Logical and philosophical literature provides different classifications of reasoning. In the Polish literature on the subject, for instance, there are three popular ones accepted by representatives of the Lvov-Warsaw School: Jan Łukasiewicz, Tadeusz Czeżowski and Kazimierz Ajdukiewicz (Ajdukiewicz in Logika pragmatyczna [Pragmatic Logic]. PWN, Warsaw (1965, 2nd ed. 1974). Translated as: Pragmatic Logic. Reidel & PWN, Dordrecht, 1975). The author of this paper, having modified those classifications, distinguished the following types of reasoning: (1) deductive and (2) non-deductive, and additionally two (...) types of them in each of the two, depending on the manner of combining their premises with the conclusion through the relation of classical logical entailment. Consequently, the four types of reasoning: unilateral deductive (incl. its sub-types: deductive inference and proof), bilateral deductive (incl. complete induction), and reductive (incl. the sub-types: explanation and verification), logically nonvaluable (incl. inference by analogy, statistic inference), correspond to four operators of derivability. They are defined formally on the ground of Tarski’s axiomatic theory of deductive systems, by means of the consequence operation _Cn_ (Tarski in Monatshefte Math Phys 37:361–404, 1930a, C R Soc Sci Lett Vars 23:22–29, 1930b). Also, certain metalogical properties of these operators are given, as well as their relations with Tarski’s consequence operations \(Cn^+\) ( \(Cn^+ = Cn\) ) and dual consequences \(Cn^{-1}\) (Słupecki in Zeszyty Naukowe Uniwersytetu Wrocławskiego Seria B Nr 3:33–40, 1959, Słupecki et al. in Stud Log 29:76–123, 1971, Wybraniec-Skardowska, in: Wybraniec-Skardowska, Bryll (eds) Z badań nad teorią zdań odrzuconych [Studies in the Theory of Rejected Propositions], Series B, Studia i Monografie, Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Opole, 1969), and \(Cn^-\) (Wójcicki in Bull Sect Log 2(2):54–57, 1973)). (shrink)

The monograph contains three works on research on the concept of a rejected sentence. This research, conducted under the supervision of Prof. Jerzy Słupecki by U. Wybraniec-Skardowska (1) "Theory of rejected sentences" and G. Bryll (2) "Some supplements of theory of rejected sentences" and (3) "Logical relations between sentences of empirical sciences" led to the construction of a theory rejected sentences and made it possible to formalize certain issues in the methodology of empirical sciences. The concept of a rejected sentence (...) was introduced by J. Łukasiewicz in the course of his inquiry into Aristotelian Syllogistic. It was essentialy generelized by J. Słupecki who, with reference to Tarski's axiomatic theory of deductive systems (1930), gave the formal definition of the rejection function Cn' according to which a sentence y is rejected on the base of a set of sentences X iff at least one sentence which belongs to X can be deduced from the sentence y. Słupecki proved (1959) that the function Cn', which assignes to the set X the set of sentences rejected on the basis of X, is additive and satisfies the axioms of Tarski's general theory of deductive systems. Work (1) uses not only Tarski's general theory of systems but also his the so-called enlarged theory of systems (1930). It constructs a theory T of rejected sentences (propositions), which is created from Tarski's richer theory of systems by adding one axiom and a number of definitions; among them the definition of the rejection function Cn' is fundamental one. Cn' is a function which assignes to every set X of false sentences the set Cn'X whose members are exclusively false senteces. A number of theorems which describes properties of defined concepts have been proved in T. Some of them are necessary for conducting considertions of the concluding section of (1) devoted to the empirical sentences. Section 2 of (1) deals with the theory of the unit consequenses. It is characterized by an axiom from which follow all axioms of Tarski's general theory of deductive systems, additivity and the property that the unit consequence of the set X of sentences is a unit consequence only one sentence of X. An example of unit consequence is the rejection consequence Cn'. In section 3 of (1) the dual theory T' equivalent to T is presented. In T' the primitive concept is the rejection function Cn'. The fundamental axiom of T' states that Cn' is a unit consequence. The concluding section brings a few examples of applications of the definitions and theorems of T in the domain of the scientific methodology. Work (2) corresonds to J. Słupecki paper (1959) and the theory T of rejected propositions given in work (1). In this paper some suplements of the theory T are given. Work (3) is an extension of the theory of rejected propositios given in the paper (1). It is an attempt of formalization some problems of methodology of the empirical sciences which concern to such concepts as: empirical sentence, similar sentence, generalization, inductive conclusion etc. (shrink)

This is the PhD dissertation, written under supervision of Professor Jerzy Słupecki, published in the book: U.Wybraniec-Skardowska i Grzegorz Bryll "Z badań nad teorią zdań odrzuconych" ( "Studies of theory of rejected sentences"), Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Seria B: Studia i Monografie nr 22, pp. 5-131. It is the first, original publication on the theory of rejected sentences on which are based, among other, papers: "Theory of rejected propositions. I"and "Theory of rejected propositions II" with Jerzy Słupecki (...) and Grzegorz Bryll. -/- -/- . (shrink)

A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences (...) in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth. (shrink)

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for implication. (...) The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic. (shrink)

CORCORAN REVIEWS THE 4 VOLUMES OF TARSKI’S COLLECTED PAPERS Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but most (...) of the Aristotelian corpus is not about logic, whereas virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework (...) recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

In this paper, two axiomatic theories T− and T′ are constructed, which are dual to Tarski’s theory T+ (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory T+ the primitive notion is the classical consequence function (entailment) Cn+, in the dual theory T− it is replaced by the notion of Słupecki’s rejection consequence Cn− and in the dual theory T′ it is replaced by the notion of the family Incons of inconsistent sets. The author (...) has proved that the theories T+, T−, and T′ are equivalent. (shrink)

The theme of this paper is Husserl’s concept of experience, through which I hope to show that and how Husserl’s description points the way toward a more adequate account of experience than traditional ones operating within realist-idealist and rationalist-empiricist frameworks.

