Pedro Teixeira Yago

In this paper, we offer an account of a paradox analogous to Fitch's paradox of knowability, elaborated around knowledge of necessary truths - such as mathematical or logical truths. In particular, we highlight at which semantic levels the paradox does and does not rise, and explain why that happens. The account employs a novel modal operator and language, which present some distinctive semantic features, such as being unable to characterise many frame properties characterisable in normal modal logic, a fact which may be used in interesting ways to study the logics of the proposed language. Following recent studies, this work may be seen as a case study of semantically insensitive non-normal modal logics.

In this paper we investigate cumulative hierarchies of functions on structures, or cumulative powers, and study their properties. Particularly, we show how they extend the preservation phenomena of reduced powers, direct powers and ultrapowers by offering a characterization of the fragment of first-order theory it preserves, and elucidate the connections between the three sorts of constructions. More precisely, we show how both direct powers and ultrapowers may be obtained from cumulative powers as quotients by appropriate equivalence relations. We address how embeddability lifts from generating structures to their cumulative powers, direct powers and ultrapowers, and under what conditions ultrapowers embed into corresponding cumulative powers or direct powers. We further offer an application of the framework to show a straightforward way of constructing Conway's surreal field.

Arbitrary objects play an important role in both ordinary and mathematical reasoning. The reason for that is their distinct behavior: an arbitrary object presents those properties common to all individual objects in its range - what, following Kit Fine, has become known as the "Principle of Generic Attribution", or simply "PGA". Many philosophers have argued this precise quality to be contradictory, therefore rejecting arbitrary objects altogether - of which the most famous is George Berkeley's argument against Locke's general ideas. However, all the arguments that have been offered assume rejecting properties of an arbitrary object to be expressible in terms of asserting something of it - a unilateral setting. This paper has two main purposes: first, to show how, in the propositional bilateral system of Weak Rejectivist Logic, introduced by Luca Incurvati and Julian Schloder (2017), the PGA does not fall prey to the counter arguments which rise in tension with the nature of the logical connectives; second, to argue understanding sentences about arbitrary objects in terms of assertions which preserve (or not) generality better frames our reasoning about them, and is reflected by the Weak Rejectivist Logic. Then, following the relation between this propositional system and arbitrary objects, a further goal of this paper is to offer a characterization result for the Weak Rejectivist Logic with respect to first-order logic.

Despite their controversial ontological status, the discussion on arbitrary objects has been reignited in recent years. According to the supporting views, they present interesting and unique qualities. Among those, two define their nature: their assuming of values, and the way in which they present properties. Leon Horsten has advanced a particular view on arbitrary objects which thoroughly describes the earlier, arguing they assume values according to a sui generis modality, which he calls afthairetic. In this paper, we offer a general method for defining the minimal system of this modality for any given first-order theory, and possible extensions of it that incorporate further aspects of Horsten’s account. The minimal system presents an unconventional inference rule, which deals with two different notions of derivability. For this reason and the failure of the Necessitation rule, in its full generality, the resulting system is non-normal. Then, we provide conditional soundness and completeness results for the minimal system and its extensions.