Under the standard modal explication of consequence, a conclusion is a consequence of some premises just in case necessarily, if the latter are true, so is the former. Notoriously, this explication yields some results that at first glance are counter-intuitive. In particular, a necessary truth is a consequence of arbitrary premises, and premises that cannot all be true together entail arbitrary conclusions. In his paper ‘On Ground and Consequence’ (Synthese, 2021), Benjamin Schnieder introduces a novel notion of web consequence, defined (...) on the basis of the concept of ground, which, he argues, fits our intuitive conception better. Building on his idea, the present paper examines the concept of web consequence in more detail. In particular, I provide three alternative, and simpler semantic characterizations of Schnieder’s propositional logic of web consequence, two within a form of truthmaker semantics, one within a many-valued setting. I then consider some natural variations on that logic and establish their connections to well-known subclassical logics such as FDE, $$\hbox {K}_3$$ K 3, and LP. Finally, I provide sound and complete tableaux-based proof systems for each of the logics of web consequence so obtained. (shrink)

This paper clarifies and defends the global approach to defining logical validity for meta- and higher-level inferences. This is contrary to an emerging consensus in favour of local validity. Prevalent recent arguments claim that global validity is either superfluous in virtue of collapsing into local, or else untenable because it overgenerates validities, compromises the formality of logic, or breaks symmetry with regular validity. Accordingly, the literature on higher inferential logic has come to focus almost exclusively on local validity. Many key (...) philosophical takeaways that have since been drawn from the field are effectively based on local monism: the view that local is the One True Validity Criterion. The present paper argues that this is a mistake, resulting largely from an imprecise understanding of global validity. I untangle some common confusions about how global generalizes beyond the metalevel, and submit that its proper formulation is schematic. These clarifications help dispel each of the aforementioned arguments against global, leaving local monism without any justification. Finally, I outline how much of the philosophical literature around higher-level inferences has been built on local monism, and sketch out the sweeping consequences of rejecting it. (shrink)

The cut-free validity theory $$\textsf{STV}$$ proposed by Barrio, Rosenblatt, and Tajer suffers from incompleteness with respect to its object language validity predicate. The validity predicate of $$\textsf{STV}$$ fails in validating some valid inferences of its underlying logic, the Strict Tolerant logic $$\textsf{ST}$$. In this paper, we will present the non-normal modal logic $$\textsf{ST}^{\Box \Diamond }$$ whose modalities $$\Box $$ and $$\Diamond $$ capture the tautologies/valid inferences and the consistent formulas of the logic $$\textsf{ST}$$, respectively. We show that $$\textsf{ST}^{\Box \Diamond }$$ (...) does not trivialize when extended with self-referential devices. We also show that such a solution poses a dilemma. If we extend $$\textsf{ST}^{\Box \Diamond }$$ in such a way that it allows iterated modal formulas among its theorems, then the resulting interpretation of $$\Box $$ as validity implies that metametainferences of $$\textsf{ST}$$ behave like classical logic. On the other hand, if we allow these modalities to receive intermediate truth values, we obtain formulas incompatible with the proposed reading of $$\Box $$. (shrink)