Certain passages in Kaplan’s ‘Demonstratives’ are often taken to show that non-vacuous sentential operators associated with a certain parameter of sentential truth require a corresponding relativism concerning assertoric contents: namely, their truth values also must vary with that parameter. Thus, for example, the non-vacuity of a temporal sentential operator ‘always’ would require some of its operands to have contents that have different truth values at different times. While making no claims about Kaplan’s intentions, we provide several reconstructions of how such (...) an argument might go, focusing on the case of time and temporal operators as an illustration. What we regard as the most plausible reconstruction of the argument establishes a conclusion similar enough to that attributed to Kaplan. However, the argument overgenerates, leading to absurd consequences. We conclude that we must distinguish assertoric contents from compositional semantic values, and argue that once they are distinguished, the argument fails to establish any substantial conclusions. We also briefly discuss a related argument commonly attributed to Lewis, and a recent variant due to Weber. (shrink)

Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but most of the Aristotelian corpus is not about logic, whereas (...) virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82. Two of his works appeared posthumously [Hist. Philos. Logic 7 (1986), no. 2, 143--154; MR0868748 (88b:03010); Tarski and Givant, A formalization of set theory without variables, Amer. Math. Soc., Providence, RI, 1987; MR0920815 (89g:03012)]. Tarski's voluminous writings were widely scattered in numerous journals, some quite rare. It has been extremely difficult to study the development of Tarski's thought and to trace the interconnections and interdependence of his various papers. Thanks to the present collection all this has changed, and it is likely that the increased accessibility of Tarski's papers will have the effect of increasing Tarski's already enormous influence. (shrink)

It might come as a surprise for someone who has only a superficial knowledge of Donald Davidson’s philosophy that he has claimed literary language to be ‘a prime test of the adequacy of any view on the nature of language’.1 The claim, however, captures well the transformation that has happened in Davidson’s thinking on language since he began in the 1960’s to develop a truth-conditional semantic theory for natural languages in the lines of Alfred Tarski’s semantic conception of truth. About (...) twenty years afterwards, this project was replaced with a view that highlights the flexible nature of language and, in consequence, the importance of the speaker’s intentions for a theory of meaning, culminating in Davidson’s staggering claim that ‘there is no such thing as a language’. (shrink)

An essay review of Gila Sher's *Logical Consequence*.

ABSTRACT: This 1974 paper builds on our 1969 paper (Corcoran-Weaver [2]). Here we present three (modal, sentential) logics which may be thought of as partial systematizations of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of these three logics coincide with one another and with those of standard formalizations of Lewis's S5. These logics, when regarded as logistic systems (cf. Corcoran [1], p. 154), are seen to be equivalent; (...) but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be as closely linked as previously thought. -/- This 1974 paper uses the linear notation for natural deduction presented in [2]: each two-dimensional deduction is represented by a unique one-dimensional string of characters. Thus obviating need for two-dimensional trees, tableaux, lists, and the like—thereby facilitating electronic communication of natural deductions. The 1969 paper presents a (modal, sentential) logic which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of this logic coincides those of standard formalizations of Lewis’s S4. Among the paper's innovations is its treatment of modal logic in the setting of natural deduction systems--as opposed to axiomatic systems. The author’s apologize for the now obsolete terminology. For example, these papers speak of “a proof of a sentence from a set of premises” where today “a deduction of a sentence from a set of premises” would be preferable. 1. Corcoran, John. 1969. Three Logical Theories, Philosophy of Science 36, 153–77. J P R -/- 2. Corcoran, John and George Weaver. 1969. Logical Consequence in Modal Logic: Natural Deduction in S5 Notre Dame Journal of Formal Logic 10, 370–84. MR0249278 (40 #2524). 3. Weaver, George and John Corcoran. 1974. Logical Consequence in Modal Logic: Some Semantic Systems for S4, Notre Dame Journal of Formal Logic 15, 370–78. MR0351765 (50 #4253). (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

Fitch’s Paradox shows that if every truth is knowable, then every truth is known. Standard diagnoses identify the factivity/negative infallibility of the knowledge operator and Moorean contradictions as the root source of the result. This paper generalises Fitch’s result to show that such diagnoses are mistaken. In place of factivity/negative infallibility, the weaker assumption of any ‘level-bridging principle’ suffices. A consequence is that the result holds for some logics in which the “Moorean contradiction” commonly thought to underlie the result (...) is in fact consistent. This generalised result improves on the current understanding of Fitch’s result and widens the range of modalities of philosophical interest to which the result might be fruitfully applied. Along the way, we also consider a semantic explanation for Fitch’s result which answers a challenge raised by Kvanvig. (shrink)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These (...) practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

In a recent paper, Michael Huemer provides a new interpretation for ‘N’, the operator that occurs in Peter van Inwagen’s Consequence Argument, and argues that, given that interpretation, the Consequence Argument is sound. I have no quarrel with Huemer’s claim that the Consequence Argument is valid. I shall argue instead that his defense of its premises—a defense that allegedly involves refuting David Lewis’s response to van Inwagen—is unsuccessful.

