On standard views, logic has as one of its goals to characterize (and give us practical means to tell apart) a peculiar set of truths, the logical truths, of which the following English sentences are paradigmatic examples: (1) If death is bad only if life is good, and death is bad, then life is good. (2) If no desire is voluntary and some beliefs are desires, then some beliefs are not voluntary. (3) If Drasha is a cat and all cats are mysterious, then Drasha is mysterious. As it turns out, it is very hard to think of universally accepted ideas about what the generic properties of logical truths are or should be.

Intuitionistic logic encompasses the general principles of logical reasoning which have been abstracted by logicians from intuitionistic mathematics, as developed by L. E. J. Brouwer beginning in his [1907] and [1908]. Because these principles also hold for Russian recursive mathematics and the constructive analysis of E. Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics. Although intuitionistic analysis conflicts with classical analysis, intuitionistic Heyting arithmetic is a subsystem of classical Peano arithmetic. It follows that intuitionistic propositional logic is a proper subsystem of classical propositional logic, and pure intuitionistic predicate logic is a proper subsystem of pure classical predicate logic.

Temporal reasoning with conditionals is more complex than both classical temporal reasoning and reasoning with timeless conditionals, and can lead to some rather counter-intuitive conclusions. For instance, Aristotle’s famous “Sea Battle Tomorrow” puzzle leads to a fatalistic conclusion: whether there will be a sea battle tomorrow or not, but that is necessarily the case now. We propose a branching-time logic LTC to formalise reasoning about temporal conditionals and provide that logic with adequate formal semantics. The logic LTC extends the Nexttime fragment of CTL , with operators for model updates, restricting the domain to only future moments where antecedent is still possible to satisfy. We provide formal semantics for these operators that implements the restrictor interpretation of antecedents of temporalized conditionals, by suitably restricting the domain of discourse. As a motivating example, we demonstrate that a naturally formalised in our logic version of the ‘Sea Battle’ argument renders it unsound, thereby providing a solution to the problem with fatalist conclusion that it entails, because its underlying reasoning per cases argument no longer applies when these cases are treated not as material implications but as temporal conditionals. On the technical side, we analyze the semantics of LTC and provide a series of reductions of LTC-formulae, first recursively eliminating the dynamic update operators and then the path quantifiers in such formulae. Using these reductions we obtain a sound and complete axiomatization for LTC, and reduce its decision problem to that of the modal logic KD.

Recent work on logical pluralism has suggested that the view is in danger of collapsing into logical nihilism, the view on which there are no valid arguments at all. The goal of the present paper is to argue that the prospects for resisting such a collapse vary quite considerably with one’s account of logical consequence. The first section will lay out four varieties of logical consequence, beginning with the approaches Etchemendy (1999) called interpretational and representational, and then adding a Quinean substitutional approach as well as the more recent universalist account given in Williamson (2013, 2017). The second section recounts how the threat of logical nihilism arises in the debate over logical pluralism. The third and final section looks at the ways the rival accounts of logical consequence are better or worse placed to resist the threat.

This paper introduces a new equilibrium concept for normal form games called dependency equilibrium; it is defined, exemplified, and compared with Nash and correlated equilibria in Sections 2–4. Its philosophical motive is to rationalize cooperation in the one shot prisoners’ dilemma. A brief discussion of its meaningfulness in Section 5 concludes the paper.

Heinrich Scholz (1884–1956) was a German Protestant theologian, philosopher, and logician who supported the neo-positivistic scientific world view, applying it, however, to a scientific metaphysics as well. He helped to establish the academic field “Mathematical Logic and Foundations”, claiming the priority of language construction and semantics in order to solve foundational problems even outside mathematics. He was also the driving force for the institutionalization of Mathematical Logic in Germany and a pioneer in the historiography of logic.

Modified numerals are expressions such as more than three, less/fewer than three, at least three, at most three, up to ten, betwen three and ten, approximately ten, about ten, exactly ten, etc. At first sight, their semantic contribution seems pretty easy to describe. However, this impression is deceptive. Modified numerals do in fact raise very serious challenges for formal semantics and pragmatics, many of which have yet to be addressed in a fully satisfactorily way. These challenges relate to two broad questions: first, what is the linguistically encoded meaning of modified numerals? Second, how can we make sense of all the inferences they give rise to, and how should we divide the work between com-positional semantics and pragmatics in order to account for all these effects? These are the two questions we will address in this chapter, focusing on a few striking puzzles.

