# FSB research seminar

These are the FSB research seminars that have been given in recent years.

## 2016-17

The Foundational Studies research seminar currently takes place on Thursdays 5pm, in G2 in Cotham House (See the University's Google precinct map).

Each week, an idea, thesis or result is presented within thirty minutes or less, leaving plenty of time for discussion. Only very limited background knowledge is assumed and the aim is to develop new research directions within foundational studies (philosophical foundations of mathematics, logic, philosophical logic, formal theories of truth, ...).

If you are interested, email foundational-studies@bristol.ac.uk to ask to be added to the mailing list.

The seminars given in previous years can be seen here.

Date | Speaker | Title |
---|---|---|

19 January 2017 |
Oliver Tatton-Brown |
Mathematical auxiliaries and quietist Platonism |

2 February 2017 |
Mathematical equivalences and the road to formalisation | |

9 February 2017 |
Leon Horsten |
Prejudice as evidence |

16 February 2017 |
Pawel Pawlowski (Ghent) |
Non-classical logic of informal provability |

2 March 2017 |
Nemo D'Qrill |
Nozick Truth Tracking - Not Necessary |

9 March 2017 |
Carlo Nicolai (Munich) |
What can we learn from reflecting on truth? |

16 March 2017 |
Øystein Linnebo (Oslo) |
Generality explained: A truth-maker semantics |

23 March 2017 |
Irina Starikova (Sao Paolo) | Visual aspects of scientific models: the case of turbulence |

30 March 2017 |
Stuart Presnell | Martin-Löf's "Meaning Explanations" |

27 April 2017 |
Kentaro Fujimoto | Grounded classical truth |

4 May 2017 |
Beau Mount (Oxford) | Approximative Indefinite Extensibility |

16 May 2017, 3pm |
Michael Sheard | A Formal Path to a Weak Theory of Truth |

19 June 2017, 3.30pm G16 |
Toby Meadows (Queensland) | You say potato, I say potato: can definition debates be worthwhile? |

1,3,8 Aug 2017 |
Stanislav Speranski (St Petersberg) | Benjamin Meaker Visiting Professor lectures |

## Abstracts

### Oliver Tatton-Brown (Bristol) - 'Mathematical auxiliaries and quietist Platonism'

Various authors, including Maddy, Tait and the Neo-Fregeans, have suggested that perhaps the problem of the existence of mathematical objects is not a deep problem at all, and that their existence can be read off of our mathematical language or mathematical practice. I outline a counterargument to this based on the use of auxiliaries in mathematics. Roughly, whether certain objects exist can matter because we can use their existence to deduce facts about other objects. It cannot be read off of our mathematical language or mathematical practice that we can reason as though certain objects exist (and obtain truths), so it also cannot be read off of our mathematical language or mathematical practice that the objects in question do exist.

### Benedict Eastaugh (Bristol) - Mathematical equivalences and the road to formalisation

Talk of equivalences is commonplace in mathematical discourse, both for theorems (“The least upper bound principle is equivalent to the Dedekind cut principle”) and conjectures (“The following are equivalents of the Riemann hypothesis . . . ”). Given two statements P and Q, such equivalences are demonstrated by proving both that P implies Q, and conversely that Q implies P. While the pragmatics of such equivalence statements are relatively clear, the semantics are harder to make sense of. We contend that equivalence statements in ordinary mathematical practice are best understood through the lens of reverse mathematics, where the equivalence of P and Q is proven in a weak base theory incapable of proving either statement. This view avoids a trivialising problem for a classical understanding of the biconditional, on which all true statements are logically equivalent. It also connects informal equivalence statements in mathematics to their formal counterparts in set theory, such as the many statements found to be equivalent over the base theory ZF to the Axiom of Choice. Finally, we consider objections arising from an argument of Paul Halmos that such equivalence statements are nonsensical.

### Leon Horsten (Bristol) - Prejudice as evidence

In a 2012 article, Leitgeb outlines a way of building revision-based models for languages containing a self-referential subjective probability predicate. The revision-component of Leitgeb’s procedure is based on a relative frequency idea.

In my talk I want to take a few steps in exploring whether in Leitgeb’s procedure the frequency-based revision step can be replaced by a form of Bayesian conditionalisation.

### Pawel Pawlowski (Ghent) - Non-classical logic of informal provability

Mathematicians prove theorems. They don’t do that in any particular axiomatic system. Rather, they reason in a semi-formal setting, providing what we’ll call informal proofs. There are quite a few reasons not to reduce informal provability to formal provability within some appropriate axiomatic theory (Marfori, 2010; Leitgeb, 2009). The main worry about identifying informal provability with formal provability starts with the following observation. We have a strong intuition that whatever is informally provable is true. Thus, we are committed to all instances of the so-called reflection schema P(⌜φ⌝) → φ (where ⌜φ⌝ is the number coding formula φ and P is the informal provability predicate).

