Rust Belt Workshop

February 8 - February 9, 2025

9:00AM - 4:00PM

386B University Hall

Add to Calendar 2025-02-08 09:00:00 2025-02-09 16:00:00 Rust Belt Workshop The Logic or Language Society and the Society for Mathematical Logic and Foundational Studies will be hosting the first Rust Belt Workshop in the Philosophy of Logic, Language, and Mathematics. Schedule:Saturday, February 8th9:00 am - 10:00 amLight Refreshment10:00 am - 10:55 amNeil Tennant, The Ohio State University"Strong Completeness of Propositional C+"We prove the strong completeness of Classical Core Logic: Suppose a set X of premises logically implies a conclusion P (classically, in the sense of Tarski).Then either there is a Classical Core proof of P from X, or there is a Core proof of the inconsistency of X. The proof is by a variation of Henkin’s method. The expansion of a consistent set is via atoms; and in place of maximality we appeal to decisiveness. The proof works courtesy of the Cut Admissibility Theorem and its corollaries.11:00 am - 11:55 amJohn Mackay, University of Wisconsin-Madison "Counterparts and Tolerance"Abstract: This paper concerns the relationship between counterpart theory and tolerance puzzles, which involve the apparent vagueness of de re modal language. Traditionally, counterpart theory has been thought to provide a distinct solution to tolerance puzzles that makes use of the idea that counterpart relations are intransitive. Dorr, Hawthorne and Yli-Vakkuri (2021), by contrast, argue that it is unclear how this solution is supposed to work and conclude that counterpart theory has no distinct role in solving the tolerance puzzles. In this paper, I argue that counterpart theory does provide a distinct solution to the puzzles, but unlike in previous accounts, I argue that making counterpart relations intransitive does not help with the solution. Rather, the solution just has to do with the context-sensitivity of counterpart relations.12:00 pm - 2:00 pmLunch2:00 pm - 2:55 pmOwain Griffin, The Ohio State University"Platonism vs. Pluralism"Abstract: Platonism about mathematical objects and pluralism about logic are two of the most popular views in contemporary philosophy. In this paper, I show that despite their popularity, these views are in tension with one another3:00 pm - 3:55 pmJoel David Hamkins, The University of Notre Dame"On Skolem's paradox and the transitive submodel theorem"Abstract: One can find in the philosophical research literature surrounding Skolem's paradox a certain claim, referred to as the transitive submodel theorem, according to which every transitive model of set theory admits a countable transitive submodel of the same theory. Although the statement may initially appear quite plausible—perhaps one thinks it follows from an application of the downward Löwenheim-Skolem theorem—nevertheless it turns out that as a mathematical claim, it is overstated, and there is no such theorem. It is a mistake, although an interesting mistake worth discussing. In this talk I shall give a full account of the countable transitive submodel proposition, taken as a principle of set theory, by showing from suitable hypotheses that counterexamples are possible, characterizing exactly the circumstances in which the principle does hold, and investigating the consistency strength of the proposition and also the consistency strength of its negation. Ultimately, the countable transitive submodel proposition should be seen as a certain anti-large cardinal principle that is equiconsistent with but independent of ZFC, refuted by all the moderately strong large cardinal notions. This is joint work with Timothy Button, with thanks to W. Hugh Woodin. Sunday, February 9th9:00 am - 10:00 amLight Refreshments10:00 am - 10:55 amStewart Shapiro, The Ohio State University"Semantic indeterminacy, concept sharpening, and set theories"Abstract: Friedrich Waismann once suggested that mathematical concepts are not subject to open-texture; they are “closed”. In other work, I have highlighted some traditional mathematical notions that were, at least at one time, open-textured. One of them is the notion of “polyhedron” following the history sketched in Imre Lakatos’s Proofs and refutations. Another is “computability”, which has now been sharpened into an arguably closed notion, via the Church-Turing thesis. There are also some mathematical notions that have longstanding, intuitive principles underlying them, principles