In at least a few of his formulations of the pragmatic maxim around 1905—those in which he sought to inoculate his brand of pragmatism against misappropriation by other pragmatists and also to supply a demonstration of its truth—Charles Peirce instructs us to look not only at the consequences of affirming some claim or concept, but also at the consequences of denying it. Referring to himself in the third person as "the author," Peirce writes: Endeavoring, as a man of that type (...) naturally would, to formulate what he so approved, he framed the theory that a conception, that is, the rational purport of a word or other expression, lies exclusively in its conceivable bearing on the conduct of life; so that, since... (shrink)

The paper suggests a revision of the notion of harmony, a major necessary condition in proof-theoretic semantics for a natural-deduction proof-system to qualify as meaning conferring, when moving to a bilateral proof-system. The latter considers both forces of assertion and denial as primitive, and is applied here to positive logics, lacking negation altogether. It is suggested that in addition to the balance between (positive) introduction and elimination rules traditionally imposed by harmony, a balance should be imposed also on: (...) (i) negative introduction and elimination rules, and (ii) positive and negative introduction rules. The paper suggests a proof-theoretical definition of duality (not referring to truthtables), using which double harmony is defined. The paper proves that in a doubly-harmonious system, the coordination rule, typical to bilateral systems, is admissible. (shrink)

The paper suggests a revision of the notion of harmony, a major necessary condition in proof-theoretic semantics for a natural-deduction proof-system to qualify as meaning conferring, when moving to a bilateral proof-system. The latter considers both forces of assertion and denial as primitive, and is applied here to positive logics, lacking negation altogether. It is suggested that in addition to the balance between introduction and elimination rules traditionally imposed by harmony, a balance should be imposed also on: negative (...) introduction and elimination rules, and positive and negative introduction rules. The paper suggests a proof-theoretical definition of duality, using which double harmony is defined. The paper proves that in a doubly-harmonious system, the coordination rule, typical to bilateral systems, is admissible. (shrink)

Ian Rumfitt (2000) developed a bilateralist account of logic in which the meaning of the connectives is given by conditions on asserted and rejected sentences. An additional set of inference rules, the coordination principles, determines the interaction of assertion and rejection. Fernando Ferreira (2008) found this account defective, as Rumfitt must state the coordination principles for arbitrary complex sentences. Rumfitt (2008) has a reply, but we argue that the problem runs deeper than he acknowledges and is in fact related to (...) the challenge of establishing proof-theoretic harmony. We motivate a distinctively bilateral criterion for harmony and show how the bilateralist can meet it. This also resolves Ferreira's complaint. (shrink)

Bilateral proof systems, which provide rules for both affirming and denying sentences, have been prominent in the development of proof-theoretic semantics for classical logic in recent years. However, such systems provide a substantial amount of freedom in the formulation of the rules, and, as a result, a number of different sets of rules have been put forward as definitive of the meanings of the classical connectives. In this paper, I argue that a single general schema for bilateral (...) proof rules has a reasonable claim to inferentially articulating the core meaning of all of the classical connectives. I propose this schema in the context of a bilateral sequent calculus in which each connective is given exactly two rules: a rule for affirmation and a rule for denial. Positive and negative rules for all of the classical connectives are given by a single rule schema, harmony between these positive and negative rules is established at the schematic level by a pair of elimination theorems, and the truth-conditions for all of the classical connectives are read off at once from the schema itself. (shrink)

I argue that one need not be an inferentialist in order to model inconsistent concepts, contrary to what some have thought. Representationalists can do so by adopting a form of bilateralism about contents. It remains unclear, however, why conceptual inconsistency would constitute a defect to be eliminated, rather than a vindication of dialetheism to be embraced. I suggest some answers to explore that involve accepting a descriptive form of dialetheism but denying its normative forms.

