We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of (...) theorems with much wider scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction. (shrink)

Section 1 contains a survey of options in constructing a formal system of dialogue rules. The distinction between material and formal systems is discussed (section 1.1). It is stressed that the material systems are, in several senses, formal as well. In section 1.2 variants as to language form (choices of logical constants and logical rules) are pointed out. Section 1.3 is concerned with options as to initial positions and the permissibility of attacks on elementary statements. (...) The problem of ending a dialogue, and of infinite dialogues, is treated in section 1.4. Other options, e.g., as to the number of attacks allowed with respect to each statement, are listed in section 1.5. Section 1.6 explains the concept of a chain of arguments.From section 2 onward four types of dialectic systems are picked out for closer study: D, E, Di and Ei. After a preliminary section on dialogue sequents and winning strategies, the equivalence of derivability in intuitionistic logic and the existence of a winning strategy (for the Proponent) on the strength of Ei is shown by simple inductive proofs. (shrink)

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking just one (...) syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence. (shrink)

Section 1 contains a survey of options in constructing a formal system of dialogue rules. The distinction between material and formal systems is discussed (section 1.1). It is stressed that the material systems are, in several senses, formal as well. In section 1.2 variants as to language form (choices of logical constants and logical rules) are pointed out. Section 1.3 is concerned with options as to initial positions and the permissibility of attacks on elementary statements. (...) The problem of ending a dialogue, and of infinite dialogues, is treated in section 1.4. Other options, e.g., as to the number of attacks allowed with respect to each statement, are listed in section 1.5. Section 1.6 explains the concept of a chain of arguments.From section 2 onward four types of dialectic systems are picked out for closer study: D, E, Di and Ei. After a preliminary section on dialogue sequents and winning strategies, the equivalence of derivability in intuitionistic logic and the existence of a winning strategy (for the Proponent) on the strength of Ei is shown by simple inductive proofs. (shrink)

This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of (...) the laws of physics governing these processes. In accordance with the physicalist understanding of mind, this is true even if the operations in question are executed in the head. A truth obtained through (mathematical) reasoning is, therefore, an observed outcome of a neuro-physiological (or other physical) experiment. Consequently, deduction is nothing but a particular case of induction. (shrink)

This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of (...) the laws of physics governing these processes. In accordance with the physicalist understanding of mind, this is true even if the operations in question are executed in the head. A truth obtained through (mathematical) reasoning is, therefore, an observed outcome of a neuro-physiological (or other physical) experiment. Consequently, deduction is nothing but a particular case of induction. (shrink)

This paper investigates the effect of the countervailing forces within organizations of formal systems that direct employees toward ethical acts and informal systems that direct employees toward fraudulent behavior. We study the effect of these forces on deception, a key component of fraud. The results provide support for an interactive effect of these formal and informal systems. The effectiveness of formal systems is greater when there is a strong informal “push” to do wrong; (...) conversely, in the absence of a strong push to do wrong, the strength of formal systems has little impact on fraudulent behavior. These results help to explain why the implementation of formal systems within organizations has been met with mixed results and identifies when formal systems designed to promote ethical behavior will be most efficacious. (shrink)

Wilfried Sieg. Formal Systems, Church Turing Thesis, and Gödel's Theorems: Three Contributions to The MIT Encyclopedias of Cognitive Science.

In the paper [8], the first author developped a topos- theoretic approach to reference and modality. (See also [5]). This approach leads naturally to modal operators on locales (or spaces without points). The aim of this paper is to develop the theory of such modal operators in the context of the theory of locales, to axiomatize the propositional modal logics arising in this context and to study completeness and decidability of the resulting systems.