Yet, not all such instances for formal provability (in standard Peano Arithmetic, henceforth PA) are provable in PA. Even worse, a sufficiently strong arithmetical theory T resulting from adding to PA (or any sufficiently strong arithmetic) all instances of the reflection schema for provability in T will be inconsistent (assuming derivability conditions for provability in T are provable in T). Thus, something else has to be done.

The main idea behind most of the current approaches (Shapiro, 1985; Horsten, 1994, 1996) is to extend the language with a new informal provability predicate or operator, and include all instances of the reflection schema for it. Contradiction is avoided at the price of dropping one of the derivability conditions. Thus, various options regarding trade-offs between various principles which all seem convincing are studied. In order to overcome some of the resulting difficulties and arbitrariness we investigate the strategy which changes the underlying logic and treats informal provability as a partial notion, just like Kripke’s theory of truth (Kripke, 1975) treats truth as a partial notion (one that clearly applies to some sentences, clearly doesn’t apply to some other sentences, but is undecided about the remaining ones). The intuition is that at a given stage, certain claims are clearly informally provable, some are clearly informaly disprovable, whereas the status of the remaining ones is undecided.

In Kripke-style truth theories strong Kleene three-valued logic is usually used – which seems adequate for interpreting truth as a partial notion. Yet, we will argue that no well-known three-valued logic can do a similar job for informal provability. The main reason is that the value of a complex formula in those logics is always a function of the values of its components. This fails to capture the fact that, for instance, some informally provable disjunctions of mathematical claims have informally provable disjuncts, while some other don’t.

We develop a non-functional many-valued logic which avoids this problem and captures our intuitions about informal provability. We describe the semantics of our logic and some of its properties. We argue that it does a better job when it comes to reasoning with informal provability predicate in formalized theories built over arithmetic.

### Nemo D'Qrill (Bristol) - Nozick Truth Tracking -- Not Necessary

Kripke (1980s), Williams (2015), and others, have attempted to show that Nozick's tracking theory for knowledge is insufficient. Murray and Adams (2003:2005:2015) have defended against these objections. This paper demonstrates, with two independent counterexamples, that even were Nozick's theory sufficient, it would be unnecessary.

### Carlo Nicolai (Munich) - What can we learn from reflecting on truth?

In recent work with Martin Fischer and Leon Horsten, we have studied the result of iterating a form of uniform reflection over a simple truth theory featuring only basic disquotational principles formulated in four-valued logic. The resulting picture can be read as (i) suggesting a conceptual analysis of the notion of truth based on disquotation (ii) a way to close the proof-theoretic gap existing between classical and non-classical theories of truth (iii) pointing at a foundational project based on weak combinatorial principles. In the talk I will mostly focus on options (ii) and (iii) and discuss some potential problems.

### Øystein Linnebo (Oslo) - Generality explained: A truth-maker semantics

What explains a true universal generalization? This paper distinguishes two kinds of explanation. While an instance-based explanation proceeds via each instance of the generalization, a generic explanation is independent of each instance, relying instead on completely general facts about the properties or operations involved in the generalization. This distinction is illuminated by means of a truth-maker semantics, which is also used to show that instance-based explanations support classical logic, while generic explanations support only intuitionistic logic.

### Irina Starikova (Sao Paolo) - Visual aspects of scientific models: the case of turbulence

Recent discussions question the role of sensory aspect in not only in proofs but also in models and thought experiments. How important are images? Are they necessary? Relevant philosophical opinions divide into camps. For example, Brown, Gendler, Nersessian argue that visualisations are essential. Norton claims that visualisations are irrelevant in thought experiments. Meanwhile, Salis & Frigg (forthcoming) suggest that images are sometimes useful for thought experiments but never necessary. My position is that in some cases visualisations are necessary, e.g. when reasoning requires mental manipulations over them of images from diagrams (most recently Giaquinto & Starikova forthcoming, Starikova 2016, De Toffoli & Giardino 2014, 2016).

This paper moves focus from pure to applied mathematics, and to the use of images in studying physical phenomena. There are still phenomena waiting for a better mathematical grip, for example, turbulence. I will argue that a visual image (of a model of physical phenomena) can play an important role in guiding the mathematicians’ research and choosing new mathematical resources. In particular, I will show that Richardson’s model of a cascading wave motivated both Kolmogorov’s statistical theory of turbulence and more recent geometric interpretation of the shape dynamics of a fluid volume. This is how an application of Riemannian geometry (Ricci flows) in mathematical description of turbulence became accessible.

The paper distinguishes "loose" geometry, which means simply visualising a phenomenon, and "strict" geometry, which means already looking at the visual representation geometrically and applying geometry to the initial problem. On the basis of this distinction one can observe from the case study that loose geometry opens up possibilities for strict geometry. Visual representations can (even in very complex mathematics) guide research in a certain (geometric) direction, when a merely linguistic / symbolic representation does not help.

