The picture of information acquisition as the elimination of possibilities has proven fruitful in many domains, serving as a foundation for formal models in philosophy, linguistics, computer science, and economics. While the picture appears simple, its formalization in dynamic epistemic logic reveals subtleties: given a valid principle of information dynamics in the language of dynamic epistemic logic, substituting complex epistemic sentences for its atomic sentences may result in an invalid principle. In this article, we explore such failures of uniform (...) substitution. First, we give epistemic examples inspired by Moore, Fitch, and Williamson. Second, we answer affirmatively a question posed by van Benthem: can we effectively decide when every substitution instance of a given dynamic epistemic principle is valid? In technical terms, we prove the decidability of this schematic validity problem for public announcement logic (PAL and PAL-RC) over models for finitely many fully introspective agents, as well as models for infinitely many arbitrary agents. The proof of this result illuminates the reasons for the failure of uniform substitution. (shrink)

Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. A distinction therefore arises between the logic and its substitution core, the set of formulas all of whose substitution instances are valid. The classic example of a non-uniform dynamic epistemic logic is Public Announcement Logic (PAL), and a well-known open problem is to axiomatize the substitution core of PAL. In this paper we solve this problem for PAL over the class (...) of all relational models with infinitely many agents, PAL-K_omega, as well as standard extensions thereof, e.g., PAL-T_omega, PAL-S4_omega, and PAL-S5_omega. We introduce a new Uniform Public Announcement Logic (UPAL), prove completeness of a deductive system with respect to UPAL semantics, and show that this system axiomatizes the substitution core of PAL. (shrink)

Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. A distinction therefore arises between the logic and its substitu- tion core, the set of formulas all of whose substitution instances are valid. The classic example of a non-uniform dynamic epistemic logic is Public Announcement Logic (PAL), and a well-known open problem is to axiomatize the substitution core of PAL. In this paper we solve this problem for PAL over the class (...) of all relational models with infinitely many agents, PAL-Kω , as well as standard extensions thereof, e.g., PAL-Tω , PAL-S4ω, and PAL-S5ω. We introduce a new Uniform Public Announcement Logic (UPAL), prove completeness of a deductive system with respect to UPAL semantics, and show that this system axiomatizes the substitution core of PAL. (shrink)

A substitutional account of logical validity for formal first‐order languages is developed and defended against competing accounts such as the model‐theoretic definition of validity. Roughly, a substitution instance of a sentence is defined as the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category and possibly relativizing quantifiers. In particular, predicate symbols can be replaced with formulae possibly containing additional free variables. A sentence is defined to be logically true iff all its (...) substitution instances are satisfied by all variable assignments. Logical consequence is defined analogously. Satisfaction is taken to be a primitive notion and axiomatized.For every set‐theoretic model in the sense of model theory there exists a corresponding substitutional interpretation in a sense to be specified. Conversely, however, there are substitutional interpretations – in particular the ‘intended’ interpretation – that lack a model‐theoretic counterpart. The substitutional definition of logical validity overcomes the weaknesses of more restrictive accounts of substitutional validity; unlike model‐theoretic logical consequence, the substitutional notion is trivially and provably truth preserving. In Kreisel's squeezing argument the formal notion of substitutional validity naturally slots into the place of intuitive validity. (shrink)

This paper explores the relation between the philosophical idea that logic is a science studying logical forms, and a mathematical feature of logical systems called the principle of uniform substitution, which is often regarded as a technical counterpart of the philosophical idea. We argue that at least in one interesting sense the principle of uniform substitution does not capture adequately the requirement that logic is a matter of form and that logical truths are formal truths. We (...) show that some specific logical expressions can produce propositions of different kinds and the resulting diversity of informational types can lead to a justified failure of uniform substitution without undermining the view that logic is a purely formal discipline. (shrink)

Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $, $\vee $, $\forall ^{\mathbb {R}}$, continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $. Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $. Then the countable length everywhere club (...) uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all $\ell \in {}^{<{\omega _1}}\kappa $ and clubs $C \subseteq D \subseteq \kappa $, $R(\ell,D)$ implies $R(\ell,C)$, there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$, $R(\ell,\Lambda (\ell ))$. In particular, under these assumptions, for all $n \in \omega $, $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization. (shrink)

Two real-valued deduction schemes are introduced, which agree on $\vdash \triangle$ but not on $\Gamma \vdash \triangle$ , where Δ and ▵ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and (...) the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other. (shrink)

This article studies seriously actualistic quantified modal logics. A key component of the language is an abstraction operator by means of which predicates can be created out of complex formulas. This facilitates proof of a uniform substitution theorem: if a sentence is logically true, then any sentence that results from substituting a predicate abstract for each occurrence of a simple predicate abstract is also logically true. This solves a problem identified by Kripke early in the modern semantic study (...) of quantified modal logic. A tableau proof system is presented and proved sound and complete with respect to logical truth. The main focus is on seriously actualistic T, an extension of T, but the results established hold also for systems based on other propositional modal logics. Following Menzel it is shown that the formal language studied also supports an actualistic account of truth simpliciter. (shrink)

We explore depth substitution invariance, or hyperformalism, and extend known results in this realm to justification logics extending weak relevant logics. We then examine the surprising invariance of justifications over formulas and restrict our attention to the substitution of proofs in the original relevant logic. The results of this paper indicate that depth invariance is a recalcitrant feature of the logic and that proof structures in hyperformal logics are quite inflexible.

