Constructive geometrical reasoning and diagrams

Modern formal accounts of the constructive nature of elementary geometry do not aim to capture the intuitive or concrete character of geometrical construction. In line with the general abstract approach of modern axiomatics, nothing is presumed of the objects that a geometric construction produces. This study explores the possibility of a formal account of geometric construction where the basic geometric objects are understood from the outset to possess certain spatial properties. The discussion is centered around Eu, a recently developed formal system of proof (presented in Mumma (Synthese 175:255?C287, 2010)) within which Euclid??s diagrammatic proofs can be represented.     

A Gentzen Calculus for Nothing but the Truth

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof.     

Analytic Tableaux for all of <Emphasis Type="Italic">SIXTEEN</Emphasis><Subscript>3</Subscript>

In this paper we give an analytic tableau calculus P L _{1 6} for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ? _t , ? _f , ? _i , and ? under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment relations will in general require developing four tableaux, while proving that they are in the ? relation may require six.     

Strategic Maneuvering in Mathematical Proofs

This paper explores applications of concepts from argumentation theory to mathematical proofs. Note is taken of the various contexts in which proofs occur and of the various objectives they may serve. Examples of strategic maneuvering are discussed when surveying, in proofs, the four stages of argumentation distinguished by pragma-dialectics. Derailments of strategies (fallacies) are seen to encompass more than logical fallacies and to occur both in alleged proofs that are completely out of bounds and in alleged proofs that are at least mathematical arguments. These considerations lead to a dialectical and rhetorical view of proofs.

Michael Wolff über Kant als Logiker. Eine Stellungnahme zu Wolffs Metakritik

In an earlier article (see J Gen Philos Sci (2009) 40: 357–372) I have discussed the arguments brought forward by Michael Wolff against the interpretation given in the commentary by Ebert and Nortmann on Aristotle’s syllogistic theory (Aristoteles Analytica Priora Buch I, übersetzt und erläutert von Theodor Ebert und Ulrich Nortmann. Berlin 2007) and against the critique of Kant’s adaption of the syllogistic logic. I have dealt with Wolff’s arguments concerning (Ebert/Nortmann’s interpretation of) Aristotle in the paper mentioned and with his attempts to defend his critique in this subsequent article (part 1; see J Gen Phils Sci (2010) 41: 215–231). Part 2 (the paper below) is concerned with Wolff’s renewed attempts to defend Kant as a logician. In particular I point out that if, as Wolff claims, the nota notae relation in Kant is restricted to subordinated concepts, then it can hardly serve as a principle for syllogistic logic, as Kant claims. Against Wolff’s attempts to defend Kant’s claim that o-propositions are simpliciter convertible, I point out two arguments: (1) Even if Kant, following the Vernunftlehre by Meier, has assumed that an o-proposition can be turned into an i-proposition, this conversion is useless for the reduction to first figure syllogisms since we are no longer dealing with three syllogistic terms but with four. (2) It is quite unlikely that Kant has a conversion of this type in mind since the texts of his students always talk of the group of either the particular propositions or else of the negative propositions. Given Kant’s mistakes concerning the convertibility simpliciter of o-propositions, it is no wonder that he overlooks the special status of the moods Baroco and Bocardo. Wolff’s attempts to provide Kant with what he claims are direct proofs for these moods can be shown to rely on a reductio ad impossibile. Kant mistook what are parts of the proofs for the validity of moods in figures two to four as parts of these moods themselves. Wolff—who tries to defend Kant on this point—is forced to an artificial and unconvincing reading of the Kantian texts.     

Quantification in Some Non-normal Modal Logics

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems by Stolpe (Logic Journal of the IGPL 11(5), 557–575, 2003), but unfortunately that work used only models and not frames.     

Understanding Negation Implicationally in the Relevant Logic R

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory (display calculus). It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework.     

The Modal Logic of Gödel Sentences

The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of ‘Essence and Accident’ proposed by Marcos (Bull Sect Log 34(1):43–56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic.     

Proof and Dialogue in Aristotle

Jan ?ukasiewicz’s analysis of Aristotle’s syllogism drew attention to the nature of syllogisms as conditionals rather than premise-conclusion arguments. His further idea that syllogisms should be understood as theorems of an axiom system seems a step too far for many logicians. But there is evidence to suggest that Aristotle’s syllogism was to regularise some of the steps made in ‘dialogue games.’ This way of seeing the syllogism is explored in the framework of modern formal dialogue systems. A modern formal syllogistic game, DLSyll, is set out and analysed in use.     

