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Philosophy of logic (logical consequence, formality, logical terms, logic and natural language)

Philosophy of language (the analytic/synthetic distinction)

Philosophy of mathematics (foundations of mathematics, optimism in mathematics)

History of the philosophy of logic (Frege, Tarski, Carnap).




Invariance Criteria as Meta-Constraints

Bulletin of Symbolic Logic (2021)

Invariance criteria are widely accepted as a means to demarcate the logical vocabulary of a language. In previous work, I proposed a framework of ``semantic constraints'' for model-theoretic consequence which does not rely on a strict distinction between logical and nonlogical terms, but rather on a range of constraints on models restricting the interpretations of terms in the language in different ways. In this paper I show how invariance criteria can be generalized so as to apply to semantic constraints on models. Some obviously unpalatable semantic constraints turn out to be invariant under isomorphisms. I shall connect our discussion to known counter-examples to invariance criteria for logical terms, and so the generalization will also shed light on the current existing debate on logicality. I analyse the failure of invariance to fulfil its role as a criterion for logicality, and argue that invariance conditions should best be thought of as merely methodological meta-constraints restricting the ways the model-theoretic apparatus should be used.

Considerations on Logical Consequence and Natural Language

Dialectica (March 2020), 74:2.

In a recent article, “Logical Consequence and Natural Language”, Michael Glanzberg claims that there is no relation of logical consequence in natural language (Glanzberg, 2015). The present paper counters that claim. I shall discuss Glanzberg’s arguments and show why they don’t hold. I further show how Glanzberg’s claims may be used to rather support the existence of logical consequence in natural language.

Logic as a Methodological Discipline

Synthese (2021), (Special Issue: Anti-Exceptionalism about Logic, Ben Martin, Maria Paola Sforza

Fogliani, and Filippo Ferrari (eds.)) doi:10.1007/s11229-021-03223-3

This essay offers a conception of logic by which logic may be considered to be exceptional among the sciences on the backdrop of a naturalistic outlook. The conception of logic focused on emphasises the traditional role of logic as a methodology for the sciences, which distinguishes it from other sciences that are not methodological. On the proposed conception, the methodological aims of logic drive its definitions and principles, rather than the description of scientific phenomena. The notion of a methodological discipline is explained as a relation between disciplines or practices. Logic serves as a methodological discipline with respect to any theoretical practice, and this generality, as well as logic’s reflexive nature, distinguish it from other methodological disciplines. Finally, the evolution of model theory is taken as a case study, with a focus on its methodological role. Following recent work by John Baldwin and Juliette Kennedy, we look at model theory from its inception in the mid-twentieth century as a foundational endeavour until developments at the end of the century, where the classification of theories has taken centre-stage.

Formalization, Semantic Constraints and Anthropocentrism

Iyyun: The Jerusalem Philosophical Quarterly (April 2021), 69: 149-169.

The paper presents a view of formalization as an interpretive endeavour carried out by a reasoner or an interlocutor by way of making explicit their logical and interpretive commitments. The view is likened to Mark Steiner’s approach to physics, by which the use of mathematics in theorizing about empirical phenomena makes physics anthropocentric. In the present approach, the use of formalization in a normative domain of interpretive commitments makes logic a properly humanistic discipline. The paper presents the model-theoretic framework of semantic constraints, extended to include formalization constraints, as a formal framework for articulating logical and interpretive commitments. The proposed approach to formalization is then applied to examples taken from Kripke and Boghossian. Contributed for a special issue of Iyyun in memory of Mark Steiner.

Logic in Natural Language: Commitments and Constraints

Disputatio (2020), 12(58): 277-308.

In his new book, Logical Form, Andrea Iacona distinguishes between two different roles that have been ascribed to the notion of logical form: the logical role and the semantic role. These two roles entail a bifurcation of the notion of logical form. Both notions of logical form, according to Iacona, are descriptive, having to do with different features of natural language sentences. I agree that the notion of logical form bifurcates, but not that the logical role is merely descriptive. In this paper, I focus on formalization, a process by which logical form, on its logical role, is attributed to natural language sentences. According to some, formalization is a form of explication, and it involves normative, pragmatic, as well as creative aspects. I present a view by which formalization involves explicit commitments on behalf of a reasoner or an interpreter, which serve the normative grounds for the evaluation of a given text. In previous work, I proposed the framework of semantic constraints for the explication of logical consequence. Here, I extend the framework to include formalization constraints. The various constraints then serve the role of commitments. I discuss specific issues raised by Iacona concerning univocality, co-reference and equivocation, and I show how our views on these matters diverge as a result of our different starting assumptions.

