the idea to consider certain complex algorithms as algebraics models of those programs. Complex algorithms are generalizations of pushdown algorithms being algebraic models of recursive procedures (see Mazurkiewicz [4]). LCA — logic of complex algorithms — was formulated in [11]. It formalizes algorithmic properties of a class of deterministic programs called here complex recursive ones or interacting stacks-programs, for which complex algorithms constitute mathematical models. LCA is in a sense an extension of algorithmic logic as initiated by Salwicki [14] and of extended algorithmic logic EAL as formulated and examined by the present author in [8], [9], [10]. In LCA — similarly as in EAL-ω ⁺ -valued logic is applied as a tool to construct control systems (stacks) occurring in corresponding algorithms. The aim of this paper is to give a complete axiomatization. of LCA and to prove a completeness theorem. Logic of complex algorithms was presented at FCT'79 (International Symposium on Fundamentals of Computation Theory, Berlin 1979)     

A Few More Useful 8-valued Logics for Reasoning with Tetralattice <Emphasis Type="Italic">EIGHT</Emphasis><Subscript>4</Subscript>

In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN ₃ with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are a — incoming data is asserted, d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT ₄. Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a–order (truth order) and d–order (falsity order) coincide with first-degree entailment. Finally logic with two kinds of operations (a–connectives and d–connectives) and consequence relation defined via a–ordering is considered. An adequate axiomatization for this logic is proposed.     

Simmering in the Soviet pot: language heterogeneity in early Soviet socio-linguistics

At the beginning of the ’30s—the period of lively debates on the relation between language and society—one of the main issues in linguistics was language heterogeneity. On the example of the texts by Boris Larin, Georgij Danilov and Lev Jakubinskij we shall compare two attitudes about unity and division of a language. If the studies by Larin and Danilov in various ways establish divisions in society and language at the end of the ’20s, in the ’30s there is a marked tendency to recognize language unity and the cohesiveness of the proletarian society, as seen in socio-linguistic analyses by Jakubinskij. The conclusion, suggested at the end of this exposition, claims that the idea of one national language grows in importance in the discourse of the Soviet linguistics at the beginning of 1930s. Disappearance of the contemporary language heterogeneity in the discourse of Soviet linguists of the period corroborates how linguistics adapts to the political conceptions of society.

Henkin Quantifiers and the Definability of Truth

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L _* ¹(H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L _* ¹(H) with respect to Boolean operations, and obtain the language L ¹(H). At the next level, we consider an extension L _* ²(H) of L ¹(H) in which every sentence is an L ¹(H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)-definability of truth – in – N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996).     

Transcendental Arguments and Practical Reason in Indian Philosophy

This paper examines some Indian philosophical arguments that are understandable as transcendental arguments—i.e., arguments whose conclusions cannot be denied without self-contradiction, insofar as the truth of the claim in question is a condition of the possibility even of any such denial. This raises the question of what kind of self-contradiction is involved—e.g., pragmatic self-contradiction, or the kind that goes with logical necessity. It is suggested that these arguments involve something like practical reason—indeed, that they just are arguments against the primacy of “theoretical reason.” This characterization illuminates a characteristically Indic appeal to ordinary language.

Horizons of hermeneutics: Intercultural hermeneutics in a globalizing world

Starting from the often-used metaphor of the “horizon of experience” this article discusses three different types of intercultural hermeneutics, which respectively conceive hermeneutic interpretation as a widening of horizons, a fusion of horizons, and a dissemination of horizons. It is argued that these subsequent stages in the history of hermeneutics have their origin in—but are not fully restricted to—respectively premodern, modern and postmodern stages of globalization. Taking some striking moments of the encounter between Western and Chinese language and philosophy as example, the particular merits and flaws of these three types of hermeneutics are being discussed. The claim defended is that although these different types of hermeneutics are mutually exclusive from a theoretical point of view, as interpreting beings in the current era we depend on each of these distinct hermeneutic practices and cannot avoid living them simultaneously.     

