This article identifies problems with regard to providing criteria that regulate the matching of logical formulae and natural language. We then take on to solve these problems by defining a necessary and sufficient criterion of adequate formalization. On the basis of this criterion we argue that logic should not be seen as an ars iudicandi capable of evaluating the validity or invalidity of informal arguments, but as an ars explicandi that renders transparent the formal structure of informal reasoning.

One of the central logical ideas in Wittgenstein’sTractatus logico-philosophicusis the elimination of the identity sign in favor of the so-called “exclusive interpretation” of names and quantifiers requiring different names to refer to different objects and (roughly) different variables to take different values. In this paper, we examine a recent development of these ideas in papers by Kai Wehmeier. We diagnose two main problems of Wehmeier’s account, the first concerning the treatment of individual constants, the second concerning so-called “pseudo-propositions” (Scheinsätze) of (...) classical logic such as$a=a$or$a=b \wedge b=c \rightarrow a=c$. We argue that overcoming these problems requires two fairly drastic departures from Wehmeier’s account: (1) Not every formula of classical first-order logic will be translatable into asingleformula of Wittgenstein’s exclusive notation. Instead, there will often be a multiplicity of possible translations, revealing the original “inclusive” formulas to beambiguous. (2) Certain formulas of first-order logic such as$a=a$will not be translatable into Wittgenstein’s notation at all, being thereby revealed as nonsensical pseudo-propositions which should be excluded from a “correct” conceptual notation. We provide translation procedures from inclusive quantifier-free logic into the exclusive notation that take these modifications into account and define a notion of logical equivalence suitable for assessing these translations. (shrink)

Newton claims to have proven the heterogeneity of light through his experimentum crucis. However, Olaf Müller has worked out in detail Goethe’s idea that one could likewise prove the heterogeneity of darkness by inverting Newton’s famous experiment. Müller concludes that this invalidates Newton’s claim of proof. Yet this conclusion only holds if the heterogeneity of light and the heterogeneity of darkness is logically incompatible. This paper shows that this is not the case. Instead, in Quine’s terms, we have two logically (...) compatible theories based on mutually irreducible theoretical terms. From a Quinean point of view, this does no harm to the provability of the corresponding statements. (shrink)

Das Buch entwickelt einen neuartigen, physikalistischen Interpretationsansatz zu Wittgensteins Tractatus Logico-Philosophicus. Das traditionelle Urteil, Wittgenstein habe im Tractatus keine klare Vorstellung der Analyse gehabt, wird widerlegt. Auf der Basis der Rekonstruktion der um die Jahrhundertwende etablierten Sinnesdatenanalysen im allgemeinen und der Farbanalysen im besonderen wird nachgewiesen, daß Wittgensteins Tractatus eine physikalische Sinnesdatenanalyse voraussetzt. Auf diesem Hintergrund werden Wittgensteins allgemeine Auffassungen zur Analyse der Welt und Sprache gedeutet, begründet und exemplifiziert. Der Tractatus liefert die philosophische Klärung des mechanistischen Weltbildes von Boltzmann (...) und Hertz. Er stellt die Mittel bereit, um den Sinn der Sätze zu analysieren und die logische Zulässigkeit von Aussagen zu prüfen. Die Anwendung dieser Mittel ist Aufgabe der Philosophie. Daß sie anwendbar sind und wie sie anzuwenden sind, demonstriert dieses Buch. (shrink)

In philosophical contexts, logical formalisms are often resorted to as a means to render the validity and invalidity of informal arguments formally transparent. Since Oliver and Massey , however, it has been recognized in the literature that identifying valid arguments is easier than identifying invalid ones. Still, any viable theory of adequate logical formalization should at least reliably identify valid arguments. This paper argues that accounts of logical formalization as developed by Blau and Brun do not meet that benchmark. The (...) paper ends by suggesting different strategies to remedy the problem. (shrink)

According to some scholars, such as Rodych and Steiner, Wittgenstein objects to Gödel’s undecidability proof of his formula $$G$$, arguing that given a proof of $$G$$, one could relinquish the meta-mathematical interpretation of $$G$$ instead of relinquishing the assumption that Principia Mathematica is correct. Most scholars agree that such an objection, be it Wittgenstein’s or not, rests on an inadequate understanding of Gödel’s proof. In this paper, I argue that there is a possible reading of such an objection that is, (...) in fact, reasonable and related to Gödel’s proof. (shrink)

Es wird anhand von Fallbeispielen aus der Geschichte der Farbenlehre inwissenschaftstheoretische Probleme eingeführt. Das Buch dient als Grundlagefür eine anwendungsbezogene Lehre und als Einführung in folgende ThemenbereicheNewton vs. Goethe; Theorie und Experiment, Colormetrie; Empfindungsmessung;Helmholtz vs. Hering; Theorienevaluation, Psychologische Farbenlehre; Phänomenologie,Farbausschluss; Beweistheorie, Farbdefinitionen; Theorien- und Begriffsbildung.Neben Aufgaben, Texten und Lösungsvorschlägen finden sich eine Bibliographiesowie Einleitungen zu den behandelten Fragen und Lösungsvorschlägen derjeweiligen Themen.