A Completeness Proof of Kiczuk’s Logic of Physical Change

In this paper the class of minimal models C ^ZI for Kiczuk’s system of physical change ZI is provided and soundness and completeness proofs of ZI with respect to these models are given. ZI logic consists of propositional logic von Wright’s And Then and six specific axioms characterizing the meaning of unary propositional operator “Zm”, read “there is a change in the fact that”. ZI is intended to be a logic which provides a formal account for describing two kinds of process change: the change from one state of the process to its other state (e.g., transmitting or absorbing energy with greater or less than the usual intensity) and the perishing of the process (e.g., cessation of the energetic activity of the sun).     

Turing–Taylor Expansions for Arithmetic Theories

Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you “hit” the target theory. Turing progressions based on n-consistency give rise to a \({\Pi_{n+1}}\) proof-theoretic ordinal \({|U|_{\Pi^0_{n+1}}}\) also denoted \({|U|_n}\). As such, to each theory U we can assign the sequence of corresponding \({\Pi_{n+1}}\) ordinals \({\langle |U|_n\rangle_{n > 0}}\). We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories of Peano Arithmetic to Ignatiev’s universal model for the closed fragment of the polymodal provability logic \({\mathsf{GLP}_\omega}\). In particular, we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expansion will define a unique point in Ignatiev’s model.     

Admissible Bases Via Stable Canonical Rules

We establish the dichotomy property for stable canonical multi-conclusionrules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.     

Pi on Earth, or Mathematics in the Real World

We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like.

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On All Strong Kleene Generalizations of Classical Logic

By using the notions of exact truth (‘true and not false’) and exact falsity (‘false and not true’), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the \({{\bf SK}^\mathcal{P}}\) and \({\bf SK}^{\mathcal{N}}\) calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, we also hint at its philosophical significance.     

<Emphasis Type="Italic">Wh</Emphasis>-Questions,Universal Statements and Free Choice Inferences in Child Mandarin

This study investigated 5-year-old Mandarin-speaking children’s comprehension of wh-questions, universal statements and free choice inferences. Previous research has found that Mandarin-speaking children assign a universal interpretation to sentences with a wh-word (e.g., shei ‘who’) followed by the adverbial quantifier dou ‘all’ (Zhou in Appl Psycholinguist 36:411–435, 2013). Children also compute free choice inferences in sentences that contain a modal verb in addition to a wh-word and dou (Zhou, in: Nakayama, Su, Huang (eds.) Studies in Chinese and Japanese language acquisition: in honour of Stephen Crain. John Benjamins Publishing Company, Amsterdam, pp 223–235, 2017). The present study used a Question-Statement Task to assess children’s interpretation of sentences containing shei + dou, both with and without the modal verb beiyunxu ‘was allowed to’, as well as the contrast between sentences with shei + dou, which are statements for adults, versus ones with dou + shei, which are wh-questions for adults. The 5-year-old Mandarin-speaking child participants exhibited adult-like linguistic knowledge of the semantics and pragmatics of wh-words, the adverbial quantifier dou, and the deontic modal verb beiyunxu.     

Principles for Object-Linguistic Consequence: from Logical to Irreflexive

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by (Beall and Murzi, Journal of Philosophy, 3 pp. 143–65 (2013)) that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion of logical consequence: we study the corresponding theory of validity (and some of its variants) by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth (strong Kleene version). Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence.     

