Various authors of logic texts are cited who either suggest or explicitly state that the Gödel incompleteness result shows that some unprovable sentence of arithmetic is true. Against this, the paper argues that the matter is one of philosophical controversy, that it is not a mathematical or logical issue.

Summarizing a surrounding 200 pages, pages 179 to 190 of Shadows of the Mind contain a future dialog between a human identified as "Albert Imperator" and an advanced robot, the "Mathematically Justified Cybersystem", allegedly Albert's creation. The two have been discussing a Gödel sentence for an algorithm by which a robot society named SMIRC certifies mathematical proofs. The sentence, referred to in mathematical notation as Omega(Q*), is to be precisely constructed from on a definition of SMIRC's algorithm. It can (...) be interpreted as stating "SMIRC's algorithm cannot certify this statement." The robot has asserted that SMIRC never makes mistakes. If so, SMIRC's algorithm cannot certify the Goedel sentence, for that would make the statement false. But, if they can't certify it, what is says is true! Humans can understand it is true, but mighty SMIRC cannot certify it. The dialog ends melodramatically as the robot, apparently unhinged by this revelation, claims to be a messenger of god, and the human shuts it down with a secret control. (shrink)

According to the received view, formalism – interpreted as the thesis that mathematical truth does not outrun the consequences of our maximal mathematical theory – has been refuted by Goedel's theorem. In support of this claim, proponents of the received view usually invoke an informal argument for the truth of the Goedel sentence, an argument which is supposed to reconstruct our reasoning in seeing its truth. Against this, Field has argued in a series of papers that the principles involved in (...) this argument – when applied to our maximal mathematical theory – are unsound. This paper defends the received view by showing that there is a way of seeing the truth of the Goedel sentence which is immune to Field's strategy. (shrink)

The presented paper takes up the attempt to analyse and specify the suspicion that some modal rules of inference are paralogical in application to non-logical reasonings (s.c. modal fallacy). The considerations have been limited to modal prepositional calculi: K and S5, which are intended to be a formal base of these non-logical reasonings - proofs of so called specific thesis on the grounds of the particular specific theories. Pointing out the properties of being permitted, being valid and being derivable in (...) case of inferences rules and also semantical relations of point, structure, frame and inferential consequence in standard semantics of possible worlds, enables to define two kinds of paralogism: point and structural. Justification of the suspicion of modal fallacy occurrence in the case of a given inference rule, depends on pointed metalogical properties of this rule and also on what kind of the notion of paralogism is being discussed. It appears that when a given rule is paralogical only pointly (and not structurally), the sufficient condition of avoiding modal fallacy is to consider the specific axioms of the given specific theory as the sentences which are structurally true (structural truth is of course not equivalent to logical truth). If we want to treat these axioms as sentences which are pointly true, we have to eliminate pointly paralogical rules. In this case it is enough to construct such axiomatisation of calculi K and S5, in which we use the notion of modal closure (it eliminates the primitive rule of Goedel and all rules derivable from it - rules which are structurally but not pointly correct). (shrink)

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6]. In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a (...) model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3, which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. The resulting model-theoretic structure is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli. We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete. In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest. (shrink)

We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of (...) the existential Yablo paradox. We also look at a formulation which employs Rosser’s provability predicate. (shrink)

The argument is a dialectical one. It is not a direct proof that the mind is something more than a machine, but a schema of disproof for any particular version of mechanism that may be put forward. If the mechanist maintains any specific thesis, I show that [146] a contradiction ensues. But only if. It depends on the mechanist making the first move and putting forward his claim for inspection. I do not think Benacerraf has quite taken the point. He (...) criticizes me both for "failing to notice" that my ability to show that the Gödel sentence of a formal system is true "depends very much on how he is given. (shrink)

The Jewish religious tradition summons its adherents to save life. For religious Jews preservation of life is the ultimate religious commandment. At the same time Jewish law recognizes that the agony of a moribund person may not be stretched. When the time to die has come this has to be respected. The process of dying should not needlessly be prolonged. We discuss the position of two prominent Orthodox Jewish authorities – the late Rabbi Moshe Feinstein and Rabbi J David Bleich (...) – towards the role of life-sustaining treatment in end-of-life care. From the review, the characteristic halachic and heterogeneous character of Jewish ethical reasoning appears. The specificity of Jewish dealing with ethical dilemmas in health care indicates the importance for contemporary healthcare professionals of providing care which is sensitive to a patient’s culture and worldview. (shrink)

The Diagnostic and Statistical Manual of Mental Disorders (5th ed.; DSM–5; American Psychiatric Association [APA], 2013) defines melancholia as “A mental state characterized by very severe depressi...

