This paper stresses the importance of identifying the nature of an author’s conception of logic when using terms from modern logic in order to avoid, as far as possible, injecting our own conception of logic in the author’s texts. Sundholm (2012) points out that inferences are staged at the epistemic level and are made out of judgments, not propositions. Since it is now standard to read Aristotelian sullogismoi as inferences, I have taken Alexander of Aphrodisias’s commentaries to Aristotle’s logical treatises as a basis for arguing that the premises and conclusions should be read as judgments rather than as propositions. Under this reading, when Alexander speaks of protaseis, we should not read the modern notion of proposition, but rather what we now call judgments. The point is not just a matter of terminology, it is about the conception of logic this terminology conveys. In this regard, insisting on judgments rather than on propositions helps bring to light Alexander’s epistemic conception of logic.
A Pure View of Ecumenical Modalities
Recent works about ecumenical systems, where connectives from classical and intuitionistic logics can co-exist in peace, warmed the discussion on proof systems for combining logics. This discussion has been extended to alethic modalities using Simpson’s meta-logical characterization: necessity is independent of the viewer, while possibility can be either intuitionistic or classical. In this work, we propose a pure, label free calculus for ecumenical modalities, nEK, where exactly one logical operator figures in introduction rules and every basic object of the calculus can be read as a formula in the language of the ecumenical modal logic EK. We prove that nEK is sound and complete w.r.t. the ecumenical birelational semantics and discuss fragments and extensions.
An ecumenical notion of entailment
Much has been said about intuitionistic and classical logical systems since Gentzen’s seminal work. Recently, Prawitz and others have been discussing how to put together Gentzen’s systems for classical and intuitionistic logic in a single unified system. We call Prawitz’ proposal the Ecumenical System, following the terminology introduced by Pereira and Rodriguez. In this work we present an Ecumenical sequent calculus, as opposed to the original natural deduction version, and state some proof theoretical properties of the system. We reason that sequent calculi are more amenable to extensive investigation using the tools of proof theory, such as cut-elimination and rule invertibility, hence allowing a full analysis of the notion of Ecumenical entailment. We then present some extensions of the Ecumenical sequent system and show that interesting systems arise when restricting such calculi to specific fragments. This approach of a unified system enabling both classical and intuitionistic features sheds some light not only on the logics themselves, but also on their semantical interpretations as well as on the proof theoretical properties that can arise from combining logical systems.
Logical reasoning: a first course
| This book describes how logical reasoning works and puts it to the test in applications. It is self-contained and presupposes no more than elementary competence in mathematics. |
The Logical Syntax of Prejudice: Oppression and the Constitutive A Priori
I argue that a thoroughgoing naturalized epistemology can easily underestimate the extent to which certain background assumptions will infl uence arguments. Instead, then, I suggest that we can borrow a conceptual tool from neo-Kantian philosophy of science, namely the constitutive a priori. This idea originates in neo-Kantian philosophers who understood, in light of Einsteinian physics, that Kantian views about the a priority of space were untenable. Frameworks that adopt some version of a constitutive a priori take certain propositions to play the role of a priori principles, without granting them the universality or necessity that such principles traditionally hold. I will argue that thinking of certain views or values as having the status of constitutive a priori principles can help us understand what would be required for an epistemic agent to change them, and thus illustrate the extent to which they are resistant to being dislodged by evidence.
The Liberation Argument for Inconsistent Mathematics
Val Plumwood charged classical logic not only with the invalidity of some of its laws, but also with the support of systemic oppression through naturalization of the logical structure of dualisms. In this paper I show that the latter charge – unlike the former – can be carried over to classical mathematics, and I propose a new conception of inconsistent mathematics – queer incomaths – as a liberatory activity meant to undermine said naturalization.
De-centering and Genderqueering Val Plumwood’s Feminist Logic
The strongest and, until recently, least-explored approach to feminist logic holds that some formal logics have structural features that perpetuate sexism and oppression, whereas other logics are helpful for resisting and opposing these social phenomena. Our choice of logics may not be purely formal on this view: for example, some logics are preferrable to others on the grounds of feminist commitments. This strong account of feminist logic was first articulated by Val Plumwood. We will critically engage salient features of her view, especially her critique of classical logic and the centering and dominating functions she believes classical negation has. We will see that her understanding of classical negation captures neither the development of Intersectional Feminism, nor the position the concept of centering holds in transformative justice. However, Plumwood’s critique of classical negation does lead us to a deeper insight regarding which logics to apply in social justice contexts. Robin Dembroff’s analysis of genderqueer as a critical gender kind helps us delineate a non-classical context in which a four-valued logic, such as FDE, can structurally account for the critical feature of this gender kind in a way classical logic cannot. We will also observe how four-valued logics precisely capture the destabilization of, and resistance to, the exclusive and exhaustive gender binary categories Dembroff describes.
Activist Epistemology
I propose a model on which epistemic frameworks are understood in terms of not only beliefs, but also sets of evidential support relations. We are generally responsive to multiple frameworks, some more compatible than others.The model allows for prioritizing certain frameworks by drawing on van Benthem and Pacuit’s work on logics for evidence-based belief. This prioritization allows us to capture the idea that some epistemic frameworks are “held come what may” with nuance and complexity.
Feminist Readings of Aristotelian Logic
Hass examines chapters devoted to Aristotle in a recent, prominent, and controversial feminist critique of logic, Andrea Nye’s Words of Power: A Feminist Reading of the History of Logic. Hass shows that Nye’s criticisms of logic in general and of Aristotle in particular are misplaced. What is crucial in Nye’s attack are alleged problems caused by overzealous “abstraction.” But Hass argues that abstraction is not problematic; instead, it is crucial (and empowering) for feminist political theory. Although she rejects Nye’s form of feminist logic critique, Hass finds more that is worthwhile in the criticisms of logic advanced by Luce lrigaray and Val Plumwood. These thinkers call for feminist alternatives to what has come to be standard deductive logic – and interestingly enough, their call is echoed in other contemporary criticisms from within the field of logic itself, for example, from intuitionist or entailment logics. The logical schemes envisaged by lrigaray and Plumwood would encompass more situated and fluid ways of using formal systems to describe and analyse reality and diverse experiences. Hass argues that, in Aristotle’s case, we can glimpse something of such an alternative by looking to his account of negation, which is richer and more complex than that allowed by most contemporary formal systems.
Social Spheres: Logic, Ranking, and Subordination
This paper uses logic – a formal language with models and a consequence relation – to think about the social and political topics of subordination and subordinative speech. I take subordination to be a matter of three things: i) ranking one person or a group of people below others, ii) depriving the lower-ranked of rights, and iii) permitting others to discriminate against them. Subordinative speech is speech – utterances in contexts – which subordinates. Section 1 introduces the topic of subordination using examples from the 1979 novel Kindred by Octavia Butler. Section 2 uses these examples to clarify and illustrate the definitions of subordination and subordinative speech. Sections 3 and 4 then develop a way of modeling subordination using a system of social spheres, an adaptation of (Lewis, 1973)’s approach to modeling the relation of comparative similarity on worlds for counterfactuals. Section 4 looks at three possible applications for this work: giving truth-conditions for social quantifiers, identifying fallacies involving such expressions, and explaining the pragmatics of subordinative speech. The last section anticipates objections and raises further questions.