that later proved to be inconsistent with each other, sometimes when the notion is applied to cases not considered previously (in which case it is perhaps an instance of open-texture). One example is “same size”, which is or was governed by the part-whole principle (one of Euclid’s Common Notions) and the one-one principle, now called “Hume’s Principle”. Another is the notion of continuity. The purpose of this paper is to explore the notion of “set” and other related notions like “class”, “totality”, and the like. I tentatively put forward a thesis that this notion, too, is or at least was subject to open-texture (or something like it) and has been sharpened in various ways. This raises some questions concerning what the purposes of a (sharpened) theory of sets are to be. And, in that context, the role of trying to give non-ad-hoc explanations or answers to various questions.11:00 am - 11:55 amLawrence Moss, Indiana University Bloomington"Logic Done as if Inference in Language Mattered"Abstract: Our topic is logical inference in natural language, as it is done by people and computers. The first main topic will be monotonicity inference, arguably the best of the simple ideas in the area. Monotonicity can be incorporated in running systems whereby one can takeparsed real-life sentences and see simple inferences in action. I will present some of the theory, related to higher-order monotonicity and the syntax-semantics interface offered by categorial grammar.In a different direction, these days monotonicity inference can be done by machines as well as humans. The talk also discusses this development along with some ongoing work on the borderline of natural logic and machine learning.The second direction in the talk will be an overview of the large number of logical systems for various linguistic phenomena. This work begins as an updating of traditional syllogistic logic, but with much greater expressive power.Overall, the goal of the talk is to persuade you that the research program of “natural logic” leads to a lively research area with connections to many areas both inside and outside of more mainstream areas of logic.12:00 pm - 2:00 pmLunch2:00 pm - 2:55 pmAnand Ekbote, The Ohio State University"Truthlikeness and Scientific Progress"Abstract: Truthlikeness, a construct based on formal logic, is widely considered to be the most developed view in literature for scientific progress, both as an account of it and as a measure of it. It considers the aim of science to be verisimilitude i.e., proximity or similarity, to truth. It is thus a realist conception of scientific progress. I argue that while truthlikeness addresses a rich, complex, and philosophically deep question in logic, that question is nevertheless not one of scientific progress. It misconstrues scientific theories in two crucial ways. First, it incorporates falsehoods as a matter of course in its conception of scientific theory, which is not consonant with realism, anti-realism, and scientific practice. Second, it understructures scientific theories. Besides, as noted by Tennant (1989) it does not offer a satisfactory account of theory change. Furthermore, as Miller (1974, 1976, 2006) has convincingly pointed out, prevalent approaches in truthlikeness are dependent on the linguistic framework used, which means that what is found to be closer (or similar) to the truth may be different depending on which language is used. This severely constrains the adoption of truthlikeness by scientific realists - the very community it purports to serve. I make the case instead for my set-theoretic Realm-based assessment of scientific progress (RASP) which I build on a variant of Popper’s content view of verisimilitude, utilizing resources of the AGM account of belief revision. RASP overcomes the drawbacks of the truthlikeness account, and delivers on two key desiderata for assessments of scientific progress, namely (i) accommodate evidence that is not a logical consequence of the theory, (ii) be applicable to realist as well as antirealist accounts of scientific progress. It offers additional advantages in being amenable to (i) incorporation of axiological values (such as simplicity) and (ii) more fine-grained accounts of scientific progress (e.g., epistemic, noetic, semantic, and functional).3:00 pm - 3:55 pmGabriel Day, University of Notre DameTBA 386B University Hall OSU ASC Drupal 8 ascwebservices@osu.edu America/New_York public

