Proof Theory

A simplified framework for first-order languages and its formalization in Mizar

by Marco Caminati

Eliminacija reza i strukturna teorija dokaza logika drugog reda

by Sandro Skansi

Exposition for the thesis summary defense

Godelov prvi teorem nepotpunosti 2

by Sandro Skansi

Godelov prvi teorem nepotpunosti 1

by Sandro Skansi

Quantitative Analysis of Pre-service Elementary Mathematics Teachers’ Opinions about Doing Mathematical Proof

by Halil Eksi

Tuba AYDOĞDU İSKENDEROĞLU Adnan BAKİ
Educational Sciences: Theory & Practice - 11(4) • Autumn • 2285-2290

Meaning and importance of proof in mathematics and education increases gradually. Therefore levels of doingmore Meaning and importance of proof in mathematics and education increases gradually. Therefore levels of doing
proof, proof-related opinions and perceptions of the teachers and pre-service teachers who will train the
students in future are of importance. Accordingly, this study aims to determine the proof-related opinions of
pre-service elementary mathematics teachers who still study at different grade levels. In line with this purpose,
a questionnaire developed under the title “Questionnaire for Constructing Mathematical Proof” was used to determine
the pre-service teachers’ opinions about proof. The questionnaire comprises 27 items based on 5 point
Likert-type. In the study, developmental research method was conducted and the questionnaire was applied
to 187 pre-service elementary mathematics teachers from different grade levels. As a result of the study, it was
revealed that pre-service teachers have positive views about proof. Also, the study revealed that confidence of
pre-service teachers in proving is lower than mental process, self-assessment and belief, and attitude factors.

An Attention Based Theory to Explore Affordances of Textual and Diagrammatic Proofs

by Peter Coppin

Coppin, P.W., Burton, J., & Hockema, S.A. (2010). An attention based theory to explore affordances of textual and diagrammatic proofs. In A. Goel, M. Jamnik, N. Narayanan (Eds.), Diagrammatic Representation and Inference, Lecture Notes in Computer Science (Vol. 6170, pp. 271-278). Berlin / Heidelberg: Springer.

Shimojima and Katagiri have demonstrated that diagrams reduce “inferential load” during reasoning by scaffolding... more Shimojima and Katagiri have demonstrated that diagrams reduce “inferential load” during reasoning by scaffolding visual-spatial aspects of memory. In response, we wondered why, if this is true, that proofs are usually text based? The purpose of this paper is to explore ergonomic affordances of text that may encourage its use in the communication of proofs by building on prior work in attention. We claim that textual notations may focus a reasoner’s “spotlight” of attention through serialized sequential chunks, whereas many diagrams may “diffuse” attention and that a diagrammatic notation system that serialized information in chunks amenable to focused attention could leverage the power of textual notations. We present such an example through a case study focused on generalized constraint diagrams, a visual logic with attributes that may support focused attention and extract ergonomic principles that may transcend each notation system.

The structure of logical consequence: proof-theoretic conceptions

by Ole Hjortland

PhD Thesis, University of St Andrews, 2009

The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple... more The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple of decades. The present work looks at an alternative approach to logical consequence where the notion of inference takes center stage. Formally, the model-theoretic framework is exchanged for a proof-theoretic framework. It is argued that contrary to the traditional view, proof-theoretic semantics is not revisionary, and should rather be seen as a formal semantics that can supplement model-theory. Specifically, there are formal resources to provide a proof-theoretic semantics for both intuitionistic and classical logic.

We develop a new perspective on proof-theoretic harmony for logical constants which incorporates elements from the substructural era of proof-theory. We show that there is a semantic lacuna in the traditional accounts of harmony. A new theory of how inference rules determine the semantic content of logical constants is developed. The theory weds proof-theoretic and model-theoretic semantics by showing how proof-theoretic rules can induce truth-conditional clauses in Boolean and many-valued settings. It is argued that such a new approach to how rules determine meaning will ultimately assist our understanding of the apriori nature of logic.