Information?rich environments are created to promote data use in schools for the purpose of self?evaluation and quality assurance. However, providing feedback does not guarantee that schools will actually put it to use. One of the main stumbling blocks relates to the interpretation and diagnosis of the information. This study examines the relationship between data literacy competences, support given in interpreting the information, actual use of the feedback and potential school improvement effect. A randomised field experiment with 188 school principals from (...) primary education was set up and a post?test was used to investigate the effects of a support initiative. The results revealed that a minority of schools invested significantly in the interpretation and diagnosis of the school performance feedback (SPF), despite the fact that most of the respondents showed an interest in the SPF report. In addition, data competence support and the subsequent use of feedback were found to be limited. (shrink)

This article develops a Gricean account for the computation of scalarimplicatures in cases where one scalar term is in the scope ofanother. It shows that a cross-product of two quantitative scalesyields the appropriate scale for many such cases. One exception iscases involving disjunction. For these, I propose an analysis that makesuse of a novel, partially ordered quantitative scale for disjunction andcapitalizes on the idea that implicatures may have different epistemic status.

Our knowledge about the world is often expressed by generic sentences, yet their meanings are far from clear. This book provides answers to central problems concerning generics: what do they mean? Which factors affect their interpretation? How can one reason with generics? Cohen proposes that the meanings of generics are probability judgments, and shows how this view accounts for many of their puzzling properties, including lawlikeness. Generics are evaluated with respect to alternatives. Cohen argues that alternatives are induced by (...) the kind as well as by the predicated property, and thus provides a uniform account of the varied interpretations of generics. He studies the formal properties of alternatives and provides a compositional account of their derivation by focus and presupposition. Cohen uses his semantics of generics to provide a formal characterization of adequate default reasoning, and proves some desirable results of this formalism. (shrink)

It is often thought that questions of reference are crucial in assessing scientific realism, construed as the view that successful theories are at least approximately true descriptions of the unobservable; realism is justified only if terms in empirically successful theories generally refer to genuinely existing entities or properties. In this paper this view is questioned. First, it is argued that there are good reasons to think that questions of realism are largely decided by convention and carry no epistemic significance. An (...) alternative conception of realism is then proposed, which focuses on the Ramsey sentences of scientific theories, constructed in the manner of David Lewis's 'How to define theoretical terms'. It is argued that because the Ramsey sentence of a theory preserves the epistemically significant part of the theory's content without generating commitments to any particular conclusions about reference, the realism issue is better addressed by asking whether Ramsey sentences of theories, rather than the theories themselves, are approximately true. (shrink)

The processing of words in sentence reading is influenced by both information from sentential context and information from previewing upcoming words, but how both effects interact during online reading is not clear. In this study, we tested the interaction of predictability effect and the preview effect in predicting reading processing. In the experiment, sentence constraint was controlled using all high-constraint sentences as materials. We manipulated both the predictability of the target word in the sentence and the semantic relationship between (...) the preview word and the target word as predictors of the semantic preview effect. The results showed that the semantic preview effect was present only when the target word had low-predictability in the sentence but was not observed when the target word had high-predictability in the sentence. The results suggest that contextual information in reading can modulate the pre-activation of words and thus influence whether the preview word has a priming effect. The results of this study provide further evidence that reading comprehension involves an interactive system of processing multiple sources of information at multiple levels. (shrink)

