This article argues that the much-maligned ‘notational variants’ of a given formal linguistic theory play a role similar to alternative numerical measurement scales. Thus, they can be used to identify the invariant components of the grammar; i.e., those features that do not depend on the choice of empirically equivalent representation. Treating these elements as the ‘meaningful’ structure of language has numerous consequences for the philosophy of science and linguistics. I offer several such examples of how linguistic theorizing can profit (...) from adopting a measurement-theoretic viewpoint. The first concerns a measurement-theoretic response to a famous criticism of Quine's. Others follow from issues of simplicity in the current biolinguistics program. An unexpected similarity with behaviorist practices is also uncovered. I then argue that managable and useful steps can be taken in this area. (shrink)

In recent years, dependency grammars have established themselves as valuable tools in theoretical and computational linguistics. To many linguists, dependency grammars and the more standard constituency-based formalisms are notational variants. We argue that, beyond considerations of formal equivalence, cognition may also serve as a background for a genuine comparison between these different views of syntax. In this paper, we review and evaluate some of the most common arguments and evidence employed to advocate for the cognitive or neural reality of (...) dependency grammars in linguistics, psycholinguistics, or neurolinguistics. We then raise the possibility that the abilities to represent and track, alternatively or in parallel, constituency and dependency structures co-exist in human cognition and are constitutive of syntactic competence. (shrink)

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of (...) compositionality these accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic. (shrink)

Gentzen’s singular sequential system of first-order logic was an alternative notation for his system of natural deductions. His multiple sequential system was his symmetric generalization that was more appropriate to classical logic. Beth’s tableaus system was a system that was derived directly from the semantic analysis of connectives and quantifiers. It was soon realized that the Beth’s system and the Gentzen’s multiple system were only notational variants of each other. Kneale’s system of multiple natural deductions was a generalization of (...) Gentzen’s system of natural deductions. We prove that Kneale’s natural deductions are also a notational variant of Beth’s tableaus. (shrink)

What does it take for two logics to be mere notational variants? The present paper proposes a variety of different ways of cashing out notational variance, in particular isolating a constraint on any reasonable account of notational variance which makes plausible that the only kinds of translations which can witness notational variance are what are sometimes called definitional translations.

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy, and in which there has been a recent resurgence of interest. In contemporary presentations, it is stated in terms of fundamental languages: as the view that such languages contain more than one quantifier. For example, one ranging over abstract objects, and another over concrete ones. A natural worry, however, is that the languages proposed by the pluralist (...) are mere notational variants of those proposed by the monist, in which case the debate between the two positions would not seem to be substantive. Jason Turner has given an ingenious response to this worry, in terms of a principle that he calls ‘logical realism’. This paper offers a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that the languages in question are notational variants. Indeed, there seems to be every reason to think that they are. (shrink)

Danielle Macbeth's purpose in Macbeth 2005 is threefold. Her monograph proposes "to provide a logical justification for all aspects of Frege's peculiar notation, to motivate and explain the developments in Frege's views over the course of his intellectual life, and to explicate his most developed, critically reflective conception of his Begriffschrift, his formula language of pure thought" (p. vii). I shall focus here on a few selected aspects of the first and third points and leave on the side the discussion (...) of the historical development of Frege's views on logic, semantics and mathematics. The book's underlying claim is that although Frege's logical language, "can of course be read as a language of quantificational logic, [it] can also be read very differently" (p. vii), so much so, as a matter of fact, that although Frege's logic is usually thought of as a "notational variant of standard quantificational logic", it is nevertheless "unknown to us" (p. 1). This naturally prompts three questions : "What is Frege's logic?", "What is quantificational logic?", and given the way we understand formal languages for first and higher order predicate logic, most notably the notions of argument 2 place (for a predicate symbol), of scope and of binding (of variables - either individual or predicate of first or higher order): "How come Frege's logic says nothing about such notions, or something so different from what we say?". (shrink)

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic , has recently been unconditionally separated from full bounded arithmetic S2. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to (...) transfer many results and methods from open to sharply bounded induction with relative ease.We provide two applications: the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes. (shrink)

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational (...) variants of each other, roughly in the same way that tableaus and Gentzen sequent calculi are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators. (shrink)

We are used to talking about the “structure” posited by a given theory of physics, such as the spacetime structure of relativity. What is “structure”? What does the mathematical structure used to formulate a theory tell us about the physical world according to the theory? What if there are different mathematical formulations of a given theory? Do different formulations posit different structures, or are they merely notational variants? I consider the case of Lagrangian and Hamiltonian classical mechanics. I argue (...) that, contrary to standard wisdom, these are not genuinely equivalent theories: they differ in statespace structure. I suggest that we should be realists about statespace structure. (shrink)

