Quantifiers and Modal Logic (Melvin Fitting)

The Logic and Metaphysics Workshop will meet on April 23rd from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Melvin Fitting (CUNY).

Title: Quantifiers and Modal Logic

Abstract: In classical logic the move from propositional to quantificational is profound but essentially takes one route, following a direction we are all familiar with.  In modal logic, such a move shoots off in many directions at once.  One can quantify over things or over intensions.  Quantifier domains can be the same from possible world to possible world, shrink or grow as one moves from a possible world to an accessible one, or follow no pattern whatsoever.  A long time ago, Kripke showed us how shrinking or growing domains related to validity of the Barcan and the converse Barcan formulas, bringing some semantic order into the situation.  But when it comes to proof theory things get somewhat strange.  Nested sequents for shrinking or growing domains, or for constant domains or completely varying domains, are relatively straightforward.  But axiomatically some oddities are quickly apparent.  A simple combination of propositional modal axioms and rules with standard quantificational axioms and rules proves the converse Barcan formula, making it impossible to investigate its absence.  Kripke showed how one could avoid this, at the cost of using a somewhat unusual axiomatization of the quantifiers.  But things can be complicated and even here an error crept into Kripke’s work that wasn’t pointed out until 20 years later, by Fine. Justification logic was started by Artemov with a system related to propositional S4, called LP.  This was extended to a quantified version by Artemov and Yavorskaya, for which a semantics was supplied by Fitting.  Recently Artemov and Yavorskaya introduced what they called bounding modalities, by transferring ideas back from quantified LP to S4.  In this paper we continue the investigation of bounding modalities, but for axiomatic K since modal details aren’t that important for what I’m interested in.  We wind up with axiomatic systems allowing for a monotonic domain condition, an anti-monotonic one, neither, or both.  We provide corresponding semantics and give direct soundness and completeness proofs.  Unlike in Kripke’s treatment, the heavy lifting is done through generalization of the modal operator, instead of restriction on quantification. (This talk continues one given earlier in the semester in Artemov’s seminar.  There are differences, but if you happened to hear that talk, you could easily skip this one since the differences are not great.)

Metaphysics Beyond Grounding (Daniel Nolan)

The Logic and Metaphysics Workshop will meet on April 16th from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Daniel Nolan (Notre Dame).

Title: Metaphysics Beyond Grounding

Abstract: Thinking about metaphysical problems in terms of grounding has its uses, but those uses are limited. I am not a sceptic either about grounding or our ability to make progress on some metaphysical puzzles by invoking it, but I will argue it only has a partial role to play in our metaphysical theories. I will discuss how grounding relates to necessity, to explanation and to parsimony in theory choice. Finally, I will discuss the connection between grounding and the proper aim, or rather aims, of metaphysics.

Isomorphisms in a Category of Proofs (Greg Restall)

The Logic and Metaphysics Workshop will meet on April 9th from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Greg Restall (Melbourne).

Title: Isomorphisms in a Category of Proofs

Abstract: In this talk, I show how a category of classical proofs can give rise to three different hyperintensional notions of sameness of content. One of these notions is very fine-grained, going so far as to distinguish p and p∧p, while identifying other distinct pairs of formulas, such as p∧q and q∧p; p and ¬¬p; or ¬(p∧q) and ¬p∨¬q. Another relation is more coarsely grained, and gives the same account of identity of content as equivalence in Angell’s logic of analytic containment. A third notion of sameness of content is defined, which is intermediate between Angell’s and Parry’s logics of analytic containment. Along the way we show how purely classical proof theory gives resources to define hyperintensional distinctions thought to be the domain of properly non-classical logics.

Slides/Handout: for those interested, the slides and handout for this talk will be made available for advance reading here.

Mathematical Truth is Historically Contingent (Chris Scambler)

The Logic and Metaphysics Workshop will meet on March 26th from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Chris Scambler (NYU).

Title: Mathematical Truth is Historically Contingent

Abstract: In this talk I will defend a view according to which certain mathematical facts depend counterfactually on certain historical facts. Specifically, I will sketch an alternative possible history for us in which (I claim) the proposition ordinarily expressed by the English sentence “there is a universal set” is true, despite its falsity in the actual world.

Admissibility of Multiple-Conclusion Rules of Logics with the Disjunction Property (Alex Citkin)

The Logic and Metaphysics Workshop will meet on March 19th from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Alex Citkin (Private Researcher).

Title: The Admissibility of Multiple-Conclusion Rules of Logics with the Disjunction Property

Abstract: I study admissible multiple-conclusion rules of logics having the meta-disjunction expressible by a finite set of formulas. I show that in such logics the bases of admissible single- and multiple-conclusion rules can be converted into each other. Since these conversions are constructive and preserve cardinality, it is possible to obtain a simple way of constructing a base of admissible single-conclusion rules, by a given base of admissible multiple-conclusion rules and vice versa. Because the proofs are purely syntactical, these results can be applied to a broad class of logics.

