In this paper we discuss the notion of “possible worlds” contrasting a philosophical idea due to Malebranche with a mathematical concept of modern logic due to Kripke from which many pseudo-philosophical discussions have arisen.

Two philosophical arguments, e.g. that the meaning of an expression transcends its use and that the human arithmetical thinking is not entirely algorithmic (the Lucas/Penrose argument) base their theses on Gödel’s first incompleteness theorem. But in both these arguments and in some of their criticisms the word “true” is often used ambiguously: it swings between a licit metamathematical use and an illicit transfer of it in a formal system. The aim of this paper is to show the way these arguments are connected, via G-type sentences (sect 2), and how we argue that the sentence G, albeit unprovable in PA, is true, by using non-conservative extensions of PA with reflections (sect 3). And this without any illicit use of “true”.

Historically, George Boole’s philosophy of logic has been regarded as the very opposite of Gottlob Frege’s, insofar as Frege is characterized as a strong logical anti-psychologist. Although there are significant differences between Frege’s and Boole’s views on logic, there are also significant similarities, which provide support for our representation of Frege’s philosophy of logic as weakly psychologistic. Both Boole and Frege aspire to capture the essence of a pure and ideally perfect language that may faithfully express correct reasoning, but Boole’s trust in the power of natural language is stronger than Frege’s. Although we cannot adequately defend this claim, we submit that a thorough examination of the differences between Frege’s and Boole’s views on symbolic and natural languages may lead to a more interesting comparison of their stances on logic and thinking than the often repeated contrast in their attitudes toward psychologism.

Following Halldén, we define S7 as the system generated by the addition of ↓ ↓ p to S3. Initial motivation for the extension comes from Halldén’s paradox. In addition to resolving the paradox, the resulting system generates a helpful framework for comparing classical propositional logic (CPL) with otherwise incommensurable logics, including paraconsistent logics such as LP. S7, although non-regular and non-normal, thus turns out to be preferable to systems such as S4 and S5 as an account of alethic modality.

Nowadays, many liar-like paradoxes have emerged in the fields of economics, sociology, linguistics, and ethics. However, none of them has been resolved satisfactorily, since they have been examined separately. The Gaifman-Koons paradox is a logical abstraction for all those puzzles, which provides a simple model for analysing their essence. Research shows that all these puzzles are concerned with “rational action”, and differ with liar-like paradoxes essentially. The rational action paradox, together with the epistemic paradox, consists of two types of pragmatical paradox. The paradox is of great significance for epistemic logic. More importantly, the acceptance of the Gaifman-Koons paradox as a rational action paradox enlightens the inner relationship between logical paradox research and philosophy of action research.

In this paper I argue for the view that the meaning of logical constants is properly understood only after taking their cognitive and communicative role into account. My contention is that logic, and the meaning of its vocabulary, are inseparable from man’s inferential and linguistic practices. In order to show this I start by comparing the formal conception of inferential validity with an informal, meaning-based account, thereby arguing that the former has precedence over the latter. Logical expressions would thus be seen as means to bring out the rules we follow in exercising our practical, pre-logical inferential abilities. Finally, I shall suggest that this role can be best appreciated by replacing the traditional truth-conditional account of the meaning of logical constants by a consequentialist one, that is, by adopting an inferentialist semantics.

The paper introduces the idea that L. Borkowski’s theory of ‘proper quantifiers’ is the salient part of the solution to the problem of the limitation of logic. The philosophical motivation of presented reasoning is grounded in L. Wittgenstein’s early philosophy.The first section is concerned with Wittgenstein’s general claim that logic is decidable, and hence it discusses the idea of logic as the calculus. According to Wittgenstein, logical constants in propositional calculus are ‘punctuation-marks’, i.e., they do not refer to any objects but they are symbols that express truth functions.The second section defines the notion of ‘proper quantifier’ that is characterized by ‘quantifier matrix’, i.e., function from the set of the sets of sequences of logical values to the set of logical values. The notion of ‘proper quantifier’ can be applied to the decision problem in the predicate calculus – it provides the zero-one decision procedure for the expressions of the first order monadic predicate calculus.The third section formulates a new definition of logical constants – logical connectives and quantifiers in the monadic predicate calculus – as functions whose ranges consist only of logical values. Since this definition is consistent with Wittgenstein’s idea of logical constants as ‘punctuation-marks’, it might be considered philosophically interesting contribution to the problem of limitation of logic.