Semanticists face substitution challenges even outside of contexts commonly recognized as opaque. Jennifer M. Saul has drawn attention to pairs of simple sentences - her term for sentences lacking a that-clause operator - of which the following are typical: -/- (1) Clark Kent went into the phone booth, and Superman came out. (1*) Clark Kent went into the phone booth, and Clark Kent came out. -/- (2) Superman is more successful with women than Clark Kent. (2*) Superman is (...) more successful with women than Superman. -/- She challenges us to explain why the upper and lower sentences in each pair differ, or at least appear to differ, in their truth-values and hence truth-conditions. This appearance of substitution failure is inherently puzzling. Moreover, it is taken by Saul to generate a dilemma for anyone hostile to direct reference accounts of that-clause constructions. Direct reference theorists regard the appearance of substitution failure in that-clause contexts as mere appearance, to be dealt with pragmatically rather than semantically. -/- Critics of such accounts need to say something about simple-sentence cases. If they choose to allow that intuitions of substitution failure can be over-ridden and explained away pragmatically in simple-sentence cases but not in that-clause cases, they lay themselves open to the charge of operating a double standard. But if they do not choose this option, they must offer a semantic explanation of apparent substitution failure in simple-sentence cases - no easy task, it turns out. Other respondents to Saul's challenge have sought to provide elaborate semantic treatments. In contrast, this paper proposes a far simpler pragmatic explanation of intuitions of substitution failure in simple sentences, an explanation that deploys no more resources than are to be found in Grice's 'Logic and Conversation'. Ironically, this proposal turns out to be incompatible with a direct reference perspective. So if it is, as I maintain, the most plausible treatment of simple-sentence cases available, Saul's initial thought gets turned around 180 degrees: the phenomenon she has drawn attention to ends up representing a challenge to supporters of direct reference theories. (shrink)

The Goedelian approach is discussed as a prime example of a science towards the origins. While mere self­referential objectification locks in to its own by­products, self­releasing objectification informs the formation of objects at hand and their different levels of interconnection. Guided by the spirit of Goedel's work a self­reflective science can open the road where old tenets see only blocked paths. “This is, as it were, an analysis of the analysis itself, but if that is done it forms the fundamental (...) of human science, as far as this kind of things is concerned.” G. Leibniz, ('Methodus Nova ...', 1673) . (shrink)

In this paper, I show that the availability of what some authors have called the weak reading and the strong reading of donkey sentences with relative clauses is systematically related to monotonicity properties of the determiner. The correlation is different from what has been observed in the literature in that it concerns not only right monotonicity, but also left monotonicity (persistence/antipersistence). I claim that the reading selected by a donkey sentence with a double monotone determiner is in fact the (...) one that validates inference based on the left monotonicity of the determiner. This accounts for the lack of strong reading in donkey sentences with MON determiners, which have been neglected in the literature. I consider the relevance of other natural forms of inference as well, but also suggest how monotonicity inference might play a central role in the actual process of interpretation. The formal theory is couched in dynamic predicate logic with generalized quantifiers. (shrink)

The classical analysis of donkey sentences like (1.a,b) in Kamp (1981) and Heim (1982) assigns them truth conditions as given in (2.a). That is, they are treated as quantifications over farmer-donkey pairs. Partee (1984) and Kadmon (1987) have pointed out that the proper reading of (1.b), and a preferred reading of (1.a), is rather a quantification over farmers, as illustrated in (2.b).

We call a semantic theory 'classical' if it includes the assertions that (I) a function V assigning semantic value maps object language proper names into some set D, (ii) V maps object language atomic sentences into some set F, and (iii) the extension of any object language unary predicate is a member of the power set of D. Two theorems can be proven which assert that any classical theory which includes certain other assumptions assigns the same member of F (...) to every true object language sentence. Many accept the following argument: (1) every plausible semantic theory is classical and contains the assertions named in the theorems, (2) if the semantic value of declarative sentences is a representation or representational then, some different true sentences differ in what they represent, hence, declarative sentences are not representational. I show how to avoid the conclusion by arguing for the falsity of (1). (shrink)

In this paper, I argue for two principles to determine the temporal interpretation of modal sentences in English, given a theory in which modals are interpreted against double conversational backgrounds and an ontology in which possible worlds branch towards the future, The Disparity Principle requires that a modal sentence makes distinctions between worlds in the modal base. The Non- disparity Principle requires that a modal sentence does not make distinctions on the basis of facts settled at speech time. Selection (...) of the modal base will set these principles against each other, or allow for their cooperative interaction. For a root modal base, there is a conflict and disparity wins. The resulting interpretation is future. For a non-root modal base, the principles cooperate. Non-disparity determines a non-future interpretation and disparity requires the sentence to go beyond what is known by the speaker. (shrink)