Gender Bender: Gender Errors in L2 Pronoun Production

To address questions about information processing at the message level, pronoun errors of second language (L2) speakers of English were studied. Some L2 pronoun errors—he/she confusions by Spanish speakers of L2 English—could be due to differences in the informational requirements of the speakers’ two languages, providing a window into the composition of the preverbal message that guides grammatical encoding during language production. To study this, Spanish and French speakers of L2 English were made to answer questions designed to elicit pronouns. Spanish speakers produced significantly more gender errors than any other type of pronoun error, and significantly more gender errors than French speakers. The results are as expected given that Spanish is a pro-drop language where many sentences would not have required gender to be encoded had they been uttered in the L1 instead of the L2. The implications for theories about the preverbal message are discussed.     

Iterated Belief Change and the Recovery Axiom

The axiom of recovery, while capturing a central intuition regarding belief change, has been the source of much controversy. We argue briefly against putative counterexamples to the axiom—while agreeing that some of their insight deserves to be preserved—and present additional recovery-like axioms in a framework that uses epistemic states, which encode preferences, as the object of revisions. This makes iterated revision possible and renders explicit the connection between iterated belief change and the axiom of recovery. We provide a representation theorem that connects the semantic conditions we impose on iterated revision and our additional syntactical properties. We show interesting similarities between our framework and that of Darwiche–Pearl (Artificial Intelligence 89:1–29 1997). In particular, we show that intuitions underlying the controversial (C2) postulate are captured by the recovery axiom and our recovery-like postulates (the latter can be seen as weakenings of (C2)). We present postulates for contraction, in the same spirit as the Darwiche–Pearl postulates for revision, and provide a theorem that connects our syntactic postulates with a set of semantic conditions. Lastly, we show a connection between the contraction postulates and a generalisation of the recovery axiom. Portions of this paper were originally presented at ECAI 2002.     

Slowed Speech Input has a Differential Impact on On-line and Off-line Processing in Children’s Comprehension of Pronouns

The central question underlying this study revolves around how children process co-reference relationships—such as those evidenced by pronouns (him) and reflexives (himself)—and how a slowed rate of speech input may critically affect this process. Previous studies of child language processing have demonstrated that typical language developing (TLD) children as young as 4 years of age process co-reference relations in a manner similar to adults on-line. In contrast, off-line measures of pronoun comprehension suggest a developmental delay for pronouns (relative to reflexives). The present study examines dependency relations in TLD children (ages 5–13) and investigates how a slowed rate of speech input affects the unconscious (on-line) and conscious (off-line) parsing of these constructions. For the on-line investigations (using a cross-modal picture priming paradigm), results indicate that at a normal rate of speech TLD children demonstrate adult-like syntactic reflexes. At a slowed rate of speech the typical language developing children displayed a breakdown in automatic syntactic parsing (again, similar to the pattern seen in unimpaired adults). As demonstrated in the literature, our off-line investigations (sentence/picture matching task) revealed that these children performed much better on reflexives than on pronouns at a regular speech rate. However, at the slow speech rate, performance on pronouns was substantially improved, whereas performance on reflexives was not different than at the regular speech rate. We interpret these results in light of a distinction between fast automatic processes (relied upon for on-line processing in real time) and conscious reflective processes (relied upon for off-line processing), such that slowed speech input disrupts the former, yet improves the latter.     

Original mind and cosmic consciousness in the co-creative process

This article will investigate the issue of accessing benxin 本心 (original mind), subsequent operation from Self and, in that process, union with the “greater universe” or benti 本体 (original substance)—a state expressed in the West as “cosmic consciousness.” It is proposed that this allows one to participate as a partner in the creative process of one’s own life and the surrounding world. The equally important question of how to gain contact with original mind will also be addressed, as well as the consequences of doing so with regard to the human condition. The concept of original thought is introduced, being important here as it is held to be that thought which is generated in the pure condition of original mind, devoid of influence from finite physical existence.