This paper aims to contribute a design-focused perspective on the ‘alternative paradigm research’ discussion. To clarify the aspect of ‘design-focus’ that we wish to refer to, we will use the term ‘areas of design practice’ to cover those activities that focus on the conception and production of artefacts, in contrast to the activities of theorizing and writing histories. The literature on academic research in areas of design practice encompasses a board range of subjects and terminology -- it refers to the (...) creative and performing arts, the arts, design, etc., however what we take to be common in the literature is an interest in research involving the creation of original artefacts. In this paper, we cluster together the currently academicized, previously vocational areas. -/- We claim that researchers in different disciplines operate in different research paradigms owing to different worldviews. A worldview is determined by a set of foundational beliefs that are taken on faith, and a research paradigm is defined by axioms and assumptions that are a consequence of that worldview. For example, the scientific view of the world is that it exists independently from the individual and the individual’s thoughts about it; and it is possible to find out objective facts about the world by following certain procedures. This worldview conditions a research paradigm with an ontology of a real external world; that methodologically the data that is collected in procedurally appropriate ways will yield objective facts about that world; and that epistemologically the researcher stands in a disengaged relation to the world that is being investigated. The procedures that are regarded as appropriate within a particular research paradigm form the research models that are acceptable in that paradigm, and these research models determine what one should do in order to extend knowledge in a particular subject. Of course, the scientific worldview is not the only one, and this paper will contrast it to the design worldview as indicating the emergence of a unique research paradigm. -/- The relationship between a community’s worldview and the academic models that it adopts may be functional or dysfunctional. We claim that the relationship in the areas of design practice is often dysfunctional. This is because the academic model has not developed authentically in relation to its fundamental beliefs, but has done so in response to external forces for academicization. When pushed into the academy, areas of design practice did not possess their own academic models that were effectively linked to its worldview. We claim that, as a result, these areas simply co-opted research models from other disciplines. -/- The paper proposes a rationalistic approach for investigating the relationship between worldview and research paradigm -- between fundamental beliefs, theoretical concepts and methodology in research models. The rationalistic approach enables a critique, which reveals that there are indeed problematic issues with adopting academic models from other research paradigms in an uncritical way, and concludes that areas of design practice should be investigated as an example of a distinct worldview. (shrink)

The thesis deals with the concept of truth and the paradoxes of truth. Philosophical theories usually consider the concept of truth from a wider perspective. They are concerned with questions such as - Is there any connection between the truth and the world? And, if there is - What is the nature of the connection? Contrary to these theories, this analysis is of a logical nature. It deals with the internal semantic structure of language, the mutual semantic connection of sentences, (...) above all the connection of sentences that speak about the truth of other sentences and sentences whose truth they speak about. Truth paradoxes show that there is a problem in our basic understanding of the language meaning and they are a test for any proposed solution. It is important to make a distinction between the normative and analytical aspect of the solution. The former tries to ensure that paradoxes will not emerge. The latter tries to explain paradoxes. Of course, the practical aspect of the solution is also important. It tries to ensure a good framework for logical foundations of knowledge, for related problems in Artificial Intelligence and for the analysis of the natural language. Tarski’s analysis emphasized the T-scheme as the basic intuitive principle for the concept of truth, but it also showed its inconsistency with the classical logic. Tarski’s solution is to preserve the classical logic and to restrict the scheme: we can talk about the truth of sentences of a language only inside another essentially richer metalanguage. This solution is in harmony with the idea of reflexivity of thinking and it has become very fertile for mathematics and science in general. But it has normative nature | truth paradoxes are avoided in a way that in such frame we cannot even express paradoxical sentences. It is also too restrictive because, for the same reason we cannot express a situation in which there is a circular reference of some sentences to other sentences, no matter how common and harmless such a situation may be. Kripke showed that there is no natural restriction to the T-scheme and we have to accept it. But then we must also accept the riskiness of sentences | the possibility that under some circumstances a sentence does not have the classical truth value but it is undetermined. This leads to languages with three-valued semantics. Kripke did not give any definite model, but he gave a theoretical frame for investigations of various models | each fixed point in each three-valued semantics can be a model for the concept of truth. The solutions also have normative nature | we can express the paradoxical sentences, but we escape a contradiction by declaring them undetermined. Such a solution could become an analytical solution only if we provide the analysis that would show in a substantial way that it is the solution that models the concept of truth. Kripke took some steps in the direction of finding an analytical solution. He preferred the strong Kleene three-valued semantics for which he wrote it was "appropriate" but did not explain why it was appropriate. One reason for such a choice is probably that Kripke finds paradoxical sentences meaningful. This eliminates the weak Kleene three valued semantics which corresponds to the idea that paradoxical sentences are meaningless, and thus indeterminate. Another reason could be that the strong Kleene three valued semantics has the so-called investigative interpretation. According to this interpretation, this semantics corresponds to the classical determination of truth, whereby all sentences that do not have an already determined value are temporarily considered indeterminate. When we determine the truth value of these sentences, then we can also determine the truth value of the sentences that are composed of them. Kripke supplemented this investigative interpretation with an intuition about learning the concept of truth. That intuition deals with how we can teach someone who is a competent user of an initial language (without the predicate of truth T) to use sentences that contain the predicate T. That person knows which sentences of the initial language are true and which are not. We give her the rule to assign the T attribute to the former and deny that attribute to the latter. In that way, some new sentences that contain the predicate of truth, and which were indeterminate until then, become determinate. So the person gets a new set of true and false sentences with which he continues the procedure. This intuition leads directly to the smallest fixed point of strong Kleene semantics as an analytically acceptable model for the logical notion of truth. However, since this process is usually saturated only on some transfinite ordinal, this intuition, by climbing on ordinals, increasingly becomes a metaphor. This thesis is an attempt to give an analytical solution to truth paradoxes. It gives an analysis of why and how some sentences lack the classical truth value. The starting point is basic intuition according to which paradoxical sentences are meaningful (because we understand what they are talking about well, moreover we use it for determining their truth values), but they witness the failure of the classical procedure of determining their truth value in some "extreme" circumstances. Paradoxes emerge because the classical procedure of the truth value determination does not always give a classically supposed (and expected) answer. The analysis shows that such an assumption is an unjustified generalization from common situations to all situations. We can accept the classical procedure of the truth value determination and consequently the internal semantic structure of the language, but we must reject the universality of the exterior assumption of a successful ending of the procedure. The consciousness of this transforms paradoxes to normal situations inherent to the classical procedure. Some sentences, although meaningful, when we evaluate them according to the classical truth conditions, the classical conditions do not assign them a unique value. We can assign to them the third value, \undetermined", as a sign of definitive failure of the classical procedure. An analysis of the propagation of the failure in the structure of sentences gives exactly the strong Kleene three-valued semantics, not as an investigative procedure, as it occurs in Kripke, but as the classical truth determination procedure accompanied by the propagation of its own failure. An analysis of the circularities in the determination of the classical truth value gives the criterion of when the classical procedure succeeds and when it fails, when the sentences will have the classical truth value and when they will not. It turns out that the truth values of sentences thus obtained give exactly the largest intrinsic fixed point of the strong Kleene three-valued semantics. In that way, the argumentation is given for that choice among all fixed points of all monotone three-valued semantics for the model of the logical concept of truth. An immediate mathematical description of the fixed point is given, too. It has also been shown how this language can be semantically completed to the classical language which in many respects appears a natural completion of the process of thinking about the truth values of the sentences of a given language. Thus the final model is a language that has one interpretation and two systems of sentence truth evaluation, primary and final evaluation. The language through the symbol T speaks of its primary truth valuation, which is precisely the largest intrinsic fixed point of the strong Kleene three valued semantics. Its final truth valuation is the semantic completion of the first in such a way that all sentences that are not true in the primary valuation are false in the final valuation. (shrink)