Confessing to a Superfluous Premise (Roy Sorensen)

The Logic and Metaphysics Workshop will meet on March 12th from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Roy Sorensen (WUSTL).

Title: Confessing to a Superfluous Premise

Abstract: In a hurried letter to beleaguered brethren, Blaise Pascal (1658) confesses to a lapse of concision: “I have made this longer than usual because I have not had time to make it shorter.”  Pascal’s confession was emulated with the same warmth as philosophers now emulate the apology introduced by D. C. Mackinson’s “The Preface Paradox”. Could Pascal’s confession of superfluity be sound? Pascal thinks his letter could be conservatively abridged; the shortened letter would be true and have the exact same content. In contrast to the Preface Paradox, where Mackinson’s author apologizes for false assertions, Pascal apologizes for an excess of true assertions. He believes at least one of his remarks could be deleted in a fashion that leaves all of its consequences entailed by the remaining assertions. Pascal’s confession of superfluity is plausible even if we count the apology as part of the letter (as we should since this is the most famous part of the letter). Yet there is an a priori refutation. Any conservative abridgement must preserve the implication that there is a superfluous assertion. This means any abridged version can itself be abridged. Since the letter is finite, we must eventually run out of conservative abridgements. Any predecessor of an unabridgeable abridgement is itself an unabridgeable.  So the original letter cannot be conservatively abridged.

Manuscript: for those interested, the manuscript has been made available for advance reading here.

The Metasemantics of Indefinite Extensibility (Vera Flocke)

The Logic and Metaphysics Workshop will meet on March 5th from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Vera Flocke (NYU).

Title: The Metasemantics of Indefinite Extensibility

Abstract: Indefinite extensibility is the thesis that any domain of quantification can always be expanded. But how is the possibility of expanding domains of quantification reflected in the semantics of quantified sentences? This paper discusses the relevant meta-semantic options within a framework that distinguishes between semantic values and assertoric contents. This choice of a framework is independently motivated, helps received accounts of indefinite extensibility to escape weighty objections and adds to the available metasemantic options. I then argue for a hitherto overlooked view according to which quantified sentences express stable semantic values but variable assertoric contents. Specifically, the semantic value of quantified sentences are sets of possible worlds that are structured by two equivalence relations, one of which models counterfactual necessity and the other one of which models objectivity. Assertoric contents however are ordinary possible worlds propositions. The advantage of this view is that it explains succinctly what’s at issue in the debate between generality-absolutists, who think that quantification over absolutely everything is possible, and generality-relativists. If the box expresses objectivity, this disagreement concerns the Barcan formula, which entails that domains do not grow as one moves to objectively-accessible worlds.

A Dynamic Solution to the Liar Paradox (Martin Pleitz)

The Logic and Metaphysics Workshop will meet on February 26th from 4:15-6:15 in room 3309 of the CUNY Graduate Center for a talk by Martin Pleitz (Muenster).

Title: A Dynamic Solution to the Liar Paradox

Abstract: The Liar paradox arises when we combine the assumption that a sentence can refer to itself with our naïve notion of truth and apply our unrevised logic. Most current approaches to the Liar paradox focus on revising our notion of truth and logic because nowadays almost everyone is convinced that there are self-referential sentences. I will argue against this conviction. My argument starts from observations about the metaphysics of expressions: A meaningful expression is based in a syntactic expression which in turn is based in a non-semiotic object, and these are pairwise distinct. As all objects of this three-fold ontology exist only relative to contexts, we can import ideas from tense logic about how existence and reference can interact in a contextualist metaphysics. Semantico-metaphysical reasoning then shows that in this dynamic setting, an object can be referred to only after it has started to exist. Hence the self-reference needed in the Liar paradox cannot occur, after all. As this solution is contextualist, it evades the expressibility problems of other proposals.

Spring 2018 Schedule

The Logic and Metaphysics Workshop will be meeting on Mondays from 4:15 to 6:15 in room 3309 of the Graduate Center, CUNY (365 5th Avenue). The following is the schedule:

Feb 12. GC CLOSED. NO MEETING

Feb 19. GC CLOSED NO MEETING

Feb 26. Martin Pleitz, Muenster

Mar 5. Vera Flocke, NYU

Mar 12. Graham Priest, CUNY Roy Sorensen, WUSTL

Mar 19. Alex Citkin, Private Researcher

Mar 26. Chris Scambler, NYU

Apr 2. SPRING RECESS. NO MEETING

Apr 9. Greg Restall, Melbourne

Apr 16. Daniel Nolan, Notre Dame

Apr 23. Mel Fitting, CUNY

Apr 30. Sungil Han, Seoul National

May 7. Andreas Ditter, NYU

May 14. Rohit Parikh, CUNY

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