As it is well known, in the forties of the past century, Curry proved that in any logic S closed under Modus Ponens, uniform substitution of propositional variables and the Contraction Law, the naïve Comprehension axiom trivializes S in the sense that all propositions are derivable in S plus CA. Not less known is the fact that, ever since Curry published his proof, theses and rules weaker than W have been shown to cause the same effect as W (...) causes. Among these, the Contraction rule or the Modus Ponens axiom, for example, are to be noted. But, moreover, as Brady has proved, even the Generalized Modus Ponens axiom or the Generalized Contraction rule give rise to “Curry’s Paradox” under the same circumstances as W does. In some previous work by us, “weak relevant model structures” are defined on “weak relevant matrices” by generalizing Brady’s model structure MCL built upon Meyer’s Crystal matrix CL. We have proved that wr-ms only verify logics with the “depth relevance condition”. The aim of this paper is to show how to falsify gMPa and gRW in certain wr-ms. In particular, it will be shown that gMPa is falisfied in any wr-ms and gRW in any wr-ms verifying Routley and Meyer’s basic positive logic B+. (shrink)

The recovery of Aristotle’s logic during the twelfth century was a great stimulus to medieval thinkers. Among their own theories developed to explain Aristotle’s theories of valid and invalid reasoning was a theory of consequence, of what arguments were valid, and why. By the fourteenth century, two main lines of thought had developed, one at Oxford, the other at Paris. Both schools distinguished formal from material consequence, but in very different ways. In Buridan and his followers in Paris, formal consequence (...) was that preserved under uniform substitution. In Oxford, in contrast, formal consequence included analytic consequences such as ‘If it’s a man, then it’s an animal’. Aristotle’s notion of syllogistic consequence was subsumed under the treatment of formal consequence. Buridan developed a general theory embracing the assertoric syllogism, the modal syllogism and syllogisms with oblique terms. The result was a thoroughly systematic and extensive treatment of logical theory and logical consequence which repays investigation. (shrink)

Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this (...) context was called. This theory gave a justification for the validity of what we todayexpress through the formulae ¬ and ¬. Hugh MacColl and, more recently, Storrs McCall searched for a formal system in which the validity ofthese formulae could be expressed. Unfortunately neither of the resulting systems is very satisfactory. In this paper we introduce dialogical games with the help of a new connexive If-Then (), the structural rules of which allow the Proponent to develop winning strategies not only for the above-mentioned connexive theses but also for ¬ and ¬. Further on, we developthe corresponding tableau systems and conclude with some remarks on possibleperspectives and consequences of the dialogical approach to connexivity including the loss of uniform substitution leading to a new concept of logical form. (shrink)

The first section (§1) of this essay defends reliance on truth values against those who, on nominalistic grounds, would uniformly substitute a truth predicate. I rehearse some practical, Carnapian advantages of working with truth values in logic. In the second section (§2), after introducing the key idea of auxiliary parameters (§2.1), I look at several cases in which logics involve, as part of their semantics, an extra auxiliary parameter to which truth is relativized, a parameter that caters to special kinds (...) of sentences. In many cases, this facility is said to produce truth values for sentences that on the face of it seem neither true nor false. Often enough, in this situation appeal is made to the method of supervaluations, which operate by “quantifying out” auxiliary parameters, and thereby produce something like a truth value. Logics of this kind exhibit striking differences. I first consider the role that Tarski gives to supervaluation in first order logic (§2.2), and then, after an interlude that asks whether neither-true-nor-false is itself a truth value (§2.3), I consider sentences with non-denoting terms (§2.4), vague sentences (§2.5), ambiguous sentences (§2.6), paradoxical sentences (§2.7), and future-tensed sentences in indeterministic tense logic (§2.8). I conclude my survey with a look at alethic modal logic considered as a cousin (§2.9), and finish with a few sentences of “advice to supervaluationists” (2.10), advice that is largely negative. The case for supervaluations as a road to truth is strong only when the auxiliary parameter that is “quantified out” is in fact irrelevant to the sentences of interest—as in Tarski’s definition of truth for classical logic. In all other cases, the best policy when reporting the results of supervaluation is to use only explicit phrases such as “settled true” or “determinately true,” never dropping the qualification. (shrink)

Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this (...) context was called συναρτησις(connection). This theory gave a justification for the validity of what we todayexpress through the formulae ¬(a → ¬ a) and ¬(¬ a → a). Hugh MacColl and, more recently, Storrs McCall (from 1877 to 1906 and from1963 to 1975 respectively) searched for a formal system in which the validity ofthese formulae could be expressed. Unfortunately neither of the resulting systems is very satisfactory. In this paper we introduce dialogical games with the help of a new connexive If-Then (“→”), the structural rules of which allow the Proponent to develop (formal) winning strategies not only for the above-mentioned connexive theses but also for (a → b) → ¬(a → ¬ b) and (a → b) → ¬(¬ a → b). Further on, we developthe corresponding tableau systems and conclude with some remarks on possibleperspectives and consequences of the dialogical approach to connexivity including the loss of uniform substitution leading to a new concept of logical form. (shrink)

As the final component of a chain of reasoning intended to take us all the way to logical nihilism, Russell (2018) presents the atomic sentence ‘prem’ which is supposed to be true when featuring as premise in an argument and false when featuring as conclusion in an argument. Such a sentence requires a non-reflexive logic and an endnote by Russell (2018) could easily leave the reader with the impression that going non-reflexive suffices for logical nihilism. This paper shows how one (...) can obtain non-reflexive logics in which ‘prem’ behaves as stipulated by Russell (2018) but which nonetheless has valid inferences supporting uniform substitution of any formula for propositional variables such as modus tollens and modus ponens. (shrink)

A typical theorem of conaexive logics is Aristotle''s Thesis(A), (AA).A cannot be added to classical logic without producing a trivial (Post-inconsistent) logic, so connexive logics typically give up one or more of the classical properties of conjunction, e.g.(A & B)A, and are thereby able to achieve not only nontriviality, but also (negation) consistency. To date, semantical modellings forA have been unintuitive. One task of this paper is to give a more intuitive modelling forA in consistent logics. In addition, while inconsistent (...) but nontrivial theories, and inconsistent nontrivial logics employing prepositional constants (for which the rule of uniform substitution US fails), have both been studied extensively within the paraconsistent programme, inconsistent nontrivial logics (closed under US) do not seem to have been. This paper gives sufficient conditions for a logic containingA to be inconsistent, and then shows that there is a class of inconsistent nontrivial logics all containingA. A second semantical modelling forA in such logics is given. Finally, some informal remarks about the kind of modellingA seems to require are made. (shrink)

In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric and (...) combinatorial nature will be introduced in order to give a mathematical description of the basic logical/algebraic constructions .We begin by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible .The central part of the paper is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F .As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic ). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane or the appendix. (shrink)

In this paper we study families of resource aware logics that explore resource restriction on rules; in particular, we study the use of controlled cut-rule and introduce three families of parameterised logics that arise from different ways of controlling the use of cut. We start with a formulation of classical logic in which cut is non-eliminable and then impose restrictions on the use of cut. Three Cut-and-Pay families of logics are presented, and it is shown that each family provides an (...) approximation process for full propositional classical logic when the control over the use of cut is progressively weakened. A sound and complete semantics is given for each component of each of the three families of approximated logics. One of these families is shown to possess the uniform substitution property, a new result for approximated reasoning. A tableau based decision procedure is presented for each element of the approximation families and the complexity of each decision procedure is studied. We show that there are families in which every component logic can be decided polynomially. (shrink)

It is shown that the implicational fragment of Anderson and Belnap's R, i.e. Church's weak implicational calculus, is not uniquely characterized by MP (modus ponens), US (uniform substitution), and WDT (Church's weak deduction theorem). It is also shown that no unique logic is characterized by these, but that the addition of further rules results in the implicational fragment of R. A similar result for E is mentioned.

We explore in an experimental spirit the prospects for extending classical propositional logic with a new operator P intended to be interpreted when prefixed to a formula as saying that formula in question is at least partly true. The paradigm case of something which is, in the sense envisaged, false though still "partly" true is a conjunction one of whose conjuncts is false while the other is true. Ideally, we should like such a logic to extend classical logic - or (...) any fragment thereof under consideration - conservatively, to be closed under uniform substitution (of arbitrary formulas for sentence letters or propositional variables), and to allow the substitutivity of provably equivalent formulas salva provabilitate. To varying degrees, we experience some difficulties only with this last ('congruentiality') desideratum in the two four-valued logics we end up giving our most extended consideration to. (shrink)

This paper is in the scope of the philosophy of modal logic; more precisely, it concerns the semantics of modal logic, when the modal elements are interpreted as logical modalities. Most authors have thought that the logic for logical modality—that is, the one to be used to formalize the notion of logical truth (and other related notions)—is to be found among logical systems in which modalities are allowed to be iterated. This has raised the problem of the adequacy, to that (...) formalization purpose, of some modal schemes, such as S4 and S5. It has been argued that the acceptance of S5 leads to non-normal modal systems, in which the uniform substitution rule fails. The thesis supported in this paper is that such a failure is rather to be attributed to what will be called “Condition of internalization.” If this is correct, there seems to be no normal modal logic system capable of formalizing logical modality, even when S5 is rejected in favor of a weaker system such as S4, as recently proposed by McKeon. (shrink)