In contrast to Hintikka’s enormously complex distributive normal forms of first- order logic, this paper shows how to generate minimized disjunctive normal forms of first-order logic. An effective algorithm for this purpose is outlined, and the benefits of using minimized disjunctive normal forms to explain the truth conditions of propo- sitions expressible within pure first-order logic are presented.

This paper systematically outlines Wittgenstein's ab-notation. The purpose of this notation is to provide a proof procedure in which ordinary logical formulas are converted into ideal symbols that identify the logical properties of the initial formulas. The general ideas underlying this procedure are in opposition to a traditional conception of axiomatic proof and are related to Peirce's iconic logic. Based on Wittgenstein's scanty remarks concerning his ab-notation, which almost all apply to propositional logic, this paper explains how to extend his (...) method to a subset of first-order formulas, namely, formulas that do not contain dyadic sentential connectives within the scope of any quantifier. (shrink)

This paper argues for a physicalistic interpretation of Wittgenstein's Tractatus Logico-Philosophicus. Wittgenstein's general conception of world and language analysis is interpreted and exemplified in relation to the historical background of the psychophysical analysis of sense data and, in particular, color analysis. Three of his main principles of analysis—the principle of independence, the context principle and the principle of atomism—are interpreted and justified on the background of physicalism. From his proof of color exclusion in the Tractatus, it is shown that Wittgenstein (...) had a detailed conception of how philosophy should fulfil the task of distinguishing between sense and nonsense using physicalistic presuppositions. (shrink)

Anhand der genaueren Analyse von Newtons experimentum crucis und der Argumentation, die er auf dieses Experiment stützt, sowie Goethes Kritik hieran sollen im Folgenden zwei verbreitete Vorurteile revidiert werden: -/- 1. Newton ist kein Dogmatiker, der methodische Ansprüche vertritt, die er nicht einlösen kann, sondern gründet seinen Anspruch, experimentelle Beweise führen zu können, auf einer vorbildlichen Methodologie kausaler Erklärungen, was seine Kritiker allerdings übersehen. 2. Goethe ist kein Antiwissenschaftler, der einen einzigartigen Kontrapunkt zur vorherrschenden wissenschaftlichen Tradition bildet, sondern steht inmitten (...) traditioneller Auffassungen zur Farbenlehre, deren experimentelle und methodologische Grundlagen bezüglich eines Erklärungsanspruches denen Newtons unterlegen sind. (shrink)

In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in (...) Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension. (shrink)

In his early philosophy as well as in his middle period, Wittgenstein holds a purely syntactic view of logic and mathematics. However, his syntactic foundation of logic and mathematics is opposed to the axiomatic approach of modern mathematical logic. The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; (...) its aim is to reduce mathematics to operations. This paper illustrates Wittgenstein’s approach by examining his discussion of irrational numbers. (shrink)

This paper illustrates what a philosophical and a logical investigation of colors amounts to in contrast to other kinds of color analysis such as physical, physiological, chemical, psychological or cultural analysis of colors. Neither a philosophical nor a logical analysis of colors is concerned with specific aspects of colors. Rather, these kinds of color analysis are concerned with what one might call “logical foundations of color theory”. I will illustrate this first by considering philosophical and then logical analysis of colors.

Es wird gezeigt, dass Wittgenstein in seiner Frühphilosophie ein nicht-axiomatisches Beweisverständnis entwickelt, für das sich das Problem der Begründung der Axiome nicht stellt. Nach Wittgensteins Beweisverständnis besteht der Beweis einer formalen Eigenschaft einer Formel – z.B. der logischen Wahrheit einer prädikatenlogischen Formel oder der Gleichheit zweier arithmetischer Ausdrücke – in der Transformation der Formel in eine andere Notation, an deren Eigenschaften sich entscheiden lässt, ob die zu beweisende formale Eigenschaft besteht oder nicht besteht. Dieses Verständnis grenzt Wittgenstein gegenüber einem axiomatischen (...) Beweisverständnis ab. Sein Beweisverständnis bedingt ein Programm der Grundlegung der Mathematik, das eine Alternative zu den Ansätzen des Logizismus, Formalismus und Konstruktivismus darstellt. Wittgensteins Ansatz steht im Widerspruch zu den Ergebnissen der Metamathematik, da er die Möglichkeit der Formulierung von Entscheidungsverfahren in der Prädikatenlogik und Arithmetik voraussetzt. Um seinem Ansatz gegenüber der traditionellen Metamathematik Recht zu geben, müsste gezeigt werden, dass sein Beweisverständnis im Bereich der Logik und Arithmetik – der traditionellen Metamathematik zum Trotz – realisierbar ist. (shrink)

This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on (...) nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs. The described algorithm illustrates how, in the case of Herbrand formulae, decision problems can be solved through a systematic search for proofs that reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I. Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction. (shrink)

This paper compares several models of formalization. It articulates criteria of correct formalization and identifies their problems. All of the discussed criteria are so called “semantic” criteria, which refer to the interpretation of logical formulas. However, as will be shown, different versions of an implicitly applied or explicitly stated criterion of correctness depend on different understandings of “interpretation” in this context.