Is native speaker variation in understanding complex sentences due to individual differences in working memory capacity or in syntactic competence? The answer to this question has very important consequences for both theoretical and applied concerns in linguistics and education. This book is distinctive in giving an historical and interdisciplinary perspective on the rule- based and experience-based debate and in supporting an integrated account. In the study reported here, variation was found to be due to differences in syntactic competence and (...) the author argues that sentence comprehension is a learned skill, displaying many of the general characteristics of cognitive skills. The book will be stimulating reading for psycholinguists, theoretical linguists, applied linguists and educators. (shrink)

The syntax of Frege's scientific language is commonly taken to be characterized by two oddities: the representation of the intended illocutionary role of sentences by a special sign, the judgement-stroke, and the treatment of sentences as a species of singular terms. In this paper, an alternative view is defended. The main theses are: the syntax of Frege's scientific language aims at an explication of the logical form of judgements; the judgement-stroke is, therefore, a truth-operator, not a pragmatic operator; (...) in Frege's first system, '⊦ Δ' expresses that the circumstance Δ is a fact, and in his second system that the truth-value - Δ is the True; in both systems, the judgement-stroke is construed as a sign sui generis, not as a genuine predicate; its counterpart in natural language is the syntactic "form of assertoric sentences", not the truth-predicate; neither in Frege's first nor in his second system sentences are treated as singular terms. (shrink)

Guaspari (J Symb Logic 48:777–789, 1983) conjectured that a modal formula is it essentially Σ1 (i.e., it is Σ1 under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form ${\square{B}}$ . This conjecture was proved first by A. Visser. Then, in (de Jongh and Pianigiani, Logic at Work: In Memory of Helena Rasiowa, Springer-Physica Verlag, Heidelberg-New York, pp. 246–255, 1999), the authors characterized essentially Σ1 formulas of languages including witness comparisons (...) using the interpretability logic ILM. In this note we give a similar characterization for formulas with a binary operator interpreted as interpretability in a finitely axiomatizable extension of IΔ 0 + Supexp and we address a similar problem for IΔ 0 + Exp. (shrink)

In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus. We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.

It is widely considered that Gödel’s and Rosser’s proofs of the incompleteness theorems are related to the Liar Paradox. Yablo’s paradox, a Liar-like paradox without self-reference, can also be used to prove Gödel’s first and second incompleteness theorems. We show that the situation with the formalization of Yablo’s paradox using Rosser’s provability predicate is different from that of Rosser’s proof. Namely, by using the technique of Guaspari and Solovay, we prove that the undecidability of each instance of Rosser-type (...) formalizations of Yablo’s paradox for each consistent but not Σ1-sound theory is dependent on the choice of a standard proof predicate. (shrink)

How are non-declarative sentences understood? How do they differ semantically from their declarative counterparts? Answers to these questions once made direct appeal to the notion of illocutionary force. When they proved unsatisfactory, the fault was diagnosed as a failure to distinguish properly between mood and force. For some years now, efforts have been under way to develop a satisfactory account of the semantics of mood. In this paper, we consider the current achievements and future prospects of the mood-based semantic (...) programme. (shrink)

There is a distinctive kind of command, namely commands to answer specific questions. An imperative sentence denoting such a command has an interrogative sentence corresponding to it-a sentence denoting the respective question. LetImp, Int, andQ be such an imperative sentence, the interrogative sentence corresponding to it, and the question denoted by the interrogative sentence, respectively. LetQ be an empirical question, i. e., and ((ητ)ω)-object. LetP be an ((ητ)ω)-construction constructingQ. Then the analysis ofImp has the form (QL).LetQ be an analytical question, (...) i. e., a molecular extensional η-construction. LetQ=C, whereC is a ℊ-object (over the extended base). Then the analysis ofImp has the form (QL′). The imperative sentences whose analyses have the form (QL) or (QL′) can be called ‘question-like imperative sentences’ or simply ‘QL-imperative sentences’. No imperative sentence whose analysis differs from (QL) and (QL\t') can be associated with a question. Therefore, no transformation of such an imperative sentence should result in a correct interrogative sentence. Fulfilling an order which is determined by a QL-imperative sentenceImp, one simultaneously gives the right answer to the questionQ (which is associated withImp). One who does not fulfil this order either wrongly answersQ or does not answerQ at all. (shrink)