I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.

It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical (...) philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the co-incidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato’s Theory of Forms and Aristotle’s metaphysics and logic. Plato’s and Aristotle’s systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a ‘principle of contradiction’ is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing “space” as the seat of stability, and “time” as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion ‘separable system’, related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of ‘non-locality’ as related to it, indicating the need to re-think the notions system, entity, as well as the implications of the operation ‘measurement’, which is seen here as an application of classical logic (including its ontological consequences) on the material world. (shrink)

Dirac’s relativistic theory of electron generally results in two possible solutions, one with positive energy and other with negative energy. Although positive energy solutions accurately represented particles such as electrons, interpretation of negative energy solution became very much controversial in the last century. By assuming the vacuum to be completely filled with a sea of negative energy electrons, Dirac tried to avoid natural transition of electron from positive to negative energy state using Pauli’s exclusion principle. However, many scientists like Bohr (...) objected to the idea of sea of electrons as it indicates infinite density of charge and electric field and consequently infinite energy. In addition, till date, there is no experimental evidence of a particle whose total energy (kinetic plus rest) is negative. In an alternative approach, Feynman, in quantum field theory, proposed that particles with negative energy are actually positive energy particles running backwards in time. This was mathematically consistent since quantum mechanical energy operator contains time in denominator and the negative sign of energy can be absorbed in it. However, concept of negative time is logically inconsistent since in this case, effect happens before the cause. To avoid above contradictions, in this paper, we try to reformulate the Dirac’s theory of electron so that neither energy needs to be negative nor the time is required to be negative. Still, in this new formulation, two different possible solutions exist for particles and antiparticles (electrons and positrons). (shrink)

Four experiments examined children's ability to reason about the causal significance of the order in which 2 events occurred (the pressing of buttons on a mechanically operated box). In Study 1, 4-year-olds were unable to make the relevant inferences, whereas 5-year-olds were successful on one version of the task. In Study 2, 3-year-olds were successful on a simplified version of the task in which they were able to observe the events although not their consequences. Study 3 found that older children (...) had difficulties with the original task even when provided with cues to attend to order information. However, 5-year-olds performed successfully in Study 4, in which the causally relevant event was made more salient. (shrink)

What is a logical constant? The question is addressed in the tradition of Tarski's definition of logical operations as operations which are invariant under permutation. The paper introduces a general setting in which invariance criteria for logical operations can be compared and argues for invariance under potential isomorphism as the most natural characterization of logical operations.

"Paraconsistent Belief Revision Based on a Formal Consistency Operator" delves into Belief Revision—a significant area of research in Formal Philosophy that uses logic to model the ways in which human and artificial agents modify their beliefs in response to new information and examines how these changes can be considered rational. -/- Originally authored as a PhD thesis (previously published in Portuguese), this work provides a novel epistemic interpretation of Paraconsistency through Paraconsistent Belief Revision systems. It explores the concept of paraconsistency (...) from the standpoint of epistemic attitudes of acceptance and rejection. -/- This work challenges the traditional notion that accepting a new belief requires retracting its negation from the current epistemic state. The author contends that such reflexive retraction goes against the principle of informational economy, which is a crucial aspect of rationality in the context of belief change. Consequently, the phenomenon of paraconsistency is further examined from this fresh perspective of belief change, shedding light on the complexities of the Logics of Formal Inconsistency (LFIs). These LFIs provide the foundational logic, offering a comprehensive framework for understanding and implementing paraconsistent principles within belief revision systems. -/- This thesis was supervised by Marcelo Esteban Coniglio and co-supervised by Márcio Moretto Ribeiro, as part of Rafael Rodrigues Testa's doctoral studies at the University of Campinas (Unicamp), Brazil. (shrink)

The focus in this essay will be upon the paradoxes, and foremostly in set theory. A central result is that the librationist set theory £ extension \Pfund $\mathscr{HR}(\mathbf{D})$ of \pounds \ accounts for \textbf{Neumann-Bernays-Gödel} set theory with the \textbf{Axiom of Choice} and \textbf{Tarski's Axiom}. Moreover, \Pfund \ succeeds with defining an impredicative manifestation set $\mathbf{W}$, \emph{die Welt}, so that \Pfund$\mathscr{H}(\mathbf{W})$ %is a model accounts for Quine's \textbf{New Foundations}. Nevertheless, the points of view developed support the view that the truth-paradoxes (...) and the set-paradoxes have common origins, so that the librationist resolutions of the set theoretic paradoxes are at the same time resolutions of the truth theoretic paradoxes. Both the librationist resolutions of the set theoretic paradoxes and the truth theoretic paradoxes have non-trivial philosophical implications: librationist set theories have the consequence that there are no absolutely uncountable sets, and librationist truth theories allow the use of syntactical modalities in ways which circumvent limitations as those of \parencite{Montague1963}, and a truth predicate which is useful for more precise philosophical discourse. (shrink)