2025-02-08 09:00:00 2025-02-09 16:00:00 Rust Belt Workshop The Logic or Language Society and the Society for Mathematical Logic and Foundational Studies will be hosting the first Rust Belt Workshop in the Philosophy of Logic, Language, and Mathematics.  Schedule:Saturday, February 8th9:00 am - 10:00 amLight Refreshment10:00 am - 10:55 amNeil Tennant, The Ohio State University"Strong Completeness of Propositional C+"We prove the strong completeness of Classical Core Logic: Suppose a set X of premises logically implies a conclusion P (classically, in the sense of Tarski).Then either there is a Classical Core proof of P from X, or there is a Core proof of the inconsistency of X. The proof is by a variation of Henkin’s method. The expansion of a consistent set is via atoms; and in place of maximality we appeal to decisiveness. The proof works courtesy of the Cut Admissibility Theorem and its corollaries.11:00 am - 11:55 amJohn Mackay, University of Wisconsin-Madison "Counterparts and Tolerance"Abstract: This paper concerns the relationship between counterpart theory and tolerance puzzles, which involve the apparent vagueness of de re modal language. Traditionally, counterpart theory has been thought to provide a distinct solution to tolerance puzzles that makes use of the idea that counterpart relations are intransitive. Dorr, Hawthorne and Yli-Vakkuri (2021), by contrast, argue that it is unclear how this solution is supposed to work and conclude that counterpart theory has no distinct role in solving the tolerance puzzles. In this paper, I argue that counterpart theory does provide a distinct solution to the puzzles, but unlike in previous accounts, I argue that making counterpart relations intransitive does not help with the solution. Rather, the solution just has to do with the context-sensitivity of counterpart relations.12:00 pm - 2:00 pmLunch2:00 pm - 2:55 pmOwain Griffin, The Ohio State University"Platonism vs. Pluralism"Abstract: Platonism about mathematical objects and pluralism about logic are two of the most popular views in contemporary philosophy. In this paper, I show that despite their popularity, these views are in tension with one another3:00 pm - 3:55 pmJoel David Hamkins, The University of Notre Dame"On Skolem's paradox and the transitive submodel theorem"Abstract: One can find in the philosophical research literature surrounding Skolem's paradox a certain claim, referred to as the transitive submodel theorem, according to which every transitive model of set theory admits a countable transitive submodel of the same theory. Although the statement may initially appear quite plausible—perhaps one thinks it follows from an application of the downward Löwenheim-Skolem theorem—nevertheless it turns out that as a mathematical claim, it is overstated, and there is no such theorem. It is a mistake, although an interesting mistake worth discussing. In this talk I shall give a full account of the countable transitive submodel proposition, taken as a principle of set theory, by showing from suitable hypotheses that counterexamples are possible, characterizing exactly the circumstances in which the principle does hold, and investigating the consistency strength of the proposition and also the consistency strength of its negation. Ultimately, the countable transitive submodel proposition should be seen as a certain anti-large cardinal principle that is equiconsistent with but independent of ZFC, refuted by all the moderately strong large cardinal notions. This is joint work with Timothy Button, with thanks to W. Hugh Woodin. Sunday, February 9th9:00 am - 10:00 amLight Refreshments10:00 am - 10:55 amStewart Shapiro, The Ohio State University"Semantic indeterminacy, concept sharpening, and set theories"Abstract: Friedrich Waismann once suggested that mathematical concepts are not subject to open-texture; they are “closed”. In other work, I have highlighted some traditional mathematical notions that were, at least at one time, open-textured. One of them is the notion of “polyhedron” following the history sketched in Imre Lakatos’s Proofs and refutations. Another is “computability”, which has now been sharpened into an arguably closed notion, via the Church-Turing thesis. There are also some mathematical notions that have longstanding, intuitive principles underlying them, principles that later proved to be inconsistent with each other, sometimes when the notion is applied to cases not considered previously (in which case it is perhaps an instance of open-texture). One example is “same size”, which is or was governed by the part-whole principle (one of Euclid’s Common Notions) and the one-one principle, now called “Hume’s Principle”. Another is the notion of continuity. The purpose of this paper is to explore the notion of “set” and other related notions like “class”, “totality”, and the like. I tentatively put forward a thesis that this notion, too, is or at least was subject to open-texture (or something like it) and has been sharpened in various ways. This raises some questions concerning what the purposes of a (sharpened) theory of sets are to be. And, in that context, the role of trying to give non-ad-hoc explanations or answers to various questions.11:00 am - 11:55 amLawrence Moss, Indiana University Bloomington"Logic Done as if Inference in Language Mattered"Abstract: Our topic is logical inference in natural language, as it is done by people and computers. The first main topic will be monotonicity inference, arguably the best of the simple ideas in the area. Monotonicity can be incorporated in running systems whereby one can takeparsed real-life sentences and see simple inferences in action. I will present some of the theory, related to higher-order monotonicity and the syntax-semantics interface offered by categorial grammar.In a different direction, these days monotonicity inference can be done by machines as well as humans. The talk also discusses this development along with some ongoing work on the borderline of natural logic and machine learning.The second direction in the talk will be an overview of the large number of logical systems for various linguistic phenomena. This work begins as an updating of traditional syllogistic logic, but with much greater expressive power.Overall, the goal of the talk is to persuade you that the research program of “natural logic” leads to a lively research area with connections to many areas both inside and outside of more mainstream areas of logic.12:00 pm - 2:00 pmLunch2:00 pm - 2:55 pmAnand Ekbote, The Ohio State University"Truthlikeness and Scientific Progress"Abstract: Truthlikeness, a construct based on formal logic, is widely considered to be the most developed view in literature for scientific progress, both as an account of it and as a measure of it. It considers the aim of science to be verisimilitude i.e., proximity or similarity, to truth. It is thus a realist conception of scientific progress. I argue that while truthlikeness addresses a rich, complex, and philosophically deep question in logic, that question is nevertheless not one of scientific progress. It misconstrues scientific theories in two crucial ways. First, it incorporates falsehoods as a matter of course in its conception of scientific theory, which is not consonant with realism, anti-realism, and scientific practice. Second, it understructures scientific theories. Besides, as noted by Tennant (1989) it does not offer a satisfactory account of theory change. Furthermore, as Miller (1974, 1976, 2006) has convincingly pointed out, prevalent approaches in truthlikeness are dependent on the linguistic framework used, which means that what is found to be closer (or similar) to the truth may be different depending on which language is used. This severely constrains the adoption of truthlikeness by scientific realists - the very community it purports to serve. I make the case instead for my set-theoretic Realm-based assessment of scientific progress (RASP) which I build on a variant of Popper’s content view of verisimilitude, utilizing resources of the AGM account of belief revision. RASP overcomes the drawbacks of the truthlikeness account, and delivers on two key desiderata for assessments of scientific progress, namely (i) accommodate evidence that is not a logical consequence of the theory, (ii) be applicable to realist as well as antirealist accounts of scientific progress. It offers additional advantages in being amenable to (i) incorporation of axiological values (such as simplicity) and (ii) more fine-grained accounts of scientific progress (e.g., epistemic, noetic, semantic, and functional).3:00 pm - 3:55 pmGabriel Day, University of Notre DameTBA  386B University Hall America/New_York public

