The aim of the conference was to clarify foundational questions about logical consequence. Should the foundations be essentially model-theoretic, proof-theoretic, or some combination of the two, or is there a third way, e.g., deflationary? Is the extensional apparatus of models adequate to capture the necessity apparently essential to logical consequence? Can adequate constraints be placed on inferentialist accounts of consequence and the logical connectives to avoid trivialization? A further issue concerns the relata of the consequence relation: are they sentences, propositions, utterances, statements or states of affairs? Are they sets of such relata, or multi-sets, or sequences? Are they finite or can they be infinite? The conference brought together top researchers in the field to share their findings, reach conclusions, and provide a stimulus to further research. What follows are brief summaries of the work presented to the conference by our nine invited speakers.
JC Beall (UConn) argued that the lesson of Curry's paradox is that there is a distinction between normal and non-normal worlds in the semantics for implication. The fact that a 'normal worlds only' constant cannot be non-trivially added to his truth theory entails that, for Beall, validity cannot be understood as truth-preservation.
Josh Dever (Texas at Austin) discussed conditionality. Apparent problems with the Lewis-Stalnaker conditional were diagnosed by Dever as tracing back to its genesis as a variation on the strict conditional. He explored a proof-theoretic route to a better theory of conditionality by using gate-keeping policies on 'subproof permeability'.
Hartry Field (NYU) pursued the moral of his earlier argument to the effect that logical consequence cannot be necessary truth preservation. Instead, Field advances an elucidation of the concept of logical consequence in terms of the connection between valid arguments and subjective probabilities.
Michael Glanzberg (Berkeley) argued that logical consequence is not a relation in natural language. While there are close analogues between natural language entailment and logical consequence, the latter is a much narrower relation which results from an explicit process of abstraction from natural language entailment.
Hannes Leitgeb (Bristol) offered an account of how absolute beliefs can be defined in terms of subjective probability. The theory shed light on the role played by closure principles in formal epistemology, and outlined a probabilistic semantics for indicative conditionals that improves Ernest Adams’s account.
Vann McGee (MIT) developed a notion of categoricity for logical expressions. He explores formal frameworks where the use of a logical expression in inferences (i.e. its proof rules) uniquely determines a truth-conditional content.
Agustín Rayo (MIT) presented a theory of identity statements of the form ‘for x to be F just is …’ within a philosophy of mathematics which he describes as “platonist, anti-Tractarian trivialism”. This realist philosophy rejects the idea of a close correspondence between semantic structure and metaphysical reality, and claims that the statements of mathematics have trivial truth- and falsity-conditions. For example, for the number of Fs to be n just is for there to be exactly n Fs. Thus the existence of mathematical entities reduces to the existence of non-mathematical things.
Dag Westerståhl (Goteborg) talked about his joint work with Denis Bonnay on logical constants. They provide a new approach to demarcating logical constants relative to a language and a consequence relation, so-called consequence mining. Their theory turns the orthodox approach on its head by determining the class of logical constants by the class of consequences.
Robbie Williams (Leeds) tackled the normativity of logic. Approaching this as a question about why our beliefs should be closed under logical consequence, he co-opted some results of James Joyce’s to argue that such a policy dominates its rivals with respect to degree of accuracy (closeness to truth).
The Conference received financial support from the Analysis Trust, the British Logic Colloquium, the Mind Association, the Scots Philosophical Association and the University of St Andrews.