This article reconsiders Husserl’s concept of Leib in light of Merleau‐Ponty’s interpretation of the human body as an ontologically significant phenomenon. I first analyze Husserl’s account of the body as a “two‐fold unity” and demonstrate the problematic nature of its four implications, namely, the ambiguous ontological status of the body as subject‐object, the view of “my body” as “my object,” the preconstitutive character of the unity of the body, and the restriction of the constitution of the body to touch alone. (...) Building on this analysis, I explain how Merleau‐Ponty resolves the difficulties raised by Husserl’s account by reversing it. According to Merleau‐Ponty, “flesh” is not a two‐fold reality comprising subjective and objective aspects, but an ontological dimension from which these aspects can be abstracted through specific cognitive operations. Consequently, all subjective and objective aspects, even beyond the boundaries of one’s body, must be understood as founded in the indivisible unity of flesh. I argue that a thorough phenomenological description of the human body requires abandoning Husserl’s concept of Leib because it contributes to perpetuating subject‐object dualism. In contrast, Merleau‐Ponty’s notion of flesh reveals the circularity between subject and object and its general ontological significance. (shrink)

Two plausible claims seem to be inconsistent with each other. One is the idea that if one reasonably believes that one ought to fi, then indeed, on pain of acting irrationally, one ought to fi. The other is the view that we are fallible with respect to our beliefs about what we ought to do. Ewing’s Problem is how to react to this apparent inconsistency. I reject two easy ways out. One is Ewing’s own solution to his problem, which is (...) to introduce two different notions of ought. The other is the view that Ewing’s Problem rests on a simple confusion regarding the scope of the ought-operator. Then, I discuss two hard ways out, which I label objectivism and subjectivism, and for which G.E. Moore and Bishop Butler are introduced as historical witnesses. These are hard ways out because both of these views have strong counterintuitive consequences. After explaining why Ewing’s Problem is so difficult, I show that there is conceptual room in-between Moore and Butler, but I remain sceptical whether Ewing’s Problem is solvable within a realist framework of normative facts. (shrink)

When a philosophical or scientific project comes of age, it finds itself possessed of a tradition, and consequently of scope for a display of its classics. If Praxeology is such a project, then Jakob Meløe’s article The Agent and His World is just such a classic. It’s often no very long step from acquiring a tradition to becoming one. But a project that suffers this transformation stands in risk of losing its character as a project. It then remains only to (...) write the history of the tradition: this is how it all went. As Praxeology is not just a tale to be told, these notes will not be taking up The Agent and His World as the original and radical statement it so patently was within the analytic philosophy of action of the sixties. Nor will they deal with its significance for subsequent praxeological thinking. The aim is rather to enter into a dialogue with Meløe’s article — as one praxeologist to another, or as one piece of praxeology to another. (shrink)

Abstract: As Nigeria’s Central Bank marks its 60th anniversary, it has become necessary to assess its performance. Consequently in this paper, we provide a comprehensive, unbiased review of the Central bank of Nigeria (CBN). The paper starts with a discussion on the role and activities of the West African Currency Board, which was the precursor of the CBN and subsequently analyzes how the Board led to the emergence of the CBN. We look at the functions, mandate and organizational structure of (...) the CBN. Second, we assess how the CBN has responded to the changing financial landscape over the years. Third, noting that the CBN has over time promoted financial reforms and played a key regulatory role, we discuss the sufficiency of the financial reforms it has promoted as well as the regulatory tools that it has deployed over the years. In this regard, we specifically highlight the role of the CBN in the global financial crises of 2007-2009 as well as the European debt crises of 2010 to 2013. Fourth, given that the primary mandate of the CBN revolves around price and exchange rate stability, we analyze its performance thus far noting the numerous challenges it has faced. Summarily, we find that the CBN has operated a multi-pronged approach by utilizing a combination of policy instruments including price-based, quantity-based and administrative policy measures. Going forward, we recommend that the CBN needs to continue to adopt avant garde mechanisms to further strengthen the Nigerian banking industry with a view to maintaining the stability and health of the economy. (shrink)

Causal Determinism (CD) entails that all of a person’s choices and actions are nomically related to events in the distant past, the approximate, but lawful, consequences of those occurrences. Assuming that history cannot be undone nor those (natural) relations altered, that whatever results from what is inescapable is itself inescapable, and the contrariety of inevitability and freedom, it follows that we are completely devoid of liberty: our choices are not freely made; our actions are not freely performed. Instead of disputing (...) the soundness of this reasoning, some philosophers prefer to maintain that we could yet have a small measure of freedom were CD true of our world: although being unable to choose or act differently, one could at least under normal circumstances truly claim to be acting ‘on one’s own’, beyond the control of ‘outside forces’, in a word, autonomous. They further argue that being free in this sense suffices for moral responsibility. Call their philosophy ‘Autonomy Compatibilism’ (AC). -/- In adopting here reactive attitudes towards an agent, one is choosing to highlight the fact that the individual in question is of sound mind, reasoning and acting free from the interference of others. These facts alone, the adherent of AC claims, justify his stance, despite the necessity of the agent’s choices. Why would we not regard a sane individual who is not being coerced, intimidated, deceived or unduly put upon as in charge of his life so as to be responsible for his activities? -/- The Manipulation Argument (MA) is supposed to cut off this line of retreat. Its authors hold that, were CD true of our world, we would be no more autonomous than a victim of “covert, non-constraining control” (CNC): manipulation whereby one person causes another, through the use of methods such as brainwashing or circumspect operant conditioning, to ‘do his bidding’ without the latter being aware of his subjugation or feeling in any way coerced. Since a CNC victim obviously lacks autonomy, then so must “persons” living in a deterministic universe. Defenders of AC have, then, the following argument with which to contend: -/- 1. Victims of CNC (obviously) lack autonomy. 2. Thus, AC would be true only if some definition of autonomy succeeds in specifying a freedom relevant difference between victims of CNC and agents whose choices/actions are necessary consequences of prior events. 3. There could be no such definition. 4. Therefore, AC must be false. -/- The challenge issued here is clear: find a way to refute the claim that being subject to natural laws would be tantamount to being a victim of CNC, to show that Nature is no manipulator. Moreover, this challenge cannot be met by responding with a Frankfurt case: a situation in which things have been surreptitiously arranged so that an agent is unable to avoid doing something that he manages to do ‘on his own’, thus, being autonomous despite his inability to act otherwise. For, even if CD is not inconsistent with autonomy because it eliminates the ability to do otherwise per se, it may yet entail that no human agent ever does act of his own accord, an implication of which would be a lack of alternatives on anyone’s part. In other words, the fact that causally determined beings could never act differently than they do does is perhaps only symptomatic of the reason why such beings would lack autonomy: forces beyond their control would have dominion over their psychological development. Thus, AC advocates must show that the way that an agent’s character would be shaped, were she (merely) subject to natural laws, would leave unimpaired an ability that CNC would destroy. What follows is a definition of this ability, which I also use to solve the Problem of Freedom and Foreknowledge. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