The Logic or Language Society and the Society for Mathematical Logic and Foundational Studies will be hosting the first Rust Belt Workshop in the Philosophy of Logic, Language, and Mathematics.

Schedule:

Saturday, February 8th

9:00 am - 10:00 am

Light Refreshment

10:00 am - 10:55 am

Neil Tennant, The Ohio State University
"Strong Completeness of Propositional C⁺"

We prove the strong completeness of Classical Core Logic:

Suppose a set X of premises logically implies a conclusion P (classically, in the sense of Tarski).

Then either there is a Classical Core proof of P from X, or there is a Core proof of the inconsistency of X.

The proof is by a variation of Henkin’s method. The expansion of a consistent set is via atoms; and in place of maximality we appeal to decisiveness.

The proof works courtesy of the Cut Admissibility Theorem and its corollaries.

11:00 am - 11:55 am

John Mackay, University of Wisconsin-Madison
"Counterparts and Tolerance"
Abstract: This paper concerns the relationship between counterpart theory and tolerance puzzles, which involve the apparent vagueness of de re modal language. Traditionally, counterpart theory has been thought to provide a distinct solution to tolerance puzzles that makes use of the idea that counterpart relations are intransitive. Dorr, Hawthorne and Yli-Vakkuri (2021), by contrast, argue that it is unclear how this solution is supposed to work and conclude that counterpart theory has no distinct role in solving the tolerance puzzles. In this paper, I argue that counterpart theory does provide a distinct solution to the puzzles, but unlike in previous accounts, I argue that making counterpart relations intransitive does not help with the solution. Rather, the solution just has to do with the context-sensitivity of counterpart relations.

12:00 pm - 2:00 pm

Lunch

2:00 pm - 2:55 pm

Owain Griffin, The Ohio State University
"Platonism vs. Pluralism"
Abstract: Platonism about mathematical objects and pluralism about logic are two of the most popular views in contemporary philosophy. In this paper, I show that despite their popularity, these views are in tension with one another

3:00 pm - 3:55 pm

Joel David Hamkins, The University of Notre Dame
"On Skolem's paradox and the transitive submodel theorem"
Abstract: One can find in the philosophical research literature surrounding Skolem's paradox a certain claim, referred to as the transitive submodel theorem, according to which every transitive model of set theory admits a countable transitive submodel of the same theory. Although the statement may initially appear quite plausible—perhaps one thinks it follows from an application of the downward Löwenheim-Skolem theorem—nevertheless it turns out that as a mathematical claim, it is overstated, and there is no such theorem. It is a mistake, although an interesting mistake worth discussing. In this talk I shall give a full account of the countable transitive submodel proposition, taken as a principle of set theory, by showing from suitable hypotheses that counterexamples are possible, characterizing exactly the circumstances in which the principle does hold, and investigating the consistency strength of the proposition and also the consistency strength of its negation. Ultimately, the countable transitive submodel proposition should be seen as a certain anti-large cardinal principle that is equiconsistent with but independent of ZFC, refuted by all the moderately strong large cardinal notions. This is joint work with Timothy Button, with thanks to W. Hugh Woodin.