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements. It will be proved that if the weaker system of [2] is consistent, thenPis also consistent.After the (...) discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions. Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all. (shrink)

In the history of Newtonian Mechanics physicists and astronomers did not rely on so-called inertial frames, indeed they were not able to identify such frames. So the usual neo-Newtonian formalism of Newtonian Mechanics contains some superfluous components. In the present paper I will formulate a formal system for classical particle mechanics in Leibnizian space-time, where a relation, a counterpart of the second law of motion, between force on bodies and derivative of their momentum will be defined relative to every, (...) inertial or not, reference frame. And I will present a view that in the research process under Newtonian mechanics the accumulation of those models of the relation that satisfied some realistic conditions determined an additional structure of non-rotating frames to Leibnizian space-time. (shrink)

In recent years CL diagrams inspired by Lange’s Cubus Logicus have been used in various contexts of diagrammatic reasoning. However, whether CL diagrams can also be used as a formal system seemed questionable. We present a CL diagram as a formal system, which is a fragment of propositional logic. Syntax and semantics are presented separately and a variant of bitstring semantics is applied to prove soundness and completeness of the system.

A formal system for Ł ${}_{\infty}$ , based on a "game-theoretic" analysis of the Łukasiewicz propositional connectives, is defined and proved to be complete. An "Herbrand theorem" for the Ł ${}_{\infty}$ predicate calculus (a variant of some work of Mostowski) and some corollaries relating to its axiomatizability are proved. The predicate calculus with equality is also considered.

Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of (...) envisaging and analyzing at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic. (shrink)

Haskell Curry’s philosophy of mathematics is really a form of “structuralism” rather than “formalism” despite Curry’s own description of it as formalist (Seldin 2011). This paper explains Curry’s actual view by a formal analysis of a simple example. This analysis is extended to solve Keränen’s (2001) identity problem for structuralism, confirming Leitgeb’s (2020a, b) solution, and further clarifies structural ontology. Curry’s methods answer philosophical questions by employing a standard mathematical method, which is a virtue of the “methodological autonomy” emphasized (...) by Curry (1951, 1963) and more recently with greater clarity by Maddy (1997, 2007). (shrink)

One of the most interesting and entertaining philosophical discussions of the last few decades is the discussion between Daniel Dennett and John Searle on the existence of intrinsic intentionality. Dennett denies the existence of phenomena with intrinsic intentionality. Searle, however, is convinced that some mental phenomena exhibit intrinsic intentionality. According to me, this discussion has been obscured by some serious misunderstandings with regard to the concept ‘intrinsic intentionality’. For instance, most philosophers fail to realize that it is possible that the (...) intentionality of a phenomenon is partly intrinsic and partly observer relative. Moreover, many philosophers are mixing up the concepts ‘original intentionality’ and ‘intrinsic intentionality’. In fact, there is, in the philosophical literature, no strict and unambiguous definition of the concept ‘intrinsic intentionality’. In this article, I will try to remedy this. I will also try to give strict and unambiguous definitions of the concepts ‘observer relative intentionality’, ‘original intentionality’, and ‘derived intentionality’. These definitions will be used for an examination of the intentionality of formal mathematical systems. In conclusion, I will make a comparison between the (intrinsic) intentionality of formal mathematical systems on the one hand, and the (intrinsic) intentionality of human beings on the other hand. (shrink)

Solving numeric, logic and language puzzles and paradoxes is common within a wide community of high school and university students, fact witnessed by the increasing number of books published by mathematicians such as Martin Gardner, Douglas Hofstadter [in one of the best popular science books on paradoxes ], inspired by Gödel’s incompleteness theorems), Patrick Hughes and George Brecht and Raymond M. Smullyan, inter alia. Books by Smullyan are, however, much more involved, since they introduce learning trajectories and strategies across several (...) subjects of mathematical logic, as difficult as combinatorial logic, computability theory, and proof theory. These books provide solutions to their suggested exercises. Both statements and their solutions are written in the natural language, introducing some informal algorithms. As an exercise in Mathematics we wonder if an easy proof system could be devised to solve the amusing equations proposed by Smullyan in his books. Moreover, university students of logic could well train themselves in constructing deductive systems to solve puzzles instead of a non-uniform treatment one by one. In this paper, addressing students, we introduce one such formal systems, a tableaux approach able to provide the solutions to the puzzles involving either propositional logic, first order logic, or aspect logic. Let the reader amuse herself or himself! (shrink)

