Jan & Lstrok;ukasiewicz, in a series of writings from 1918 to 1946, argued against logical determinism-the idea that the truth or falsity of a statement predetermines future events, thus limiting free will. To defend an indeterministic worldview, he rejected the principle of bivalence and introduced a third value, the 'indeterminate', pioneering polyvalent logics. This problem dates back to Aristotle's De interpretatione and the Hellenistic disputes involving Stoics, Epicureans, and Academics. As an expert historian of ancient logic, & Lstrok;ukasiewicz was well acquainted with these debates and employed classical patterns of reasoning in his argument. However, while & Lstrok;ukasiewicz believed that classical logic entails determinism, the author of this text claims that bivalence has no bearing on free will. He argues that the problem arises from a failure to distinguish between different types of truth-bearers (propositions, sentences, assertions) and from a conceptual confusion between causality and logical implication, as Gilbert Ryle pointed out. The author concludes that logical determinism is ultimately a pseudo-problem that has persisted for centuries due to a lack of clear definitions of the key concepts involved in it.
This paper proposes a novel approach to Stoic and Aristotelian-Epicurean modal reasoning by replacing traditional Kripkean relational semantics with a topological framework. In the Stoic account, causality and modality are associated with the density of a space, whereas in the Epicurean tradition they are grounded in discreteness and atomic structure. Motivated by this distinction, we argue that causality and modality can be interpreted through the topology of neighborhoods: the Stoic framework corresponds to a density-based notion of logical determinism, while the Aristotelian-Epicurean framework corresponds to a discreteness-based notion of logical contingency. This perspective allows modal operators to be defined via limit-point and interior structures, rather than Kripke's accessibility relations. Also, to capture Stoic logical reasoning, we introduce the notion of logemes, which we model geometrically as simplicial complexes. This representation reflects the flexible, non-linear, and competitive character of ancient inference rules. The proposed topological interpretation not only clarifies the internal structure of Stoic logic but also enables systematic comparison with Aristotelian and Epicurean modal frameworks. In addition, the Stoic interpretation of modality admits concrete applications in data analysis: in particular, it leads to the construction of modal filters that implement a deterministic, topology-based mechanism for noise rejection.
Tarski's characteristics of colloquial language (CL) give rise to a sceptical mode. The mode posits that it is not possible to ascribe a logical value to CL sentences. The text identifies two standard interpretations of Tarski's mode: the optimistic view asserts that it is merely an inconsequential mistake; and the pessimistic view, which suggests that Tarski recognized the inherent limitations of everyday language and shifted his focus to metamathematics. In contrast, I put forward an interpretation which argues that Tarski's mode represents a critical, though unsystematic, elaboration of the CL properties (such as universality, inconsistency, and pseudo non-monocinity). These properties render it impossible to define truth for CL. Consequently, all philosophical claims in CL remain beyond the scope of truth and falsehood.
McCall's 4-valued logic CC1 is one of the earliest and most influential systems of connexive logic. CC1 has been criticised because its truth values lack an intuitive interpretation. In this paper, we propose that the semantic value of a sentence in this logic is represented as an ordered pair consisting of a classical truth value (true or false) and a content polarity (positive or negative). In particular, the truth value of a connexive entailment is a function both of the truth values and of the content polarities of its antecedent and consequent. We develop this informal idea into a formal semantics and prove a completeness theorem for CC1 with respect to a class of algebraic models obtained via a certain construction on Boolean algebras.
The aim of this paper is to make the philosophical point explicit that the sequent calculus as such is a formal metatheory for classical reasoning about the derivability relation in natural deduction. As a consequence, it is neither suitable for foundational intuitionistic research on intuitionistic logic(s) nor adequate as a basis of intuitionistic proof-theoretic semantics.
The history of the search for a minimal set of inference rules for syllogistic reasoning starts with Aristotle, with one of the metatheorems proved in Prior Analytics stating that all syllogistic deductions are ultimately reducible to two universal moods in the first figure: Barbara and Celarent. In the present work, we will consider what is the greatest reduction one can achieve when proving all the moods indirectly. The question we shall answer is whether, when proved indirectly, syllogisms behave the same way and, consequently, whether the Aristotelian reduction to Barbara and Celarent holds for indirect proofs as well. To achieve this, we shall consider four different scenarios of applying single-premise inference rules to syllogism. The results for each scenario will then be analyzed and displayed as tables. Considering the tables, we shall claim that in certain cases, the Aristotelian reduction does not hold and further propose our division of all the syllogistic moods, which will be different from the usual figure arrangement. Finally, we will analyze the results quantitatively and show that they can provide insight into the nature of the division to regular and the so-called subalternated moods.
