Logic’s Tower of Babel

Late in his life, Jung speculated that the natural numbers, the integers, “contain the whole of mathematics and everything yet to be discovered in this field.” This article presents the attempts by mathematicians to address this question in their terms; that is, whether arithmetic (the mathematics of the natural numbers) was complete and consistent.

Early in the twentieth century, mathematicians began to seek a formalism that could provide a solid foundation for mathematics. The first important product of this new formalism was Giuseppe Peano’s Postulates: five axioms from which the full arithmetic of the natural numbers or integers (i.e., 0, 1, 2, 3, …) can be derived. Inspired by Peano’s achievement, philosopher and mathematician Bertrand Russell began a project to show that mathematics could be reduced to logic. His overweening aim was to eventually show that all science could be reduced to logic.

Logician Kurt Gödel realized that the goal of the formalists and logicians was impossible. He produced a logically impeccable proof that no system at least as complex as arithmetic could be proved both complete and consistent within the system. In essence, he proved that the core of mathematical discovery must be intuitive: direct perception of reality, which then clothes itself in mathematical garb. This accords closely with Jung’s own insight, which was based on the idea that each number is qualitatively different from every other number. To this day, Gödel’s proof stands unchallenged.     

Hegel’s Secular Theology

This essay attempts to present Hegel as a secular theologian and to argue that the theological dimension of Hegel’s thought is central to his entire philosophy and is, in fact, the leitmotif that draws together all of his work. The task of overcoming the dualism between the sacred and the secular provides the driving spirit of all Hegel’s endeavors, from his juvenilia to the mature thought of his Heidelberg and Berlin periods. A secular theology demonstrates its commitment to secularity through three main affirmations: (1) the full reality and significance of this world, (2) the autonomy of the different fields of culture and knowledge besides that of religion, and (3) the epistemological authority of reason and shared experience in determining the real and the true. Hegel’s secular theology, however, has an ambiguous relationship with most forms of theism, including panentheism.     

Chateaubriand’s Realist Conception of Logic

Axiomathes - I present the realist conception of logic supported by Oswaldo Chateaubriand which integrates ontological and epistemological aspects, opposing it to mathematical and linguistic...     

Some Modifications of Carnap’s Modal Logic

In this paper, Carnap??s modal logic C is reconstructed. It is shown that the Carnapian approach enables us to create some epistemic logics in a relatively straight-forward way. These epistemic modifications of C are axiomatized and one of them is compared with intuitionistic logic. At the end of the paper, some connections between this epistemic logic and Medvedev??s logic of finite problems and inquisitive semantics are shortly discussed.     

Rosenkranz’s Logic of Justification and Unprovability

Journal of Philosophical Logic - Rosenkranz has recently proposed a logic for propositional, non-factive, all-things-considered justification, which is based on a logic for the notion of being in a...     

Benedykt Bornstein’s Philosophy of Logic and Mathematics

The aim of this paper is to present and discuss main philosophical ideas concerning logic and mathematics of a significant but forgotten Polish philosopher Benedykt Bornstein. He received his doctoral degree with Kazimierz Twardowski but is not included into the Lvov–Warsaw School of Philosophy founded by the latter. His philosophical views were unique and quite different from the views of main representatives of Lvov–Warsaw School. We shall discuss Bornstein’s considerations on the philosophy of geometry, on the infinity, on the foundations of set theory and his polemics with Stanis?aw Le?niewski as well as his conception of a geometrization of logic, of the categorial logic and of the mathematics of quality.     

Carroll’s Regress and the Epistemology of Logic

On an internalist account of logical inference, we are warranted in drawing conclusions from accepted premises on the basis of our knowledge of logical laws. Lewis Carroll’s regress challenges internalism by purporting to show that this kind of warrant cannot ground the move from premises to conclusion. Carroll’s regress vindicates a repudiation of internalism and leads to the espousal of a standpoint that regards our inferential practice as not being grounded on our knowledge of logical laws. Such a standpoint can take two forms. One can adopt either a broadly externalist model of inference or a sceptical stance. I will attempt, in what follows, to defend a version of internalism which is not affected by the regress. The main strategy will be to show that externalism and scepticism are not satisfying standpoints to adopt with regard to our inferential practice, and then to suggest an internalist alternative.     

An Intuitionistic Reformulation of Mally’s Deontic Logic

In 1926, Ernst Mally proposed a number of deontic postulates. He added them as axioms to classical propositional logic. The resulting system was unsatisfactory because it had the consequence that A is the case if and only if it is obligatory that A. We present an intuitionistic reformulation of Mally’s deontic logic. We show that this system does not provide the just-mentioned objectionable theorem while most of the theorems that Mally considered acceptable are still derivable. The resulting system is unacceptable as a deontic logic, but it does make sense as a lax logic in the modern sense of the word.     

On Axiomatizing Shramko-Wansing’s Logic

This work treats the problem of axiomatizing the truth and falsity consequence relations, ⊨ _t and ⊨ _f , determined via truth and falsity orderings on the trilattice SIXTEEN ₃ (Shramko and Wansing, 2005). The approach is based on a representation of SIXTEEN ₃ as a twist-structure over the two-element Boolean algebra.     

