Abstracts

Koray Akcaguner and Elias Yuan (University of Calgary): The Equivalence of the Law of Excluded Middle and Double Negation Elimination

Abstract. Intuitionistic logic is generally understood as classical logic without certain rules, namely the law of excluded middle (LEM) and double negation elimination (DNE). LEM and DNE are commonly understood to be equivalent, where removal of one or the other from the rules of classical logic gives us intuitionistic logic. However, there is a sense in which they are not equivalent, and LEM is more powerful than DNE. Brouwer has argued that DNE is a corollary of LEM, but LEM is not a corollary of DNE. Building from Brouwer’s argument, we provide formal proofs, both semantically and syntactically, showing that LEM and DNE are equivalent as schemas but not as formulas. We then present Brouwer's example and tie this with the semantic proof that shows DNE does not entail LEM. We then interpret LEM, DNE, and the question of their equivalence through a constructivist lens. We show how in the constructivist’s case, the salient question becomes the possibility of transforming two 'omniscient' algorithms into each other. Finally, we argue that this understanding of LEM and DNE may better capture Brouwer's original conception of intuitionistic mathematics.

Tyler D. P. Brunet (University of Exeter) and Gillman Payette (University of Calgary): Adjointness in modal semantics

Abstract: By treating the semantic structures for modal logics as coalgebras for a functor, coalgebraic modal logic has captured the traditional notions of frames and models as well as many types of semantic structures unfamiliar in traditional philosophical logic. The coalgebraic approach also deepens our understanding of the traditional cases: Kripke frames and models. In this talk I will present the coalgebraic persepctice on Kripke frames, and show how this perspective can lead us in helpful new directions. Once we adopt a coalgebraic perspective on Kripke frames, we can see that the central invariance result about modal definability---that all formulas of the basic modal language are invariant under bounded morphisms---follows from a single, fundamental fact about functions: that for every function, its associated direct image function is left adjoint to its inverse image function. However, there is also a right adjoint to the inverse image, the ``codirect image''. If we use codirect images to produce morphisms of Kripke frames, we obtain another, non-traditional, notion of invariance and another modal language. This talk will explain how this adjoint sequence arises from a coalgebraic perspective on Kripke frames, and discuss its implications for modal definability. 

Allen Hazen (University of Alberta): Type Theory and Single-sorted Languages 

Abstract. Simple Type Theory, as a theory of sets, is typically thought of as formulated in a many-sorted language, one alphabet of variables per type, both in its strictly typed (sets can have members only of the immediately preceding type) and cumulative (sets can have objects of any lower type as members) variants.  The two variants are mutually interpretable, and each, since they can interpret number theory when strengthened with an axiom of infinity, is undecidable.  Given, instead, an axiom saying there is a particular finite number of individuals, they yield complete, and so decidable, theories.  Each can be reformulated, in the standard way, in a language with a single sort of variable, construed as ranging over the objects in the union of the infinitely many types.  (Quine[1], and somewhat more elegantly, Resnik[2], gave axiomatizations of the single-sorted version of the strictly typed variant.)  The single-sorted versions are interestingly different from the many-sorted.  Even with an axiom giving a finite size to the type of individuals, both are undecidable.  The strictly typed version is interpretable in the cumulative, but not the other way around, giving an interesting example of a phenomenon noted by Hook[3].

   References: [1] W.V. Quine, Set Theory and its Logic [2] Michael D. Resnik, "A set theoretic approach to the simple theory of types," Theoria 35 (1969), pp. 239-258 [3] Julian L. Hook, "A note on interpretations of many-sorted theories," Journal of Symbolic Logic 50 (1985), pp. 372-374.

Nicolas Fillion (Simon Fraser University): Doing logic with partially indeterminate operators: the case of deontic logic

Abstract. Doing logic in the accepted rigorous manner--with symbolic languages, functional semantics, and a notion of validity defined by quantifying over all possible models--is a thing of beauty. This being said, not even the most enthusiastic formalists pretend that all formal systems are equally worthy of attention. That is because we (or at least many of us) want logic not to be just a thing of beauty, but one that earns its keep. We want to apply logical analysis to the understanding of notions that appear outside of logic, e.g., in ethics, epistemology, metaphysics, science, etc. But which system should we choose to do this work? Taking the case of "ought" as an example. A successful analysis of this operator would lead to a formal system with an operator that captures all and only the logical properties "ought". But in this analytic process, we often want to say that an inference is valid, although we don't really know what the key operator even means, exactly. The way we define validity doesn't seem to readily apply to sentences containing operators that are not fully well-defined. This talk discusses some aspects of how it is that we can proceed with such theorizing.  

Sophia Kimiagari (University of Calgary): Feminist Critiques of Logic and the Possibility of Feminist Logic

Abstract: I aim to investigate the possibility of a feminist logic by analyzing the Western feminist critiques of rationality and logic. Through the lens of third-world feminism, I intend to demonstrate that these critiques are not universally applicable, particularly when examining rationality within the Islamic tradition. By highlighting the distinctions between Islamic and Western conceptualizations of rationality, I seek to challenge the assumptions made by Western feminists about the nature of rationality. Building upon these insights, I aspire to defend the possibility of a feminist epistemology of logic. 

Mark McCormack (University of Alberta): Universal Logic: Logic that Logically Proves Itself  

Abstract: This abstract posits that logic, often perceived as mundane, is crucial for solving complex problems. The essence of logic was lost following Aristotle, as demonstrated by Bertrand Russell's dissection of Frege's paradox in the "set of sets that contains itself", and echoed in Kant's unity of apperception and Wittgenstein's "duck-rabbit" ambiguity. These examples illustrate logic's true circularity, distinguishing between 'bad infinite' tautological loops and genuinely infinite circles. This study argues that genuine science, as envisioned by Hegel, is a system of these self-reflexive loops. It asserts that Hegel's Encyclopedia of the Philosophical Sciences exemplifies this with a Universal Logic that integrates all knowledge domains into a self-proving science. This approach addresses long-standing issues like the Symbol Grounding Problem and the Demarcation Problem of Science, providing a unified ontology and epistemology for the first time in over two millennia.

(Maybe) Gillman Payette (University of Calgary): Peter Schotch's 'What is this thing called "Logic"?'

Abstract: In this talk, I will read a---mostly---completed paper by Peter Schotch that overlaps with some of our joint work and my work, from a number of years ago. I will try to fill in some of the details and point to related work.    

Kent Peacock (University of Lethbridge): "The characteristic trait of quantum mechanics":  The Puzzle of Entanglement and Why It Matters Today

Abstract: Erwin Schrödinger coined the term "entanglement" as a picturesque way of describing the mysterious non-factorizability of tensor product states in quantum mechanics, and said that it was "the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought."  I will sketch the basics of entanglement for an audience that is mathematically sophisticated but not necessarily conversant in quantum mechanics, and review some of the deep puzzles about entanglement from Bell's Theorem to its implications for cosmology.