The young Wittgenstein called his conception of logic “New Logic” and opposed it to the “Old Logic”, i.e. Frege’s and Russell’s systems of logic. In this paper the basic objects of Wittgenstein’s conception of a New Logic are outlined in contrast to classical logic. The detailed elaboration of Wittgenstein’s conception depends on the realization of his ab-notation for first order logic.

Paul Engelmann hat über zwanzig Jahre seines Lebens an einer systematischen Darstellung der Psychologie mittels einer von ihm entwickelten graphischen Methode gearbeitet. Das Resultat dieser Arbeit bildet seine Psychologie graphisch dargestellt, die sich in seinem Nachlaß befindet. In diesem Werk will Engelmann die Klärung geistiger Aufgabengebiete, wie sie seine Lehrer Karl Kraus, Adolf Loos und Ludwig Wittgenstein betrieben haben, in der Psychologie fortsetzen. Hierbei fiihrt er Freuds Methode weiter, psychische Erscheinungen räumlich darzustellen, und wendet die Bildtheorie Wittgensteins auf seine Theorie (...) psychischer Vorgänge an. (shrink)

Paul Engelmann hat über zwanzig Jahre seines Lebens an einer systematischen Darstellung der Psychologie mittels einer von ihm entwickelten graphischen Methode gearbeitet. Das Resultat dieser Arbeit bildet seine Psychologie graphisch dargestellt, die sich in seinem Nachlaß befindet. In diesem Werk will Engelmann die Klärung geistiger Aufgabengebiete, wie sie seine Lehrer Karl Kraus, Adolf Loos und Ludwig Wittgenstein betrieben haben, in der Psychologie fortsetzen. Hierbei fiihrt er Freuds Methode weiter, psychische Erscheinungen räumlich darzustellen, und wendet die Bildtheorie Wittgensteins auf seine Theorie (...) psychischer Vorgänge an. (shrink)

Das Buch vermittelt die Grundlagen der Aussagen- und erweiterten Prädikatenlogik in 12 Lektionen. Neben Techniken zum überprüfen der Schlüssigkeit von Argumenten bilden die Kunst des Formalisierens wissenschaftlicher Argumente und metalogische Fragen den Inhalt des Buches. Das Buch eignet sich in Verbindung mit begleitenden interaktiven Übungseinheiten und Klausuren, die ber Internet zugänglich sind, sowohl zum Selbststudium als auch für Einführungskurse in die Logik. Die zweite berarbeitete Auflage erscheint in einem größeren und besser lesbaren Format.

This paper discusses a definition of logical validity of arguments that is provided by Christoph Schamberger in his book Logik der Umgangssprache (2016). The paper identifies some problems of the given definition. It argues that any definition reducing the logical validity of ordinary arguments to the logical validity of their formalization does not meet the criteria of non-circularity, unambiguousness, correctness and completeness. Finally, the paper shows that Schamberger’s filter logic brings about that his definition does also not meet the standards (...) of formal precision and explanatory power. (shrink)

Newton claims that his theorems in the Opticks are derived from experiments alone. The paper explains this dictum by relating Newton’s proof method to an iconic conception of proof as opposed to a symbolic one. Theorems are not derived from hypotheses; instead properties of light are identified by experimental properties based on rules of inductive reasoning.

In this paper Richard’s Paradox and the Proof of Cantor’s Theorem are compared. It is argued that there is no conclusive reason to treat them differently such as to call the one a Paradox and the other a Proof.

This paper reveals two fallacies in Turing's undecidability proof of first-order logic (FOL), namely, (i) an 'extensional fallacy': from the fact that a sentence is an instance of a provable FOL formula, it is inferred that a meaningful sentence is proven, and (ii) a 'fallacy of substitution': from the fact that a sentence is an instance of a provable FOL formula, it is inferred that a true sentence is proven. The first fallacy erroneously suggests that Turing's proof of the non-existence (...) of a circle-free machine that decides whether an arbitrary machine is circular proves a significant proposition. The second fallacy suggests that FOL is undecidable. (shrink)

In vol. 2 of Grundlagen der Mathematik Hilbert and Bernays carry out their undecid- ability proof of predicate logic basing it on their undecidability proof of the arithmeti- cal systemZ00. In this paper, the latter proof is reconstructed and summarized within a formal derivation schema. Formalizing the proof makes the presumed use of a meta language explicit by employing formal predicates as propositional functions, with ex- pressions as their arguments. In the final section of the paper, the proof is analyzed (...) critically by applying Wittgenstein’s view on meta language, which does not lead to a questioning of the assumptions on which the proof is based but of its presumed use of meta language. Finally, it will be argued that determining if it is the undecidability proof or Wittgenstein’s analysis of meta language that is right depends on whether a decision procedure for a predicate that undecidability proofs have seemingly proven undecidable can nonetheless be defined. Thus serious attempts to define a procedure for such a predicate should not be ruled out without studying them. (shrink)