A Completeness Proof of Kiczuk’s Logic of Physical Change

In this paper the class of minimal models C ^ZI for Kiczuk’s system of physical change ZI is provided and soundness and completeness proofs of ZI with respect to these models are given. ZI logic consists of propositional logic von Wright’s And Then and six specific axioms characterizing the meaning of unary propositional operator “Zm”, read “there is a change in the fact that”. ZI is intended to be a logic which provides a formal account for describing two kinds of process change: the change from one state of the process to its other state (e.g., transmitting or absorbing energy with greater or less than the usual intensity) and the perishing of the process (e.g., cessation of the energetic activity of the sun).     

Hintikka’s Independence-Friendly Logic Meets Nelson’s Realizability

Inspired by Hintikka’s ideas on constructivism, we are going to ‘effectivize’ the game-theoretic semantics (abbreviated GTS) for independence-friendly first-order logic (IF-FOL), but in a somewhat different way than he did in the monograph ‘The Principles of Mathematics Revisited’. First we show that Nelson’s realizability interpretation—which extends the famous Kleene’s realizability interpretation by adding ‘strong negation’—restricted to the implication-free first-order formulas can be viewed as an effective version of GTS for FOL. Then we propose a realizability interpretation for IF-FOL, inspired by the so-called ‘trump semantics’ which was discovered by Hodges, and show that this trump realizability interpretation can be viewed as an effective version of GTS for IF-FOL. Finally we prove that the trump realizability interpretation for IF-FOL appropriately generalises Nelson’s restricted realizability interpretation for the implication-free first-order formulas.     

A Peculiarity in Pearl’s Logic of Interventionist Counterfactuals

We examine a formal semantics for counterfactual conditionals due to Judea Pearl, which formalizes the interventionist interpretation of counterfactuals central to the interventionist accounts of causation and explanation. We show that a characteristic principle validated by Pearl’s semantics, known as the principle of reversibility, states a kind of irreversibility: counterfactual dependence (in David Lewis’s sense) between two distinct events is irreversible. Moreover, we show that Pearl’s semantics rules out only mutual counterfactual dependence, not cyclic dependence in general. This, we argue, suggests that Pearl’s logic is either too weak or too strong.     

Franco Montagna’s Work on Provability Logic and Many-valued Logic

Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic.     

An Interpretation of Łukasiewicz’s 4-Valued Modal Logic

A simple, bivalent semantics is defined for ?ukasiewicz’s 4-valued modal logic ?m4. It is shown that according to this semantics, the essential presupposition underlying ?m4 is the following: A is a theorem iff A is true conforming to both the reductionist (rt) and possibilist (pt) theses defined as follows: rt: the value (in a bivalent sense) of modal formulas is equivalent to the value of their respective argument (that is, ‘ A is necessary’ is true (false) iff A is true (false), etc.); pt: everything is possible. This presupposition highlights and explains all oddities arising in ?m4.     

Husserl and the Algebra of Logic: Husserl’s 1896 Lectures

In his 1896 lecture course on logic–reportedly a blueprint for the Prolegomena to Pure Logic–Husserl develops an explicit account of logic as an independent and purely theoretical discipline. According to Husserl, such a theory is needed for the foundations of logic (in a more general sense) to avoid psychologism in logic. The present paper shows that Husserl’s conception of logic (in a strict sense) belongs to the algebra of logic tradition. Husserl’s conception is modeled after arithmetic, and respectively logical inferences are viewed as analogical to arithmetical calculation. The paper ends with an examination of Husserl’s involvement with the key characters of the algebra of logic tradition. It is concluded that Ernst Schr?der, but presumably also Hermann and Robert Grassmann influenced Husserl most in his turn away from psychologism.     

Compositionality,Relevance, and Peirce’s Logic of Existential Graphs

Charles S. Peirce’s pragmatist theory of logic teaches us to take the context of utterances as an indispensable logical notion without which there is no meaning. This is not a spat against compositionality per se , since it is possible to posit extra arguments to the meaning function that composes complex meaning. However, that method would be inappropriate for a realistic notion of the meaning of assertions. To accomplish a realistic notion of meaning (as opposed e.g. to algebraic meaning), Sperber and Wilson’s Relevance Theory (RT) may be applied in the spirit of Peirce’s Pragmatic Maxim (PM): the weighing of information depends on (i) the practical consequences of accommodating the chosen piece of information introduced in communication, and (ii) what will ensue in actually using that piece in further cycles of discourse. Peirce’s unpublished papers suggest a relevance-like approach to meaning. Contextual features influenced his logic of Existential Graphs (EG). Arguments are presented pro and con the view in which EGs endorse non-compositionality of meaning.     

Giles’s Game and the Proof Theory of Łukasiewicz Logic

In the 1970s, Robin Giles introduced a game combining Lorenzen-style dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in Łukasiewicz logic. In this paper, it is shown that ‘disjunctive strategies’ for Giles’s game, combining ordinary strategies for all instances of the game played on the same formula, may be interpreted as derivations in a corresponding proof system. In particular, such strategies mirror derivations in a hypersequent calculus developed in recent work on the proof theory of Łukasiewicz logic. Presented by Daniele Mundici     

Priestley Duality for Paraconsistent Nelson’s Logic

The variety of N4^{^}{{\bf N4}^\perp}-lattices provides an algebraic semantics for the logic N4^{^}{{\bf N4}^\perp} , a version of Nelson’s logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of N4^{^}{{\bf N4}^\perp}-lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski.     

Frege’s Anti-Psychologism about Logic : the Relationship between Logic and Judgment

Philosophia - Frege is an anti-psychologist about logic who takes logic to be sharply distinguished from psychology. However, Frege also takes judgment, which seems to be a subject of psychology,...