This paper provides a proof-theoretic account of imperative logical consequence by generalising Greg Restall’s multiple conclusion bilateralism for declarative logic. According to imperative bilateralism, a sequent Γ ⊢ Δ is valid iff jointly commanding all the imperatives Φ ∈ Γ and prohibiting all the imperatives Ψ ∈ Δ clashes. This account has three main virtues: (1) it provides a proof-theoretic account of imperatives; (2) it does not rely on the controversial notion of imperative inference; and (3) it is neutral regarding (...) cognitivism about imperatives. (shrink)

Proof-theoretic semantics is an inferentialist theory of meaning originally developed in a unilateral framework. Its extension to bilateral systems opens both opportunities and problems. The problems are caused especially by Coordination Principles (a kind of rule that is not present in unilateral systems) and mismatches between rules for assertion and rules for rejection. In this paper, a solution is proposed for two major issues: the availability of a reduction procedure for tonk and the existence of harmonious rules for (...) the paradoxical zero-ary connective \(\bullet\). The solution is based on a reinterpretation of bilateral rules as complex rules, that is, rules that introduce or eliminate connectives in a subordinate position. Looking at bilateral rules from this perspective, the problems faced by bilateralism can be seen as special cases of general problems of complex systems, which have been already analyzed in the literature. In the end, a comparison with other proposed solutions underlines the need for further investigation in order to complete the picture of bilateral proof-theoretic semantics. (shrink)

The theory of mental models considers sentential connectives to refer to semantic possibilities. The theory also proposes that the possibilities can be amended by virtue of modulation processes, that is, processes in which semantics or pragmatics can have an influence. The definition of modulation the theory of mental models gives has been deemed as a scientific definition within Carnap’s framework. This is because it is easy to find reduction sentences corresponding to it. What seems to be harder is to (...) think about bilateral reduction sentences for that definition. This paper addresses this issue. It shows that at least a bilateral reduction sentence is possible for modulation. (shrink)

The acquisition and evolution of speech production, discourse and communication can be negatively impacted by brain malformations. We describe, for the first time, a case of developmental dynamic dysphasia (DDD) in a right-handed adolescent boy (subject D) with cortical malformations involving language-eloquent regions (inferior frontal gyrus) in both the left and the right hemispheres. Language evaluation revealed a markedly reduced verbal output affecting phonemic and semantic fluency, phrase and sentence generation and verbal communication in everyday life. Auditory comprehension, repetition, naming, (...) reading and spelling were relatively preserved, but executive function was impaired. Multimodal neuroimaging showed a malformed cerebral cortex with atypical configuration and placement of white matter tracts bilaterally and abnormal callosal fibers. Dichotic listening showed right hemisphere dominance for language and functional magnetic resonance imaging (fMRI) additionally revealed dissociated hemispheric language representation with right frontal activation for phonology and bilateral dominance for semantic processing. Moreover, subject D also had congenital mirror movements (CMM), defined as involuntary movements of one side of the body that mirror intentional movements of the other side. Transcranial magnetic stimulation and fMRI during voluntary unimanual (left and right) hand movements showed bilateral motor cortex recruitment and tractography revealed a lack of decussation of bilateral corticospinal tracts. Genetic testing aimed to detect mutations that disrupt the development of commissural tracts correlating with CMM (e.g., Germline DCC mutations) was negative. Overall, our findings suggest that DDD in subject D resulted from the underdevelopment of the left inferior frontal gyrus with limited capacity for plastic reorganization by its homologous counterpart in the right hemisphere. Corpus callosum anomalies probably contributed to hinder interhemispheric connectivity necessary to compensate language and communication deficits after left frontal involvement. (shrink)

Ian Rumfitt has proposed systems of bilateral logic for primitive speech acts of assertion and denial, with the purpose of ‘exploring the possibility of specifying the classically intended senses for the connectives in terms of their deductive use’ : 810f). Rumfitt formalises two systems of bilateral logic and gives two arguments for their classical nature. I assess both arguments and conclude that only one system satisfies the meaning-theoretical requirements Rumfitt imposes in his arguments. I then formalise an intuitionist (...) system of bilateral logic which also meets those requirements. Thus Rumfitt cannot claim that only classical bilateral rules of inference succeed in imparting a coherent sense onto the connectives. My system can be extended to classical logic by adding the intuitionistically unacceptable half of a structural rule Rumfitt uses to codify the relation between assertion and denial. Thus there is a clear sense in which, in the bilateral framework, the difference between classicism and intuitionism is not one of the rules of inference governing negation, but rather one of the relation between assertion and denial. (shrink)

