An Infinitary Extension of Jankov’s Theorem

It is known that for any subdirectly irreducible finite Heyting algebra A and any Heyting algebra B, A is embeddable into a quotient algebra of B, if and only if Jankov’s formula χ _A for A is refuted in B. In this paper, we present an infinitary extension of the above theorem given by Jankov. More precisely, for any cardinal number κ, we present Jankov’s theorem for homomorphisms preserving infinite meets and joins, a class of subdirectly irreducible complete κ-Heyting algebras and κ-infinitary logic, where a κ-Heyting algebra is a Heyting algebra A with # ≥  κ and κ-infinitary logic is the infinitary logic such that for any set Θ of formulas with # Θ ≥  κ, ∨Θ and ∧Θ are well defined formulas.     

证候存在的逻辑回答

认为存在的就是客观的,包含了自然客观、思维客观和理性客观;把疾病现象和疾病本质划等号,在症状层面规范证候标准,抛弃中医辨证思维,就等于否定中医。物质不能等于客观,不能取代存在。证候属于理性客观,发生于中医学,是中医“阴阳神气”观念临床实在化(还原)的必然。     

Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity

Two classes of π are studied whose properties are similar to those of the protoalgebraic deductive systems of Blok and Pigozzi. The first is the class of N-protoalgebraic π-institutions and the second is the wider class of N-prealgebraic π-institutions. Several characterizations are provided. For instance, N-prealgebraic π-institutions are exactly those π-institutions that satisfy monotonicity of the N-Leibniz operator on theory systems and N-protoalgebraic π-institutions those that satisfy monotonicity of the N-Leibniz operator on theory families. Analogs of the correspondence property of Blok and Pigozzi for π-institutions are also introduced and their connections with preand protoalgebraicity are explored. Finally, relations of these two classes with the (, N)-algebraic systems, introduced previously by the author as an analog of the -algebras of Font and Jansana, and with an analog of the Suszko operator of Czelakowski for π-institutions are also investigated. Presented by Josep Maria Font     

A Note on Graded Modal Logic

We introduce a notion of bisimulation for graded modal logic. Using this notion, the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property and proving invariance and definability results.     

Intuitionistic Propositional Logic without 'Contraction' but with 'Reductio'

Routley-Meyer type relational complete semantics are constructed for intuitionistic contractionless logic with reductio. Different negation completions of positive intuitionistic logic without contraction are treated in a systematical, unified and semantically complete setting.     

The "Natural" and the "Formal"

The paper presents an argument against a metaphysical conception of logic according to which logic spells out a specific kind of mathematical structure that is somehow inherently related to our factual reasoning. In contrast, it is argued that it is always an empirical question as to whether a given mathematical structure really does captures a principle of reasoning. (More generally, it is argued that it is not meaningful to replace an empirical investigation of a thing by an investigation of its a priori analyzable structure without paying due attention to the question of whether it really is the structure of the thing in question.) It is proposed to elucidate the situation by distinguishing two essentially different realms with which our reason must deal: the realm of the natural, constituted by the things of our empirical world, and the realm of the formal, constituted by the structures that we use as prisms to view, to make sense of, and to reconstruct the world. It is suggested that this vantage point may throw light on many foundational problems of logic.     

Partiality and Its Dual

This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity".     

Hegel,modal logic,and the social nature of mind

ABSTRACT

Hegel's Phenomenology of Spirit provides a fascinating picture of individual minds caught up in “recognitive” relations so as to constitute a realm—“spirit”—which, while necessarily embedded in nature, is not reducible to it. In this essay I suggest a contemporary path for developing Hegel's suggestive ideas in a way that broadly conforms to the demands of his own system, such that one moves from logic to a philosophy of mind. Hence I draw on Hegel's “subjective logic”, understood in the light of modern modal logic, in an attempt to model the way minds might be thought as connected by way of shared intentional contents. Here, we should not be surprised at some of the parallels that emerge between the approaches of Hegel and the modal logician Arthur Prior, as Prior had testified to the influence of his teacher, John N. Findlay, who himself had strong Hegelian leanings. In the final section, Robert Stalnaker's version of possible-world semantics is suggested as a framework within which Hegel's recognitive account of the mind might be understood.     

Disagreements Over Analogies

This essay presents a dialogical framework for treating philosophical disagreements as persuasion dialogues with analogical argumentation, with the aim of recasting philosophical disputes as disagreements over analogies. This has two benefits: (1) it allows us to temporarily bypass conflicting metaphysical intuitions by focusing on paradigmatic examples, (dis)similarities, and the plausibility of conclusions for or against a given point of view; and (2) it can reveal new avenues of argumentation regarding a given issue. This approach to philosophical disagreements is illustrated by studying the role of argumentation by analogy in disputes over Lockean justifications of intellectual property rights, and disagreements over whether artificial agents should be treated as moral agents. Finally, a nonexhaustive list of possible global and local points of contention over analogical arguments is identified in order to highlight potential avenues of argumentation in disagreements over analogies.     

Philosophical foundations of partial belief models

This paper is an attempt to put forward a new kind of partial model for representing belief states. I first introduce some philosophical motivations for working with partial models. Then, I present the standard (total) model proposed by Hintikka, and the partial models studied by Humberstone and Holliday. I then show how to reduce Hintikka’s semantics in order to obtain a partial model which, however, differs from Humberstone’s and Holliday’s. The nature of such differences is assessed, and I provide motivations for using the newly proposed semantics rather than the existing ones. Finally, I review some promising philosophical applications of the ideas developed throughout the discussion.