The book Das Interpretationsproblem der Formalisierten Zahlentheorie und ihre Formale Widerspruchsfreiheit by Erik Stenius published in 1952 contains a consistency proof for infinite ω-arithmetic based on a semantical interpretation. Despite the proof’s reference to semantics the truth definition is in fact equivalent to a syntactical derivability or reduction condition. Based on this reduction condition Stenius proves that the complexity of formulas in a derivation can be limited by the complexity of the conclusion. This independent result can also be proved by cut elimination for ω-arithmetic which was done by Schütte in 1951.

In this paper we interpret the syntactic reduction in Stenius’ work as a method for cut elimination based on invertibility of the logical rules. Through this interpretation the constructivity of Stenius’ proof becomes apparent. This improvement was explicitly requested from Stenius by Paul Bernays in private correspondence (In a letter from Bernays begun on the 19th of September 1952 (Stenius & Bernays, 1951–75)). Bernays, who took a deep interest in Stenius’ manuscript, applied the described method in a proof Herbrand’s theorem. In this paper we prove Herbrand’s theorem, as an application of Stenius’ work, based on lecture notes of Bernays (Bernays, 1961). The main result completely resolves Bernays’ suggestions for improvement by eliminating references to Stenius’ semantics and by showing the constructive nature of the proof. A comparison with Schütte’s cut elimination proof shows how Stenius’ simplification of the reduction of universal cut formulas, which in Schütte’s proof requires duplication and repositioning of the cuts, shifts the problematic case of reduction to implications.

Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened.

A deductive system for Lewis counterfactuals is presented, based directly on the influential generalisation of relational semantics through ternary similarity relations introduced by Lewis. This deductive system builds on a method of enriching the syntax of sequent calculus by labels for possible worlds. The resulting labelled sequent calculus is shown to be equivalent to the axiomatic system VC of Lewis. It is further shown to have the structural properties that are needed for an analytic proof system that supports root-first proof search. Completeness of the calculus is proved in a direct way, such that for any given sequent either a formal derivation or a countermodel is provided; it is also shown how finite countermodels for unprovable sequents can be extracted from failed proof search, by which the completeness proof turns into a proof of decidability.

Causal models defined in terms of structural equations have proved to be quite a powerful way of representing knowledge regarding causality. However, a number of authors have given examples that seem to show that the Halpern–Pearl (HP) definition of causality (Halpern & Pearl, 2005) gives intuitively unreasonable answers. Here it is shown that, for each of these examples, we can give two stories consistent with the description in the example, such that intuitions regarding causality are quite different for each story. By adding additional variables, we can disambiguate the stories. Moreover, in the resulting causal models, the HP definition of causality gives the intuitively correct answer. It is also shown that, by adding extra variables, a modification to the original HP definition made to deal with an example of Hopkins & Pearl (2003) may not be necessary. Given how much can be done by adding extra variables, there might be a concern that the notion of causality is somewhat unstable. Can adding extra variables in a “conservative” way (i.e., maintaining all the relations between the variables in the original model) cause the answer to the question “Is X = x a cause of Y = y?” to alternate between “yes” and “no”? It is shown that we can have such alternation infinitely often, but if we take normality into consideration, we cannot. Indeed, under appropriate normality assumptions. Adding an extra variable can change the answer from “yes’ to “no”, but after that, it cannot change back to “yes”.

Say that two sentences are factually equivalent when they describe the same facts or situations, understood as worldly items, i.e. as bits of reality rather than as representations of reality. The notion of factual equivalence is certainly of central interest to philosophical semantics, but it plays a role in a much wider range of philosophical areas. What is the logic of factual equivalence? This paper attempts to give a partial answer to this question, by providing an answer the following, more specific question: Given a standard propositional language with negation, conjunction and disjunction as primitive operators, which sentences of the language should be taken to be factually equivalent by virtue of their logical form? The system for factual equivalence advocated in this paper is a proper fragment of the first-degree system for the logic of analytic equivalence put forward in the late seventies by R. B. Angell. I provide the system with two semantics, both formulated in terms of the notion of a situation’s being fittingly described by a linguistic item. In the final part of the paper I argue, contra a view I defended in my “Grounding and Truth-Functions” (2010), that the logic for factual equivalence I advocate here should be preferred to Angell’s logic if one wishes to follow the general conception of the relationships between factual equivalence and the notion of grounding put forward in the 2010 paper.

According to propositional contingentism, it is contingent what propositions there are. This paper presents two ways of modeling contingency in what propositions there are using two classes of possible worlds models. The two classes of models are shown to be equivalent as models of contingency in what propositions there are, although they differ as to which other aspects of reality they represent. These constructions are based on recent work by Robert Stalnaker; the aim of this paper is to explain, expand, and, in one aspect, correct Stalnaker’s discussion.

Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulations. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.

We analyze a class of modal logics rendered insensitive to reflexivity by way of a modification to the semantic definition of the modal operator. We explore the extent to which these logics can be characterized, and prove a general completeness theorem on the basis of a translation between normal modal logics and their reflexive-insensitive counterparts. Lastly, we provide a sufficient semantic condition describing when a similarly general soundness result is also available.

This paper extends Kripke’s theory of truth to a language with a variably strict conditional operator, of the kind that Stalnaker and others have used to represent ordinary indicative conditionals of English. It then shows how to combine this with a different and independently motivated conditional operator, to get a substantial logic of restricted quantification within naive truth theory.