It is shown that the Nominalism of much of Modern Logic is what has given rise to many of its problems, especially The Liar Paradox. Shifting to a Realist Logic, in which ‘that’-clauses have a central place, overcomes these problems. The move involved, from the study of mentioned sentences to the employment of ‘that’-clauses, reveals the indexicality of referring phrases, and it is that which enables an escape from The Liar. But also it is shown that no parallel paradox is obtainable with the propositions that ‘that’-clauses designate. Several theoretical issues, however, then need to be faced about what ‘that’-clauses designate, primarily what has been called the problem of ‘The Unity of the Proposition’. The term ‘proposition’, it turns out, is not always appropriate for what ‘that’-clauses designate, so a richer language is introduced to handle all cases. But the problem of the unity of what is designated arises in all cases, and is solved by noting a parallel with Russell’s treatment of individuals and descriptive phrases.

I have two main objectives in this paper. First, I attempt to make sense of the Aristotelian notion of infinity or the notion of potential infinity. I show that the so-called notion of “potential infinity” should be understood as the notion of infinite potentialities. Second, I consider how well different theories of modality may fare as accounts of the notion of potential infinity (i.e. the notion of infinite potentialities). I discuss well-known theories of modality such as genuine modal realism, ersatz modal realism, and modalism. Then I introduce my own theory of modality, modal quantificationalism. I argue that only modal quantificationalism offers an adequate formulation of the notion of potential infinity.

In this paper, I will argue from a historical perspective that (1) Gentzen’s 1935 consistency proof of 1st order Peano Arithmetic PA ([Gentzen 1974]) principally aimed to give a finitist interpretation of implication and (2) this aspect of the 1935 proof emerged as the attempt to cope with the non-finiteness in BHK-interpretation of implication. My argument consists of two parts. First, I will explain that the fundamental idea of the 1935 proof is to show the soundness of PA on some finitist interpretation and Gentzen had attempted to use BHK-interpretation as such an interpretation. It is claimed that he eventually thought that there is a ‘circularity’ which is due to the non-finiteness of BHK-interpretation of implication, so it cannot be used as a finitist interpretation. Secondly, I will argue that the principal aim of the 1935 proof is to give his own finitist interpretation of implication which avoids such a ‘circularity’. Finally, this paper’s conclusion and an open question are presented.

This paper analyzes the temporal structures that are appropriate to study the notion of point of view. For understanding time, the image of a cable is used, instead of the classical one of a tree. It is examined how to understand time in this way and how to formalize it. A logical approach to the notion of point of view is also shown. When we analyze the points of view and their structure, it seems clear that we must take into account the time t in which a point of view is attributed to a subject. A two-dimensional temporal logic which combines a modal dimension for possibilities and a temporal one for the flow of time, offers a clear view of the temporary location of a point of view. In this logic, we have histories, thanks to the temporal dimension, and evaluation is in two indices, time and history. These stories can be seen as different scenarios, providing a clear advantage when applying to the analysis of the notion of point of view. The conclusion is that, in order to give a proper approach for the notion of point of view, all these aspects should be combined.

“Surrender; therefore, surrender or fight” is apparently an argument corresponding to an inference from an imperative to an imperative. Several philosophers, however (Williams 1963; Wedeking 1970; Harrison 1991), have denied that imperative inferences exist, arguing that (1) no such inferences occur in everyday life, (2) imperatives cannot be premises or conclusions of inferences because it makes no sense to say, for example, “since surrender” or “it follows that surrender or fight”, and (3) distinct imperatives have conflicting permissive presuppositions (“surrender or fight” permits you to fight without surrendering, but “surrender” does not), so issuing distinct imperatives amounts to changing one’s mind and thus cannot be construed as making an inference. In response I argue inter alia that, on a reasonable understanding of ‘inference’, some everyday-life inferences do have imperatives as premises and conclusions, and that issuing imperatives with conflicting permissive presuppositions does not amount to changing one’s mind.

Notre étude part de l’hypothèse que l’analyse logique des valeurs spirituelles exige une forme particulière de l’identité, à savoir l’identité symbolique. Il est également considéré que la logique adéquate pour ce domaine est une forme de la logique partitive, où les opérations valablès sont l’opération de potentiation et l’opération de compénétration entre parties et entiers. La mereologie (Lesniewski) et l’holologie (Brentano) sont limitées, parce que elles n’expliquent pas la logique des opérations de potentiation et de compénétration. L’auteur, sur la base des suggestions du philosophe roumain Constantin Noica propose une nouvelle forme de la logique partitive – l’holomerie, cas dans l’auquel la partie a le pouvoir de l’entier. Un holomer est une partie qui est équipotent avec l’entier. Pour expliquer le fonctionnement des holomers s’utilise comme un outil logique les propriétés formelles de la relation de l’identité symbolique.

