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.

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)

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)

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)

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)

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)

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 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)

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)

‘Epistemic structural realism’ (ESR) insists that all that we know of the world is its structure, and that the ‘nature’ of the underlying elements remains hidden. With structure represented via Ramsey sentences, the question arises as to how ‘hidden natures’ might also be represented. If the Ramsey sentence describes a class of realisers for the relevant theory, one way of answering this question is through the notion of multiple realisability. We explore this answer in the context of the work (...) of Carnap, Hintikka and Lewis. Both Carnap and Hintikka offer clear structuralist perspectives which, crucially, accommodate the openness inherent in theory change. Unfortunately there is little purchase for a viable form of realism in either case. Lewis’s approach, on the other hand, offers more scope for realism but, as we shall see, concerns arise as to whether a relevant form of structuralism can be maintained. In particular his thesis of Ramseyan humility undermines certain conceptions of scientific laws that the structural realist might naturally cleave to. Our overall conclusion is that the representational device of Ramsey sentence plus multiple realisability can accommodate either the structuralist or realist aspects of ESR but has difficulties capturing both. (shrink)

Frege-Geach worries about embedding and composition have plagued metaethical theories like emotivism, prescriptivism and expressivism. The sharpened point of such criticism has come to focus on whether negation and inconsistency have to be understood in descriptivist terms. Because they reject descriptivism, these theories must offer a non-standard account of the meanings of ethical and normative sentences as well as related semantic facts, such as why certain sentences are inconsistent with each other. This paper fills out such a solution (...) to the negation problems, following some of the original emotivist ideas about the interplay of interests in conversation. We communicate both to share information and coordinate our actions, and we use distinctively normative language like deontic ‘must’ and ‘may’ to negotiate what people are to do. The kinds of disagreement involved in such negotiation can illuminate the issues with negation and inconsistency. This paper outlines a dynamic semantic system in which these ideas can bear fruit, developing the scorekeeping model of conversation. The result is clarification about what Frege-Geach worries can mean for nondescriptive semantics. (shrink)

This paper is about what we may provisionally call “indicative” conditionals. It aims to describe one use of the word ‘if’, by giving the truth conditions of sentences using ‘if’ in the way in question. Here are some sentences that, on their natural interpretations, illustrate the target use.

Keith Donnellan’s distinction between the attributive and referential uses of definite descriptions has spurred debates regarding the truth conditions of the utterance “the F is G” with definite descriptions used referentially. In this article, I present a semantic account of referential descriptions, grounded in the contextual factors of the utterance, including the speaker’s intention and presupposition as well as the interlocutor’s recognition of them. This account is called the IPR-semantic account, according to which the speaker’s intention (I), presupposition (P), and (...) the interlocutor's recognition (R) jointly determine whether “the F” in an utterance “the F is G” is used referentially or attributively, and the meaning of “the F” is determined by whether it is used referentially or attributively. Moreover, I argue that the meaning of the referential description “the F” is the intended object e, embodied with a property H that has prompted the speaker to presuppose that e is F and to intend to use “the F” to refer to e, as well as the interlocutor to recognize the presupposition and intention. According to the IPR-semantic account, the utterance “the F is G” with “the F” used referentially expresses a singular proposition, namely, that e is G, and it is true if and only if the intended object e is G. Additionally, I argue that the IPR-semantic account not only surpasses some alternative semantic accounts but also outperforms Kripke’s pragmatic account. (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)

It is typically assumed that actions are events, but there is a growing consensus that negative actions, like omissions and refrainments, are not events, but absences thereof. If so, then we must either deny the obvious, that we can exercise our agency by omitting and refrainment, or give up on event-based theories of agency. I trace the consensus to the assumption that negative action sentences are negative-existentials, and argue that this is false. The best analysis of negative action (...) class='Hi'>sentences treats them as quantifying over omissions and refrainments, conceived of as events. (shrink)

Dogs display vast phenotypic diversity, including differences in height, skull shape, tail, etc. Yet, humans are almost always able to quickly recognize a dog, despite no single feature or group of features are critical to distinguish dogs from other objects/animals. In search of the mental activities leading human individuals to state “I see a dog”, we hypothesize that the brain might extract meaningful information from the environment using Ramsey sentences-like procedures. To turn the proposition “I see a dog” in (...) a Ramsey sentence, the term dog must be replaced by a long and complex assertion consisting only of observational terms, existential quantifiers and operational rules. The Ramsey sentence for “I see a dog” sounds: “There is at least an entity called dog which satisfies the following conditions: it is an animal, it has four legs, …, etc, …, and is something that I have in my sight”. We discuss the biological plausibility and the putative neural correlates of a Ramsey-like mechanism in the central nervous system. We accomplish a brain-inspired, theoretical neural architecture consisting of a parallel network that requires virtually no memory, is devoid of probabilistic choices and can analyze huge but finite amounts of unique visual details, combining them into a single conceptual output. In sum, Ramsey sentence stands for a versatile tool that can be used not just as a methodological device to cope with biophysical affairs, but also for a model to describe the real functioning of cognitive operations such as sensation and perception. (shrink)

In this paper I distinguish normative and descriptive reasons for attempting to construct a logic for belief sentences, and argue that because the interpretation of the content of an attribution of belief is context sensitive and ambiguous, no simple logic is adequate.

