We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n -universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.

In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that (...) these are the formulas preserved in monotonic images of frames and that ONNILLI-formulas are stable formulas as introduced by Bezhanishvili and Bezhanishvili in 2013. Thus, ONNILLI is a syntactically defined set of formulas axiomatizing all stable logics. This resolves a problem left open in 2013. (shrink)

We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n-universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.

Here, we provide a detailed description of the mutual relation of formulas with finite propositional variables p1, …, pm in modal logic S4. Our description contains more information on S4 than those given in Shehtman (1978) and Moss (2007); however, Shehtman (1978) also treated Grzegorczyk logic and Moss (2007) treated many other normal modal logics. Specifically, we construct normal forms, which behave like the principal conjunctive normal forms in the classical propositional logic. The results include finite and effective methods (...) to find a normal form equivalent to a given formula A by clarifying the behavior of connectives and giving a finite method to list all exact models. (shrink)

The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for the (...) infinitary languages $\mathcal {L}_{\text {Ord},\omega }$, allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega,\omega }$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$. Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$. (shrink)

Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes $$\textrm{E}_k$$ and $$\textrm{U}_k$$ introduced in [1] are exactly the classes induced by $$\Sigma _k$$ and $$\Pi _k$$ respectively via the transformation procedure (...) in any first-order theory. (shrink)

Normal forms for wide classes of closed IL formulas were given in Čačić and Vuković. Here we quantify asymptotically, in exact numbers, how wide those classes are. As a consequence, we show that the “majority” of closed IL formulas have GL-equivalents, and by that, they have the same normal forms as GL formulas. Our approach is entirely syntactical, except for applying the results of Čačić and Vuković. As a byproduct we devise a convenient way of computing (...) asymptotic behaviors of somewhat general classes of formulas given by their grammar rules. Its applications do not require any knowledge of the recurrence relations, generating functions, or the asymptotic enumeration methods, as all these are incorporated into two fundamental parameters. (shrink)

In [60] N. Belnap presented an 8-element matrix for the relevant logic R with the following property: if in an implication A → B the formulas A and B do not have a common variable then there exists a valuation v such that v(A → B) does not belong to the set of designated elements of this matrix. A 6-element matrix of this kind can be found in: R. Routley, R.K. Meyer, V. Plumwood and R.T. Brady [82]. Below we (...) prove that the logics generated by these two matrices are the only maximal extensions of the relevant logic R which have the relevance property: if A → B is provable in such a logic then A and B have a common propositional variable. (shrink)

It is shown that the pure implication fragment of the modal logic [3], pp. 385--387) has finitely many non-equivalent formulae in one variable. The exact number of such formulae is not known. We show that this finiteness result is the best possible, since the analogous fragment of S4, and therefore of [3], in two variables has infinitely many non-equivalent formulae.

Fundamental Theory has been called an "unfinished symphony" and "a challenge to the musicians among natural philosophers of the future". This book, written in 1944 but left unfinished because Eddington died too soon, proved to be his final effort at a vision for harmonization of quantum physics and relativity. The work is less connected and internally integrated than 'Protons and Electrons' while representing a later point in the author's thought arc. The really interested student should read both books together.The physical (...) and mathematical reasoning are very deep, but it is possible to enjoy much of the flavor of the book even without following everything. The tidbits given below illustrate Eddington's beautiful and mellifluous English style. This edition contains a clickable table of contents and some interesting illustrations.WARNING: Due to the complexity of the mathematical typesetting, the only way to reproduce this book has been as a series of scanned page images. We believe this work is scientifically and culturally important, and despite a few imperfections and the inevitably small print, have brought it back into print as part of our continuing commitment to the preservation of Eddington's work. On a portable electronic reader the pages come out pretty small, but they are legible. On a desktop computer running the Amazon app, they are perfectly fine. We appreciate your understanding, and hope you enjoy this valuable book.TIDBITS:The number of protons and electrons in the universe would, in itself, be merely a matter of idle curiosity. But N has a more general significance as a fundamental constant which enters into many physical formulae. Its special interpretation as the number of particles in the universe arises in the following way. If we consider a distribution of hydrogen in equilibrium at zero temperature, the presence of the matter produces a curvature of space, and the curvature causes the space to close when the number of particles contained in it reaches a certain total; that number is N. The present investigation seeks to determine N directly from the principles of measurement. We have to show, not that there are N particles in the universe, but that anyone who accepts certain elementary principles of measurement must, if he is consistent, think there are. We have to express in mathematical symbolism what we think we are doing when we measure things; for if we had no conception of what we were doing, the results of the measurements would not persuade us to believe anything in particular. All our results are derived from the condition that the conceptual interpretation which we place on the results of measurement must be consistent with our conceptual interpretation of the process of measurement.If we play about with a pin and a meter standard, we do not become aware of any number in particular unless the standard has been graduated, or unless we use a process of displacement which we interpret as adding pin-extensions. The fact is that, although measurement is primarily a process involving four entities, the conceptual interpretation of measurement postulates in addition the existence (in the structure contemplated) as something referred to as Z. The existence of Z is the condition that makes graduation an exact concept. In the actual universe there exists a basis which is uniform to a very high approximation, and this serves for almost all purposes; but to the much higher approximation required in the calculation of N the ideal exact basis of uniformity does not exist. The finitude of N is, in fact, the cause of the failure of the approximation.It must be remembered that we are now working to what would ordinarily be considered a fantastically high approximation. For ordinary purposes a measurable has 16 eigenvalues; but under the super-magnification here employed (which is not content to overlook 1 part in 10E39) there is a fine structure which splits each of them into 16 components. (shrink)

