Fine-grained Mathematical Justifications
Jesse Alama

From a corpus of formalized mathematical knowledge (definitions, theorems, and proofs), one can extract fine-grained information about what principles are sufficient for certain linguistic and justificatory tasks: for a certain expressions to be well-defined, for a theorem to be a well-formed formula, and for a proof to be successful. That they are sufficient is clear: the proof checker has accepted a proof without error. We report on some initial experiments going the other way around, of computing necessary principles for mathematical knowledge. The task here is evidently of a different character than simply checking a proof. We report on our work on decomposing texts in the Mizar proof-checking system to extract suitable information for computing necessary principles.

An Empirical Perspective on Logical Constants
Denis Bonnay

The project of delineating a special class of logical constants is usually pursued from the perspective of "pure" philosophy of logic. In this respect, it is not clear whether being a logical constant constitue a natural kind. One might rather think that the joint characterization of logical consequence and of logical constancy is nothing but the output of a certain reflective equilibrium involving general theoretical considerations as well as particular judgments about validity.

However, from a broader linguistic and cognitive perspective, logical constants do seem to belong to a natural kind, namely the class of functional expressions in natural languages. Functional words are grammatical expressions that "glue" the different constituents of a sentence together and they share a wide number of linguistic and psycholinguistic properties (eg, functional categories are not productive, psycholinguistic evidence suggests that access to functional words is different from access to lexical words, etc.)

In this talk, my aim will be to clarify the connections between the logico-philosophical project of providing a principled chacterization of the class of logical constants and the empirical project in linguistics of studying functional words and functional categories as such. As a tentative attempt, I will suggest that the hypothesis of the innateness of grammatical notions and the hypothesis that (a generalized version of) permutation invariance is a distinctive property of logical constant provide mutual support to each other.

Multiplicative Quantifiers in First-Order Fuzzy Logics
Petr Cintula

Mathematical fuzzy logics can be viewed as special class of substructural logics, and the most prominent ones lack the rule of contraction. Therefore there are two conjunctions: “additive" ^ (also called lattice) and “multiplicative" & (also called residuated). Informally speaking, additive conjunction allows using either (or any) of its conjuncts as a premise for further inference, while multiplicative conjunction allows using both (or all) conjuncts. There is a long-standing problem of extending this distinction to first-order fuzzy logics, which should analogously contain an additive universal quantifier (any, ) and a multiplicative one (all, ): while the additive quantifier can easily be defined (semantically as the infimum in the lattice of truth values and axiomatically by following Hajek's original approach), the same cannot easily be done for the multiplicative quantifier. Formally, a multiplicative quantifier should satisfy the following natural conditions:
  1. from φ→ψ infer (Qx)φ → (Qx)ψ,
  2. from φ infer (Qx)φ,
  3. (Qx) φ → &{ φ(t) ; t in M} for each finite multiset M of terms.
We give several natural examples of multiplicative quantifiers, provide characterizations of the prominent ones, and show their relation to the notion of exponential, which can be viewed as special kind of truth stressers. We also try to motivate different multiplicative quantifiers by a series of (naive) linguistic examples, but the bulk of the work is mathematical.

Relevant Agents
Marta Bílková, Ondrej Majer, Michael Peliš, Greg Restall

In [4], Majer and Peliš proposed a relevant logic for epistemic agents, providing a novel extension of the relevant logic R with a distinctive epistemic modality K, which is at the one and the same time factive and an existential normal modal operator. The intended interpretation is that Kφ holds (relative to a situation s) if there is a resource available at s, confirming φ. In this article we expand the class of models to the broader class of egeneral epistemic framesf. With this generalisation we provide a sound and complete axiomatisation for the logic of general relevant epistemic frames. We also show, that each of the modal axioms characterises some natural subclasses of general frames.

A new approach to fuzzy logics with truth stressers and depressers
Francesc Esteva, Lluìs Godo, Carles Noguera

A number of papers have considered fuzzy logics with truth hedges, as unary connectives (vt and st, for very true and slightly true), which allow to stress or depress the truth value of any given proposition. The equivalent algebraic semantics for these logics turned out to be, in all cases, a variety, i.e. an equational class of algebras. This nice result was obtained at the cost of adding the axiom: vt(φ→ψ)→vt(φ)→vt(ψ) (and analogously for st). This amounts to a strong restriction for the algebraic semantics which has no natural interpretation. For instance, it implies that over Lukasiewicz logic the only possible non-Boolean function to interpret a depressing hedge is the identity function. In this talk we will generalize the approach of these previous works obtaining weaker fuzzy logics that will overcome this drawback. Given a core fuzzy logic L we define an expansion Lh with a new unary connective h defined by the following additional axioms in the case of a truth stresser:

(VTL1) hφ→φ

(VTL2) h1,

or the following axioms in the case of a truth depresser:

(STL1) φ→ hφ

(STL2) ~h0

and, in both cases, the following additional inference rule:

(MON) from (φ→ψ) or χ infer (hφ→hψ) or χ.