Early in Plato’s Republic, two cities are depicted, one healthy and one with “a fever”—the so- called luxurious city. The operative difference between these two cities is that the citizens of the latter “have surrendered themselves to the endless acquisition of money and have overstepped the limit of their necessities” (373d).i The luxury of this latter city requires the seizure of neighboring lands and consequently a standing army to defend those lands and the city’s wealth. According to the main character, (...) Socrates, war thus finds its origin in communities living beyond the natural limits of necessity. In short, the healthy or true city is sustainable, limiting its consumption to actual needs, while the luxurious city seems not to be sustainable, living beyond its needs in a perpetual quest for more. Plato spends the rest of the Republic in the attempt to reveal the political organization and virtues—in particular, the virtue of moderation—necessary for the luxurious city to be just, healthy and thereby sustainable. The contrasting images of the two cities and Plato’s subsequent discussion raise important questions about the interrelations between justice, consumption and sustainability. In this paper we appropriate Plato’s images of these metaphorical cities to discuss over-consumption and unsustainable practices by the nations of the contemporary “first world”, which in turn perpetuate poverty and environmental degradation in the nations of the “two-thirds world”. Matters of equity and justice are preeminent in our discussion. (shrink)

Robert Sparrow (among others) claims that if an autonomous weapon were to commit a war crime, it would cause harm for which no one could reasonably be blamed. Since no one would bear responsibility for the soldier’s share of killing in such cases, he argues that they would necessarily violate the requirements of jus in bello, and should be prohibited by international law. I argue this view is mistaken and that our moral understanding of war is sufficient to determine blame (...) for any wrongful killing done by autonomous weapons. Analyzing moral responsibility for autonomous weapons starts by recognizing that although they are capable of causing moral consequences, they are neither praiseworthy nor blameworthy in the moral sense. As such, their military role is that of a tool, albeit a rather sophisticated one, and responsibility for their use is roughly analogous to that of existing “smart” weapons. There will likely be some difficulty in managing these systems as they become more intelligent and more prone to unpredicted behavior, but the moral notion of shared responsibility and the legal notion of command responsibility are sufficient to locate responsibility for their use. (shrink)

When Adam Smith published his celebrated writings on economics and moral philosophy he famously referred to the operation of an invisible hand. Adam Smith's Political Philosophy makes visible the invisible hand by examining its significance in Smith's political philosophy and relating it to similar concepts used by other philosophers, revealing a distinctive approach to social theory that stresses the significance of the unintended consequences of human action. This book introduces greater conceptual clarity to the discussion of the invisible hand and (...) the related concept of unintended order in the work of Smith and in political theory more generally. By examining the application of spontaneous order ideas in the work of Smith, Hume, Hayek and Popper, Adam Smith's Political Philosophy traces similarities in approach and from these builds a conceptual, composite model of an invisible hand argument. While setting out a clear model of the idea of spontaneous order the book also builds the case for using the idea of spontaneous order as an explanatory social theory, with chapters on its application in the fields of science, moral philosophy, law and government. (shrink)

The sublime is an aspect of experience that has attracted a great deal of scholarship, not only for scholarly reasons but because it connotes aspects of experience not exhausted by what Descartes once called clear distinct perception. That is, the sublime is an experience of the world which involves us in orientating ourselves within it, and this orientation, our human orientation, elevates us in comparison to the non-human world according to traditional accounts of the sublime. The sublime tells us something (...) about our relation to the world rather than anything about the world per se. Nonetheless there is an objective sense of the sublime in that the narratives involved are culturally endorsed rather than invented by an individual. This means that objects can be judged worthy or not of evoking experiences of the sublime. In other words, it is not an idiosyncratic matter. Immanuel Kant’s formulation of this involved explaining how such an experience is possible in terms of his system of the mind. Jane Forsey notes that Kant takes the features of the sublime as given and extrapolates from them certain features of the mind as if any concept of the sublime must implicate the mental architecture of his account (2007). Further to this she argues that in fact the concept of the sublime does implicate a particular system of the mind but neither Kant nor anyone else can successfully formulate it because the concept itself frames certain contradictions. According to Forsey, two consequences follow. First she argues that Kant’s system of the mind does not support the features of the sublime; and secondly that no system could as the very concept is incoherent. If Forsey can show that Kant was mistaken in presenting his account as coherent given his commitments, this would be of interest in its own right. However, her stronger claim is that we cannot separate any concept of the sublime out from Kant’s theoretical underpinnings. That the way the features of the mind are meant to operate in experiences of the sublime are contradictory simply points to the fatal flaws in the whole concept. Her conclusion is that there is no coherent account of the sublime available to us. I will argue that Forsey bases her reasoning on the assumption that a foundational empiricist or direct perception holds; and she interprets Kant’s notions of imagination, understanding and reason as though they are grounded in just such an account of perception. This is revealed in her interpretation of Kant’s phrase “beyond cognition”. Once this foundationalism is replaced with an account of perception more aligned with current research on perception, both philosophical and empirical, then an account of the sublime is available. Further to this however, I argue that what constitutes the narrative of the sublime is historically contingent. Before setting out my arguments, I consider Forsey’s argument in more detail. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. (...) It is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