Fundamental Theory has been called an "unfinished symphony" and "a challenge to the musicians among natural philosophers of the future". This book, written in 1944 but left unfinished because Eddington died too soon, proved to be his final effort at a vision for harmonization of quantum physics and relativity. The work is less connected and internally integrated than 'Protons and Electrons' while representing a later point in the author's thought arc. The really interested student should read both books together.The physical (...) and mathematical reasoning are very deep, but it is possible to enjoy much of the flavor of the book even without following everything. The tidbits given below illustrate Eddington's beautiful and mellifluous English style. This edition contains a clickable table of contents and some interesting illustrations.WARNING: Due to the complexity of the mathematical typesetting, the only way to reproduce this book has been as a series of scanned page images. We believe this work is scientifically and culturally important, and despite a few imperfections and the inevitably small print, have brought it back into print as part of our continuing commitment to the preservation of Eddington's work. On a portable electronic reader the pages come out pretty small, but they are legible. On a desktop computer running the Amazon app, they are perfectly fine. We appreciate your understanding, and hope you enjoy this valuable book.TIDBITS:The number of protons and electrons in the universe would, in itself, be merely a matter of idle curiosity. But N has a more general significance as a fundamental constant which enters into many physical formulae. Its special interpretation as the number of particles in the universe arises in the following way. If we consider a distribution of hydrogen in equilibrium at zero temperature, the presence of the matter produces a curvature of space, and the curvature causes the space to close when the number of particles contained in it reaches a certain total; that number is N. The present investigation seeks to determine N directly from the principles of measurement. We have to show, not that there are N particles in the universe, but that anyone who accepts certain elementary principles of measurement must, if he is consistent, think there are. We have to express in mathematical symbolism what we think we are doing when we measure things; for if we had no conception of what we were doing, the results of the measurements would not persuade us to believe anything in particular. All our results are derived from the condition that the conceptual interpretation which we place on the results of measurement must be consistent with our conceptual interpretation of the process of measurement.If we play about with a pin and a meter standard, we do not become aware of any number in particular unless the standard has been graduated, or unless we use a process of displacement which we interpret as adding pin-extensions. The fact is that, although measurement is primarily a process involving four entities, the conceptual interpretation of measurement postulates in addition the existence (in the structure contemplated) as something referred to as Z. The existence of Z is the condition that makes graduation an exact concept. In the actual universe there exists a basis which is uniform to a very high approximation, and this serves for almost all purposes; but to the much higher approximation required in the calculation of N the ideal exact basis of uniformity does not exist. The finitude of N is, in fact, the cause of the failure of the approximation.It must be remembered that we are now working to what would ordinarily be considered a fantastically high approximation. For ordinary purposes a measurable has 16 eigenvalues; but under the super-magnification here employed (which is not content to overlook 1 part in 10E39) there is a fine structure which splits each of them into 16 components. (shrink)

One of the most striking differences between Frege's Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted , that is, which are being used to express judgements . There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role of (...) the judgement stroke in a way that renders it readily comprehensible why Frege insisted that this symbol was an essential part of his logical system. The key point here is that Frege viewed logic as the study of inference relations amongst acts of judgement , rather than – as in the typical contemporary view – of consequence relations amongst certain objects (propositions or well-formed formulae). The paper also explains why Frege's use of the judgement stroke is not in conflict with his avowed anti-psychologism, and why Wittgenstein's criticism of the judgement stroke as 'logically quite meaningless' is unfounded. The key point here is that while the judgement stroke has no content , its use in logic and mathematics is subject to a very stringent norm of assertion. (shrink)

The ArgumentIn Germany during the 1870s and 1880s a number of important scientific innovations in chemistry and biology emerged that were linked to advances in the new technology of synthetic dyestuffs. In particular, the rapid development of classical organic chemistry was a consequence of programs in which chemists devised new theories and experimental strategies that were applicable to the processes and products of the burgeoning dye factories. Thereafter, the novel products became the means to examine and measure biological systems. This (...) took place as a result of two trends. The first was a move toward diversification in the dye industry – made possible by the extensive range of products – which in turn was stimulated by economic and political conditions. The second was the increasing availability of techniques, substances, and processes used in industry. This made possible a concrete program of introducing the qualitative and quantitative methods of chemistry into the domain of laboratory experimentation on biological materials, thereby realizing the abstract desire to transform cell biology into an exact science.Moreover, the conceptualization of biological systems that emerged from this endeavor leaned heavily on a theory of dye chemistry that indicated which particular arrangements of atoms performed specific functions. This biological modeling used the imagery of chemical structural formulae to transform chemical nuclei and their side chains into adequate representations of protoplasmic structure. (shrink)