From this presentation one can easily prove that the logic is complete with respect to a semantics of linearly ordered algebras where hedges are interpreted as any subdiagonal (superdiagonal) monotonic function mapping the maximum (minimum) element to itself. Moreover, if L extends BL we can prove that the corresponding class of algebras is a variety. Finally we show these expansions with h preserve standard completeness properties, i.e. if L is complete with respect to chains defined over the real unit interval, then so it is Lh.

Towards an experimental philosophy of uncertain reasoning
Niki Pfeifer

Experimental philosophy is a recent trend in philosophy that applies empirical methods to investigate philosophical intuitions. The main research topics include intuitions on morality, consciousness, epistemology and causation. The aim of my talk is to extend the domain of experimental philosophy to uncertain reasoning. Specifically, I critically survey previous philosophical and empirical work on nonmonotonic reasoning and uncertain conditionals. I discuss how "armchair philosophy" and experimental work can fruitfully interact and illustrate my position with recent experimental results on how people interpret and reason with conditionals.

A Simulation Based Analysis of Logico-Probabilistic Reasoning Systems
Paul Thorn and Gerhard Schurz

Systems of logico-probabilistic (LP) reasoning characterize inference from conditional assertions that are taken (semantically) to express high conditional probabilities. There are several existent LP systems. These systems differ in the number and type of inferences they licence. An LP system that licenses a greater number of inferences offers the opportunity of deriving more true informative conclusions. But with this possible reward comes the risk of drawing more false conclusions. By means of computer simulations, we investigated four well known LP systems, systems O, P, Z and QC, with the goal of determining which system provides the best balance of reward versus risk. In this talk, we explain why each of the four systems (O, P, Z and QC) is a prima facie contender to be the correct prescriptive theory of LP reasoning. We then present data which suggests that (of the four systems) system Z has the best claim to be the correct prescriptive theory of LP reasoning, since it offers the best balance of reward versus risk.

Faithfulness in formal modelling
Sara Uckelman

There is a strong analogy between the process of mathematical modeling used in science and engineering and the process of modeling used in the construction of formal models for historical logical theories, and this can be contrasted with the use of philosophical modeling found more generally in main-stream philosophy, such as the use of thought experiments. Both processes are primarily descriptive, and only derivatively prescriptive, because what is being modeled are objective facts in the world, as opposed to, e.g., our pre-theoretic intuitions. The descriptive nature of the process gives rise to the question of faithfulness: When are we allowed to say that our logical model is a faithful representation of the historical logical theory?

We discuss two different ways that a model or a description can be faithful: It can be faithful to the content and it can be faithful to the context. We focus on faithfulness to content, and discuss two benchmarks that can be used to determine the degree of faithfulness of a model: its level of generation and what we call structural or procedural agreement. We illustrate this discussions with examples of both good and bad logical models of historical theories: The ontological argument of Anselm of Canterbury, the formalization of Aristotelian syllogistics in first-order logic, and models of medieval theories of obligationes.

Be tolerant about vagueness, because it is unavoidable!
Robert van Rooij

Vagueness is a pervasive feature of natural language. Although logicians don't like it, because it gives rise to (Sorites) paradox, there is hardly a term in natural language that is not vague. To save language from paradox, most logicians have proposed that natural language terms are not tolerant: the tolerance principle which states that `if x has property P and y is indistinguishable from x, y has to have property P as well' is giving up. But this is unfortunate, because the tolerance principle seems to be constitutive of what it means to be vague. In the first part of this talk it will be proposed that we should face the paradox by accepting tolerance, rather than simply avoiding it. A new logical treatment will be presented and defended.

In the second part of the talk we seek to explain why vagueness is such a pervasive feature of natural language in the first place. Making use of evolutionary game theory and a new stochastic equilibrium concept, it will be shown that vagueness is unavoidable once we take our agents to be realistic bounded rational agents.