The paper focuses on the epistemology developed by Richard Burthogge, the lesser-known seventeenth-century English philosopher, and author, among other works, of Organum Vetus & Novum (1678) and An Essay upon Reason and the Nature of Spirits (1694). Although his ideas had a minimal impact on the philosophy of his time, and have hitherto not been the subject of a detailed study, Burthogge’s writings contain a highly original concept of idealistic constructivism, anticipating (relatively speaking) Kant’s idealism. At the same time, some (...) interesting implications for the issue of self-interpretation and self-identity can be derived from his account of the mind. At the core of Burthogge’s epistemological position is a remarkable idea of structural and functional similarity between the three human cognitive faculties, that is between reasoning, sensation and imagination. Accordingly, every cognition, intellectual as well as sensuous, contains in its structure an object-directed operation of apprehending and thus can be viewed as an intentional act in a broad sense of the term. An immediate consequence of this claim for both mind and object is that they cannot be reduced to a pure stream of sense data, but on the contrary are to be considered ontologically independent from the content of a cognitive act. Furthermore, an object of cognition is never presented to the mind directly, as it is in itself, but always under the modus concipiendi, a particular form or manner of conceiving, specific to human cognitive powers due to their internal structure. As a result, Burthogge clearly anticipates Kant in claiming that it is impossible to know reality in itself, and in situating human consciousness as if in between two unknowable spheres: the external and internal one. Since these restrictions are imposed not only on external, but also on internal cognition, that is on self-knowledge, they lead to considerable difficulties regarding the mind’s identity. Basically, two principal strategies to address this issue are available in the post-Cartesian framework of seeing the mind as an ontologically independent entity, and both of them appear to be ineffective when employed in Burthogge’s epistemology. First, the mind identity can be grounded in some extra-mental factors, preferably in matter. However, with no access to the external world in itself, and with the scope of knowledge restricted to mental phenomena, the mind’s identity cannot be defined in terms of relationship with the reality of which it knows nothing except that it exits. The second option is to explain personal identity by referring it to the intrinsic properties of the mind. Nevertheless, this strategy also faces insuperable obstacles, and for the same reasons as the former does. Since self-knowledge is no exception to the structural-functional principles imposed on cognition, the mind can also be known only under the subjective conditions of conceiving. It is, therefore, just as unknowable in itself as the external things are. The main purpose of the paper is to explore this aspect of Burthogge’s philosophy. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

In this paper, we will motivate the application of specific rules of inference from the propositional calculus to natural language sentences. Specifically, we will analyse De Morgan’s laws, which pertain to the interaction of two central topics in syntactic research: negation and coordination. We will argue that the applicability of De Morgan’s laws to natural language structures can be derived from independently motivated operations of grammar and principles restricting the application of these operations. This has direct empirical consequences for the (...) hypothesised relations between natural language and logic. (shrink)

Analytic philosophy of language has often criticized classical pragmatism for holding to an unwarranted notion of experience which lapses into epistemological foundationalism; defenders of the classics have denied such a consequence. The paper tries to move this debate forward by pointing out that the criticism of the empiricist “given” is not wedded to a specific philosophical method, be it linguistic or pragmatist. From a broader historical perspective drawing in particular on Kant, antifoundationalism turns out to be deeply rooted in (...) modern western philosophy and its ambivalent attitude towards the success of the empirical sciences. This diagnosis allows to reassess classical pragmatism beyond the perceived alternative “language vs. experience”, and to concentrate on antifoundationalism as the real challenge to any modern, epistemologically oriented philosophy. In that perspective, classical pragmatism’s genuine contribution is to do justice to antifoundationalism by focusing on the experimental dynamic of scientific practice, which is most commonly ignored by the analytic tradition. Pragmatism identifies rationality with the practical operation of reflexively determining and articulating what is being experienced. With this approach, it is argued, experiential pragmatism serves modern antifoundationalism ends better than its analytic siblings. (shrink)

In this paper I discuss the nature of consent in general, and as it applies to Carlos Nino’s consensual theory of punishment. For Nino the criminal’s consent to change her legal-normative status is a form of implied consent. I distinguish three types of implied consent: 1) implied consent which is based on an operative convention (i.e. tacit consent); 2) implied consent where there is no operative convention; 3) “direct consent” to the legal-normative consequences of a proscribed act – this is (...) the consent which Nino employs. I argue that Nino’s conception of consent in crime exhibits many common features of “everyday” consent, which justify that it be classed as a form of (implied) consent. h us, Nino is right to claim that the consent in crime is similar to the consent in contracts and to the consent to assume a risk in tort law. (shrink)

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski- Grothendieck set theory TG [1]-[3] It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which implies the (...) existence of inaccessible cardinals [1].Non-conservative extension of ZFC based on an generalized quantifiers considered in [4]. In this paper we look at a set theory NC_{∞^{#}}^{#},based on bivalent gyper infinitary logic with restricted Modus Ponens Rule [5]-[8]. In this paper we deal with set theory NC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule. Set theory NC_{∞^{}}^{} contains Aczel's anti-foundation axiom [9]. We present a new approach to the invariant subspace problem for Hilbert spaces. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace. Non-conservative extension based on set theory NC_{∞}^{#} of the model theoretical nonstandard analysis [10]-[12] also is considered. (shrink)

