This paper explores trivalent truth conditions for indicative conditionals, examining the "defective" truth table proposed by de Finetti (1936) and Reichenbach (1935, 1944). On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti's table come with some problems, and highlight two ways out of the predicament: one pairs de Finetti's conditional (DF) with validity as the preservation of non-false values (TT-validity), but at the expense of Modus Ponens; the other modifies de Finetti's table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti's table (CC) proposed by Cooper (

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by (Beall and Murzi, Journal of Philosophy, 3 pp. 143-65 (2013)) that yield a version of Curry's paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion of logical consequence: we study the corresponding theory of validity (and some of its variants) by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. (strong Kleene version). Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence.

Since Saul Kripke's influential work in the 1970s, the revisionary approach to semantic paradox-the idea that semantic paradoxes must be solved by weakening classical logic-has been increasingly popular. In this paper, we present a new revenge argument to the effect that the main revisionary approaches breed new paradoxes that they are unable to block. A multiset is just like a set, except that repetitions count. We use { } as brackets for sets, and [ ] as brackets for multisets. Thus, {w, c, c} and {w, c} are the same set, but [w, c, c] and [w, c] are distinct multisets. We omit brackets from multisets in sequents-e.g. writing w , wrc instead of [w, w] rc. 9 This suffices for the purposes of this paper: the results in section 5 only require propositional logical rules. For simplicity, we have opted for a single-conclusion natural deduction calculus in sequent-style in which structural rules are explicitly formulated. 10 A double line indicates that a rule can be read in both directions.

Beall and Murzi (J Philos 110(3):143-165, 2013) introduce an objectlinguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules (over sufficiently expressive base theories). As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field (Notre Dame J Form Log 58(1):1-19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a non-classical solution to the semantic paradoxes need not be substructural. The aim of this paper is to respond to Field's objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi's principles: grounded validity. The notion, first introduced by Nicolai and Rossi (J Philos Log. doi:10.1007/s10992-017-9438-x, 2017), is a generalisation of Kripke's notion of grounded truth (J Philos 72:690-716, 1975), and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic (nor, more generally, of a revisionary approach to semantic paradox), we take the notion of naïve