# Blog Archives

These are the items posted in this seminar, currently ordered by their post-date, rather than by the event date. We will create improved views in the future. In the meantime, please click on the Seminar menu item above to find the page associated with this seminar, which does have a more useful view order.# Justification Logics

Gödel inaugurated a project of finding an arithmetic semantics for intuitionistic logic, but did not complete it. It was finished by Sergei Artemov, in the 1990’s. As part of this work, Artemov introduced the first *justification logic*, LP, (standing for *logic of proofs*). This is a propositional modal-like logic, with an infinite family of *proof *or *justification* *terms*, and can be seen as an explicit version of the well-known modal logic S4. There is a possible world semantics for LP (due to me). Since then, many other justification logic/modal logic pairs have been investigated, and justification logic has become a subject of independent interest, going beyond the original connection with intuitionistic logic. It is now known that there are infinitely many justification logics, but the exact extent of the family is not known. Justification logics are connected with their corresponding modal logics via *Realization Theorems*. A Realization Theorem connecting LP and S4 has a constructive proof, but there are other cases for which realization holds, but it is not known if a constructive proof exists. More recently, a first order version of LP has been developed, but I will not talk about it in detail. I will present a sketch of the basic propositional ideas.

# Plural Logic and Sensitivity to Order

David Nicolas, Institut Jean Nicod, Paris

Sentences that exhibit sensitivity to order (e.g. “John and Mary arrived at school in that order” and “Mary and John arrived at school in that order”) present a challenge for the standard formulation of plural logic. In response, some authors have advocated new versions of plural logic based on more fine-grained notions of plural reference, such as serial reference (Hewitt 2012) and articulated reference (Ben-Yami 2013). The aim of this work is to show that sensitivity to order should be accounted for without altering the standard formulation of plural logic. In particular, sensitivity to order does not call for a more fine-grained notion of plural reference. We point out that the phenomenon in question is quite broad and that current proposals are not equipped to deal with the full range of cases in which order plays a role. Then we develop an alternative, unified account, which locates the phenomenon not in the way in which plural terms can refer, but in the meaning of special expressions such as “in that order” and “respectively”.

# Does definiteness-of-truth follow from definiteness-of-objects?

This talk — a mix of mathematics and philosophy — concerns the extent to which we may infer definiteness of truth in a mathematical context from definiteness of the underlying objects and structure of that context. The philosophical analysis is based in part on the mathematical observation that the satisfaction relation for model-theoretic truth is less absolute than often supposed. Specifically, two models of set theory can have the same natural numbers and the same structure of arithmetic in common, yet disagree about whether a particular arithmetic sentence is true in that structure. In other words, two models can have the same arithmetic objects and the same formulas and sentences in the language of arithmetic, yet disagree on their corresponding theories of truth for those objects. Similarly, two models of set theory can have the same natural numbers, the same arithmetic structure, and the same arithmetic truth, yet disagree on their truths-about-truth, and so on at any desired level of the iterated truth-predicate hierarchy. These mathematical observations, for which I shall strive to give a very gentle proof in the talk (using only elementary classical methods), suggest that a philosophical commitment to the determinate nature of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure N = {0,1,2,…} itself, but rather seems to be an additional higher-order commitment requiring its own analysis and justification.

This work is based on my recent paper, Satisfaction is not absolute, joint with Ruizhi Yang (Fudan University, Shanghai).

# Strict Truthmaking Logic

The notion of strict truthmaking is central to the logic of truthmaking and to the project of giving semantics in terms of truthmakers. A strict truthmaker for A is a state α in virtue of which A is true, so that states with irrelevant parts do not count as strict truthmakers for A. The notion of strict truthmaking is therefore non-monotonic. This semantics produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the logic highly unusual and worth investigating on purely logical grounds. But the investigation of strict truthmaker logic also has interesting applications in metaphysics.

Strict truthmaking logic has received very little attention in the technical literature. Fine notes in passing that ‘the notion of exact [i.e., strict] consequence is of great interest in its own right’ (Fine 2013, 21), but says nothing further about the notion. In this talk, I’ll set out systems of formal semantics for strict truthmaking, which have parallels with Fine’s (2013) truthmaking semantics for intuitionistic logic. I’ll spend some time investigating the resulting notions of entailment, building up to some representation theorems. I’ll then give sequent-style proof systems, establish a number of results about them, and prove soundness and completeness results.

I’ll finish by discussing applications of these semantic systems to various metaphysical debates about truthmaking. One application concerns the status of Rodriguez-Pereyra’s conjunction and disjunction theses; another concerns Armstrong’s entailment thesis. I’ll argue that truthmaking semantics (including the strict semantics presented here) help us to systematise philosophical intuitions about these important metaphysical concepts.

# On the Methodology of Paraconsistent Logic

In this talk I will critically discuss some widely shared methodological assumptions about paraconsistent logic. I will argue that there exist several reasons not to consider classical logic as the reference logic for developing systems of paraconsistent logic and will suggest to weaken a certain maximality condition. Moreover, I will argue that the guiding motivation for the development of paraconsistent logics should be neither epistemological nor ontological, but informational, and I will, from this perspective, discuss the idea of “ex contradictione nihil sequitur”.