Many of the formalisms used in Attribute Value grammar are notational variants of languages of propositional modal logic, and testing whether two Attribute Value Structures unify amounts to testing for modal satisfiability. In this paper we put this observation to work. We study the complexity of the satisfiability problem for nine modal languages which mirror different aspects of AVS description formalisms, including the ability to express re-entrancy, the ability to express generalisations, and the ability to express recursive constraints. Two (...) main techniques are used: either Kripke models with desirable properties are constructed, or modalities are used to simulate fragments of Propositional Dynamic Logic. Further possibilities for the application of modal logic in computational linguistics are noted. (shrink)

That past patterns may continue in many different ways has long been identified as a problem for accounts of induction. The novelty of Goodman’s ”new riddle of induction” lies in a meta-argument that purports to show that no account of induction can discriminate between incompatible continuations. That meta-argument depends on the perfect symmetry of the definitions of grue/bleen and green/blue, so that any evidence that favors the ordinary continuation must equally favor the grue-ified continuation. I argue that this very dependence (...) on the perfect symmetry defeats the novelty of the new riddle. The symmetry can be obtained in contrived circumstances, such as when we grue-ify our total science. However, in all such cases, we cannot preclude the possibility that the original and grue-ified descriptions are merely notationally variant descriptions of the same physical facts; or if there are facts that separate them, these facts are ineffable, so that no account of induction should be expected to pick between them. In ordinary circumstances, there are facts that distinguish the regular and grue-ified descriptions. Since accounts of induction can and do call upon these facts, Goodman’s meta-argument cannot provide principled grounds for the failure of all accounts of induction. It assures us only of the failure of accounts of induction, such as unaugmented enumerative induction, that cannot exploit these symmetry breaking facts. (shrink)

As part of her argument that relativism and contextualism are nothing but notational variants of each other, Stojanovic holds that contextualism is flexible enough to achieve whatever relativism might do if the matter is what truth-value is assigned to each pair of sentence and context. In this paper, I reply to this statement by arguing that contextualism cannot be made as flexible as relativism without in fact turning it into a version of relativism. The key to my response to (...) Stojanovic is that, while relativism relativizes utterance truth, contextualism does not, so parameters that are not fixed at the context of utterance will be accessible for the relativist, but not for the contextualist. Although the relativity of utterance truth follows as soon as propositional truth is relativized to contexts of assessment, as the relativist does, it is easy to lose sight of this fact if we identify the context of assessment with the assessor’s context. Hence, the point of this paper is that the difference between relativism and contextualism is not one as to whose parameters play a role in determining the sentence’s truth-value. If it were, contextualism could indeed be made just as flexible as relativism. (shrink)

The aim of this article is to argue against the real possibility of languages without subject‐predicate structure, so‐called feature‐placing languages. They were first introduced by Strawson (1959/1990), later given formal expression through Quine's Predicate Functor Logic (Quine, 1960, Quine, 1971/Quine, 1976, Quine, 1992), and further elaboration in (Hawthorne & Cortens, 1995). I argue that, on the presumption that feature‐placing languages are not mere notational variants on first‐order languages, the idea of such languages is incoherent. The argument for this view (...) rests on two highly plausible views about the conditions for the possibility of sentential meaning in natural languages. If accepted, these two views jointly imply that the sentences of feature‐placing languages lack sufficient internal structure to meet the conditions for sentential meaning. Feature‐placing languages cannot then count as genuine languages. (shrink)

In this paper, I take issue with an idea that has emerged from recent relativist proposals, and, in particular, from Lasersohn, according to which the correct semantics for taste predicates must use contents that are functions of a judge parameter rather than implicit arguments lexically associated with such predicates. I argue that the relativist account and the contextualist implicit argument-account are, from the viewpoint of semantics, not much more than notational variants of one another. In other words, given any (...) sentence containing a taste predicate, and given any assignment of values to the relevant parameters, the two accounts predict the same truth value and are, in that sense, equivalent. I also look at possible reasons for preferring one account over the other. The phenomenon of "faultless disagreement" is often believed to be one such reason. I argue, against Kölbel and Lasersohn, that disagreement is never faultless: either the two parties genuinely disagree, hence if the one is right then the other is wrong, or the two parties are both right, but their apparent disagreement boils down to a misunderstanding. What is more, even if there were faultless disagreement, I argue that relativism would fail to account for it. The upshot of my paper, then, is to show that there is not much disagreement between a contextualist account that models the judge parameter as an implicit argument to the taste predicate, and a relativist account that models it as a parameter of the circumstances of evaluation. The choice between the two accounts, at least when talking about taste, is thus, to a large extent, a matter of taste. (shrink)