The transition from form to meaning is not neatly layered: there is no point where form ends and content sets in. Rather, there is an almost continuous process that converts form into meaning. That process cannot always take a straight line. Very often we hit barriers in our mind, due to the inability to understand the exact content of the sentence just heard. The standard division between formula and interpretation (or value) should therefore be given up when talking about (...) the process of understanding. Interestingly, when we do this it turns out that there are 'easy' formulae, those we can understand without further help, and 'difficult' ones, which we cannot. (shrink)

In this work we propose the partial Wiener index as one possible measure of branching in phylogenetic evolutionary trees. We establish the connection between the generalized Robinson–Foulds (RF) metric for measuring the similarity of phylogenetic trees and partial Wiener indices by expressing the number of conflicting pairs of edges in the generalized RF metric in terms of partial Wiener indices. To do so we compute the minimum and maximum value of the partial Wiener index WT,r,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W\left(T,r, n\right)$$\end{document}, where T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T$$\end{document} is a binary rooted tree with root r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document} and n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} leaves. Moreover, under the Yule probabilistic model, we show how to compute the expected value of WT,r,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W\left(T,r, n\right)$$\end{document}. As a direct consequence, we give exact formulas for the upper bound and the expected number of conflicting pairs. By doing so we provide a better theoretical understanding of the computational complexity of the generalized RF metric. (shrink)

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” Not only is (...) the Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s (...) found one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L. One of the nice things about AD is that the theory ZF + AD + V = L appears as a choiceless “completion” of ZF in that any interesting question seems to find an at least attractive answer in that theory. Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension. (shrink)

Since being brought to light in 1853 by Adalbert Kuhn, the fact that the Homeric expressionκλέος ἄφθιτονhas an exact parallel in the Veda has played an extremely important role in formulating the hypothesis that Greek epic poetry is of Indo-European origin. Yet only with Milman Parry's analysis of the formulaic character of Homeric composition did it become possible to test the antiquity ofκλέος ἄφθιτονon the internal grounds of Homeric diction.It is generally agreed that the conservative character of oral composition (...) entails a high degree of correlation between the antiquity of a Homeric expression and its formulaic character. In other words, although not all Homeric formulae are necessarily of ancient origin, it is nevertheless in the formulaic stock of the epic diction that archaic and backward-looking expressions should be sought. Consequently, demonstration thatκλέος ἄφθιτον is a Homeric formula would constitute valuable evidence for its origin in Indo-European heroic poetry. Strangely enough, however, as Parry's analysis won the recognition of scholars,κλέος ἄφθιτονwas identified as a Homeric formula simply because of its agreement with the Vedicśráva ákṣitam. Yet examination ofκλέος ἄφθιτονfrom the internal standpoint of the Greek epic casts serious doubts on the formulaic and traditional character of this Homeric expression. (shrink)

I develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes and use it to find exact formulas for expressions which are intuitively of the form $\sum_{s=0}^t\phi(\omega,dl_{s},s)$ and $\prod_{s=0}^t\psi(\omega,dl_{s},s)$ , where l is a Lévy process. These formulas are then applied to geometric Lévy processes, infinitesimal transformations of hyperfinite Lévy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic calculus for (...) processes with jumps on manifolds, and the paper may be regarded as a reworking of his ideas in a different setting and with totally different techniques. (shrink)

We focus on the dichotomous choice model, which goes back as far as Condorcet (1785; Essai sur l'application de l'analyse a la probabilité des décisions rendues a la pluralité des voix, Paris). A group of experts is required to select one of two alternatives, of which exactly one is regarded as correct. The alternatives may be related to a wide variety of areas. A decision rule translates the individual opinions of the members into a group decision. A decision rule is (...) optimal if it maximizes the probability of the group to make a correct choice. In this paper we assume the correctness probabilities of the experts to be independent random variables, selected from some given distribution. Moreover, the ranking of the members in the team is (at least partly) known. Thus, one can follow rules based on this ranking. The polar different rules are the expert and the majority rules. The probabilities of the two polar rules being optimal were compared in a series of papers. The main purpose of this paper is to outline the results, providing exact formulas or estimates for these probabilities. We consider a variety of distributions and show that for all of these distributions the asymptotic behaviour of the probabilities of the two polar rules follows the same patterns. (shrink)