### Stuart Presnell - Martin-Löf's "Meaning Explanations"

In "On the Meanings of the Logical Constants and the Justifications of the Logical Laws" Per Martin-Löf presents an analysis of the notions of 'proposition' and 'judgement', from which he derives an account of intuitionistic/constructive logic. I'll start by giving a summary of this account, which forms the conceptual core of Homotopy Type Theory. I'll then make some connections to Øystein Linnebo's recent "truth-maker semantics" talk, and consider some ways that the idea can be broadened to new applications (as suggested by Ben Eva).

### Kentaro Fujimoto - Grounded classical truth

I'll present a new theory of truth that I like very much: it is self-referential, compositional, and classical.

### Beau Mount (Oxford) - Approximative Indefinite Extensibility

Michael Dummett (1993) argued that the classical paradoxes (Russell, Burali-Forti, the Liar, and so on) show that some of our mathematical concepts, such as set, ordinal number, and true sentence of arithmetic are indefinitely extensible. This conception is normally taken to support generality relativism, but Timothy Williamson (1998) and Gabriel Uzquiano (2015) have developed an alternative, linguistic account of indefinite extensibility (IE) that is compatible with generality absolutism. I argue that, although the linguistic conception of IE has much to recommend it, the specific proposals by Uzquiano and Williamson are flawed. I develop an alternative account of linguistic IE, approximative indefinite extensibility (AIE). Roughly, if a concept Σ is subject to AIE, then Σ is naïvely characterized by an abstraction principle A that posits the existence of a total function μ that cannot exist for cardinality reasons. Nonetheless, there exists an open-ended sequence of approximants to μ: I contend that these approximants correspond to candidate extensions of the indefinitely extensible concept.

### Michael Sheard - A Formal Path to a Weak Theory of Truth

I will begin by outlining a new formal framework for approaching the proof theory of the axiomatic system FS and related systems, one that provides unified and substantially simplified proofs for known results. While this approach should be of independent interest, I will also use it as a means to identify and explore an especially weak theory of type-free truth which nonetheless has several desirable features. More specifically, I will suggest a solution to this analogy among formal systems: ??? / FS ≡ PKF / KF. Unlike most talks in logic, this presentation will become less technical as it proceeds.

### Toby Meadows (Queensland) - You say potato, I say potato: can definition debates be worthwhile?

In the last few years a new literature has grown up around assessing whether certain disputes are merely verbal. The basic idea is to capture what is going on when people argue past each other and to show that in some cases they aren't really arguing at all. This is part of a relatively new area of metaphysics somewhat pretentiously known as metametaphysics.

In this paper, I want to draw on some tools and examples from mathematical logic in the attempt to develop a more fine-grained map of the landscape in which these disputes take place. However, I would like to stress that the paper is intended for a general audience, so when technicalities emerge I'll try to draw a helpful picture rather than scribbling up a mess of hieroglyphics. The output of the paper will be more in the line of a series of observations and an advertisement for a particular methodology than a sustained and focused argument.

I'll start the paper with a simple example and a little history. Then we'll generalize the approach to deeper and more troubling examples. At this point, logic will make its entrance allowing us to observe - quite precisely - some interesting variations on the theme. The paper will close with some features of the approach that bother me. This will allow me to then illustrate the essential role of philosophical interpretation in these matters and suggest a necessary condition for the meaningfulness of quibbles about semantics.

### Stanislav Speranski (St Petersberg) - Benjamin Meaker Visiting Professorship lectures

*Measuring complexity of Kripkean truth predicates*

*On the discovery of incomputability and unsolvability*

*A constructive realizability interpretation for Hintikka's independence-friendly first-order logic*(joint work with S.P. Odintsov and I.Yu. Shevchenko)

Date | Speaker | Title/Subject |
---|---|---|

21 October 2016 |
Johannes Stern |
The Sky is the Limit: Reconsidering the Equivalence Scheme |

28 October 2016 |
Sam Roberts |
Modal structuralism and the access problem |

11 November 2016 |
Catrin Campbell-Moore |
Non-classical probabilities |

18 November 2016 |
Max Jones |
Numerical Perception and the Access Problem |

25 November 2016 |
Dan Saattrup Nielsen |
Determinacy of games |

2 December 2016 |
Alex Jones |
What is deflationism about truth? |

9 December 2016 |
Yang Liu |
A Simpler and More Realistic Subjective Decision Theory |

16 December 2016 |
Johannes Stern |
The Mind Cannot be Mechanized |

## Abstracts

### Johannes Stern (Bristol) - 'The Sky is the Limit: Reconsidering the Equivalence Scheme'

In this talk we reconsider the role of the Equivalence schema against the backdrop of the paradoxes. The most prominent reaction to the paradoxes within the boundaries of classical logic is to restrict the Equivalence scheme to a class of permissible instances. We argue that this strategy is not without problems and that it might be preferable to give up the Equivalence scheme altogether and seek for weaker principles of truth. To this end we propose a criterion for when such weaker principles of truth aptly characterize the notion of truth. The guiding idea will be that the non-naive truth predicate should be maximally truthlike.