In recent work, Michael Tye has criticized a certain sort of representationalist view of experience for holding that phenomenal characters are properties of experiences. Instead, Tye holds that phenomenal character is 'out there in the world.' This paper has two aims. One is to argue for the somewhat surprising conclusion that Tye’s apparently radical new view is not a change in view at all, but a notational variant of a standard representationalist theory. My more general aim, though, is (...) to lay out a bit more clearly than is usually done the basic metaphysics of representationalist views of consciousness. In the end, what I argue is that unclarity on this score is what has permitted the choice between standard representationalist theories and Tye’s new position to appear to be something more than a verbal question. I conclude by discussing the bearing of this on the question of whether we can be aware of phenomenal character in introspection, and the relative importance of properties of subjects and properties of experiences in the theory of consciousness. (shrink)

It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version (...) of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic, in a sense to be made precise. Our main target theory, the intuitionistic small set theory is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally. (shrink)

This article considers two different ways of formulating a desire-satisfaction theory of prudential value. The first version of the theory (the object view) assigns basic prudential value to the state of affairs that is the object of a person’s desire. The second version (the combo view) assigns basic prudential value to the compound state of affairs in which (a) a person desires some state of affairs and (b) this state of affairs obtains. My aims in this article are twofold. First, (...) I aim to highlight that these are not mere notational variants, but in fact have quite different implications, so that this distinction is not one that the theorist of prudential value should ignore. More positively, I argue that the object view is better able to capture what is distinctive and appealing about subjective theories of prudential value, on any plausible account of what the central subjectivist insight is. (shrink)

This essay shows that Derrida’s discussion of “Differance,” is remarkably parallel to Plato’s discussion of Difference in the Parmenides. Plato’s presentation of “Parmenides’” discussion of generation from a One which Is is a version of Derrida’s preconceptual spacing. Derrida’s implicit reference to Plato both interprets Plato and explains the obscure features of “Differance.” Derrida’s paradoxical remarks about Differance are very like what Plato implies about Difference.Derrida’s Differance addresses the puzzle that concepts are required to construct the beings in a plurality (...) of objects, but concepts cannot differentiate unless there is already a plurality of objects. Plato’s version of the same problem is a notational variant of Derrida’s Husserlian dilemma.Derrida, following Davidson, is not only skeptical about the project of founding metaphysics on simple entities, but also holds that necessity has no foundation in the “world.” Plato, on the other hand, retains the idea that necessity has an objective basis in the self-evident truths of mereology. (shrink)

Kriegel has revived adverbialism as a theory of consciousness. But recent attacks have shed doubt on the viability of the theory. To save adverbialism, I propose that the adverbialist take a stance on the nature of adverbial modification. On one leading theory, adverbial modification turns on the instantiation by a substance of a psychological type. But the resulting formulation of adverbialism turns out to be a mere notational variant on the relationalist approaches against which Kriegel dialectically situates adverbialism. (...) By contrast, I argue that the way to be an adverbialist is to adopt an event ontology, emphasizing the active contribution of the mind to the phenomenology of experience. My close examination of the semantics of adverbial modification throws this metaphysical distinction into sharp relief. The event-based semantics overcomes recent objections in a way superior to the methods that would have been obviously available in the absence of a sophisticated semantics. (shrink)

In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether second-order logic can count as logic in some proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the case, it might be argued that it does not (...) have the "epistemic innocence" which would be needed for, e.g., foundational programmes in mathematics for which second-order logic is sometimes used. I suggest a Deductivist conception of logic, that characterises logical consequence by means of inference rules, and argue that on this conception second-order logic should count as logic in the proper sense. (shrink)