In a recent article Margalit Finkelberg raises the question of whether or not the phrase κλοσ π;θιτον at Iliad 9.413 is indeed a Homeric formula: λετο μν μοι νóατοσ, τρ κλοσ π;θιτον σται Her purpose is to ‘test the antiquity of κλοσ π;θιτον on the internal grounds of Homeric diction’ .1 Proposing to use specifically the analytic techniques of oral theory, she argues that this phrase does not represent a survival from an Indo-European heroic poetry, as has been suggested from (...) the occurrence of its exact cognate, śápos;rdvas áksitam, in Vedic poetry. To this end Finkelberg presents a precise and carefully organized argument. I briefly summarize its two branches as follows: It is the formulae that comprise the oldest stratum of Homeric diction, and so it is here that one would find survivals of Indo-European poetic diction. κλοσ π;θιτον σται , however, cannot be judged a Homeric formula by the criterion of repetition since it is a unique phrase in Homer. Nor can it be judged a formula by the ‘functional’ criterion since the better attested κλοσ οὒ ποτ óλεται expresses the same essential idea in the same metrical shape. A unique phrase such as that in question might nonetheless be ancient. The development of the use of π;θιτοσ, first to modify concrete nouns, and only later with abstracts, however, would indicate that its use with κλοσ is late. The demonstration that κλοσ θιτον σται is a ‘formulaic expression’, moreover, argues that it was coined for this specific context in Iliad 9 by analogy with other Homeric formulae, and so does not preserve an Indo-European formula. (shrink)

Kurt Gödel’s Incompleteness theorem is well known in Mathematics/Logic/Philosophy circles. Gödel was able to find a way for any given P (UTM), (read as, “P of UTM” for “Program of Universal Truth Machine”), actually to write down a complicated polynomial that has a solution iff (=if and only if), G is true, where G stands for a Gödel-sentence. So, if G’s truth is a necessary condition for the truth of a given polynomial, then P (UTM) has to answer first that (...) G is true in order to secure the truth of the said polynomial. But, interestingly, P (UTM) could never answer that G was true. This necessarily implies that there is at least one truth a P (UTM), however large it may be, cannot speak out. Daya Krishna and Karl Potter’s controversy regarding the construal of India’s Philosophies dates back to the time of Potter’s publication of “Presuppositions of India’s Philosophies” (1963, Englewood Cliffs Prentice-Hall Inc.) In attacking many of India’s philosophies, Daya Krishna appears to have unwittingly touched a crucial point: how can there be the knowledge of a ‘non-cognitive’ mokṣa? [‘mokṣa’ is the final state of existence of an individual away from Social Contract]—See this author’s Indian Social Contract and its Dissolution (2008) mokṣa does not permit the knowledge of one’s own self in the ordinary way with threefold distinction, i.e., subject–knowledge-object or knower–knowledge–known. But what is important is to demonstrate whether such ‘knowledge’ of non-cognitive mokṣa state can be logically shown, in a language, to be possible to attain, and that there is no contradiction involved in such demonstration, because, no one can possibly express the ‘experience-itself’ in language. Hence, if such ‘knowledge’ can be shown to be logically not impossible in language, then, not only Daya Krishna’s arguments against ‘non-cognitive mokṣa’ get refuted but also it would show the possibility of achieving ‘completeness’ in its truest sense, as opposed to Gödel’s ‘Incompleteness’. In such circumstances, man would himself become a Universal Truth Machine. This is because the final state of mokṣa is construed as the state of complete knowledge in Advaita. This possibility of ‘completeness’ is set in this paper in the backdrop of Śrī Śaṅkarācārya’s Advaitic (Non-dualistic) claim involved in the mahāvākyas (extra-ordinary propositions). (Mahāvākyas that Śaṅkara refers to are basically taken from different Upaniṣads. For example, “Aham Brahmāsmi” is from Bṛhadāraṇyaka Upanisad, and “Tattvamasi” is from Chāndogya Upaniṣad. Śrī Śaṅkarācārya has written extensively. His main works include his Commentary on Brahma-Sūtras, on major Upaniṣads, and on ŚrīmadBhagavadGītā, called Bhāṣyas of them, respectively. Almost all these works are available in English translation published by Advaita Ashrama, 5 Dehi Entally Road, Calcutta, 700014.) On the other hand, the ‘Incompleteness’ of Gödel is due to the intervening G-sentence, which has an adverse self-referential element. Gödel’s incompleteness theorem in its mathematical form with an elaborate introduction by R.W. Braithwaite can be found in Meltzer (Kurt Gödel: on formally undecidable propositions of principia mathematica and related systems. Oliver & Boyd, Edinburgh, 1962). The present author believes first that semantic content cannot be substituted by any amount of arithmoquining, (Arithmoquining or arithmatization means, as Braithwaite says,—“Gödel’s novel metamathematical method is that of attaching numbers to the signs, to the series of signs (formulae) and to the series of series of signs (“proof-schemata”) which occur in his formal system…Gödel invented what might be called co-ordinate metamathematics…”) Meltzer (1962 p. 7). In Antone (2006) it is said “The problem is that he (Gödel) tries to replace an abstract version of the number (which can exist) with the concept of a real number version of that abstract notion. We can state the abstraction of what the number needs to be, [the arithmoquining of a number cannot be a proof-pair and an arithmoquine] but that is a concept that cannot be turned into a specific number, because by definition no such number can exist.”.), especially so where first-hand personal experience is called for. Therefore, what ultimately rules is the semanticity as in a first-hand experience. Similar points are voiced, albeit implicitly, in Antone (Who understands Gödel’s incompleteness theorem, 2006). (“…it is so important to understand that Gödel’s theorem only is true with respect to formal systems—which is the exact opposite of the analogous UTM (Antone (2006) webpage 2. And galatomic says in the same discussion chain that “saying” that it ((is)) only true for formal systems is more significant… We only know the world through “formal” categories of understanding… If the world as it is in itself has no incompleteness problem, which I am sure is true, it does not mean much, because that is not the world of time and space that we experience. So it is more significant that formal systems are incomplete than the inexperiencable ‘World in Itself’ has no such problem.—galatomic”) Antone (2006) webpage 2. Nevertheless galatomic certainly, but unwittingly succeeds in highlighting the possibility of experiencing the ‘completeness’ Second, even if any formal system including the system of Advaita of Śaṅkara is to be subsumed or interpreted under Gödel’s theorem, or Tarski’s semantic unprovability theses, the ultimate appeal would lie to the point of human involvement in realizing completeness since any formal system is ‘Incomplete’ always by its very nature as ‘objectual’, and fails to comprehend the ‘subject’ within its fold. (shrink)