### Sam Roberts (Bristol) - 'Modal structuralism and the access problem'

In this talk, I will look at modal structuralism -- the view that mathematics is about possible non-abstract structures. The main consideration in its favour is that it might solve the access problem -- the problem of explaining how we come to know all of mathematical facts that we do. I will try to get clear on the extent to which modal structuralism solves the access problem by investigating the presuppositions underlying it.

### Catrin Campbell-Moore (Bristol) - 'Non-classical probabilities'

This talk is an introduction to non-classical probabilities. When one typically presents the probability axioms, as they apply to sentences, they embed assumptions of classical logic; for example they require that the probability of Pv¬P is 1. In this talk we present an overview of non-classical probabilities which say what the probabilities should look like if these assumptions of classical logic aren't in play but instead there is some non-classical logic in the background. We will particularly focus on the case of supervaluational probabilities and explain how they connect to a model of belief that has been very popular in recent years: that of imprecise probabilities where an agent's belief state is modeled by a set of probability functions. Time permitting we will also present some connections with epistemic utility arguments for rationality constraints on agents in a non-classical framework.

### Max Jones (Bristol) - 'Numerical Perception and the Access Problem'

Over the last twenty years a wealth of evidence from the cognitive sciences has emerged that suggests that humans (and a wide range of other species) possess the capacity to perceive numerical properties or numerosities. However, within the philosophy of mathematics the notion that we perceive number is relatively unpopular. Those who support the idea that perception provides us with access to mathematical content, such as (early) Maddy, Kitcher, and Resnik, take this to provide a response to Benacerraf's access problem, which supports some form of realism. I'll briefly present some of the evidence for numerical perception and discuss it's impact on a more generalised version of Benacerraf's access problem, before arguing that, while numerical perception fails to support a realist solution, it may still have significant consequences for the metaphysics of number.

### Dan Saattrup Nielsen (Bristol) - 'Determinacy of games'

This is going to be a brief glimpse of how game theory can affect logic and Mathematics, where determinacy of a game is when there exists a winning strategy for one of the players in the game. I will present determinacy in a syntactical way as a special case of de Morgan's law in a certain infinitary logic, explore how much determinacy our axioms of set theory allows us to prove and show a sample application.

### Alex Jones (Bristol) - What is deflationism about truth?

Deflationism about truth has seen many advocates in recent years, those who argue that truth is insubstantial. These philosophers agree on this broad point, but still hold different theories of truth. In my talk I shall be looking at the question of what it is that these theories have in common, what makes a theory of truth deflationary/insubstantial. I shall examine some proposals that have been put forward and argue that these are deficient. I shall then introduce my own proposal, making use of the metaphysical notion of grounds, which at least improves over these deficiencies and I will argue is the correct way of deciding whether a theory of truth is deflationary or not. This will be heavy on philosophy, but lacking in formal technical details, so should be accessible to all!

### Yang Liu (Cambridge) - '**A Simpler and More Realistic Subjective Decision Theory'**

In his seminal work “the Foundations of Statistics,” Savage put forward a theory of subjective probabilities. The theory is based on a well-developed axiomatic system of rational decision making. In establishing this system of decision making, additional problematic assumptions are however required. First, there is a Boolean algebra of events on which subjective probabilities are defined. Savage's proof requires that this algebra be a σ-algebra. However, on Savage's view, one should not require the probability to be σ-additive. He, therefore, finds the insistence on a σ-algebra peculiar and unsatisfactory. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every given consequence there exists a constant act which has that consequence in every state. This assumption is known to be highly counterintuitive. The paper on which this talk is based includes two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the plausible, much weaker assumption that there are at least two non-equivalent constant acts.

In this talk, I will first provide an overview of Savage’s theory of expected utilities, I will then outline the new technique of tripartition trees we developed in the paper which leads to the definition of quantitative probabilities without the σ-algebra assumption. During the talk, I will also discuss the notion of "idealized agent" that underlies Savage's approach and argue that our simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent.

The talk is based on a joint work with Professor Haim Gaifman.

### Johannes Stern (Bristol) - 'The Mind Cannot be Mechanized'

Gödel's disjunction is the famous thesis that either the human mind cannot be mechanized or that there exist absolutely undecidable statements. Authors such as Lucas and Penrose have brought forward arguments purporting to show the first disjunct, namely, that the human mind cannot be mechanized. In this talk I shall focus on one particuar argument by Penrose to this effect, for which Peter Koellner has recently proposed a reconstruction using a self-applicable truth and a self-applicable absolute provability predicate. We investigate whether there are reasonable theories of truth and absolute provability in which Penrose's argument can be carried out and whether the addition of the truth predicate has any interesting philosophical consequences.