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

In this essay, I will look into Arthur Schopenhauer’s pessimism, which culminates in the view that since life is not worth living, it is better for us to deny it than try to affirm it. I will argue that his pessimism is not sustainable, and that it fails on its own propositions. In section 1, I will look at the importance of suffering as the central point in Schopenhauer’s pessimism. For Schopenhauer, the essence of the world, which he calls will, (...) is a never-ending, blind and purposeless striving that objectifies itself in the organic and inorganic world. Will, which can also be called the will to live, is constantly driving us from one desire to another, leaving us almost without rest in the constant state of need, hence suffering. I will argue against his position that only suffering is of a positive nature, while happiness can only be negative. According to Schopenhauer, the rare moments of happiness we experience consist only in the absence of suffering. I will also look into his view that absolute happiness is impossible to achieve and question this statement as a weak argument for pessimism. -/- Section 2 will focus on Schopenhauer’s use of the concepts of pain and suffering. As it seems that he uses them as co-extensive terms, I will explain why they are not co-extensive. -/- In section 3, I will discuss Schopenhauer’s view that the world has a moral meaning. Because of inevitable suffering, according to Schopenhauer, our only moral solution to the problem of suffering will be to deny suffering, consequently, life itself. And this is, for him, the hidden moral meaning of the world that only philosophy can uncover. -/- In section 4, I will look at the importance of aesthetic experience, which for Schopenhauer, represents the proof that the will can be abandoned and that the short periods of liberation from the will/suffering can be reached. As for Kant, for Schopenhauer, the aesthetic experience must be purposeless and unintentional and never contemplated or purposeful. Since our contemplated activities belong to the knowledge driven by the principle of practical reason that operates on the basis of our innate categories of time, space and causality and is related only to the phenomenal world, ideas that are the essence of artistic experience cannot be grasped by such knowledge. They can only be contemplated by objective knowledge by genius, who, in the process of artistic creation, abandons his individuality and disconnects from the phenomenal world completely, grasping the idea of a perceived object purely by its rare gift and not by reason. -/- In the final section, I will look at compassion, which can help us overcome suffering, according to Schopenhauer. By abandoning our selfishness through compassion, we can identify with the suffering of others and understand that we are the same as others. That will help us comprehend that our individuality is the main cause of suffering and trigger our attempt to overcome suffering through compassion. I will argue that compassion cannot be devoid of self-interest and, as such, still requires a strong presence of the will. I will also show why asceticism is, as the ultimate attempt to completely redeem suffering, antipodal to compassion. (shrink)

Many commentators on Alfred Tarski have, following Hartry Field, claimed that Tarski's truth-definition was motivated by physicalism—the doctrine that all facts, including semantic facts, must be reducible to physical facts. I claim, instead, that Tarski did not aim to reduce semantic facts to physical ones. Thus, Field's criticism that Tarski's truth-definition fails to fulfill physicalist ambitions does not reveal Tarski to be inconsistent, since Tarski's goal is not to vindicate physicalism. I argue that Tarski's only published (...) remarks that speak approvingly of physicalism were written in unusual circumstances: Tarski was likely attempting to appease an audience of physicalists that he viewed as hostile to his ideas. In later sections I develop positive accounts of: (1) Tarski's reduction of semantic concepts; (2) Tarski's motivation to develop formal semantics in the particular way he does; and (3) the role physicalism plays in Tarski's thought. (shrink)

The aim of this study is to discuss in what sense one can speak about universal character of logic. The authors argue that the role of logic stands mainly in the generality of its language and its unrestricted applications to any field of knowledge and normal human life. The authors try to precise that universality of logic tends in: (a) general character of inference rules and the possibility of using those rules as a tool of justification of theorems of every (...) science, (b) principles of correct formulating of thoughts and using language, and (c) general deductive method of reasoning, i.e in the study of language syntax and semantics. The authors present two theories of language syntax and semantics as exemplifications based on the Tarski’s famous results in metalogic: (1) a formalization of universal syntax system by Andrzej Grzegorczyk based on a new primitive notion of quotation-marks operator, and (2) an enlarged system of axiomatic theory of syntax of any categorial language of Urszula Wybraniec-Skardowska’s in which the main role plays the relation of concatenation understood as a set-theoretical function. The study concludes in arguing that the importance of the result (2) lies not only in solving some philosophical problems but also in different fields of knowledge and spheres of human life as ex. in discussion and education as well. (shrink)

A Husserlian phenomenological approach to logic treats concepts in terms of their experiential meaning rather than in terms of reference, sets of individuals, and sentences. The present article applies such an approach in turn to the reasoning operative in various paradoxes: the simple Liar, the complex Liar paradoxes, the Grelling-type paradoxes, and Gödel’s Theorem. It finds that in each case a meaningless statement, one generated by circular definition, is treated as if were meaningful, and consequently as either true or false, (...) although in fact it is neither. The situation is further complicated by the fact that the sentence used to express the meaningless statement is ambiguous, and may also be used to express a meaningful statement. The paradoxes result from a failure to distinguish between the two meanings the sentence may have. (shrink)

The notion of validity for modal languages could be defined in two slightly different ways. The first is the original definition given by S. Kripke, for which a formula φ of a modal language L is valid if and only if it is true in every actual world of every interpretation of L. The second is the definition that has become standard in most textbook presentations of modal logic, for which a formula φ of L is valid if and only (...) if it is true in every world in every interpretation of L. For simple modal languages, “Kripkean validity” and “Textbook validity” are extensionally equivalent. According to E. Zalta, however, Textbook validity is an “incorrect” definition of validity, because: (i) it is not in full compliance with Tarski’s notion of truth; (ii) in expressively richer languages, enriched by the actuality operator, some obviously true formulas count as valid only if the Kripkean notion is used. The purpose of this paper is to show that (i) and (ii) are not good reasons to favor Kripkean valid- ity over Textbook validity. On the one hand, I will claim that the difference between the two should rather be seen as the result of two different conceptions on how a modal logic should be built from a non-modal basis; on the other, I will show the advantages, for the question at issue, of seeing the actuality operator as belonging to the family of two-dimensional operators. (shrink)