In this paper, the notion of structural completeness is explored in the context of a generalized class of superintuitionistic logics involving also systems that are not closed under uniform substitution. We just require that each logic must be closed under D-substitutions assigning to atomic formulas only ∨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vee $$\end{document}-free formulas. For these systems we introduce four different notions of structural completeness and study how they are related. We focus on superintuitionistic (...) inquisitive logics that validate a schema called Split and have the disjunction property. In these logics disjunction can be interpreted in the sense of inquisitive semantics as a question forming operator. It is shown that a logic is structurally complete with respect to D-substitutions if and only if it includes the weakest superintuitionistic inquisitive logic. Various consequences of this result are explored. For example, it is shown that every superintuitionistic inquisitive logic can be characterized by a Kripke model built up from D-substitutions. Additionally, we resolve a conjecture concerning superintuitionistic inquisitive logics due to Miglioli et al.. (shrink)

We present a method for integrating rippling-based rewriting into matrix-based theorem proving as a means for automating inductive specification proofs. The selection of connections in an inductive matrix proof is guided by symmetries between induction hypothesis and induction conclusion. Unification is extended by decision procedures and a rippling/reverse-rippling heuristic. Conditional substitutions are generated whenever a uniform substitution is impossible. We illustrate the integrated method by discussing several inductive proofs for the integer square root problem as well as the (...) algorithms extracted from these proofs. (shrink)

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary (...) set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas and , C(X{{a}}) iff P(, ) AC(X). [P(, ) denotes the set of formulas which result by the simultaneous substitution of for p and for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian. (shrink)

The knowledge account of assertion construes assertion as subject to constitutive norms. In its standard version, it combines a wide scope obligation not to assert p without knowing p, with narrow scope principles specifying conditions under which it is permissible to assert p, where the notions of obligation and permission are duals and behave uniformly for variable p. It is argued that, given natural assumptions about the logic of ‘ought’, the account proves incoherent. The argument generalizes to accounts that substitute (...) other factive notions for knowledge. A recent non‐standard version of the knowledge account employs proposition‐relative norms and circumvents the problem. However, it still leads to intolerable combinations of verdicts. Again, the problem arises because knowledge is factive, and it generalizes to other factive notions. It is shown that non‐factive accounts face none of the diagnosed difficulties and can do much of the explanatory work that the knowledge account is alleged to do. (shrink)

The logical form of a discourse—such as a proposition, a set of propositions, an argument, or an argumentation—is obtained by abstracting from the subject-matter of its content terms or by regarding the content terms as mere place-holders or blanks in a form. In a logically perfect language the logical form of a proposition, a set of propositions, an argument, or an argumentation is determined by the grammatical form of the sentence, the set of sentences, the argument-text, or the argumentation-text expressing (...) it. Two such discourse-texts are said to have the same grammatical form, in this sense, if a uniform one-one substitution of content words transforms the one exactly into the other. The sentence ‘Abe properly respects every agent who respects himself’ may be regarded as having the same grammatical form as the sentence ‘Ben generously assists every patient who assists himself’. (shrink)

One way to theorize about we-identities—the identities that individual subjects have as ‘one of us’—is in terms of the uniformity, interchangeability, and prototypicality of group members. The social-psychological theory of self-categorization epitomizes this approach, which has strongly influenced contemporary phenomenological research on the we. This paper argues that this approach has one important and largely overlooked limitation: the we-identities tied to close personal relationships—exemplified by long-term friendships and romantic partnerships—are based on patterns of interpersonal interaction and integration through which individuals (...) tend to grasp their non-substitutability and complementarity. This limitation suggests that another approach is needed to tackle the we-identities characteristic of close personal relationships. I outline such an approach, by combining resources from classical phenomenology and ongoing research on the socially extended mind. (shrink)

This paper proposes to present a novel method of generating cryptographic dynamic substitution-boxes, which makes use of the combined effect of discrete hyperchaos mapping and algebraic group theory. Firstly, an improved 2D hyperchaotic map is proposed, which consists of better dynamical behaviour in terms of large Lyapunov exponents, excellent bifurcation, phase attractor, high entropy, and unpredictability. Secondly, a hyperchaotic key-dependent substitution-box generation process is designed, which is based on the bijectivity-preserving effect of multiplication with permutation matrix to obtain (...) satisfactory configuration of substitution-box matrix over the enormously large problem space of 256!. Lastly, the security strength of obtained S-box is further elevated through the action of proposed algebraic group structure. The standard set of performance parameters such as nonlinearity, strict avalanche criterion, bits independent criterion, differential uniformity, and linear approximation probability is quantified to assess the security and robustness of proposed S-box. The simulation and comparison results demonstrate the effectiveness of proposed method for the construction of cryptographically sound S-boxes. (shrink)