This paper is devoted to the topic of indexicality in relation to the problem of cognitive significance. I undertake a critical examination of what I call the Millian Notational Variance Claim; this is the claim that those versions of a neo-Fregean semantics for demonstratives and other indexicals which rest upon the notion of a de re sense are eventually notational variants of a directly referential or Millian semantics for indexicals. I try to show that several lines of reasoning (...) that might be pursued by Millian theorists with a view to establishing the Millian Notational Variance Claim are inconclusive, and hence that the claim is in general unsound. The problem of cognitive significance is tackled in connection with those categories of indexicals concerning which neo-Fregeanism and Millianism are alleged to yield similar results, viz. temporal indexicals, spatial indexicals, and perceptual demonstratives. I argue towards the conclusion that the notions the Millian theorist might invoke to accommodate the phenomena of cognitive significance in this area of indexicality are hardly adequate to the effect, and hence that senses are indispensable also here. (shrink)

The hidden strength of Goodman's ingenious "new riddle of induction" lies in the perfect symmetry of grue/bleen and green/blue. The very same sentence forms used to define grue/bleen in terms of green/blue can be used to define green/blue in terms of grue/bleen by permutation of terms. Therein lies its undoing. In the artificially restricted case in which there are no additional facts that can break the symmetry, grue/bleen and green/blue are merely notational variants of the same facts; or, if (...) they represent different facts, the differences are ineffable, and no account of induction should be expected to pick between them. This still obtains in the more interesting case in which we embed grue/bleen in a grue-ified total science; the grue-ified and regular total sciences are merely equivalent descriptions of the same facts. In the most realistic case, we allow additional facts that break the symmetry and then we can also evade Goodman's new riddle by employing an account of induction rich enough to exploit these facts. Unaugmented enumerative induction is not such an account and it is the primary casualty of Goodman's new riddle. (shrink)

In “Truth – A Traditional Debate Reviewed”, Crispin Wright proposed an inductive definition of “coherence truth” for arithmetic relative to an arithmetic base theory B. Wright’s definition is in fact a notational variant of the usual Tarskian inductive definition, except for the basis clause for atomic sentences. This paper provides a model-theoretic characterization of the resulting sets of sentences "cohering" with a given base theory B. These sets are denoted WB. Roughly, if B satisfies a certain minimal condition, (...) then WB is the Th, where M is the canonical model of the set At of atomic sentences provable in B. The paper also shows that the disquotational T-scheme is provable from Wright’s inductive definition just in case the base theory B is sound and complete for arithmetic atomic sentences. (shrink)

This paper attempts a comprehensive account of the comparative merits of paracomplete and dialetheic approaches to the semantic paradoxes. It argues that aside from issues about conditionals, there can be no strong case for paracomplete approaches over dialetheic, or dialetheic over paracomplete, and indeed that in absence of conditionals, the two approaches are plausibly seen as notational variants. Graham Priest disagrees: many of his arguments favoring dialetheic solutions over paracomplete do not turn on issues about conditionals. The paper discusses (...) his arguments on these points in some detail. On the matter of conditionals, it argues that extant paracomplete approaches are far better than extant dialetheic approaches, a fact that traces in part to dialetheists having focused too heavily on “relevant logics”. It holds out hope for better dialetheic treatments of conditionals, but it also suggests that this is the one area where paracomplete approaches are inevitably better. (shrink)

According to ontological nihilism there are, fundamentally, no individuals. Both natural languages and standard predicate logic, however, appear to be committed to a picture of the world as containing individual objects. This leads to what I call the \emph{expressibility challenge} for ontological nihilism: what language can the ontological nihilist use to express her account of how matters fundamentally stand? One promising suggestion is for the nihilist to use a form of \emph{predicate functorese}, a language developed by Quine. This proposal faces (...) a difficult objection, according to which any theory in predicate functorese will be a notational variant of the corresponding theory stated in standard predicate logic. Jason Turner (2011) has provided the most detailed and convincing version of this objection. In the present paper, I argue that Turner's case for the notational variance thesis relies on a faulty metasemantic principle and, consequently, that an objection long thought devastating is in fact misguided. (shrink)

Danielle Macbeth's purpose in Macbeth 2005 is threefold. Her monograph proposes "to provide a logical justification for all aspects of Frege's peculiar notation, to motivate and explain the developments in Frege's views over the course of his intellectual life, and to explicate his most developed, critically reflective conception of his Begriffschrift, his formula language of pure thought". I shall focus here on a few selected aspects of the first and third points and leave on the side the discussion of the (...) historical development of Frege's views on logic, semantics and mathematics. The book's underlying claim is that although Frege's logical language, "can of course be read as a language of quantificational logic, [it] can also be read very differently", so much so, as a matter of fact, that although Frege's logic is usually thought of as a "notational variant of standard quantificational logic", it is nevertheless "unknown to us". This naturally prompts three questions : "What is Frege's logic?", "What is quantificational logic?", and given the way we understand formal languages for first and higher order predicate logic, most notably the notions of argument 2 place, of scope and of binding : "How come Frege's logic says nothing about such notions, or something so different from what we say?". (shrink)