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬ BOX p, read as “p is true but you do not know p.” Not (...) only is the Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

We show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬ BOX p, read as “p is true but you do not know p.” Not (...) only is the Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

In mathematical practice certain formulas φ are believed to essentially decide all other natural properties of the object x. The purpose of this paper is to exactly quantify such a belief for four formulas φ, namely “x is a Ramsey ultrafilter”, “x is a free Souslin tree”, “x is an extendible strong Lusin set” and “x is a good diamond sequence”.

Predicate modal formulas with non-modalized quantifiers (call them Q-formulas) are considered as schemata of arithmetical formulas, where is interpreted as the provability predicate of some fixed correct extension T of arithmetic. A method of constructing 1) non-provable in T and 2) false arithmetical examples for Q-formulas by Kripke-like countermodels of certain type is given. Assuming the means of T to be strong enough to solve the (undecidable) problem of derivability in QGL, the Q-fragment of the predicate (...) version of the logic GL, we prove the recursive enumerability of the sets of Q-formulas all arithmetical examples of which are: 1) T-provable, 2) true. In. particular, the first one is shown to be exactly QGL and the second one to be exactly the Q-fragment of the predicate version of Solovay's logic S. (shrink)

An important component in the interventionist account of causal explanation is an interpretation of counterfactual conditionals as statements about consequences of hypothetical interventions. The interpretation receives a formal treatment in the framework of functional causal models. In Judea Pearl’s influential formulation, functional causal models are assumed to satisfy a “unique-solution” property; this class of Pearlian causal models includes the ones called recursive. Joseph Halpern showed that every recursive causal model is Lewisian, in the sense that from the causal model one (...) can construct a possible worlds model in David Lewis’s well-known semantics that satisfies the exact same formulas in a certain language. Moreover, he demonstrated that some Pearlian (non-recursive) models are not Lewisian in this sense. This raises the question regarding the exact contour of Lewisian causal models. In this paper, we provide a characterization of the class of Lewisian causal models and a complete axiomatization with respect to this class. Our results have philosophically interesting consequences, two of which are especially worth noting. First, the class of Stalnakerian causal models, a subclass of Lewisian causal models, is precisely the class of Pearlian models that do not contain any cycle of counterfactual dependence (in a sense of counterfactual dependence akin to Lewis’s famous relation between distinct events). Second, a more natural class of causal models is actually a superclass of Lewisian causal models, the logic of which respects only weak centering rather than centering. (shrink)

In this paper, I will describe a technique for generating a novel kind of semantics for a logic, and explore some of its consequences. It would be natural to call the semantics produced by the technique in question ‘many-valued'; but that name is, of course, already taken. I call them, instead, ‘plurivalent'. In standard logical semantics, formulas take exactly one of a bunch of semantic values. I call such semantics ‘univalent'. In a plurivalent semantics, by contrast, formulas may (...) take one or more such values. The construction I shall describe can be applied to any univalent semantics to produce a corresponding plurivalent one. In the paper I will be concerned with the application of the technique to propositional many-valued logics. Sometimes going plurivalent does not change the consequence relation; sometimes it does. I investigate the possibilities in detail with respect to small family of many-valued logics. (shrink)