Do I contradict myself? Very well, then, I contradict myself. (I am large, I contain multitudes.) —Walt Whitman, “Song of Myself” Vorrei e non vorrei.    —Zerlina, “Là ci darem la mano”, Don Giovanni This entry outlines the role of the law of non-contradiction (LNC) as the foremost among the first (indemonstrable) principles of Aristotelian philosophy and its heirs, and depicts the relation between LNC and LEM (the law of excluded middle) in establishing the nature of contradictory and contrary opposition. §1 presents the classical treatment of LNC as an axiom in Aristotle's “First Philosophy” and reviews the status of contradictory and contrary opposition as schematized on the Square of Opposition.

Ein kritisches Anliegen dieses Aufsatzes ist es deutlich zu machen, wie unbewältigt die Unklarheiten der drei Grundbegriffe „Wahrheit“, Glauben“, „Gerechtfertigtheit“ der Platonischen Wissensanalyse sind, und ebenso die Unklarheiten der modalen Wissensanalysen mit ihren Bezugnahmen auf die (kontrafaktische) Konditionale und auf Normalbedingungen. Ein konstruktives Anliegen ist es, parallel wenigstens aufzuzeigen, wie sehr die Rangtheorie bei der Bewältigung dieser Unklarheiten helfen kann. Das mündet in eine mögliche Antwort auf die Frage, worin der Mehrwert des Wissens gegenüber bloßen wahren Überzeugungen bestehen könnte.

I prove that invoking the univalence axiom is equivalent to arguing ‘without loss of generality’ within Propositional Univalent Foundations (PropUF), the fragment of Univalent Foundations (UF) in which all homotopy types are mere propositions. As a consequence, I argue that practicing mathematicians, in accepting ‘without loss of generality’ (WLOG) as a valid form of argument, implicitly accept the univalence axiom and that UF rightly serves as a Foundation for Mathematical Practice. By contrast, ZFC is inconsistent with WLOG as it is applied, and therefore cannot serve as a foundation for practice.

Disjunctions scoping under possibility modals give rise to the free choice e↵ect. The e↵ect also arises if the disjunction takes wide scope over possibility modals; it is independent of the modal flavor at play (deontic, epistemic, and so on); and it arises even if disjunctions scope under or over necessity modals. At the same time, free choice e↵ects disappear in the scope of negation or if the speaker signals ignorance or unwillingness to cooperate. I show how we can account for this wide variety of free choice observations without unwelcome side-e↵ects in an update-based framework whose key innovations consist in (i) a refined test semantics for necessity modals and (ii) a generalized conception of narrow and wide scope free choice e↵ects as arising from lexically or pragmatically generated prohibitions against the absurd state (an inconsistent information carrier) serving as an update relatum. The fact that some of these prohibitions are defeasible together with a binary semantics that distinguishes between positive and negative update relata accounts for free choice cancellation e↵ects.

According to logical inferentialists, the meanings of logical expressions are fully determined by the rules for their correct use. Two key proof-theoretic requirements on admissible logical rules, harmony and separability, directly stem from this thesis—requirements, however, that standard single-conclusion and assertion-based formalizations of classical logic provably fail to satisfy (Dummett in The logical basis of metaphysics, Harvard University Press, Harvard, MA, 1991; Prawitz in Theoria, 43:1–40, 1977; Tennant in The taming of the true, Oxford University Press, Oxford, ; Humberstone and Makinson in Mind 120(480):1035–1051, 2011). On the plausible assumption that our logical practice is both single-conclusion and assertion-based, it seemingly follows that classical logic, unlike intuitionistic logic, can’t be accounted for in inferentialist terms. In this paper, I challenge orthodoxy and introduce an assertion-based and single-conclusion formalization of classical propositional logic that is both harmonious and separable. In the framework I propose, classicality emerges as a structural feature of the logic.

In 1933 the Polish logician Alfred Tarski published a paper in which he discussed the criteria that a definition of ‘true sentence’ should meet, and gave examples of several such definitions for particular formal languages. In 1956 he and his colleague Robert Vaught published a revision of one of the 1933 truth definitions, to serve as a truth definition for model-theoretic languages. This entry will simply review the definitions and make no attempt to explore the implications of Tarski’s work for semantics (natural language or programming languages) or for the philosophical study of truth. (For those implications, see the entries on truth and Alfred Tarski.)

Ibn Sīnā [hereafter: Avicenna] (980–1037 CE) is—directly or indirectly—the most influential logician in the Arabic tradition. His work is central in the re-definition of a family of problems and doctrines inherited from ancient and late ancient logic, especially Aristotle and the Peripatetic tradition. While, in general terms, Avicenna squarely falls into a logical tradition that it is reasonable to characterize as Aristotelian, the trove of innovations he introduces establishes him as a genuinely new canonical figure. Every later logician in this tradition confronts him, either as a critic or as a follower, to the extent that, with few exceptions, Aristotle and the Peripatetic tradition almost entirely disappear from the scene.