Though many kinds of multi-agent systems based on argumentation have been proposed where only rule-based knowledge is taken into account, they have been unable to handle the ontological knowledge so far. In our daily life, however, there are a lot of human argumentation where both ontological and rule knowledges are used. For example, in e-commerce, a seller and a buyer usually use ontologies about products along with their respective strategic rules for buying and selling. Recent progress of the Semantic Web (...) technology provides expressive ontology languages. In this paper, we demonstrate integration of the Semantic Web reasoning and argument-based reasoning. We have implemented the integrated system such that Logic of Multiple-valued Argumentation-based agent system (specialized to two values { f, t }) can be accessible to the Semantic Web reasoning established as the description logic reasoning system, given ontologies expressed by OWL DL or its notational variant the DL SHOIN (D). An interesting argumentation result using both ontologies and rules about the university curriculum is shown as an example executed by our system. (shrink)

In a review of Frege's Puzzle1, Graeme Forbes makes the claim that Salmon's account of belief might be seen, under certain conditions, as a mere notational variant of a neo-Fregean theory; and thus that such an account might be reduced to a neo-Fregean one simply by rewriting it in terms of Fregean terminology. With a view to supporting his claim, Forbes offers an outline of an account of belief which, according to him, would satisfy the following conditions: (i) (...) it could be directly obtained from Salmon's own analysis by means of a certain set of substitutions, which presumably would not affect the essential features of Salmon's view; (ii) it could naturally be described as Fregean, in the sense that it would preserve, (at least) the spirit of Frege's doctrines, especially his fundamental intuitions about belief. Of course, the upshot of Forbes's argument is that Salmon's theory would not, at bottom, constitute a genuine alternative to a Fregean semantics for belief ascriptions. In this paper I argue to the effect that Forbes's claim is not in general sound. It seems to me that the sort of indirect argument used by Forbes - that of trying to undermine Salmon's theory by showing that it is just a version of a neo-Fregean account - does not provide someone working within a Fregean framework with an adequate strategy to counter Salmon's neo-Russellian views. It would perhaps be better to concentrate a Fregean attack on certain apparently dubious and highly controversial theses and results which are constitutive of Salmon's view, e.g. the counterintuitive character of a substantial set of consequences which follow from his theory of belief, as well as the associated revisionist stand he is forced to take towards our current patterns of speaking about belief. (shrink)

This essay shows that Derrida's discussion of "Differance," is remarkably parallel to Plato's discussion of Difference in the Parmenides. Plato's presentation of "Parmenides'" discussion of generation from a One which Is is a version of Derrida's preconceptual spacing. Derrida's implicit reference to Plato both interprets Plato and explains the obscure features of "Differance." Derrida's paradoxical remarks about Differance are very like what Plato implies about Difference. Derrida's Differance addresses the puzzle that concepts are required to construct the beings in a (...) plurality of objects, but concepts cannot differentiate unless there is already a plurality of objects. Plato's version of the same problem is a notational variant of Derrida's Husserlian dilemma. Derrida, following Davidson, is not only skeptical about the project of founding metaphysics on simple entities, but also holds that necessity has no foundation in the "world." Plato, on the other hand, retains the idea that necessity has an objective basis in the self-evident truths of mereology. (shrink)

The paper is a reply to Bassi and Rasin ( 2018 ) on the treatment of sentences like [_The gifted mathematician that you claim to be_] _should have solved this task without problems_ by Grosu and Krifka ( 2007 ), which was published in _Linguistics and Philosophy_. Grosu and Krifka provide an analysis of the _de dicto_ interpretation in which the bracketed expression refers to an individual concept. Bassi and Rasin question this because equivalent expressions in Hebrew, in which the (...) gap is rendered by a resumptive pronoun, do not exhibit the _de dicto_ interpretation, but only the _de re_ interpretation. We provide independent evidence that person-marked pronouns in post-copular position are incompatible with antecedents that denote individual concepts, thus explaining Bassi and Rasin’s observation within Grosu and Krifka’s framework. We furthermore point out two new problems for the analysis of Bassi and Rasin presented in the body of their article, which does not make use of individual concepts or of some alternative mechanism that can deal with substitutivity in opaque contexts. We also note that an alternative analysis they present in an appendix amounts to a notational variant of ours. (shrink)