Date | Speaker | Title/Subject |
---|---|---|

28 June 2016 | Leon Horsten | The prime in the street |

21 June 2016 | Luca Oliva | Kantian Intuitionism |

17 May 2016 | Alex Jones | Semantic Theories and Plural Conceptions of Truth |

3 May 2016 | Stuart Presnell, James Ladyman | The Philosophy of Homotopy Type Theory: Identity, Intensionality and Univalence (Part III) |

26 April 2016 | Alf Coles | A relational view of the concept of number; pedagogical, neuroacientific and philosophical considerations |

15 March 2016 | Snezana Lawrence | Mathematicians and their Gods |

1 March 2016 | Sam Roberts | A strong reflection principle |

23 February 2016 | Stuart Presnell, James Ladyman | The Philosophy of Homotopy Type Theory: Identity, Intensionality and Univalence (Part II) |

12 January 2016 | Monika Gruber | Is there pragmatic encroachment on epistemic notions? |

## Abstracts

### Leon Horsten (Bristol) - 'The prime in the street'

In a few articles and a book in the early 1980s, Kit Fine proposed a theory for reasoning with arbitrary objects. The aim of my project is to develop a theory of arbitrary objects that has more and deeper structure than Fine's and has wider and more obvious applications. In the proposed seminar, I want to try to take the first few steps in this direction.

### Luca Oliva (Houston) - 'Kantian Intuitionism'

According to the ‘Hintikka-Parsons argument’, Kantian Intuitions (KI) behave like variables, for instance, in first-order quantificational logic. I will defend this thesis by showing 1) the mereological nature of KI, 2) their epistemic role in synthetic and analytic claims, which I will reduce to the basic class relations of inclusion and exclusion, and 3) the shared features of KI and mathematical induction, like the logical operations of generalization (EG and UG, respectively) and proving processes based on variables.

### Alex Jones (Bristol) - 'Semantic Theories and Plural Conceptions of Truth'

I will be presenting my thoughts in process on how the behaviour of semantic theories of truth for arithmetic supports a plural conception of the nature of truth. I will run through theories of plural conceptions of truth, and their philosophical features, and then look at how theories of truth for arithmetic might support this, at least for arithmetic.

### Alf Coles (Bristol) - 'A relational view of the concept of number; pedagogical, neuroacientific and philosophical considerations'

In this seminar, I will move from pedagogical to neuroscientific to philosophical considerations of the concept of 'number'. Two sources of evidence converge to bring into question the orthodoxy that learning mathematics entails a movement from the concrete to abstract. The first source comes from reflection on commonalities across the pedagogy of three of the great mathematics educators of the twentieth century (Caleb Gattegno, Vasily Davydov and Bob Davis). The second source is recent neuroscientific research into number sense and the previously unacknowledged role of ordinality (cf the work of Ian Lyons). Drawing these sources together suggests the potential, for children's learning, of approaching number as a relation, which then leads to philosophical implications for what numbers are.

### Snezana Lawrence (Bath) - 'Mathematicians and their Gods'

This talk will look at instances where mathematicians, either as individuals, or in groups, looked at the religious or theological arguments to make sense of their work, and of mathematics. Snezana will present a few such instances, and discuss in particular the case of the history of the fourth dimension.

### Sam Roberts (Bristol) - 'A strong reflection principle'

In this talk I will introduce a new reflection principle and argue that it improves substantially on previously formulated reflection principles.

### James Ladyman, Stuart Presnell (Bristol) - 'The Philosophy of Homotopy Type Theory: Identity, Intensionality and Univalence'

In this talk we will explain the basic features of HoTT which is a proposed new foundation for logic and mathematics. We will explain the interpretation of types, the unusual features of identity types that give the theory much of its power and interest, and the correct understanding of the content, role and justification of the univalence axiom. We will sketch some connections with structuralism in the philosophy of mathematics and applications to the foundations of physics.

### Monika Gruber (Bristol) - 'Is there pragmatic encroachment on epistemic notions?'

Pragmatic encroachment is the idea that epistemic notions, like belief, knowledge, judgement, or desire, are not “purely epistemic”: they depend upon other, non-epistemic factors, for example, on their relevance to action. Philosophers propounding pragmatic encroachment hold that epistemic notions by themselves have a pragmatic dimension, and hence are in a way “practical”.

In my talk, I will argue that there is, in fact, no such thing as pragmatic encroachment on epistemic notions and that the arguments its proponents hold are very off the track.