We discuss a well-known puzzle about the lexicalization of logical operators in natural language, in particular connectives and quantifiers. Of the many logically possible operators, only few appear in the lexicon of natural languages: the connectives in English, for example, are conjunction _and_, disjunction _or_, and negated disjunction _nor_; the lexical quantifiers are _all, some_ and _no_. The logically possible nand (negated conjunction) and Nall (negated universal) are not expressed by lexical entries in English, nor in any natural language. Moreover, (...) the lexicalized operators are all upward or downward monotone, an observation known as the Monotonicity Universal. We propose a logical explanation of lexical gaps and of the Monotonicity Universal, based on the dynamic behaviour of connectives and quantifiers. We define update potentials for logical operators as procedures to modify the context, under the assumption that an update by \( \phi \) depends on the logical form of \( \phi \) and on the speech act performed: assertion or rejection. We conjecture that the adequacy of update potentials determines the limits of lexicalizability for logical operators in natural language. Finally, we show that on this framework the Monotonicity Universal follows from the logical properties of the updates that correspond to each operator. (shrink)

Contents: 1. Introduction , 2. Overviews , 3.History and major works, 3.1 Gerhard Gentzen and proof-theory, 3.2 Wilfrid Sellars, 3.3 Gilbert Harman, 3.4 Christopher Peacocke, 3.5 Robert Brandom , 3.6 Paul Horwich, 3.7 Major works by other authors, 4. Mental content first vs. linguistic meaning first, 4.1 Content-first views, 4.2 Meaning-first views, 5. Wide vs. narrow CRS, 5.1 Overviews and major works about externalism/internalism, 5.2 Discussions about externalism within CRS, 6. Descriptive vs. normative CRS, 6.1 Overviews and major works about (...) the normativity of meaning, 6.2 The debate on normativity within CRS, 7. Holistic vs. non-holistic CRS, 7.1 Overviews of meaning holism and compositionality, 7.2 Overviews and major works about analyticity, 7.3 Criticisms against holism , 7.4 Replies to the criticisms, 8. Unilateral vs. bilateral CRS , 9. Connections between CRS and meta-ethics. (shrink)

In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic \(\mathbf{K}\), whose values lie in the closed interval \([0,1]\) of rational numbers [14]. In this paper, after clarifying our conception of bilateralism – dubbed “soft bilateralism” – we generalize the fractional method to encompass extensions and weakenings of \(\mathbf{K}\). Specifically, we introduce well-behaved hypersequent calculi for the deontic logic \(\mathbf{D}\) and the non-normal modal (...) logics \(\mathbf{E}\) and \(\mathbf{M}\) and thoroughly investigate their structural properties. (shrink)

Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material (...) inference and Greg Restall’s bilateralist interpretation of the multiple conclusion sequent calculus, I present a proof-theoretic semantics for atomic sentences and their constituent names and predicates. The resulting system has several interesting features: (1) the rules are harmonious and stable; (2) the rules create a structure analogous to familiar model-theoretic semantics; and (3) the semantics is compositional, in that the rules for atomic sentences are determined by those for their constituent names and predicates. (shrink)

Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material (...) inference and Greg Restall’s bilateralist interpretation of the multiple conclusion sequent calculus, I present a proof-theoretic semantics for atomic sentences and their constituent names and predicates. The resulting system has several interesting features: (1) the rules are harmonious and stable; (2) the rules create a structure analogous to familiar model-theoretic semantics; and (3) the semantics is compositional, in that the rules for atomic sentences are determined by those for their constituent names and predicates. (shrink)