In this paper I will be concerned with the question of the extent to which semantics can be thought of as a purely formal exercise, which we can engage in in a way that is neutral with respect to how our formal system is to be interpreted. I will be arguing, to the contrary, that the features of the formal systems which we use to do semantics are closely linked, in several different ways, to the interpretation that we give to (...) those formal systems. The occasion for this question, and the main example that I will use to illustrate my answer to it, is the close relationship between the formal systems employed in recent statements of apparently competing accounts of epistemic modals with the dynamic, expressivist, and relativist theoretical paradigms. The structure of the paper will be straightforward. In part 1, I will briefly introduce four theories of epistemic modals – one dynamic theory, two expressivist theories, and one relativist theory. Then in part 2 I’ll show that one expressivist theory is formally equivalent to the dynamic theory, that the other is formally equivalent to the relativist theory, and that the two expressivist theories are themselves essentially notational variants. I’ll use these facts to pose our central question: if these theories have so much formally in common, then doesn’t that suggest that we can separate the task of constructing a formal semantics from the task of deciding between competing interpretations of it? Finally, in part 3 I’ll answer that question in the negative. There are at least three reasons why formal semantics cannot be separated from questions of interpretation that are illustrated by the theories I introduce in part 1. (shrink)

In “Semantics of Fictional Terms,” Garcia-Carpintero critically surveys the most recent literature on the topic of fictional names. One of his targets is realism about fictional discourse. Realists about fictional discourse believe that: (a) it contains true sentences that have fictional names as their subjects; (b) sentences containing names can be true only if those names have referents; (c) fictional names have fictional characters – abstract objects – as their referents. The fundamental problem that arises for realists is that not (...) all true sentences containing fictional names are plausibly about abstract objects. This leads to the need to introduce disjunctive conceptions of property attribution that Garcia-Carpintero claims are implausible, and that realism should therefore be rejected. He also maintains, however, that (a) is correct. I agree. Furthermore, I am also committed to anti-realism about fictional discourse – that fictional names have no referents. Garcia-Carpintero claims that my view is simply a notational variant of realism. I argue that this is false – that my view could not possibly be a notational variant of any extant realist theory. (shrink)

In Bermúdez 2013 I argued against David Lewis’s well-known and widely accepted claim that Newcomb’s problem and the prisoner’s dilemma are really notational variants of a single problem. Mark Walker’s paper in this journal takes issue with my argument. In this note I show how Walker’s criticisms are misplaced. The problems with Walker’s argument point to more general and independently interesting conclusions about, first, the relation between deliberation and decision and, second, the differences between the prisoner’s dilemma, which is (...) a strategic choice problem, and Newcomb’s problem, which is a parametric choice problem. (shrink)

I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.

According to an influential view, when it comes to representing reality, some words are better suited for the job than others. This is elitism. There is reason to believe that the set of the best, or elite, words should not be redundant or arbitrary. However, we are often forced to choose between these two theoretical vices, especially in cases involving theories that seem to be mere notational variants. This is the riddle of redundancy: both redundancy and arbitrariness are vicious, (...) but there are cases in which one must be picked. Logical realists admit that there are some logical constants among the elite words. This leads to awkward questions, such as which among conjunction and disjunction is elite. In this paper, I show how the riddle of redundancy arises for logical realists, and offer a solution. This approach requires us to change how we represent negation. Instead of using some particular symbol, we represent negation by flipping formulae over the horizontal axis. (shrink)

This essay shows that Derrida's discussion of "Differance," is remarkably parallel to Plato's discussion of Difference in the Parmenides. Plato's presentation of "Parmenides'" discussion of generation from a One which Is is a version of Derrida's preconceptual spacing. Derrida's implicit reference to Plato both interprets Plato and explains the obscure features of "Differance." Derrida's paradoxical remarks about Differance are very like what Plato implies about Difference. Derrida's Differance addresses the puzzle that concepts are required to construct the beings in a (...) plurality of objects, but concepts cannot differentiate unless there is already a plurality of objects. Plato's version of the same problem is a notational variant of Derrida's Husserlian dilemma. Derrida, following Davidson, is not only skeptical about the project of founding metaphysics on simple entities, but also holds that necessity has no foundation in the "world." Plato, on the other hand, retains the idea that necessity has an objective basis in the self-evident truths of mereology. (shrink)