In [15], [16] G. Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated ε-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals Φ A of order type 0 which realize the Herbrand normal form A H of A. Subsequently more perspicuous proofs of this fact via functional interpretation (combined with normalization) and cut-elimination were found. These proofs however (...) do not carry out the no-counterexample interpretation as a local proof interpretation and don't respect the modus ponens on the level of the no-counterexample interpretation of formulas A and A → B. Closely related to this phenomenon is the fact that both proofs do not establish the condition (δ) and--at least not constructively-- (γ) which are part of the definition of an 'interpretation of a formal system' as formulated in [15]. In this paper we determine the complexity of the no-counterexample interpretation of the modus ponens rule for (i) PA-provable sentences, (ii) for arbitrary sentences A, B ∈ L(PA) uniformly in functionals satisfying the no-counterexample interpretation of (prenex normal forms of) A and A → B, and (iii) for arbitrary A, B ∈ L(PA) pointwise in given α( 0 ) -recursive functionals satisfying the no-counterexample interpretation of A and A → B. This yields in particular perspicuous proofs of new uniform versions of the conditions (γ), (δ). Finally we discuss a variant of the concept of an interpretation presented in [17] and show that it is incomparable with the concept studied in [15], [16]. In particular we show that the no-counterexample interpretation of PA n by α( n (ω))-recursive functionals (n ≥ 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (δ). (shrink)

In this paper I argue that a variety of consequence relations can be subsumed under a common core. The reduction proceeds by taking the unconditional consequence, or judgment, as basic and deriving the conditional consequence via a uniform abstraction scheme. A specific outcome is that it is better not to base such a scheme on the semantic notion of a matrix and valuation but rather on theories and substitutions. I will also briefly look at consequence relations that are not (...) reducible in this way. (shrink)

It is widely believed that some a priori necessary truths are not analytic in the sense of transformable by substitution of synonyms into logical truths. One much-cited example comes from the supposed incompatibility between colour predicates. The idea is that sentences like “Nothing is both blue all over (or uniformly or at a point) and also red” are not transformable into a logical truth in the same way as “Nothing is both a bachelor and married” because the requisite conceptual (...) link between “bachelor” and “not married” is absent between “blue” and “not red”. This paper examines whether colour exclusion may be more like the bachelor case than it initially appears. It transpires that the most promising line of thought is not, however, as has been argued at length in the literature, that “blue” in some more or less convoluted way manages to mean “not red”. Instead it is suggested that the requisite conceptual link may reside in the oft-ignored qualifications (“all over”, “uniformly”, etc.), without which there is no incompatibility in the first place. (shrink)

Equivalent deductive systems were introduced in [4] with the goal of treating 1-deductive systems and algebraic 2-deductive systems in a uniform way. Results of [3], appropriately translated and strengthened, show that two deductive systems over the same language type are equivalent if and only if their lattices of theories are isomorphic via an isomorphism that commutes with substitutions. Deductive equivalence of π-institutions [14, 15] generalizes the notion of equivalence of deductive systems. In [15, Theorem 10.26] this criterion for the (...) equivalence of deductive systems was generalized to a criterion for the deductive equivalence of term π-institutions, forming a subclass of all π-institutions that contains those π-institutions directly corresponding to deductive systems. This criterion is generalized here to cover the case of arbitrary π-institutions. (shrink)

In proof theory one distinguishes sequent proofs with cut and cut-free sequent proofs, while for proof complexity one distinguishes Frege systems and extended Frege systems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cut-free systems with extension, which is neither possible with Frege systems, nor with the sequent calculus. We show that the propositional pigeonhole principle admits polynomial-size proofs in a cut-free system with extension. We also (...) define cut-free systems with substitution and show that the cut-free system with extension p-simulates the cut-free system with substitution. (shrink)

While recent studies have shed some light on the significance of the electrical activity of the nervous system, there has been no adequate explanation for the wave formation or synchronization of this electrical activity. Adrian sums up the problem. “The origin of the 10-a-second rhythm is still uncertain, though the evidence points to some widespread organization, probably involving the central masses as well as the cortex. There are abundant nervous connexions for coordinating the beat, and when the rhythm is well (...) developed it is possible to record impulse discharges in phase with it passing to and fro in the nerve fibers of the white matter below the cortex. But in some areas no impulse could be detected entering or leaving the cell layers, though the potential waves keep more or less in step with those elsewhere. There is no proof that the waves are kept in phase by some means which does not involve nervous signalling, for impulses in axons of small diameter might well be missed, but it is not altogether unlikely that the synchronization is due in part to a direct electrical influence of one group of nerve cells on another. Gerard and Libet have shown that synchronization can occur between the two halves of a frog's brain cut in two and then placed in contact, and more recently Arvanitaki has shown that individual nerve cells may influence one another if they are brought close together in a conducting medium. At all events there are many examples of synchronized rhythmic activity in large collections of nerve-or-muscle cells. In the optic ganglion of the water beetle, for instance, there may be both a slow rhythm when the eye is in darkness and a rapid one when the eye is exposed to a bright light. The change from a slow to a fast rhythm is curiously reminiscent of that in the cerebral cortex when a sensory stimulus abolishes the 10-a-second rhythm and substitutes one at 40–60 a second. Such resemblances in preparations of quite different structures may be quite fortuitous, but they make it difficult to resist the suggestion that the rhythms of the brain may be dependent on the general properties of cell masses rather than on any special anatomical arrangement of them. Whatever its origin, the 10-a-second rhythm of the cortex corresponds to the resting, drowsy, or inattentive state and there is no such uniform pulsation when the brain is alert.”. (shrink)