ZusammenfassungIn einem typenfreien formalen System, das widerspruchsfrei ist und alle Begriffe der klassischen Mathematik darzustellen vermag, können nicht alle Gesetze der klassischen Aussagenlogik gelten. Es entsteht daher das Problem, die aussagenlogischen Eigenschaften formaler Systeme allgemein zu untersuchen. Unter gewissen Voraussetzungen in aussagenlogischer Hinsicht wird die « aussagenlogische Vollständigkeit » und die « aussagenlogische Widerspruchsfreiheit » eines formalen Systems durch das Tertium non datur beziehungsweise durch die Schlussregel des Ex falso quodlibet charakterisiert. Es werden Entscheidungsverfahren für diejenigen syntaktischen Schlussregeln entwickelt, (...) die unter den vorliegenden aussagenlogischen Voraus‐setzungen in jedem formalen System beziehungsweise in jedem aussagen‐logisch vollständigen formalen System beziehungsweise in jedem aussagen‐logisch widerspruchsfreien formalen System gelten.RésuméDans un système formel sans types sans contradictions et qui pent représenter toutes les notions des mathkmatiques classiques, toutes les lois de la logiqrie propositionelle classique ne peuvent pas êvalables. C'est de là que s'ensuit le problème de discuter généralement les propriétés des systèmes formels en ce qu'elles concernent la logique des propositions. Sous certaines suppositions à I'Cgard des expressions propositionelles le « complet propositionel » et la « non‐contradiction propositionelle » se caractérisent par le tertium non datur ou plutôt par la rkgle d'inférence du ex falso quodlibet. On développe des méthodes de décision pour ces règles d'inférence syntactiques qui sont valables dans chaque système formel sous les suppositions données et pour ces régles d'inférence qui sont valables dans chaque système complet ou consistarit au point de vue de la logiqire des propositions.In a formal system without types which contains no contradictions and is able to represent all notions of classical mathematics, all the laws of classical propositional calculus cannot be valid. From this fact ensues the problem generally to explore the properties of formal systems from the point of view of the propositional calculus. Under certain assumptions conrerninq the propositional expressions the « propositional completeness » and the « propositional consistency » of a formal system is characterized through the tertium Ron datur, respectively through the inference rule of the ex falso quodlibet. There are decision procedures for the syntactical inference rules which in every formal system are valid under the given assumptions and for the syntactical inference rules of every formal system which is complete or consistent with respect to the propositional calculus. (shrink)

Lorenzen, in his book Einführung in die operative Logik und Mathematik has given a relatively precise form of syntactical system which he calls a calculus. The present paper deals with the relationship of Lorenzen's notion of calculus with the notion of formal system . It is shown that the obs of a formal system can be represented as the theses of a calculus of a certain type just when the calculus has a property called the tectonic property, and (...) conditions are given under which one form of system can be transformed into the other.ZusammenfassungLorenzen hat in seinem Buch, Einführung in die operative Logik und Mathematik, eine ziemlich genaue Auffassung eines syntaktischen Systems angegeben, die er Kalkül nennt. Die vorliegende Abhandlung betrifft die Verwandtschaft dieses Lorenzenschen Begriffs mit dem eines formalen Systems . Es wird bewiesen, dass die « Obs » eines formalen Systems sich als die Thesen eines passenden Kalküls genau dann repräsentieren lassen, wenn dieser Kalkül eine gewisse Eigenschaft, die « tectonic property » heisst, besitzt; die Bedingungen für die Überführung einer Art von System in die andere werden aufgezeigt.RésuméLorenzen, dans son livre Einführung in die operative Logik und Mathematik, a exposé une forme assez précise de systéme syntaxique qu'il a appelée un calcul. L'article actuel traite les relations entre cette idée et celle d'un système formel . L'auteur dèmontre que les « obs » d'un système formel peuvent se représenter comme les thèses d'un calcul d'une certaine espèce justement quand le calcul a une propriété que l'on appelle « the tectonic property »; et it énonce des conditions pour qu'un type de système soit transformable en l'autre. (shrink)

A formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. The absolute consistency of the system is proved. A completeness proof for the system is also constructed. This proof is relative to a concept of logical validity provided by a semantics, which assumes as its philosophical background an approach to sortals from a modern form of conceptualism.