Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...) From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties. (shrink)
This is part one of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic and show that (...) the theory is a proof-theoretically conservative extension of the theory PT of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground. (shrink)
This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is (...) a proof-theoretically conservative extension of the ramified theory of positive truth up to. (shrink)
We develop an exact truthmaker semantics for permission and obligation. The idea is that with every singular act, we associate a sphere of permissions and a sphere of requirements: the acts that are rendered permissible and the acts that are rendered required by the act. We propose the following clauses for permissions and obligations: -/- - a singular act is an exact truthmaker of Pφ iff every exact truthmaker of φ is in the sphere of permissibility of the act, and (...) -/- - a singular act is an exact truthmaker of Oφ iff some exact truthmaker of φ is in the sphere of requirements of the act. -/- We show that this semantics is hyperintensional, and that it can deal with some of the so-called paradoxes of deontic logic in a natural way. Finally, we give a sound and complete axiomatization of the semantics. (shrink)
We show that any predicational theory of partial ground that extends a standard theory of syntax and that proves some commonly accepted principles for partial ground is inconsistent. We suggest a way to obtain a consistent predicational theory of ground.
We present a series of proof systems for exact entailment (i.e. relevant truthmaker preservation from premises to conclusion) and prove soundness and completeness. Using the proof systems, we observe that exact entailment is not only hyperintensional in the sense of Cresswell but also in the sense recently proposed by Odintsov and Wansing.