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, (...) we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question. (shrink)

Philosophical logic has its problem-children; and among these the Principle of Substitutivity of codesignating expressions — the linguistic spawn of Leibniz's law—has achieved a place of prominence. It has become increasingly apparent that a certain style of linguistic analysis, which seeks to impose formal regimentation ruled by the constraints of classical quantification theory, does not yield results with the kind of uniformity and elegance one should hope for from a satisfyi.ng theory. The root of the difficulty, I believe, bears upon (...) the answers to fundamental questions concerning the nature of cognitive agents and the purposes which their use of language is designed to further. If this is the case, one might expect to find some of the trouble surfacing at precisely those points where the language is used to convey information about the cognitive activities associated with the use of language by such agents. So it is.The central problem of this essay is the interpretation of referential expressions in statements involving propositional attitudes, with the focus on belief contexts. But I shall begin by placing the discussion within a broader theoretical framework. (shrink)

Equivalent deductive systems were introduced in [4] with the goal of treating 1‐deductive systems and algebraic 2‐deductive systems in a uniform way. Results of [3], appropriately translated and strengthened, show that two deductive systems over the same language type are equivalent if and only if their lattices of theories are isomorphic via an isomorphism that commutes with substitutions. Deductive equivalence of π‐institutions [14, 15] generalizes the notion of equivalence of deductive systems. In [15, Theorem 10.26] this criterion for the (...) equivalence of deductive systems was generalized to a criterion for the deductive equivalence of term π‐institutions, forming a subclass of all π‐institutions that contains those π‐institutions directly corresponding to deductive systems. This criterion is generalized here to cover the case of arbitrary π‐institutions. (shrink)

The author discusses two questions, the relation between liberalism and democracy, and the relation between ethics, morality and law. As to the first question, she argues that neither liberalism nor democracy are merely formal. Roughly spoken, it can be said that liberalism stands for negative liberties, whereas democracy stands for positive ones. She observes a non-contingent tension between the ethos of liberalism (personal freedom) and the ethos of democracy (equality; majority rule). It is the task of morality to maintain and (...) restore the balance between these two kinds of ethos. As to the second question, she is worried about the balance between law (legal regulation), ethics, and morality. On the one hand, abolishing legal regulations would amount to abolishing the freedom of the moderns. On the other hand, the substitution of legal regulations for ethical regulations would lead to a similar result: the end of the freedom of the moderns through the homogenisation of life. In the former case, personal support, charity, magnanimity, and caring would get lost, while in the latter there would be no escape from community pressure towards uniformity. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Witness Against the Beast, William Blake and the Moral LawDan LatimerWitness Against the Beast, William Blake and the Moral Law, by E. P. Thompson; xxi & 324 pp. New York: The New Press, 1993, $30.00.The social context from which William Blake arose was fundamentally hostile to the grandiose projects of Court and official Church. So modest were the ambitions of Blake’s working-class forebears that their historical oblivion would (...) be assured, had they not professed theological views irksome to those whose job it was to enforce uniformity. The eccentric theology of this nonconformist class earned them the name antinomian, from “anti-nomos,” “against the law,” a term which carried the sense of both “heretic” and “terrorist.”Antinomians emerged from the Puritan Interregnum of the 1650s. They went quiet during the Restoration of Charles II, then were given a second life in the 1790s when the energies of Jacobinism refocused working-class frustrations on the hegemonic alliance of Court and Church. Eventually, sedition laws drove the antinomians underground to stay. Abhorring all official places of worship, they met over pints of beer in public houses, singing songs composed by themselves out of their home-made hymnals set to such unthreatening melodies as “God Save the King” to delude the casual passer-by.One such sect was the Muggletonians, whose very name ensured groans of genteel ridicule from the educated and powerful. They never numbered more than two hundred members in all of England. Taking their name from a seventeenth-century sectary, Ludowick Muggleton, they kept largely to themselves, did not evangelize, and did not pray, since they considered prayer servile. They met for fellowship, song, and hours of intricate theological discussion. For three hundred years, the faithful preserved the manuscripts of their prophets, records of discussion and debate, song books and membership lists, until Thompson tracked it all down in 1975 to a furniture depository in Tunbridge Wells. What Thompson wanted to show was some empirical connection between Muggletonian theology and the eccentric mythos of William Blake, since the two systems had so many already obvious antinomian similarities. It must have been discouraging for him to get no closer than “Hermitage,” the name of Blake’s mother’s first husband, a hosier named Thomas. In the Muggletonian song books are compositions from the 1750s by a George Hermitage, who in Thompson’s working hypothesis is a possible uncle of Blake. Thompson admits, however, that the image of Blake’s mother singing nonconformist family songs over the cradle of her infant son will have to remain a pretty fiction.Why does Thompson specifically want the Muggletonians in the Blakean picture? There are four pertinent articles of Muggletonian belief. The first is that they repudiate moral law, that is, the notion of religion as a system of discipline and punishment for sin. They substitute love and generosity as [End Page 412] characteristic of the divine principle. The second is that Reason is explicitly identified as a Satanic item, binding, constraining, and inhibiting the imaginative vision of God. The third article is the doctrine of the Two Seeds, according to which Satan, in the form of the serpent, had carnal relations with Eve, who subsequently gave birth to Cain, the serpent’s child and half-brother to Abel and Seth, the legitimate offspring of Adam. Henceforth, though, the Satanic and divine seeds are mixed in all human beings. The persistent Satanic element, the humanum maternum, gives our worshippers occasion, not only for an indictment of the rich and powerful, serpent-spawn all, but for a gnostic hostility for all material creation, including the body, implying, one need hardly say, a robust sense of woman’s addiction to the secret and forbidden, the worm that flies in the night.The fourth article deals with the divine influx of God into man on the occasion of Christ’s birth. For the Muggletonians, this incarnation (and eventual extinction) was entire. There weren’t any other parts of God left over somewhere else. Thompson sees here elements of a radical humanism which may have motivated such notions as “the human form divine” in Blake. On the other hand, the doctrine of the... (shrink)

