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.     

Hobson’s Conception of Definable Numbers

In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability.     

Revising Carnap’s Semantic Conception of Modality

I provide a tableau system and completeness proof for a revised version of Carnap??s semantics for quantified modal logic. For Carnap, a sentence is possible if it is true in some first order model. However, in a similar fashion to second order logic, no sound and complete proof theory can be provided for this semantics. This factor contributed to the ultimate disappearance of Carnapian modal logic from contemporary philosophical discussion. The proof theory I discuss comes close to Carnap??s semantic vision and provides an interesting counterpoint to mainstream approaches to modal logic. Despite its historical origins, my intention is to demonstrate that this approach to modal logic is worthy of contemporary attention and that current debate is the poorer for its absence.     

Buridan’s Secular Demarcation of Logic

Philosophia - In his Tractatus de Consequentiis, the 14th century logician John Buridan states a theorem, which reads like a refutation of Aristotelian Syllogistics. Syllogistics in its usual form,...     

Thinking Through Kant’s Conception of Pedagogy

Studies in Philosophy and Education -     

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.     

Mental Causation and Searle’s Impossible Conception of Unconscious Intentionality

In my article I evaluate Searle’s account of mental causation, in particular his account of the causal efficacy of unconscious intentional states. I argue that top-down causation and overdetermination are unsolved problems in Searle’s philosophy of mind, despite his assurances to the contrary. I also argue that there are conflicting claims involved in his account of mental causation and his account of the unconscious. As a result, it becomes impossible to understand how unconscious intentional states can be causally efficacious. My conclusion will be that if Searle’s conception of unconscious intentionality is to play a genuine role in the causal explanation of human action, it needs to be rethought.     

Tragic Recognition: Revisiting Hegel’s Conception of Ethical Life

This paper interprets Hegel’s engagement with tragedy and especially tragic action as an interpretive model for understanding ethical life in complex societies in which independent value spheres collide. Tragic recognition, in contrast to the kind of recognition introduced in the master and slave dialectic, is not based on desire, but arises from the suffering deriving from clashing value spheres. As a coming to terms with one’s finitude, tragic recognition presents an important corrective to the account of mutual recognition that has been the reference point of contemporary interpretations of Hegel’s social and political philosophy. The paper concludes by pointing to some of the limits of tragedy as a universal interpretive framework for modern societies.     

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).     

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.     

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.     

Review of G. Felicitas Munzel’s Kant’s Conception of Pedagogy: Toward Education for Freedom

Studies in Philosophy and Education -     

C. S. Peirce’s Dialogical Conception of Sign Processes

This article examines the contention that the central concepts of C. S. Peirce’s semeiotic are inherently communicational. It is argued that the Peircean approach avoids the pitfalls of objectivism and constructivism, rendering the sign-user neither a passive recipient nor an omnipotent creator of meaning. Consequently, semeiotic may serve as a useful general framework for studies of learning processes.     

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.