The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of 'true but unprovable' sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk's Logic, where modality can be interpreted as 'true and provable'. As we show, GS and Grzegorczyk's Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of 'Essence and Accident' (...) proposed by Marcos (Bull Sect Log 34(1):43-56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic. (shrink)

In this chapter, I am going to discuss a very interesting case brought to our attention by Saul and references therein: NP-related substitution failure in simple sentences. Whereas it is well known that opacity occurs in intensional contexts and that in such contexts it is not licit to replace an NP with a co-referential one, one would not expect that substitution failure should also be exhibited by simple sentences in the context of stories about Superman. The suggested explanation (...) of these cases is to posit an embedding explicature, that is to say the insertion of structure that ipso facto creates an intensional context capable of blocking substitution. I consider various complications to this story in the light of important objections by García-Carpintero and, finally, I consider how this story fares when one applies constraints on explicatures along the lines of those proposed by Hall in an interesting paper.In general, this chapter exploits interesting considerations by Norrick on the structural similarities between stories and indirect reports. Norrick believes there are important differences, but he is inclined to concede that we could study structural similarities. An important similarity, brought out by the examples discussed by Saul, is that the narrative frame, once it is inserted into the interaction, can be left implicit and, during the act of narrating or referring to the story, one need not repeat the words ‘the story says’ or ‘we are told that…’ every time. Although implicit, these words are heard because they do some work at the structural level, as is shown by this attempt to resolve an otherwise intractable philosophical problem. The explicatures of simple sentences are perceived because they are integrated into the speakers’/hearers’ perception of the overall plan of discourse, as Haugh most interestingly notes:As Haugh and Jaszczolt note, this means that any putative “communicative intention of A is embedded within his higher-order intention”. In other words, to figure out the implicature that evidently arises here, the participants are necessarily making inferences about some kind of overall aim. According to this view, then, inferences about the intended implicature arise concomitant with inferences about the overall aim of the speaker.It follows from the considerations by Haugh that, since the explicature connected with simple sentences depends on the perception of the overall aim or plan of the conversation, it is not easily cancelled. Readers can check by themselves that the explicatures due to simple sentences cannot be cancelled, as cancelling them would involve returning to illogical discourses.. Haugh’s considerations about the overall aim of the discourse are precious in explaining how the embedding explicatures I posit are calculated once and for all for the whole stretch of the discourse framed by the narrative act. (shrink)

I distinguish between instantial sentences (_There are elephants that swim_), particular quantification, and predication of existence in natural language. I explore the logical relation between the first two, while the last one is shown independent of either. I continue to consider the incorporation of the three kinds of sentence in the Quantified Argument Calculus (Quarc). I provide formalisations that preserve the logical relations specified earlier. I also extend the analysis to quantified instantial sentences (_There are_ five _elephants that (...) swim_) and modal ones. A conclusion of this work is that quantification and instantial sentences provide no guide to ontology. (shrink)

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)

Some Trinitarians, such as Thomas Aquinas, wish to claim that God is mereologically simple; that is, God has no parts distinct from Himself. In this paper, I present Simple Trinitarianism, a view that takes God to be simple but, diverging from Aquinas, does not identify the Father, Son, and Holy Spirit with anything in our ontology. Nonetheless, Simple Trinitarians would like Trinitarian sentences to be true; thus, they must give a non-standard semantics for those sentences. I will focus (...) on one possible semantics a Simple Trinitarian may give: taking Trinitarian claims to be translatable into feature-placing sentences, which posit property instantiation without requiring commitment to any objects that instantiate those properties. (shrink)