Date | Speaker | Title/Subject |
---|---|---|

15 December 2015 | James Ladyman, Stuart Presnell | The Philosophy of Homotopy Type Theory: Identity, Intensionality and Univalence |

8 December 2015 | Aaron Guthrie | Can Accuracy Accounts of Probabilism Explain Truthlikeness? |

1 December 2015 | Can Baskent | Game Theoretical Semantics for Some Paraconsistent Logics |

24 November 2015 | Sam Pollock | The problem of unambiguously communicating/understanding non-algrebraic mathematical theories: Just a special case of Kripkenstein? |

10 November 2015 | Alex Jones | An Introduction to Formal Truth |

27 October 2015 | Michael Shanks | Brouwer, Intuitionism and Society |

20 October 2015 | Leon Horsten | The Logic of Truth |

13 October 2015 | Oliver Tatton-Brown | Frege and the Neo-Fregeans |

29 September 2015 | Monika Gruber | Ramsey on truth, belief and success |

## Abstracts

### James Ladyman, Stuart Presnell (Bristol) - 'The Philosophy of Homotopy Type Theory: Identity, Intensionality and Univalence'

In this talk we will explain the basic features of HoTT which is a proposed new foundation for logic and mathematics. We will explain the interpretation of types, the unusual features of identity types that give the theory much of its power and interest, and the correct understanding of the content, role and justification of the univalence axiom. We will sketch some connections with structuralism in the philosophy of mathematics and applications to the foundations of physics.

### Aaron Guthrie (Bristol) - 'Can Accuracy Accounts of Probabilism Explain Truthlikeness?'

Some false beliefs are better than others; false beliefs can be more or less truthlike. Accuracy accounts of probabilism seem in a good position to explain truthlikeness, as they give orderings on beliefs at worlds (or at least we can interpret them as doing such). Joyce has pointed out, however, that there are problems with doing such, and suggested a solution. I will discuss his solution and the problems it encounters.

### Can Baskent (Bath) - 'Game Theoretical Semantics for Some Paraconsistent Logics'

In this talk, I will give a Hintikka-style game theoretical semantics for a variety of paraconsistent and non-classical logics. I will discuss Priest’s Logic of Paradox, Dunn’s First-Degree Entailment, Routleys’ Relevant Logics, McCall’s Connexive Logic and Belnap’s Four-Valued Logic. I will underline how non-classical logics require different verification games and how different logics require different game theoretical conditions in such games.

### Sam Pollock (Bristol) - 'The problem of unambiguously communicating/understanding non-algrebraic mathematical theories: Just a special case of Kripkenstein?'

Philosophers have made efforts in recent years either to secure or to explain unambiguous communication and/or unambiguous "intuiting" of certain mathematical theories. Whilst some, like Shapiro, give an answer in terms of categoricity to the purely explanatory question, others have looked also at the sceptical question and have given answers in terms of open-ended schemas, eschewing model theory. It looks as though we can interpret these sceptical and explanatory problems for non-algebraic theories as a special case of the famous Kripkenstein problem from philosophy of language, concerning private languages and rule-following. I will give an overview of the Kripkenstein argument and show how we can frame a handful of arguments about the uniqueness of the natural numbers/sets as instances of Kripkenstein. Depending on time, I'd like to explore what consequences this interpretation has if one finds Kripkenstein convincing -- what does it mean for Shapiro's, Parsons' and Martin's arguments for our ability to uniquely communicate/intuit those mathematical subject matters?

This talk will be pretty sketchy, and won't assume any prior knowledge. It will *not* go into matters of how best to interpret Wittgenstein and Kripke (that is a whole other can of worms for another day!).

### Alex Jones (Bristol) - 'An Introduction to Formal Truth'

This talk will be an introduction to the technical background of my PhD proposal. I will introduce first order Peano Arithmetic and then look at some theories of truth which have been defined over the theory and key related theorems. I aim for this to be approachable and informal and will intersperse philosophical questions that I am interested in exploring throughout.

### Michael Shanks (Bristol) - 'Brouwer, Intuitionism and Society'

This talk will present a broad introduction to LEJ Brouwer and Intuitionism in general. I will first give an overview of his life and some of the distinguishing features of Intuitionist mathematics before offering some remarks about the nature of Intuisionist mathematics as an activity. This talk is largely based on material from my Undergraduate final year project covering the same topic.

### Leon Horsten (Bristol) - 'The Logic of Truth'

I want to discuss recent results by Graham Leigh (concerning disquotational truth and reflection) and by Carlo Nicolai (concerning the inner logic of KF). The question I will be addressing is whether these results suggest that certain compositional theories of truth are in some sense "complete".

### Oliver Tatton-Brown (Bristol) - 'Frege and the Neo-Fregeans'

I will discuss the merits of Neo-Fregeanism (of the variety espoused by Wright and Hale) as an interpretation of Frege. This will hopefully serve as an introduction to Frege’s position in the Grundlagen, and the Neo-Fregean point of view, for those not familiar with these. If there’s time I may also discuss other aspects of the Neo-Fregean point of view, such as their assumption of mathematical realism and the Julius Caesar problem.

