The idea at the basis of this special issue is that reopening the old debate about the logical status of Hegel’s dialectics is extremely interesting, for various reasons.1 The first reason is that...

The tools of logic are used properly or improperly relative to two interrelated purposes. Logic is both a symbolism for the expression of the formal structures of thought and an inference mechanism. Formalization in philosophical logic is justified to the extent that it contributes to our understanding of logical properties and the conceptual problems they may help to state, clarify, or resolve. This view of the value and limits of formalization in logic affords a pragmatic perspective that in (...) principle should selectively support the development of particular formal systems, while excluding others as unjustified. Yet the pragmatic grounds are so liberal as to disallow virtually no exercises in formalization as entirely useless. Logic is abused in another sense, when it is wrongly used, not by violating logical canons generally, but by presupposing that substantive metaphysical, normative, or scientific content can be derived from purely formal logical relations. (shrink)

This paper can be treated as a simplification of the Gentzen formalization of SCI-tautologies presented by A. Michaels in [1].

The article addresses two closely related questions: What are the criteria of adequacy of logical formalization of natural language arguments, and what gives logic the authority to decide which arguments are good and which are bad? Our point of departure is the criticism of the conception of logical formalization put forth, in a recent paper, by M. Baumgartner and T. Lampert. We argue that their account of formalization as a kind of semantic analysis brings about more problems (...) than it solves. We also argue that the criteria of adequate formalization need not be based on truth conditions associated with logical formulas; in our view, they are better based on structural (inferential) grounds. We then put forward our own version of the criteria. The upshot of the discussion that follows is that the quest for an adequate formalization in a suitable logical language is best conceived of as the search for a Goodmanian reflective equilibrium. (shrink)

This paper contains a careful derivation of principles of Interpretability Logic valid in extensions of I0+1.

This article discusses some of Chateaubriand’s views on the connections between the ideas of formalization and infinity, as presented in chapters 19 and 20 of Logical Forms. We basically agree with his criticisms of the standard construal of these connections, a view we named “formal proofs as ultimate provings”, but we suggest an alternative way of picturing that connection based on some ideas of the late Wittgenstein.

We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which (...) are algebraic and even metamathematical. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem. Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity. (shrink)

This paper deals with Tarski's first axiomatic presentations of the syntax of deductive system. Andrzej Grzegorczyk's significant results which laid the foundations for the formalization of metalogic, are touched upon briefly. The results relate to Tarski's theory of concatenation, also called the theory of strings, and to Tarski's ideas on the formalization of metamathematics. There is a short mention of author's research in the field. The main part of the paper surveys research on the theory of deductive systems (...) initiated by Tarski, in particular research on the axiomatization of the general notion of consequence operation, axiom systems for the theories of classic consequence and for some equivalent theories, and axiom systems for the theories of nonclassic consequence. In this paper the results of Jerzy Supecki's research are taken into account, and also the author's and other people belonging to his circle of scientific research. Particular study is made of his dual characterization of deductive systems, both as systems in regard to acceptance and systems in regard to rejection . Comparison is made, therefore, with axiomatizations of the theories of rejection and dual consequence, and the theory of the usual consequence operation. (shrink)

Although Kant (1998) envisaged a prominent role for logic in the argumentative structure of his Critique of Pure Reason, logicians and philosophers have generally judged Kantgeneralformaltranscendental logics is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first-order logic. The main technical application of the formalism developed here is a formal proof that Kants logic is after all a distinguished subsystem of first-order logic, namely what (...) is known as geometric logic. (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 paper offers a historical survey of the emergence of logical formalization in twentieth-century analytically oriented philosophy of religion. This development is taken to have passed through three main ?stages?: a pioneering stage in the late nineteenth and early twentieth centuries (led by Frege and Russell), a stage of crisis in the 1920s and early 1930s (occasioned by Wittgenstein, logical positivists such as Carnap, and neo-Thomists such as Maritain), and a stage of rehabilitation in the 1930s, 1940s, and 1950s (...) (led by the Cracow Circle and Quine). (shrink)

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent’s knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the logic: one (...) based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to a notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz’s solution. (shrink)

Explication is the conceptual cornerstone of Carnap’s approach to the methodology of scientific analysis. From a philosophical point of view, it gives rise to a number of questions that need to be addressed, but which do not seem to have been fully addressed by Carnap himself. This paper reconsiders Carnapian explication by comparing it to a different approach: the ‘formalisms as cognitive tools’ conception. The comparison allows us to discuss a number of aspects of the Carnapian methodology, as well as (...) issues pertaining to formalization in general. We start by introducing Carnap’s conception of explication, arguing that there is a tension between his proposed criteria of fruitfulness and similarity; we also argue that his further desideratum of exactness is less crucial than might appear at first. We then bring in the general idea of formalisms as cognitive tools, mainly by discussing the reliability of so-called statistical prediction rules, i.e. simple algorithms used to make predictions across a range of areas. SPRs allow for a concrete instantiation of Carnap’s fruitfulness desideratum, which is arguably the most important desideratum for him. Finally, we elaborate on what we call the ‘paradox of adequate formalization’, which for the Carnapian corresponds to the tension between similarity and fruitfulness. We conclude by noting that formalization is an inherently paradoxical enterprise in general, but one worth engaging in given the ‘cognitive boost’ it affords as a tool for discovery. (shrink)

This paper explores the political import of Husserl’s critical discussion of the epistemic effects of the formalization of rational thinking. More specifically, it argues that this discussion is of direct relevance to make sense of the pervasive processes of ‘technization’, that is, of a mechanistic and superficial generation and use of knowledge, to be observed in current contexts of governance. Building upon Husserl’s understanding of formalization as a symbolic technique for abstraction in the thinking with and about numbers, (...) I argue that processes of technization, while being necessary and legitimate procedures for the reduction of complexities, also may give rise to politically unresponsive and ultimately dysfunctional ‘economies of thinking.’ This paper is structured in three parts. In the first part I outline Husserl’s account of the formalization and technization of thought and knowledge. In the second part I make my case for the political import of this account, departing in this context from positions that (a) regard Husserl’s discussions of formalization and its effects as merely epistemological, or that (b) try to mobilize Husserl for a one-sided critique of instrumental reason. In the final part I address a major shortcoming of Husserl’s account, namely its neglect of the concrete and historically evolving technological infrastructures of processes of formalization/technization. (shrink)

Mainstream economics has been accused of excessive mathematization, whereas the rejection of mathematical and other formal methods is often cited as a crucial trait of Austrian economics. Based on a systematic discussion of potential benefits and drawbacks of formalization, this paper corroborates legitimate concerns that predominant types of mathematization induce a shift of attention away from the key concepts of Austrian economics. Taking this shift to the extreme, predominant modes of mathematization tend to accompany a detachment from ‘reality’ incompatible (...) with Austrian pleas for realisticness. Contrary to popular prejudice however, the most prominent representatives of the Austrian School including Carl Menger, Ludwig Mises, Friedrich Hayek, Israel Kirzner, and Peter Boettke neither provide a justification for a wholesale rejection of formalization nor actually reject it. Adequate formalization can serve as a remedy for lacking logical and semantic rigor in standard mathematical economics as well as in murky verbal chains of reasoning. (shrink)

The mathematical proof is the most important form of justification in mathematics. It is not, however, the only kind of justification for mathematical propositions. The existence of other forms, some of very significant strength, places a question mark over the prominence given to proof within mathematics. This collection of essays, by leading figures working within the philosophy of mathematics, is a response to the challenge of understanding the nature and role of the proof.