Logical Pluralism, Meaning-Variance, and Verbal Disputes

by Ole Hjortland

Journal of Australasian Philosophy (2012)

Logical pluralism has been in vogue since JC Beall and Greg Restall 2006 articulated and defended a new pluralist... more Logical pluralism has been in vogue since JC Beall and Greg Restall 2006 articulated and defended a new pluralist thesis. Recent criticisms such as Priest 2006a and Field 2009 have suggested that there is a relationship between their type of logical pluralism and the meaning-variance thesis for logic. This is the claim, often associated with Quine 1970, that a change of logic entails a change of meaning. Here we explore the connection between logical pluralism and meaning-variance, both in general and for Beall and Restall's theory specifically. We argue that contrary to what Beall and Restall claim, their type of pluralism is wedded to meaning-variance. We then develop an alternative form of logical pluralism that circumvents at least some forms of meaning-variance.

Inferentialism and the categoricity problem: reply to Raatikainen

by Ole Hjortland

Analysis, July 2009, with Julien Murzi.

Harmony, Normality and Stability

by Nils Kürbis

As the title says: my account of proof-theoretic harmony, normality and stability!

Gentzen mentions that it should be possible to specify a function that maps introduction rules onto elimination rules... more Gentzen mentions that it should be possible to specify a function that maps introduction rules onto elimination rules in systems of natural deduction. This paper specifies such a function. I specify two kinds of rules, one in which it is more natural to assume an introduction rule to be given and elimination rules are determined from it, and another kind in which it is an elimination rule which is given and the introduction rules are determined from it. The process also works the other way round, so that it doesn't really matter which rules are supposed to be given first. The process is very general and applies to a large class of logics. The paper begins with a discussion of the philosophical importance of this in connection with the notion of harmony. I discuss Dummett's ideas on harmony and stability, which is supposed to be stronger than harmony. Dummett suggests that normalisability is a formal criterion of harmony. However, he seems to aim at something else, and this criterion does not give an independent formally precise notion of stability. I propose formally precise definitions of harmony and stability, which are distinct from normalisability. My aim is not exegetical, and according to my definitions, classical as well as intuitionist logics count as governed by stable (and hence harmonious) rules of inference.

Internalization: The case of hybrid logics

by Jeremy Seligman

SELIGMAN, J. ‘Internalization: The Case of Hybrid Logics’ Journal of Logic and Computation, Vol. 11, 671-689, 2001.

A sequent calculus for hybrid logics is developed from a calculus for classical predicate logic by a series of... more A sequent calculus for hybrid logics is developed from a calculus for classical predicate logic by a series of transformations. We formalize the semantic theory of hybrid logic using a sequent calculus for predicate logic plus axioms. This works, but it is ugly. The unattractive features are removed one-by-one, until the final vestiges of the metalanguage can be set aside to reveal a fully internalized calculus. The techniques are quite general and can be applied to a wide range of hybrid and modal logics.

The Four-Color Map Theorem: "Kempe’s Fallacious Proof Repaired"

by Ibrahim Cahit

Further results can be found in the first 6 pages of my paper at arXiv :
http://arxiv.org/abs/0903.4108

A new non-computer direct algorithmic proof for the famous four color theorem based on new concept spiral-chain... more A new non-computer direct algorithmic proof for the famous four color theorem based on new concept spiral-chain coloring of maximal planar graphs has been proposed by the author in 2004 [6],[13]. Historical fallacious inductive proof of Kempe have been re-considered by many mathematicians whether it could be repaired. All attemps so far have been either modification of Kempe color switching argument or trying to show that random second-time coloring would not produce an impasse. In this
note we have shown that when Kempe’s argument fails by the trap of the incomplete four-coloring there is always a simple re-coloring of the nodes of a planar graph so that the undecided node colored properly. Hence our method may be considerd as an completion of fallacious Kempe’s inductive proof. Interesting enough, when we have resolved the impasse in the four coloring of the graphs, the solution end up again with two spirals (double-spirals) Kempe chains that cover all of the nodes.

A cognitive exploration of the “non-visual” nature of geometric proofs

by Peter Coppin

Coppin, P.W., & Hockema, S.A. (2009). A cognitive exploration of the “non-visual” nature of geometric proofs. In P. Cox, A. Fish, J. Howse (Eds.), Visual Languages and Logic, CEUR Workshop Proceedings (Vol. 510, pp. 81-95). Aachen, Germany: RWTH Aachen University.