### Monika Gruber (Bristol) - 'Ramsey on truth, belief and success.'

Regarding truth, Ramsey is best known for a redundancy theory he laid down in "Facts and Propositions” (1927), and which has been ever since falsely attributed to him. This theory became an inspiration for modern deflationary truth theories. Moreover, on Ramsey's account it is possible to bring into harmony the deflationary attitudes confined within his ideas with his correspondence theoretical intuitions, thus constructing a new theory of truth. However, a definition of truth was not what Ramsey was after. His goal was to present a logical analysis of the terms “judgement” or “belief" which did not presuppose the concept of truth, for as he said “if we have analysed judgement we have solved the problem of truth”.

I will discuss Ramsey’s attitude towards truth and the role played here by the notion of “belief”. I will also consider Ramsey’s Principle and its role in his truth theory as well as its impact on decision theory and success semantics.

Date | Speaker | Title/Subject |
---|---|---|

27 January 2015 | Mona Simion, Sam Roberts, Markus Eronen, Sam Pollock, Stefan Buijsman | Bristol-Leuven Workshop |

3 February 2015 | Peter Koepke | Natural Formalism |

10 February 2015 | Alexander Jones | Satisfaction Classes and Deflationary Truth |

17 February 2015 | Oliver Tatton-Brown | Counting without numbers |

17 March 2015 | Andrew Brooke-Taylor | Categoricity in AECs from large cardinals via category theory |

24 March 2015 | Can Baskent | Epistemic Game Theory and Paraconsistency |

21 April 2015 | Ali Enayat | Tarskian Satisfaction Predicates, Revisited |

26 May 2015 | Michael Sheard |

## Abstracts

### Bristol-Leuven Workshop

### Peter Koepke- 'Natural Formalism' (16.30 Howard House)

The language of mathematics as found in textbooks and research articles is a variant of natural language which admits symbolic terms and formulas within argumentative natural language contexts. As it is written to be unambiguously understandable by experts it is susceptible to definite translations into (first-order) formal statements by standard (computer-)linguistic methods. Conversely those methods determine a class of accepted statement and hence implement a controlled natural mathematical language that can be seen as an enriched formal language.

Formal mathematics, which aims at complete formalisations of mathematical statements and proofs, has recently seen considerable advances using powerful software like automatic theorem provers for bridging proof steps. There is now a large selection of computer-checked proofs, including major results like the Four-Colour Theorem or the Feit-Thompson Theorem. Computer proof systems can be viewed as implementing strong formal calculi which allow sophisticated deductions known from ordinary mathematical proofs.

The combination of controlled natural language and argumention will lead to rich natural logics, which are completely formal extensions of first-order logic or some variant, but which accept proofs in the familiar language of mathematics.

In my talk I want to discuss some aspects of that development:

- relevant systems in formal mathematics and linguistics;

- associated work in the foundations of mathematics;

- implications for the practice of mathematics;

- implications for mathematical formalism.

### Alexander Jones - 'Satisfaction Classes and Deflationary Truth'

Satisfaction classes are an axiomatic definition of truth for nonstandard sentences of arithmetic. I will run through the definition of a satisfaction class and argue that we should be interested in the truth of these sentences and that a satisfaction class is the minimum such structure we should consider. I will then discuss the implication that this has for deflationist conceptions of truth and the questions that it raises.

### Oliver Tatton-Brown - 'Counting without numbers'

I will discuss a way of phrasing statements of finite cardinality (including additions and multiplications of cardinality) that does not require any objects, numbers, to do the counting (actually I'll discuss two).

### Andrew Brooke-Taylor - 'Categoricity in AECs from large cardinals via category theory'

Abstract elementary classes are a framework proposed by Shelah for generalising classical model theory to non-first order cases. A key test question for whether this is a good generalisation was whether the classical categoricity results generalise; it turns out that under suitable large cardinal hypotheses they (more or less) do. In joint work with Jiri Rosicky, we have shown how to reduce the large cardinal assumption from the one originally used, and notably our proof goes by way of category theory.

### Can Baskent- 'Epistemic Game Theory and Paraconsistency'

A two-person Russell’s Paradox can be given within the context of epistemic game theory. In this talk, I will discuss the well-known paradox, called the Brandenburger - Keisler paradox, and suggest an inconsistency friendly approach, which can allow us to reason with inconsistencies in a non-trivial fashion. At the end, I will construct a counter-model that can satisfy the “paradoxical” sentence.

### Ali Enayat - 'Tarskian Satisfaction Predicates, Revisited' (14.30 4th Floor Seminar Room, Howard House)

This talk reports on joint work with Albert Visser. Suppose we start with a foundational 'base theory' B formulated in a language L (such as B = Peano arithmetic, or B = Zermelo-Fraenkel theory), and we extend B to a new theory B+ := B ∪ Σ, where Σ is a set of sentences formulated in the language L+ := L ∪ {S}, such that Σ describes certain prominent features of a Tarskian satisfaction predicate. I will give an overview of the status of our current knowledge of the relationship between B and B+ in connection with the following four questions (for various canonical choices of B and Σ) :

1. Is B+ semantically conservative over B? In other words, does every model of B expand to a model of B+?

2. Is B+ syntactically conservative over B? In other words, if some L-sentence φ is provabe from B+, then is φ also provable from B?