Klasyczny rachunek zdań w ujęciu semantycznym, uważany niekiedy za idealizację opisowej części języka naturalnego czy też za teorię spójników zdaniowych, traktuje negację jako czystą funkcję prawdziwościową. Przyzwyczajeni do takiego ujęcia, uważamy je za adekwatne, ściśle oddające sens, jaki ma ten funktor w zdaniach opisowych. Tymczasem okazuje się, że w refleksji filozoficznej przypisywano negacji i inne, bardzo różnorodne role. Na przykładzie tego, co o negacji i zdaniach przeczących mówią: Parmenides, sofiści, Platon, Arystoteles, a z filozofów nowożytnych i współczesnych Kant, Hegel, Bergson, (...) fenomenologowie Reinach i Ingarden, Ludwig Wittgenstein, przedstawiciele brytyjskiej filozofii analitycznej J.L. Austin i A.J. Ayer czy wreszcie Martin Heidegger, można zauważyć zależność funkcji semantycznych przypisywanych negacji od takich czy innych rozstrzygnięć ontologicznych dokonywanych przez tych myślicieli. Pamiętając o tym, warto jednak wyróżnić interpretację negacji jako funkcji prawdziwościowej - nie dlatego, żeby była ona jedynie poprawna, lecz ze względu na to, że interpretacja taka jako najmniej zaangażowana ontologicznie daje się uzgodnić z większością innych tu wymienionych. W tym i tylko tym znaczeniu może ona być uważana za uprzywilejowaną w stosunku do innych. (shrink)

In this dissertation I provide a tenseless account of tensed sentences and tensed belief. I begin by distinguishing tensed theories of time from tenseless theories of time. Tensed theories of time hold that there is a time that is objectively present, and that the moment that is objectively present changes from one moment to the next. I reject tensed theories of time. I deny that there is a time that is objectively present that changes from one moment to the (...) next. Instead, time is in many ways analogous to space and 'now' is in many ways analogous to 'here'. Just as there is no one location that is objectively here, there is no one moment that is objectively now. Nothing metaphysically distinguishes the present moment from past and future moments. ^ There are two major challenges facing the tenseless theory of time. The first challenge is: can a tenseless theory of time provide a plausible account of the truth-conditions for tensed sentences? Tensed sentences such as 'The meeting is now!' provide a prima facie reason in favor of irreducibly tensed properties: the sentence is true if and only if the meeting has the tensed property of being now. I examine existing attempts by tenseless theorists for providing tenseless truth-conditions for tensed sentences. I argue that they are unable to provide a complete account of what makes tensed sentences true. I formulate and defend a new tenseless account of the truth-conditions for tensed sentences, the type-context theory, which makes use of David Kaplan's theory of indexicals. ^ The second challenge facing the tenseless theory of time is: can a tenseless theory of time provide a plausible account of our tensed psychological attitudes? Most of our psychological attitudes such as belief, desire, relief, regret and anticipation are irreducibly tensed: we cannot adequate describe such attitudes without appealing to tensed language. I do not take relief in the tenseless fact that I got my wisdom teeth out in 1998. I take relief in the tensed fact that my wisdom teeth were extracted in the past. I do not run to the bus stop merely because I believe that the bus arrives at 8:45. I run to the bus stop because I believe that the bus is arriving now. How can the tenseless theory of time explain the central role that tense plays in our psychological attitudes? I claim that the objects of tensed beliefs are properties. The account I defend is a version of the self-ascription accounts defended by Roderick Chisholm and David Lewis. I argue that such a view can provide a plausible tenseless account of tensed belief. ^. (shrink)

The main guiding assumption of this paper is that there is need for a taxonomy of ethical sentences that does not overgenerate, yet can make useful contributions to debates about certain controversial sentences . After surveying the recent literature and concluding that no extant taxonomy that satisfies both of these conditions is available to us, I propose and explain a novel taxonomy which does satisfy them. I then defend my proposal from five potential objections.

It is argued that Loar's paper overestimates the importance of the distinction between 'functionalist' and 'representationalist' theories of the propositional attitudes; specifically, that the only version of functionalism which appears likely to provide an adequate account of the attitudes is one which treats them as relations to mental representations. The paper ends with a brief discussion of some of Loar's objection to 'ideal indicator' theories of the relation between beliefs and their truth conditions. It is argued that these objections are (...) not decisive. (shrink)

This book proposes a new theory of the origins of human language ability and presents an original account of the early evolution of language. It explains why humans are the only language-using animals, challenges the assumption that language is a consequence of intelligence, and offers a new perspective on human uniqueness. The author draws on evidence from archaeology, linguistics, cognitive science, and evolutionary biology. Making no assumptions about the reader's prior knowledge he first provides an introductory but critical survey of (...) all sources of evidence for language evolution. He then considers what language itself reveals about its own and human origins and evolution. He shows that certain central aspects of language do not, unexpectedly, reflect what they are used for, are maladapted for their function. He considers why this should be, and argues that these odd aspects of language reveal important clues about its evolutionary origin. The Origins of Complex Language fulfils the promise of its title. In doing so it turns upside down conventional theories about the relation between cognition and expression, truth and reference, and the co-evolution of mind and body. Original in conception, brilliantly executed, stylishly written, this book will attract a wide range of readers interested in the evolution and origins of language. (shrink)