A complementary system for a given logic is a proof system whose theorems are exactly the formulas that are not valid according to the logic in question. This article is a contribution to the complementary proof theory of classical propositional logic. In particular, we present a complementary proof-net system, $$\textsf{CPN}$$ CPN, that is sound and complete with respect to the set of all classically invalid (one-side) sequents. We also show that cut elimination in $$\textsf{CPN}$$ CPN enjoys strong normalization along (...) with strong confluence (and, hence, uniqueness of normal forms). (shrink)

This article begins by exploring a lost topic in the philosophy of science:the properties of the relations evidence confirming h confirmsh'' and, more generally, evidence confirming each ofh1, h2, ..., hm confirms at least one of h1, h2,ldots;, hn''.The Bayesian understanding of confirmation as positive evidential relevanceis employed throughout. The resulting formal system is, to say the least, oddlybehaved. Some aspects of this odd behaviour the system has in common withsome of the non-classical logics developed in the twentieth century. Oneaspect (...) – its ``parasitism'''' on classical logic – it does not, and it is this featurethat makes the system an interesting focus for discussion of questions in thephilosophy of logic. We gain some purchase on an answer to a recently prominentquestion, namely, what is a logical system? More exactly, we ask whether satisfaction of formal constraints is sufficient for a relation to be considered a (logical) consequence relation. The question whether confirmation transfer yields a logical system is answered in the negative, despite confirmation transfer having the standard properties of a consequence relation, on the grounds that validity of sequents in the system is not determined by the meanings of the connectives that occur in formulas. Developing the system in a different direction, we find it bears on the project of ``proof-theoretic semantics'''': conferring meaning on connectives by means of introduction (and possibly elimination) rules is not an autonomous activity, rather it presupposes a prior, non-formal,notion of consequence. Some historical ramifications are alsoaddressed briefly. (shrink)

In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, (...) we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of theTractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in theTractatuswhen taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.5.503. (shrink)

We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E1, a theory axiomatized by T → ⊥. The intersection CPC ∩ E1 is axiomatizable by the Principle of the Excluded Middle A V ∨ ⌝A. If B is a formula (...) such that → B is not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula → B is derivable exactly when B is provably equivalent to a formula of the form → A) →. (shrink)

Orthomodular logic is a weakening of quantum logic in the sense of Birkhoff and von Neumann. Orthomodular logic is shown to be a nonlinear noncommutative logic. Sequents are given a physically motivated semantics that is consistent with exactly one semantics for propositional formulas that use negation, conjunction, and implication. In particular, implication must be interpreted as the Sasaki arrow, which satisfies the deduction theorem in this logic. As an application, this deductive system is extended to two systems of predicate (...) logic: the first is sound for Takeuti’s quantum set theory, and the second is sound for a variant of Weaver’s quantum logic. (shrink)

Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the first order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindstrom and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models (...) nor elementary chains. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Anger of Achilles: Mēnis in Greek EpicJenny Strauss ClayLeonard Muellner. The Anger of Achilles: Mēnis in Greek Epic. Ithaca: Cornell University Press, 1996. ix + 219 pp. Cloth, $39.95.At the beginning of Greek literature, and hence the whole classical tradition, stands an enigmatic word: mēnis. Usually translated as "wrath" or "anger," mēnis constitutes the subject of the Iliad, but its precise meaning and implications remain elusive. Muellner's (...) long–awaited study offers a wide–ranging and provocative examination not only of the word itself but of its context and the underlying conceptual universe it presupposes. Observing that words for emotions are not universal and hence must be examined from within a specific cultural milieu, Muellner acknowledges the impasse to which previous semantic and lexicographical studies have led, and he proposes a fresh approach to the study of mēnis within the context of Lord's notions of epic composition by theme. The meaning of mēnis cannot be grasped by focusing narrowly on the individual word, but can only be discovered by examining the thematic complex in which it appears—and even sometimes where it does not appear. Like Nagy and some other scholars, Muellner substantially extends the notion of theme from Lord's "repeated passage with a fair amount of verbal repetition" to a cluster of motifs. (The extension of the "strict" Parryist definition of the formula to the so–called structural formula offers a parallel; at one point Muellner even speaks of "deep structure.") But the thematic complex, whose components may never be exactly the same in its multiple appearances, implies the presence of a unified underlying concept. Muellner's goal is to go "beyond a desire to 'redefine' mēnis" and instead to rebuild "the poetic function of the mēnis theme in the epic tradition" (133).In the first two chapters he examines the instances of mēnis in Homer, including the Homeric Hymns, but excluding occurrences in the first book of the Iliad and references to the mēnis of Achilles.Mēnis scenes do not contain easily recognizable formulas or formulaic sequences, as for instance in arming scenes or even in larger thematic patterns like "the hero returns." The remarkable variety of contexts in which mēnis appears renders the attempt to discover a common denominator elusive, especially since Muellner begins his investigation obliquely, with a scene in which the word mēnis does not even occur. In the course of his investigation, he offers a series of definitions of the compositional theme of mēnis: it involves "a cosmic sanction... a social force whose activation brings drastic consequences on the whole community," and it "is incurred by the breaking of basic religious or social tabus" that lead to the indiscriminate punishment of the whole community or social group (8). Even more abstract is his [End Page 631] statement that mēnis is "the irrevocable cosmic sanction that prohibits some characters from taking their superiors for equals and others for taking their equals for inferiors" (31).The fundamental rules which govern behavior and whose violation provokes mēnis are the themistes that regulate interaction vertically between hierarchical groups (gods and men) and horizontally between, say, members of the same social group. An easily recognizable category of actions that unleash mēnis occurs when human beings try to act like gods either on the battlefield or by sleeping with goddesses. Likewise the themistes involving hospitality, ransom, supplication, or the treatment of beggars are founded on fundamental notions of status, exchange, and reciprocity, all sanctioned by Zeus. As Muellner summarizes: "when a person dies, or a hero surpasses himself, or a god ignores the rank of another god, or a goddess sleeps with a mortal, or a king rejects an offer of exchange, the value of a person and the continuity of the world is at stake because the hierarchy of persons based on the themistes of exchange is being breached" (51). Yet not all such violations elicit mēnis or indiscriminate punishment. Sometimes nothing happens: for instance, not all rejected supplications provoke mēnis. Leaving the dead unburied is clearly tabu, but Muellner's attempt... (shrink)