Obligation as Optimal Goal Satisfaction

Formalising deontic concepts, such as obligation, prohibition and permission, is normally carried out in a modal logic with a possible world semantics, in which some worlds are better than others. The main focus in these logics is on inferring logical consequences, for example inferring that the obligation O q is a logical consequence of the obligations O p and O (p → q). In this paper we propose a non-modal approach in which obligations are preferred ways of satisfying goals expressed in first-order logic. To say that p is obligatory, but may be violated, resulting in a less than ideal situation s, means that the task is to satisfy the goal p ∨ s, and that models in which p is true are preferred to models in which s is true. Whereas, in modal logic, the preference relation between possible worlds is part of the semantics of the logic, in this non-modal approach, the preference relation between first-order models is external to the logic. Although our main focus is on satisfying goals, we also formulate a notion of logical consequence, which is comparable to the notion of logical consequence in modal deontic logic. In this formalisation, an obligation O p is a logical consequence of goals G, when p is true in all best models of G. We show how this non-modal approach to the treatment of deontic concepts deals with problems of contrary-to-duty obligations and normative conflicts, and argue that the approach is useful for many other applications, including abductive explanations, defeasible reasoning, combinatorial optimisation, and reactive systems of the production system variety.     

Learning multiple rules simultaneously: Affixes are more salient than reduplications

Language learners encounter numerous opportunities to learn regularities, but need to decide which of these regularities to learn, because some are not productive in their native language. Here, we present an account of rule learning based on perceptual and memory primitives (Endress, Dehaene-Lambertz, & Mehler, Cognition, 105(3), 577–614, 2007; Endress, Nespor, & Mehler, Trends in Cognitive Sciences, 13(8), 348–353, 2009), suggesting that learners preferentially learn regularities that are more salient to them, and that the pattern of salience reflects the frequency of language features across languages. We contrast this view with previous artificial grammar learning research, which suggests that infants “choose” the regularities they learn based on rational, Bayesian criteria (Frank & Tenenbaum, Cognition, 120(3), 360–371, 2013; Gerken, Cognition, 98(3)B67–B74, 2006, Cognition, 115(2), 362–366, 2010). In our experiments, adult participants listened to syllable strings starting with a syllable reduplication and always ending with the same “affix” syllable, or to syllable strings starting with this “affix” syllable and ending with the “reduplication”. Both affixation and reduplication are frequently used for morphological marking across languages. We find three crucial results. First, participants learned both regularities simultaneously. Second, affixation regularities seemed easier to learn than reduplication regularities. Third, regularities in sequence offsets were easier to learn than regularities at sequence onsets. We show that these results are inconsistent with previous Bayesian rule learning models, but mesh well with the perceptual or memory primitives view. Further, we show that the pattern of salience revealed in our experiments reflects the distribution of regularities across languages. Ease of acquisition might thus be one determinant of the frequency of regularities across languages.     

Handling Inconsistencies in the Early Calculus

The early calculus is a popular example of an inconsistent but fruitful scientific theory. This paper is concerned with the formalisation of reasoning processes based on this inconsistent theory. First it is shown how a formal reconstruction in terms of a sub-classical negation leads to triviality. This is followed by the evaluation of the chunk and permeate mechanism (C&P) proposed by Brown and Priest in (Journal of Philosophical Logic, 33(4), 379–388, 2004) to obtain a non-trivial formalisation of the early infinitesimal calculus. Different shortcomings of this application of C&P as an explication of inconsistency tolerant reasoning are pointed out, both conceptual and technical. To remedy these shortcomings, an adaptive logic is proposed that allows for conditional permeations of formulas under the assumption of consistency preservation. First the adaptive logic is defined and explained and thereafter it is demonstrated how this adaptive logic remedies the defects C&P suffered from.     

Grammatical Aspect in Early Child Mandarin: Evidence from a Preferential Looking Experiment

The study assessed 30-month-old Mandarin-speaking children’s awareness of aspectual distinctions involving the perfective marker le and the imperfective marker zhe in a preferential looking experiment. In the experiment, we presented our child subjects with a choice between two video clips (one depicting a closed event and the other depicting an on-going event), in the presence of an auditory stimulus (either the le sentence, the zhe sentence or the control sentence without any aspect marker). Children’s looking behavior in the task was recorded and analyzed. The results revealed 30-month-old children’s emerging sensitivity to the aspectual contrast between le and zhe. This was manifest by an increase in looking to the closed event when hearing the le sentence and an increase in looking to the on-going event when hearing the zhe sentence. The absence of le or zhe in the control sentence did not result in any increase or decrease in looking to either event. We also found that the effect of le on children’s looking behavior was immediate whereas the effect of zhe was late. We attributed this difference to the facilitative role of le in children’s sentence processing as well as their preference for the event boundary. The results lend support to the continuity view that functional morphemes like aspect markers are available to children early in language development.