We consider an approach to propositional synonymy in proof-theoretic semantics that is defined with respect to a bilateral G3-style sequent calculus \(\mathtt{SC2Int}\) for the bi-intuitionistic logic \(\mathtt{2Int}\). A distinctive feature of \(\mathtt{SC2Int}\) is that it makes use of two kind of sequents, one representing proofs, the other representing refutations. The structural rules of \(\mathtt{SC2Int}\), in particular its cut rules, are shown to be admissible. Next, interaction rules are defined that allow transitions from proofs to refutations, and vice versa, (...) mediated through two different negation connectives, the well-known implies-falsity negation and the less well-known coimplies-truth negation of \(\mathtt{2Int}\). By assuming that the interaction rules have no impact on the identity of derivations, the concept of inherited identity between derivations in \(\mathtt{SC2Int}\) is introduced and the notions of positive and negative synonymy of formulas are defined. Several examples are given of distinct formulas that are either positively or negatively synonymous. It is conjectured that the two conditions cannot be satisfied simultaneously. (shrink)

Natural language contains simple lexical items for some but not all Boolean operators. English, for example, contains conjunction and, disjunction or, negated disjunction nor, but no word to express negated conjunction *nand nor any other Boolean connective. Natural language grammar can be described by a logic that expresses what the lexicon can express by its primitives, and the rest compositionally. Such logic for propositional connectives is described here as a bilateral extension of update semantics. The basic intuition is (...) that a context can be updated by assertion or by rejection, and by one or multiple propositions at once. These distinctions suffice to characterize the logic of the lexicon. (shrink)

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections (...) of definite formulae and use the operational view of them as provided by uniform proof-search—the proof-theoretic foundation of logic programming (LP)—to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS. (shrink)

Two approaches to the study of diagnostic thinking are compared, one mainly propositional, namely that of Patel and Groen (1986), the other mainly semantic, that of Lemieux and Bordage (1986). Patel and Groen analyzed the linear dimension of cardiologists' discourses while solving a case of acute bacterial endocarditis, that is, the before and after propositional rules. A secondary analysis of two of their pothophysiological protocols is done using structural semantic techniques from Lemieux and Bordage where the vertical dimension of the (...) discourses is analyzed, that is, the levels of meaning or semantic axes. Contrary to Patel and Groen's position, making an accurate diagnosis is not explained in terms of pure forward reasoning through networks of causal rules, but by means of networks of semantic qualities abstracted from the symptoms and signs. The semantic qualities operate in the clinician's mind in terms of binary oppositions (e.g., sudden-gradual, unilateral-bilateral) whereby each pair of properties constitutes a semantic axis (e.g., sudden-gradual). The successful diagnosticians are those who use the most diversified and pertinent set of semantic axes and, therefore, have a deeper representation of the problem. (shrink)

Reading disability has been considered as a disconnection syndrome. Recently, an increasing number of studies have emphasized the role of subcortical regions in reading. However, the majority of research on reading disability has focused on the connections amongst brain regions within the classic cortical reading network. Here, we used graph theoretical analysis to investigate whether subcortical regions serve as hubs during reading both in Chinese children with reading disability and in age-matched typically developing children using a visual rhyming judgment task (...) and a visual meaning judgment task. We found that the bilateral thalami were the unique hubs for typically developing children across both tasks. Additionally, subcortical regions were also unique hubs for typically developing children but only in the rhyming task. Among these subcortical hub regions, the left pallidum showed reduced connectivity with inferior frontal regions in the rhyming judgment but not semantic task in reading disabled compared with typically developing children. These results suggest that subcortical-cortical disconnection, which may be particularly relevant to the phonological and phonology-related learning process, may be associated with Chinese reading disability. (shrink)

This book presents a philosophical conception of logic -- "logical expressivism"-- according to which the role of logic is to make explicit reason relations, which are often neither monotonic nor transitive. It reveals new perspectives on inferential roles, sequent calculi, representation, truthmakers, and many extant logical theories.