Sunday, February 9th

9:00 am - 10:00 am

Light Refreshments

10:00 am - 10:55 am

Stewart Shapiro, The Ohio State University
"Semantic indeterminacy, concept sharpening, and set theories"
Abstract: Friedrich Waismann once suggested that mathematical concepts are not subject to open-texture; they are “closed”. In other work, I have highlighted some traditional mathematical notions that were, at least at one time, open-textured. One of them is the notion of “polyhedron” following the history sketched in Imre Lakatos’s Proofs and refutations. Another is “computability”, which has now been sharpened into an arguably closed notion, via the Church-Turing thesis.

There are also some mathematical notions that have longstanding, intuitive principles underlying them, principles that later proved to be inconsistent with each other, sometimes when the notion is applied to cases not considered previously (in which case it is perhaps an instance of open-texture). One example is “same size”, which is or was governed by the part-whole principle (one of Euclid’s Common Notions) and the one-one principle, now called “Hume’s Principle”. Another is the notion of continuity.

The purpose of this paper is to explore the notion of “set” and other related notions like “class”, “totality”, and the like. I tentatively put forward a thesis that this notion, too, is or at least was subject to open-texture (or something like it) and has been sharpened in various ways.

This raises some questions concerning what the purposes of a (sharpened) theory of sets are to be. And, in that context, the role of trying to give non-ad-hoc explanations or answers to various questions.

11:00 am - 11:55 am

Lawrence Moss, Indiana University Bloomington
"Logic Done as if Inference in Language Mattered"
Abstract: Our topic is logical inference in natural language, as it is done by people and computers. The first main topic will be monotonicity inference, arguably the best of the simple ideas in the area. Monotonicity can be incorporated in running systems whereby one can take

parsed real-life sentences and see simple inferences in action. I will present some of the theory, related to higher-order monotonicity and the syntax-semantics interface offered by categorial grammar.

In a different direction, these days monotonicity inference can be done by machines as well as humans. The talk also discusses this development along with some ongoing work on the borderline of natural logic and machine learning.

The second direction in the talk will be an overview of the large number of logical systems for various linguistic phenomena. This work begins as an updating of traditional syllogistic logic, but with much greater expressive power.

Overall, the goal of the talk is to persuade you that the research program of “natural logic” leads to a lively research area with connections to many areas both inside and outside of more mainstream areas of logic.

12:00 pm - 2:00 pm

Lunch

2:00 pm - 2:55 pm

Anand Ekbote, The Ohio State University
"Truthlikeness and Scientific Progress"
Abstract: Truthlikeness, a construct based on formal logic, is widely considered to be the most developed view in literature for scientific progress, both as an account of it and as a measure of it. It considers the aim of science to be verisimilitude i.e., proximity or similarity, to truth. It is thus a realist conception of scientific progress. I argue that while truthlikeness addresses a rich, complex, and philosophically deep question in logic, that question is nevertheless not one of scientific progress. It misconstrues scientific theories in two crucial ways. First, it incorporates falsehoods as a matter of course in its conception of scientific theory, which is not consonant with realism, anti-realism, and scientific practice. Second, it understructures scientific theories. Besides, as noted by Tennant (1989) it does not offer a satisfactory account of theory change. Furthermore, as Miller (1974, 1976, 2006) has convincingly pointed out, prevalent approaches in truthlikeness are dependent on the linguistic framework used, which means that what is found to be closer (or similar) to the truth may be different depending on which language is used. This severely constrains the adoption of truthlikeness by scientific realists - the very community it purports to serve. I make the case instead for my set-theoretic Realm-based assessment of scientific progress (RASP) which I build on a variant of Popper’s content view of verisimilitude, utilizing resources of the AGM account of belief revision. RASP overcomes the drawbacks of the truthlikeness account, and delivers on two key desiderata for assessments of scientific progress, namely (i) accommodate evidence that is not a logical consequence of the theory, (ii) be applicable to realist as well as antirealist accounts of scientific progress. It offers additional advantages in being amenable to (i) incorporation of axiological values (such as simplicity) and (ii) more fine-grained accounts of scientific progress (e.g., epistemic, noetic, semantic, and functional).

3:00 pm - 3:55 pm

Gabriel Day, University of Notre Dame

TBA