This paper discusses two notational variance views with respect to indexical singular reference and content: the view that certain forms of Millianism are at bottom notational variants of a Fregean theory of reference, the Fregean Notational Variance Claim; and the view that certain forms of Fregeanism are at bottom notational variants of a direct reference theory, the Millian Notational Variance Claim. While the former claim rests on the supposition that a direct reference theory could be (...) easily turned into a particular version of a neo-Fregean one by showing that it is bound to acknowledge certain senselike entities, the latter claim is based upon the supposition that a neo-Fregean theory could be easily turned into a particular version of a Millian one by showing that De Re senses are theoretically superfluous and hence eliminable. The question how many accounts of singular reference and content are we confronted with here — Two different (and mutually antagonistic) theories? Or just two versions of what is in essence the same theory? — is surely of importance to anyone interested in the topic. And this question should be answered by means of a careful assessment of the soundness of each of the above claims. Before trying to adjudicate between the two accounts, one would naturally want to know whether or not there are indeed two substantially disparate accounts. Grosso modo, if the Fregean Claim were sound then we would have a single general conception of singular reference to deal with, viz. Fregeanism; likewise, if the Millian Claim were sound we would be facing a single general conception of singular reference, viz. Millianism. My view is that both the Fregean Notational Variance Claim and its Millian counterpart are wrong, though naturally on different grounds. I have argued elsewhere that the Fregean Notational Variance Claim - considered in its application to the semantics of propositional-attitude reports involving proper names — is unsound. I intend tosupplement in this paper such a result by trying to show that the Millian Claim - taken in its application to the semantics of indexical expressions — should also be rated as incorrect. I focus on a certain set of arguments for the Millian Claim, arguments which I take as adequately representing the general outlook of the Millian theorist with respect to neo-Fregeanism about indexicals and which involve issues about the cognitive significance of sentences containing indexical terms. (shrink)

Three long papers are collected here which constitute Chomsky’s major theoretical work on syntax and semantics subsequent to the "standard theory" of Aspects of the Theory of Syntax. Since 1965, transformational-generative linguists have suggested various changes in "standard theory," centering on the relationship between the syntactic and semantic components in natural language grammars. In these papers Chomsky explains several specific problems that require the extension of standard theory and he criticizes the proposals and arguments of the generative semanticists, attempting to (...) show that many of the specific features of language that are claimed to support the generativist position are as well or better explained by extended standard theory, and emphasizing that most of the generativist’s proposals turn out to be mere "notational variants," evidentially indistinguishable from extended standard theory. Aside from appealing to particular data favoring his "lexicalist" defense of syntactic deep structure, Chomsky makes a Popperean appeal that his position is preferable because it is more constraining and more falsifiable. (shrink)

Ontological pluralism is the doctrine that there are different ways or modes of being. In contemporary guise, it is the doctrine that a logically perspicuous description of reality will use multiple quantifiers which cannot be thought of as ranging over a single domain. Although thought defeated for some time, recent defenses have shown a number of arguments against the view unsound. However, another worry looms: that despite looking like an attractive alternative, ontological pluralism is really no different than its counterpart, (...) ontological monism. In this paper, after explaining the worry in detail, I argue that considerations dealing with the nature of the logic ontological pluralists ought to endorse, coupled with an attractive philosophical thesis about the relationship between logic and metaphysics, show this worry to be unfounded. (shrink)

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e., within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A lim-computablefunction is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. (...) Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "not-so-nearly" minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight. (shrink)

We investigate a variant of the variable convention proposed at Tractatus 5.53ff for the purpose of eliminating the identity sign from logical notation. The variant in question is what Hintikka has called the strongly exclusive interpretation of the variables, and turns out to be what Ramsey initially (and erroneously) took to be Wittgenstein's intended method. We provide a tableau calculus for this identity-free logic, together with soundness and completeness proofs, as well as a proof of mutual interpretability with (...) first-order logic with identity. (shrink)

Goodstein’s principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to diverge, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we provide a self-contained and modern analysis of Goodstein’s principle, obtaining some variations and improvements. We explore notions of optimality for notation systems and apply them to the classical (...) Goodstein process and to a weaker variant based on multiplication rather than exponentiation. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein’s result. We moreover show that by varying the initial base of the Goodstein process, one readily obtains independence results for each of the fragments $\mathsf {I}\Sigma _n$ of Peano arithmetic. (shrink)