Standard logic consists of a univocal vocabulary and rigorous rules of derivation, through which syntactically coherent or semantically true conclusions are reached, starting from axioms; Hegel's speculative logic, in the Science of Logic, possesses characteristics irreducible to it. Hegelian logic cannot be expressed even through non-classical-standard formalizations, not even paraconsistent ones (the author essentially considers dialetheism). It is fundamentally an ontological logic, which does not admit the principle of non-contradiction, nor that of the excluded middle, nor that of identity, as understood by Verstand (as a principle according to which identity and diversity are different from each other - but if identity is different from diversity, then it too is in itself difference). The author attempts to express this dialectical logic symbolically, to show how incomparable it is with standard logics and with (at least some) non-standard logics. He uses a plurivocal vocabulary, which cannot follow rules extrinsic to the dialectical self-constitution of concepts and, ultimately, of speculative truth.
This paper extends Benacerraf's problem - concerning the tension between Platonism of mathematical objects and epistemological access of mathematic truth - to Frege's 'third realm' of thoughts, identifying key flaws in Frege's theory: the language-independence, epistemological inaccessibility, and structural incoherence of thoughts. It proposes a revised account that construes thoughts as linguistically mediated, intersubjective entities, which resolves Frege's predicaments while retaining his core anti-psychologism and addressing the accessibility challenge posed by Benacerraf's problem.
The infinite regress problem involves the identification of a set of foundational problems in logic, deductive reasoning, and the epistemology of understanding through an allegorical dialogue between Achilles and the tortoise. In this paper, the infinite regress problem, traditionally referred to as Carroll's paradox, will be re-identified as the Bolzano-Carroll-Wilson paradox, in recognition of its multiple provenances. Furthermore, we shall investigate the nature and scope of the paradox and attempt to identify the real moral behind what the tortoise said to Achilles. In particular, we shall engage with the Russell-Grice point concerning the need to distinguish between a material implication relation ( $ \rightarrow $ ->) and an inferential relation ( $ \vdash $ proves), the further need to distinguish between first-order logical statements and higher-order statements (metalogical, meta-metalogical, and so on), and the Ryle-Wittgenstein point concerning the nature of proper inference as knowledge-how and rule-following.
This paper proposes a modern logical re-reading of a system for arithmetic proposed by Mario Pieri in a paper titled 'Sopra gli assiomi aritmetici' in 1907.
This paper offers a comprehensive reconstruction of the only example of contraposition of a negative particular judgement that Kant provides in his writings on logic, in Reflexion 3187. The interpretation focuses on the two main consequences of applying contraposition to a judgement for Kant: the generation of negative terms and the shift of the logical modality of a judgement from assertoric to apodictic. The paper provides a reconstruction that excludes the presence of infinite judgements and makes sense of the entire course of reasoning in Reflexion 3187 from the perspective of Kant's pure general logic. It shows on what grounds the course of reasoning in said note on logic contains an instance of contraposition of a negative particular judgement for Kant. This is the case even if the sequence of judgements does not fully respect the modal constraints set out by Kant, since the conclusion can still be validly derived within Kant's pure general logic.
In the Prior Analytics, book 2, chapter 25, Aristotle presents a strange type of argument called apagoge. Some, such as Ross, consider the situation in this chapter problematic, and some, such as Peirce, do not. Ross believes that apagoge is a semi-demonstrative, semi-dialectical syllogism, in the form of the first figure, with a probable conclusion that is obtained from a more probable minor premise with an apodictic major premise. Peirce says that apagoge is the very abduction or -in a more recent term-inference to the best explanation. Al-F & amacr;r & amacr;b & imacr;, however, without explicitly discussing apagoge, replaces it with the Arabic translation of epagoge, i.e. ''istiqr & amacr;' (induction)', which inspires the hypothesis that apagoge is a miswriting of epagoge. Inspired by al-F & amacr;r & amacr;b & imacr;'s words, we formulate another abduction, based on which the strange and problematic situation of Chapter 25 is explained by accepting the hypothesis that apagoge is a miswriting of epagoge. However, this reading is neither economical nor consistent.
We demonstrate that, in itself and in the absence of extra premises, the following argument scheme is fallacious: The sentence A says about itself that it has a certain property F, and A does in fact have the property F; therefore A is true. We then examine an argument of this form in G & ouml;del's introduction to his classic paper on incompleteness and examine some auxiliary premises which might have been at work in that context. Philosophically significant as it may be, that particular informal argument plays no r & ocirc;le in G & ouml;del's technical results. Going deeper into the issue and investigating truth conditions of G & ouml;delian sentences (i.e. those sentences which are provably equivalent to their own unprovability) will provide us with insights regarding the philosophical debate on the truth of G & ouml;delian sentences of systems - a debate which goes back at least to Dummett in the 1960s.