Logicality and Meaning

The Review of Symbolic Logic (2018), doi:10.1017/S1755020317000247. 

In standard model-theoretic semantics, the meaning of logical terms is said to be fixed in the system while that of nonlogical terms remains variable. Much effort has been devoted to characterizing logical terms, those terms that should be fixed, but little has been said on their role in logical systems: on what fixing their meaning precisely amounts to. My proposal is that when a term is considered logical in model theory, what gets fixed is its intension rather than its extension. I provide a rigorous way of spelling out this idea, and show that it leads to a graded account of logicality: the less structure a term requires in order for its intension to be fixed, the more logical it is. Finally, I focus on the class of terms that are invariant under isomorphisms, as they render themselves more easily to mathematical treatment. I propose a mathematical measure for the logicality of such terms based on their associated Löwenheim numbers.

Extensionality and Logicality

Synthese, 198: 1095-1119, 2021 (Special Issue: Intensionality in Mathematics, Marianna Antonutti and

Paula Quinon (eds.), published online 2017).

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension—it is a logic of cardinality (or, more accurately, of “isomorphism type”). In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, the more coarse-grained and less “intensional” it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things “in the world”. Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus’s principles of explicit and implicit extensionality, we are able to characterize purely logical languages as “sub-extensional”, namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality.

Contextualism, Relativism and the Liar 

Erkenntnis (2017), 82(4):913-928 doi:10.1007/s10670-016-9850-6.

Contextualist theories of truth appeal to context to solve the liar paradox: different stages of reasoning occur in different contexts, and so the contradiction is dispelled. The word `true' is relativized by the contextualists to contexts of use. This paper shows that contextualists approaches to the liar are committed to a form of semantic relativism: that the truth value of some sentences depends on the context of assessment, as well as the context of use. In particular, it is shown how Simmons's and Glanzberg's contextualist approaches entail relativism. In both cases, the liar sentence receives different semantic evaluations as uttered in a fixed context of use but assessed from different contexts. Shift in context of use alone cannot provide the full explanation of the liar. These contextualist approaches, as originally presented, were thus mischaracterised and they should be re-evaluated according to their full implications. 

The Modal and Epistemic Arguments Against the Invariance Criterion

for Logical Terms

Journal of Philosophy (March 2015)112(3):159-167.

The essay discusses a recurrent criticism of the isomorphism-invariance criterion for logical terms, according to which the criterion pertains only to the extension of logical terms, and neglects the meaning, or the way the extension is fixed. A term, so claim the critics, can be invariant under isomorphisms and yet involve a contingent or a posteriori component in its meaning, thus compromising the necessity or apriority of logical truth and logical consequence. This essay shows that the arguments underlying the criticism are flawed since they rely on an invalid inference from the modal or epistemic status of statements in the metalanguage to that of statements in the object-language. The essay focuses on McCarthy’s version of the argument, but refers to Hanson and McGee’s versions as well.

Formality in Logic: From Logical Terms to Semantic Constraints

Logique et Analyse (2014), 57(227):259-276.

In the paper I discuss a prevailing view by which logical terms determine forms of sentences and arguments and therefore the logical validity of arguments. This view is common to those who hold that there is a principled distinction between logical and nonlogical terms and those holding relativistic accounts. I adopt the Tarskian tradition by which logical validity is determined by form, but reject the centrality of logical terms.

I propose an alternative framework for logic where logical terms no longer play a distinctive role. This account employs a new notion of semantic constraints. The paper includes some preliminary definitions and results in the new framework. 

Models and Logical Consequence

 Journal of Philosophical Logic (2014) 43(5):943-964.

This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical consequence. Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro's blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed.


Sher and Shapiro on Logical Terms 

M. Peliš and V. Puncˇochárˇ (eds):  The Logica Yearbook 2010, College Publications, London, 2011.


Logical Consequence: Between Formal and Natural Language


Work in Progress
Invariance Criteria: Terms and Constraints
Logic and Natural Language
Logicality and Natural Language
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