Describes how the author became an on-call substitute teacher in pursuit of the realities of American public education, describing his complex difficulties with helping educate today's students in spite of flawed curriculums and interpersonal challenges.

This paper defines the form of prior knowledge that is required for sound inferences by analogy and single-instance generalizations, in both logical and probabilistic reasoning. In the logical case, the first order determination rule defined in Davies (1985) is shown to solve both the justification and non-redundancy problems for analogical inference. The statistical analogue of determination that is put forward is termed 'uniformity'. Based on the semantics of determination and uniformity, a third notion of "relevance" is defined, both logically and (...) probabilistically. The statistical relevance of one function in determining another is put forward as a way of defining the value of information: The statistical relevance of a function F to a function G is the absolute value of the change in one's information about the value of G afforded by specifying the value of F. This theory provides normative justifications for conclusions projected by analogy from one case to another, and for generalization from an instance to a rule. The soundness of such conclusions, in either the logical or the probabilistic case, can be identified with the extent to which the corresponding criteria (determination and uniformity) actually hold for the features being related. (shrink)

An increasing amount of twenty-first century metaphysics is couched in explicitly hyperintensional terms. A prerequisite of hyperintensional metaphysics is that reality itself be hyperintensional: at the metaphysical level, propositions, properties, operators, and other elements of the type hierarchy, must be more fine-grained than functions from possible worlds to extensions. In this paper I develop, in the setting of type theory, a general framework for reasoning about the granularity of propositions and properties. The theory takes as primitive the notion of a (...) substitution on a proposition (property, etc.) and, among other things, uses this idea to elucidate a number of theoretically important distinctions. A class of structures are identified which can be used to model a wide range of positions about the granularity of reality; certain of these structures are seen to receive a natural treatment in the category of M-sets. (shrink)

When a user integrates a sensory substitution device into her life, the process involves perceptual learning, that is, ‘relatively long-lasting changes to an organism’s perceptual system that improve its ability to respond to its environment’ (Goldstone 1998: 585). In this paper, I explore ways in which the extensive literature on perceptual learning can be applied to help improve sensory substitution devices. I then use these findings to answer a philosophical question. Much of the philosophical debate surrounding sensory (...) class='Hi'>substitution devices concerns what happens after perceptual learning occurs. In particular, should the resultant perceptual experience be classified in the substituted modality (as vision), in the substituting modality (as auditory or tactile), or in a new sense modality? I propose a novel empirical test to help resolve this philosophical debate. (shrink)

The paper argues that philosophers commonly misidentify the substitutivity principle involved in Russell’s puzzle about substitutivity in “On Denoting”. This matters because when that principle is properly identified the puzzle becomes considerably sharper and more interesting than it is often taken to be. This article describes both the puzzle itself and Russell's solution to it, which involves resources beyond the theory of descriptions. It then explores the epistemological and metaphysical consequences of that solution. One such consequence, it argues, is that (...) Russell must abandon his commitment to propositions. (shrink)

Goodman and Lederman (2020) argue that the traditional Fregean strategy for preserving the validity of Leibniz’s Law of substitution fails when confronted with apparent counterexamples involving proper names embedded under propositional attitude verbs. We argue, on the contrary, that the Fregean strategy succeeds and that Goodman and Lederman’s argument misfires.