It has been observed (e.g. Cooper (1979), Chierchia (1993), von Fintel (1994), Marti (2003)) that the interpretation of natural language variables (overt or covert) can depend on a quantifier. The standard analysis of this phenomenon is to assume a hidden structure inside the variable, part of which is semantically bound by the quantifier. In this paper I argue that the presupposition of the adverb 'again' and other similar presuppositions depend on a variable that gives rise to the same phenomenon.

Karl Popper developed a theory of deductive logic in the late 1940s. In his approach, logic is a metalinguistic theory of deducibility relations that are based on certain purely structural rules. Logical constants are then characterized in terms of deducibility relations. Characterizations of this kind are also called inferential definitions by Popper. In this paper, we expound his theory and elaborate some of his ideas and results that in some cases were only sketched by him. Our focus is on Popper’s notion of duality, his theory of modalities, and his treatment of different kinds of negation. This allows us to show how his works on logic anticipate some later developments and discussions in philosophical logic, pertaining to trivializing (tonk-like) connectives, the duality of logical constants, dual-intuitionistic logic, the (non-)conservativeness of language extensions, the existence of a bi-intuitionistic logic, the non-logicality of minimal negation, and to the problem of logicality in general.

The conception of implications as rules is interpreted in Lorenzen-style dialogical semantics. Implications-as-rules are given attack and defense principles, which are asymmetric between proponent and opponent. Whereas on the proponent’s side, these principles have the usual form, on the opponent’s side implications function as database entries that can be used by the proponent to defend assertions independent of their logical form. The resulting system, which also comprises a principle of cut, is equivalent to the sequent-style system for implications-as-rules. It is argued that the asymmetries arising in the dialogical setting are not deficiencies but reflect the pre-logical (‘structural’) character of the notion of rule.

Atomic systems, that is, sets of rules containing only atomic formulas, play an important role in proof-theoretic notions of logical validity. We consider a view of atomic systems as definitions that allows us to discuss a proposal made by Prawitz (2016). The implementation of this view in the base case of an inductive definition of validity leads to the problem that derivability of atomic formulas in an atomic system does not coincide with the validity of these formulas. This is due to the fact that, on the definitional view of atomic systems, there are not just production rules, but both introduction and elimination rules for atoms, which may even generate non-normalizable atomic derivations. This shows that the way atomic systems are handled is a fundamental issue of proof-theoretic semantics.

The BHK interpretation of logical constants is analyzed in terms of a systematic account given by Prawitz, resulting in a reformulation of the BHK interpretation in which the assertability of atomic propositions is determined by Post systems. It is shown that the reformulated BHK interpretation renders more propositions assertable than are provable in intuitionistic propositional logic. Mints’ law is examined as an example of such a proposition. Intuitionistic propositional logic would thus have to be considered incomplete. We conclude with a discussion on the adequacy of the BHK interpretation of implication.

The inversion principle expresses a relationship between left and right introduction rules for logical constants. Hallnas and Schroeder- Heister [2] presented the principle of definitional reflection as a means of capturing the idea embodied in the inversion principle. Using the principle of definitional reflection, we show for minimal propositional logic that the left introduction rules are admissible when the right introduction rules are given as the definition of logical constants, and vice versa. Keywords: Proof theory, inversion principle, admissibility, logical rules.

The quantum query complexity of approximate counting was one of the first topics studied in quantum algorithms. Given a nonempty finite set S ⊆ [N ] (here and throughout, [N ] = {1, . . . , N }), suppose we want to estimate its cardinality, |S|, to within some multiplicative accuracy ε. This is a fundamental task in theoretical computer science, used as a subroutine for countless other tasks. As is standard in quantum algorithms, we work in the so-called black-box model (see [10]), where we assume only that we’re given a membership oracle for S: an oracle that, for any i ∈ [N ], tells us whether i ∈ S. We can, however, query the oracle in quantum superposition. How many queries must a quantum computer make, as a function of both N and |S|, to solve this problem with high probability?

In a state-based semantics formulas are interpreted with respect to states rather than possible worlds. States are less determinate entities than worlds and can be identified with truthmakers (van Fraassen, 1969; Fine, 2017), possibilities (Humberstone, 1981; Holliday, 2015), situations (Barwise and Perry, 1983), information states (Veltman, 1985, 1996; Dekker, 2012) and more. The partial nature of a state makes a state-based semantics particularly suitable to capture various aspects of disjunctive words in natural language, including their indeterminate, epistemic and choice-offering nature.

Let f be a function from A to 73(A). Let B : {1: E A|:z: §§ f(.r)}. Suppose there exists y 6 A such that B : f(y). If y E B then y ¢ f(y) : B, and conversely if y ¢ B : f(y) then y E B. So there isn’t y E A such that B : f(y) and therefore f is not surjective.