Although we believe the results reported below to have direct philosophical import, we shall for the most part confine our remarks to the realm of mathematics. The reader is referred to [4] for a philosophically oriented discussion, comprehensible to mathematicians, of tense logic.The “minimal” tense logicT0is the system having connectives ∼, →,F(“at some future time”), andP(“at some past time”); the following axioms:(whereGandHabbreviate ∼F∼ and ∼P∼ respectively); and the following rules:(8) fromαandα → β, inferβ,(9) fromα, infer any substitution instance ofα,(10) fromα, (...) inferGα,(11) fromα, inferHα.A tense logic is a systemTwhose language is that ofT0and whose axioms and rules include (1)–(11). The axioms and rules ofTother than (1)–(11) are calledproperaxioms and rules.We shall investigate three systems of semantics for tense logics, i.e. three notions ofstructureand three relations ⊧ which stand between structures and formulas. One reads⊧αas “αis valid in.” A structureis amodelof a tense logicTif every formula provable inTis valid in. A semantics isadequateforTif the set of models ofTin the semantics is characteristic forT, i.e. if wheneverT ∀ αthen there is a modelofTin the semantics such that∀ α. Two structuresand, possibly from different semantics, are calledequivalent(∼) if exactly the same formulas are valid inas in. (shrink)

The logical calculus for SAT are not valid for MaxSAT and MinSAT because they preserve satisfiability but not the number of unsatisfied clauses. To overcome this drawback, a MaxSAT resolution rule preserving the number of unsatisfied clauses was defined in the literature. This rule is complete for MaxSAT when it is applied following a certain strategy. In this paper we first prove that the MaxSAT resolution rule also provides a complete calculus for MinSAT if it is applied following the strategy (...) proposed here. We then describe an exact variable elimination algorithm for MinSAT based on that rule. Finally, we show how the results for Boolean MinSAT can be extended to solve the MinSAT problem of the multiple-valued clausal forms known as signed conjunctive normal form formulas. (shrink)

The systems T N and T M show that necessity can be consistently construed as a predicate of syntactical objects, if the expressive/deductive power of the system is deliberately engineered to reflect the power of the original object language operator. The system T N relies on salient limitations on the expressive power of the language L N through the construction of a quotational hierarchy, while the system T Mrelies on limiting the scope of the modal axioms schemas to the sublanguage (...) L infM +, which corresponds exactly with the restrictive hierarchy of L N. The fact that L infM + is identical to the image of the metalinguistic mapping C + from the normal operator system into L M reveals that iterated operator modality is implicitly hierarchical, and that inconsistency is produced by applying the principles of the modal logic to formulas which have no natural analogues in the operator development. Thus the contradiction discovered by Montague can be diagnosed as the result of instantiating the axiom schemas with modally ungrounded formulas, and thereby adding radically new modal axioms to the predicate system.The predicate treatment of necessity differs significantly from that of the operator in that the cumulative models for the predicate system are strictly first-order. Possible worlds are not used as model-theoretic primitives, but rather alternate models are appealed to in order to specify the extension of N, which is semantically construed as a first-order predicate. In this manner, the intensional aspects of modality are built into the mode of specifying the particular set of objects which the denotation function assigns to N, rather than in the specification of the basic truth conditions for modal formulas. Intensional phenomena are thereby localised to the special requirements for determining the extension of a particular predicate, and this does not constitute a structural modification of the first-order models, but rather limits the relevant class of models to those which possess an appropriate denotation function. (shrink)

Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization (...) of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized. (shrink)

We characterize the proof-theoretic strength of systems of explicit mathematics with a general principle asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the Σ12-axiom of choice and Π12-comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a (...) stable ordinal. In all cases, the exact strength depends on what forms of induction are admitted in the respective systems. (shrink)

Given a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $, we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that (...) is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory, that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic. (shrink)

Dzhaparidze, G., A generalized notion of weak interpretability and the corresponding modal logic, Annals of Pure and Applied Logic 61 113-160. A tree Tr of theories T1,...,Tn is called tolerant, if there are consistent extensions T+1,...,T+n of T1,...,Tn, where each T+i interprets its successors in the tree Tr. We consider a propositional language with the following modal formation rule: if Tr is a tree of formulas, then Tr is a formula, and axiomatically define in this language the decidable logics (...) TLR and TLRω. It is proved that TLR yields exactly the schemata of PA-provable sentences, if Tr is understood as “Tr is tolerant”. In fact, TLR axiomatizes a considerable fragment of provability logic with quantifiers over ∑1-sentences, and many relations that have been studied in the literature can be expressed in terms of tolerance. We introduce and study two more relations between theories: cointerpretability and cotolerance which are, in a sense, dual to interpretability and tolerance. Cointerpretability is a characterization of ∑1-conservativity for essentially reflexive theories in terms of translations. (shrink)

In Of Quadrature by Ordinates (1695), Isaac Newton tried two methods for obtaining the Newton–Cotes formulae. The first method is extrapolation and the second one is the method of undetermined coefficients using the quadrature of monomials. The first method provides \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-ordinate Newton–Cotes formulae only for cases in which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=3,4$$\end{document} and 5. However this method provides another important formulae if the ratios of errors (...) are corrected. It is proved that the second method is correct and provides the Newton–Cotes formulae. Present significance of each of the methods is given. (shrink)

In this paper we shall first show that for every weak DeMorgan algebra $L$ of order $n$ , there is a quotient weak DeMorgan algebra $L{\sim}$ which is embeddable in the finite WDM-$n$ algebra $\Omega $. We then demonstrate that the finite WDM-$n$ algebra $\Omega $ is functionally free for the class $CL$ of WDM-$n$ algebras. That is, we show that any formulas $f$ and $g$ are identically equal in each algebra in $CL$ if and only if they are (...) identically equal in $\Omega $. Finally we establish that there is no weak DeMorgan algebra whose quotient algebra by a maximal filter has exactly seven elements. (shrink)

A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form , or of the form , and all the active formulas of its premises belong to the set . In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive coherence (...) criterion for such systems, and show that a quasi-canonical system of the type we investigate is coherent iff it is strongly paraconsistent or strongly paracomplete (in a sense defined in the paper), iff it has a trivalent, non-deterministic semantics of a special type (also defined in the paper) for which it is sound and complete. Finally, we determine when a system of this sort admits cut-elimination, and provide a simple procedure for transforming one which does not into one which does. (shrink)

This paper contains a logic enabling us to reason in the presence of vague- ness phenomena. We consider an epistemological vagueness of concepts caused by the unavailability of total information about a continuous world which we describe in observational terms. Lack of information is manifested by the existence of borderline cases for concepts. Since we are unable to perceive concepts exactly, we cannot establish a sharp boundary between an extension of a concept and its complement. Some results for reasoning about (...) vague concepts are already known in the literature [1], [2], [3], [4], [10]. Usually, a semantics of vague concepts is developed within the theory of fuzzy sets or the theory of supertruth [9]. Following the idea that reasoning about vague concepts requires a special logic, we introduce a propositional language with sentential operators en- abling us to dene positive, negative, and borderline regions of any vague concept. A semantics of the language is rened within a framework of the theory of rough sets . The admitted semantics provides means for dening extensions of concepts in a formal way. We show that these extensions re ect properly our intuition connected with vagueness. For ex- ample, the law of excluded middle is not valid on the level of extensions. Moreover, for any concept, the extensions of formulas representing its pos- itive, negative, and borderline regions provide a partition of a universe of discourse. (shrink)

Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that the domain of interpretation is not empty, every name denotes exactly one object in the domain and the quantifiers have existential import. Free logics usually reject the claim that names need to denote in, and of the systems considered in this paper, the positive free logic concedes that (...) some atomic formulas containing non-denoting names are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject, are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics. The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework. (shrink)

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the (...) bounded proper forcing axiom is exactly the existence of a Σ 1 -reflecting cardinal (which is less than the existence of a Mahlo cardinal). We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure. (shrink)

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 (...) we show that for the pure logic of one of these implicational connectives two – in general distinct – consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the ‘propositional operations’ associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives. (shrink)