3. Is B+ interpretable in B?

4. What type of speed-up (if any) does B+ have over B?

Date | Speaker | Title/Subject |
---|---|---|

21 October 2014 | Leon Horsten | Revising the concept of truth |

28 October 2014 | Hazel Brickhill | Vagueness and Continuity: how gnomes can help us understand the Sorites paradox |

4 November 2014 | Pavel Janda | Accuracy-Difficulty of a Single-Number Credence Representation in Belnap’s Four-Valued Logic |

11 November 2014 | Sam Roberts | Reinhardt's view of the universe of sets |

25 November 2014 | Kate Hodesdon | Discernibility Relations |

9 December 2014 | Kentaro Fujimoto | Second-order arithmetic and set theory, and theories of truth over arithmetic and set theory |

16 December 2014 | Philip Welch | Informal Categoricity |

## Abstracts:

### Leon Horsten - 'Revising the concept of truth'

I will introduce the basic ideas of the revision theory of truth by Gupta, Belnap, Herzberger. Then I will hint at a possible new revision idea, where the revision process is given not as a linear ordering but as a tree.

### Hazel Brickhill - 'Vagueness and Continuity: how gnomes can help us understand the Sorites paradox'

When looking at the phenomena of vagueness there is so much to think about- language, meaning, epistemology, philosophy of mind etc. So I construct a simple toy world, with gnomes as agents, including things I thought key to the phenomena of vagueness (continuity in the underlying properties, restrictions on epistemic access, communication, transience). As we build up this model, something very like vagueness in human languages emerges. The insights gained here can then be used to look more closely at the sorities paradox, in particular the gnomes shed light on the notion of being "indistinguishable". The result of this (if you are convinced!) is a solution to the sorites, but perhaps surprisingly we don't come much closer to understanding the logic of vague predicates.

There is nothing very complicated in this talk, I will introduce the sorites paradox and the maths involved is just intervals on the real line. It should be of fairly broad interest.

View the handout (PDF, 131kB) for this talk.

### Pavel Janda - 'Accuracy-Difficulty of a Single-Number Credence Representation in Belnap’s Four-Valued Logic'

An alternative philosophical approach to the representation of uncertain doxastic states is suggested, by considering how to model an agent who is concerned about the accuracy of her credences in Belnap’s four-valued logic B4. The paper has three parts. The first part motivates the whole project by showing that the single-number representation of credences is not appropriate for measuring an agent’s accuracy in B4. In the second part, we will introduce ordered pairs to represent an agent’s uncertain doxastic states, and we will show how they can solve the problem faced by the single-number representation. In the third part, we will show how Joyce’s idea of non-pragmatic vindication of probabilism works for ordered pairs in the classical two-valued and non-classical logical systems.

### Sam Roberts - 'Reinhardt's view of the universe of sets'

I will introduce a view in the philosophy of set theory due to William Reinhardt. It is based on the idea that the universe of sets is inherently potential, and can be used to justify very strong set-theoretic large cardinal axioms. I will discuss the significance of this view for some debates in the philosophy of set theory.

### Kate Hodesdon - 'Discernibility Relations'

I will discuss formal methods for discerning between uncountably many objects with a countable language, building on recent work of James Ladyman, Øystein Linnebo and Richard Pettigrew. In particular, I will show how stability theory in model theory provides the resources to characterize theories in which this is possible, and discuss the limitations of the stability theoretic approach.

### Kentaro Fujimoto - 'Second-order arithmetic and set theory, and theories of truth over arithmetic and set theory'

There is a certain strong disanalogy between second-order theories of numbers and those of sets. A parallel disanalogy exists between theories of truth over arithmetic and set theory. I'll explain some of the differences. If time permits, I'll touch upon some issues related to deflationism of truth.

### Philip Welch - 'Informal Categoricity'

We outline an extension of Martin’s view of a conceptual realism, to a Cantorian realm of absolute infinities, focussing on, essentially, the Zermelian categoricity arguments.

Date | Speaker | Title/Subject |
---|---|---|

8 October 2013 | Benedict Eastaugh | A tutorial on countable coded omega-models |

15 October 2013 | Benedict Eastaugh | Computable Entailment |

22 October 2013 | Aadil Kurji | Formality |

29 October 2013 | Leon Horsten | Description terms for languages with intensional operators |

5 November 2013 | Aaron Guthrie | |

12 November 2013 | Benedict Eastaugh | |

26 November 2013 | Samantha Pollock | Meadows (2013), 'What can a categoricity theorem tell us?' |

10 December 2013 | Richard Pettigrew | |

17 December 2013 | Ben Eva |