Why are Geometric Proofs (Usually) “Non-Visual”? We asked this question as a way to explore the similarities and... more Why are Geometric Proofs (Usually) “Non-Visual”? We asked this question as a way to explore the similarities and differences between diagrams and text (visual thinking versus language thinking). Traditional text-based proofs are considered (by many to be) more rigorous than diagrams alone. In this paper we focus on human perceptual-cognitive characteristics that may encourage textual modes for proofs because of the ergonomic affordances of text relative to diagrams. We sug- gest that visual-spatial perception of physical objects, where an object is perceived with greater acuity through foveal vision rather than peripheral vision, is similar to attention navigating a conceptual visual-spatial structure. We suggest that at- tention has foveal-like and peripheral-like characteristics and that textual modes appeal to what we refer to here as foveal-focal attention, an extension of prior work in focused attention.

«A Philosophical Justification of Many-Valued Extensions of Classical Logic», apud Philosophie et culture, comp. por Venant Cauchy

by Lorenzo Peña

What is Wrong with Classical Negation?

by Nils Kürbis

The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on... more The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on consideration about the meanings of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett’s outlook on the theory of meaning. In particular, I shall grant Dummett his general bias towards verificationism or justificationism, encapsulated in the slogan ‘meaning is use’. The second general assumption I see no need to question is Dummett’s particular breed of molecularism. Some of Dummett’s assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inferences in the Dummettian framework.

Pluralism and the Logical Basis of Metaphysics

by Nils Kürbis

A paper I gave at Logica 2006.

Quantification in ordinary language: from a critic of set-theoretic approaches to a proof-theoretic proposal

by Christian Retoré

with Vito Michele Abrusci
14th Congress of Logic, Mthodology and Philosophy of Science 2011

ON THE SYSTEM F AS A GLUE LANGUAGE FOR NATURAL-LANGUAGE COMPOSITIONAL-SEMANTICS

by Christian Retoré

Some arguments supporting the quantification over types that we use in many papers in compositional and lexical samentics and pragmatics.
To be included into a paper, to become a squib... don't know yet.

In order to model in a compositional framework some phenomena of lexical pragmatics
and in particular the ones... more In order to model in a compositional framework some phenomena of lexical pragmatics
and in particular the ones studied by Nicholas Asher
several contributions developed in our team did use the system F of Jean-Yves Girard
to construct logical formulae expressing the meaning of sentences
--- while other authors prefer to use Per Martin-Löf's type theory.
In this note we explain the motivations supporting our preference for system F.

On the Interpretation of the Propositional Calculus

by Tristan Haze

A fairly polished draft

The question considered is 'How can formulae of the propositional calculus be brought into a representational relation... more The question considered is 'How can formulae of the propositional calculus be brought into a representational relation with the world?'. Four approaches are discussed: (1) the denotational approach, on which formulae are taken to denote objects, (2) the abbreviational approach, on which formulae and connectives are taken to abbreviate natural-language expressions, (3) the truth-conditional approach, on which truth-conditions are stipulated for formulae, and (4) the modelling approach, on which formulae, together with either valuation- or proof-theory, are regarded as an abstract structure capable of bearing (via stipulation) a representational relation to the world.

The modelling approach is developed here for the first time. The simple technical apparatus used for this is then applied to two issues in the philosophy of logic. (1) I demonstrate a corollary or converse to Carnap's result that certain 'non-normal' valuation-functions can be added to the set of admissible valuations of formulae without destroying the soundness and completeness of standard proof-theories. This sheds considerable light on a recent thread of the inferentialism debate which involves dialectical use of Carnap's result. (2) I show how the approach can be extended to quantification theory, by defining a model-theoretic notion of validity equivalent to the usual one, but making use of a proof-theoretic apparatus in place of the device of assigning values to formulae. This sheds light on the close relationship between proof- and valuation-theory.

Spiraling the Proof: A Tale of Two Conjectures

by Ibrahim Cahit

Abstract and a figure only.