La logique des Défauts propose une formalisation des inférences faites sur la base d’information incomplète: ce qu’on nomme génériquement “des inférences non-monotoniques”. Cependant, si l’on considère les extensions des théories de base, on assume méthodologiquement le fait que les mondes sur lesquels portent ces théories doivent être clos sous les hypothèses des défauts; au sens où l’élargissement d’une théorie de base par l’ajout d’un ensemble d’hypothèses de défauts revient à compléter ladite théorie. Ce qui naturellement induit l’idée que, ces extensions deviennent des bases nouvelles d’information cette fois ci complètes, et sur lesquelles de nouvelles inférences seront faites de manière monotonique. Nous allons ici montrer le paradoxe idéologique interne à la logique des défauts quant à sa prétention à prendre commodément en charge les diverses inférences non-monotoniques que n’arrive pas à rendre la logique classique. Paradoxe interne dès lors que le principe directeur de la logique des défauts est de modéliser le raisonnement défaisable, non-monotone, qui est de fait plus naturel au système de rationalité humain. Or l’effectuation des extensions sous les hypothèses de clôture revient à rechercher la monotonicité de l’inférence classique, étant donné que ces extensions ne sont autre chose que des bases d’information complète.

One of the fundamental subjects of logic philosophy has been the question of inference formal validity independence in regard to psychology and subjectivity. This paper proposes and examines the thesis formed by the following statements: (a) the anti-psychologism position, defended by the Frege-Husserl tradition, meant as a foundation for the formal-symbolic approach as logic essence. Anti-psychologism main arguments, which affirm that logic is not reducible to psychology, are precisely the arguments that defend the idea of logic being, and having to be, only and essentially formal. Therefore, necessity, accuracy, prescription, and universality all depend on formality; (b) based on some current cognitive science approaches, especially those of Stennings and van Lambalgen, the idea that the nature of logic is exclusively founded on formality and symbolism is critically examined.

In standard logical definition of the proof the concept of truth is used. To prove some thesis – means logically to bring it out of other true judgments. But there are the statements which haven’t been connected with truth, not having a truth conditional assessment: these are questions, requests, councils, promises, estimates, etc. It is obvious that, operating with them, we need also to be logical and evidential. Thus, there is a question of expansion of concept of the proof, that is of creation of wider model of the argument. Logical modeling of the argument generates a set of its models in which it is reproduced by means of logical systems: formal and semi-formal. Problem of logical modeling is creation of formally correct system of dependences between arguments and the thesis. It is also necessary to consider and reproduce its pragmatical aspects in model and argument rules: orientation to the addressee, providing acceptability and clearness of arguments and conclusion procedure. But there is also one more moment – social – which we also have to consider at a choice of model of the argument. The changes happening in society, find close connection of language with forms of life, human existence. This circumstance caused increasing interest to the argument, as forms of human rationality, the critical thinking capable to an independent and free reasoning, a way of research of mistakes and delusions, illusions and conscious manipulations, justification of own views, the points of view and ability to convince of them others.

Logic as a phenomenon of science has been historically denoted by the set of close synonym terms – “canon”, “dialectics”, “epagoge”, “maieutics”, “synagoge”, “analytics”, “epilogizmos”, “organon”, “logic” – while the word “logic” is only a homonym. Since, in relation to logic itself, the logical ideal of unambiguity in natural language is not achievable, the limitations of logical science, imposed by it primarily upon itself, are always important. Especially this concerns the use of the terms “formal logic” and “non-classical logic”, as well a broad understanding of logic that makes it equivalent to (and not only) thinking as such. The paradigm of modern post-nonclassical rationality suggests that logic should be considered responsible for the appearance of non-classical thinking and non-classical science. Thus, rationality even in science, acquires an almost exclusively praxeological nuance. As a consequence, the basic trends of modern science – interdisciplinarity, synergy, syncretism – and the reduction of rationality to pragmatics lead to the abandonment of the undelying principles of rational thinking, the essence of which lies in differentiating analytics, and sometimes – even to the abandonment of thinking as such. This gives rise to a paradoxical situation: the rationality of science, in fact, becomes the negation of the rationalist approach and the rationalistic thinking, which initially determined the nature of scientific knowledge, not only ceases to be the main value of science, but is no more included among its essential features.