The power of Wittgenstein's N operator described in the Tractatus is that every proposition which can be expressed in the Russellian variant of the predicate calculus familiar to him has an equivalent proposition in an extended variant of his N operator notation. This remains true if the bound variables are understood in the usual inclusive sense or in Wittgenstein's restrictive exclusive sense. The problematic limit of Wittgenstein's N operator comes from his claim that symbols alone reveal the logical (...) status of propositions. This would require knowledge of the size of the Tractarian domain of unchangeable, simple objects, and this, Wittgenstein maintained, cannot be known. (shrink)

This paper is dedicated to developing a formalism that takes rejection seriously. Bilateral notation of signed formulas with force indicators is adopted to define signed consequences which can be viewed as the bilateral counterpart of Tarskian consequence relations. Its relation to some other bilateral approaches is discussed. It is shown how David Nelson’s logic N4 can be characterized bilaterally and the corresponding completeness result is proved. Further, bilateral variants of three familiar notions are considered and investigated: that of a fragment, (...) of definitional equivalence, and of a conservative extension. (shrink)

Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, (...) the axiom of choice, Zorn’s lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process. (shrink)

Kruskal proved that finite trees are well-quasi-ordered by hom(e)omorphic embeddability. Friedman observed that this statement is not provable in predicative analysis. Friedman also proposed (see in [Simpson]) some stronger variants of the Kruskal theorem dealing with finite labeled trees under home(e)omorphic embeddability with a certain gap-condition, where labels are arbitrary finite ordinals from a fixed initial segment of ω. The corresponding limit statement, expressing that for all initial segments of ω these labeled trees are well-quasi-ordered, is provable in Π 1 (...) 1 -CA, but not in the analogous theory Π 1 1 -CA 0 with induction restricted to sets. Schutte and Simpson proved that the one-dimensional case of Friedman's limit statement dealing with finite labeled intervals is not provable in Peano arithmetic. However, Friedman's gap-condition fails for finite trees labeled with transfinite ordinals. In [Gordeev 1] I proposed another gap-condition and proved the resulting one-dimensional modified statements for all (countable) transfinite ordinal-labels. The corresponding universal modified one-dimensional statement UM 1 is provable in (in fact, is equivalent to) the familiar theory ATR 0 whose proof-theoretic ordinal is Γ 0 . In [Gordeev 1] I also announced that, in the general case of arbitrarily-branching finite trees labeled with transfinite ordinals, in the proof-theoretic sense the hierarchy of the limit modified statements $\mathbf{M}_{ (which are denoted by LM λ in the present note) is as strong as the hierarchy of the familiar theories of iterated inductive definitions (more precisely, see [Gordeev 1, Concluding Remark 3]). In this note I present a "positive" proof of the full universal modified statement UM, together with a short proof of the crucial "reverse" results which is based on Okada's interpretation of the well-established ordinal notations of Buchholz corresponding to the theories of iterated inductive definitions. Formally the results are summarized in $\S5$ below. (shrink)

The first beginning logic text to employ the tree method--a complete formal system of first-order logic that is remarkably easy to understand and use--this text allows students to take control of the nuts and bolts of formal logic quickly, and to move on to more complex and abstract problems. The tree method is elaborated in manageable steps over five chapters, in each of which its adequacy is reviewed; soundness and completeness proofs are extended at each step, and the decidability proof (...) is extended at the step from truth functions to the logic of nonoverlapping quantifiers with a single variable, after which undecidability is demonstrated by example. The first three chapters are bilingual, with arguments presented twice, in logical notation and in English. The last three chapters consider the discoveries defining the scope and limits of formal methods that marked logic’s coming of age in the 20th century: Godel’s completeness and incompleteness theorems for first and second-order logic, and the Church-Turing theorem on the undecidability of first-order logic. This new edition provides additional problems, solutions to selected problems, and two new Supplements: Truth-Functional Equivalence reinstates material on that topic from the second edition that was omitted in the third, and Variant Methods, in which John Burgess provides a proof regarding the possibility of modifying the tree method so that it will always find a finite model when there is one, and another, which shows that a different modification—once contemplated by Jeffrey--can result in a dramatic speed--up of certain proofs. (shrink)

I am going to describe a variant formulation of classical extensional first-order logic and contrast it with the standard formulation. The formulation I will give is in one clear sense equivalent to the standard one, and it is a routine task to show that it is equivalent to it in this sense. So one might regard my formulation as a mere notational variation. But there are also ways in which the two formulations I will contrast are not equivalent, (...) and I will argue that these ways reflect an important conceptual difference between them. The differences become more important as one generalizes or extends classical logic in various ways, but they are there even in the classical systems. (shrink)