A uniform theory of conditionals is one which compositionally captures the behavior of both indicative and subjunctive conditionals without positing ambiguities. This paper raises new problems for the closest thing to a uniform analysis in the literature (Stalnaker, Philosophia, 5, 269–286 (1975)) and develops a new theory which solves them. I also show that this new analysis provides an improved treatment of three phenomena (the import-export equivalence, reverse Sobel-sequences and disjunctive antecedents). While these results concern central issues in (...) the study of conditionals, broader themes in the philosophy of language and formal semantics are also engaged here. This new analysis exploits a dynamic conception of meaning where the meaning of a symbol is its potential to change an agent’s mental state (or the state of a conversation) rather than being the symbol’s content (e.g. the proposition it expresses). The analysis of conditionals is also built on the idea that the contrast between subjunctive and indicative conditionals parallels a contrast between revising and consistently extending some body of information. (shrink)

Let ℋ be a finite-dimensional complex Hilbert space and D the set of density matrices on ℋ, i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure u on D that can be regarded as the uniform distribution over D. We propose a measure on D, argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to (...) this measure. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.2 (2001) 259-288 [Access article in PDF] Substitutes for Wisdom:Kant's Practical Thought and the Tradition of the Temperaments Mark Larrimore [Appendix]For much of Western history, the theory of the four temperaments played a vital part in medicine, anthropology, and moral reflection. The Hippocratic foursome of sanguine, choleric, melancholy, and phlegmatic survives on the margins of modernity, but its role in moral theory and (...) practice has been largely forgotten. Premodern understandings of human diversity based in climate, temperament, and politics collapsed with the Galenic medical tradition with which they harmonized. Yet temperament continued to be a feature of moral philosophies and philosophical anthropologies even into the twentieth century, and arguably has had a symbiotic relationship with the egalitarian aspirations of modern moral and political thought. This essay surveys the development and practical consequences of the theory of the temperaments developed by one of the greatest prophets of modern values, Immanuel Kant.Temperament was a feature in Kant's course in anthropology from the 1760s until the end of his life. Kant's teachings on temperament are by turns creative and conservative, and bespeak a deep understanding of the logic and rationale of humoral characterology in ethics. But as Kant moves from an ethics based in feeling to an ethics based on the possibility of autonomy, his theory of temperament also changes. In 1764's Observations on the Feeling of the Beautiful and the Sublime, Kant eloquently celebrates the melancholy as the temperament capable of "genuine virtue" (Obs2:217-18/60); by the time of Anthropology from a Pragmatic Point of View (1798), he is commending instead "phlegma as strength" (a variant of a temperament he had earlier dismissed), which can serve as a "substitute for wisdom" (Ant7:290/155). 1 Both of these [End Page 259] positions are distinctive, especially for an eighteenth-century thinker. As we will see, however, they are not without precedent in the tradition of temperament theory. Whether Kant knew he was developing countertraditions of temperament or not, his arguments in each case creatively exploit tensions in the humoral system which go back to its origins in ancient Greece.The mature Kant defended the classic foursome at a time when it had come under fire from many quarters. The early nineteenth century historian of temperament theories Harro Wilhelm Dircksen thought Kant played a decisive role in saving the classical theory from oblivion. 2 Why did Kant think it worth retrieving? One would expect him to reject a category which eighteenth-century German Popularphilosophie understood as a proto-psychological category bridging the gap between body and mind, nature and freedom. Yet, as I shall argue, the very things which might seem to render a theory of temperament incompatible with his mature ethics of autonomy make it important to [End Page 260] Kant. An understanding of human embodiment and diversity like that of the tradition of the temperaments forms part of the background to Kant's practical thought, and ethical formalism makes more rather than less sense if difference is taken seriously. For the same reason, it is important to have the right account of diversity. The complicated view of temperament in the Anthropology furnishes a microcosm of Kant's understanding of the fraught role of empirical considerations in recognizing and promoting freedom in rational agents embodied in human form(s).In this essay I will (1) provide a brief survey of the tradition of temperament theorizing before Kant, and expound Kant's (2) early and (3) mature views of temperament in the context of this tradition, as well as in the context of his emerging practical philosophy. 3 Some well-known parts of Kant's ethics make a new kind of sense when read in the light of the fact that their author thought humanity subdivided into temperaments. 1. The Tradition of the Temperaments 1.1. The history of moralizing about and through the temperaments is complex and—once one moves beyond the melancholy—largely uncharted. Here I will... (shrink)