DOCSLIB.ORG

Modal Logic As Metalogic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modal Logic As Metalogic Modal Logic as Metalogic KOSTA DOZEN Matemati~ki lnstitut, Knez Mihailova 35, 11001 Belgrade, p f. 367, Yugoslavia (Received 24 September, 1991; in final form 14 July, 1992) Abstract. The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's. First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. In this family one finds systems that correspond to the associative Lambek calculus, linear logic, relevant logics, BCK logic and intuitionistic logic. Above these basic systems, sequent systems parallel to the basic systems are constructed, which formalize various notions of derived rules for the basic systems. The deduction theorem is provable for the basic systems if, and only if, they are at least as strong as systems corresponding to linear logic, or BCK logic, depending on the language, and their deductive metalogic is not stronger than they are. However, though we do not always have the deduction theorem, we may always obtain a modal analogue of the deduction theorem for conservative modal extensions of the basic systems. Modal postulates which are necessary and sufficient for that are postulates of $4 plus modal postulates which mimic structural rules. For example, the modal postulates which Girard has recently considered in linear logic are necessary and sufficient for the modal analogue of the deduction theorem. All this may lead towards results about functional completeness in categories. When functional completeness, which is analogous to the deduction theorem, fails, we may perhaps envisage a modal analogue of functional completeness in a modal category, of which our original category is a full subcategory. Key words: modal logic, substructural logics, derived rules, deduction theorem, functional completeness in categories. 1. INTRODUCTION Logicians sometimes make a distinction between derived rules of a logical system and rules admissible for a logical system. A rule is derived if we can pass from the premises to the conclusion by using the axioms and primitive rules of the system. To say that a rule is admissible means that if the premises are theorems, then the conclusion is also a theorem; i.e., that adding this rule to the system would not increase the stock of theorems. An altemative way to say that a rule is admissible for a system is to say that the system is closed under this rule. Every derived rule is admissible, but not the other way round. A famous example of an admissible rule that is not derived is cut in Gentzen's cut-free sequent formulations of classical and intuitionistic logic. The distinction between derived and admissible rules is important when we Journal of Logic, Language, ana Information 1:173-201, 1992. (~) 1992 Kluwer Academic Publishers. Printed in the Netherlands. 174 KOSTADO~EN want to prove a deduction theorem. Roughly speaking, a deduction theorem says that a derived, but not any admissible role, can be recorded as an arrow in the system. A derived rule is one we wish to hold not only for our system but for all its extensions as well, whereas our system may just happen to be closed under an admissible rule, and this need not hold for all its extensions. The notion of an admissible rule should not cause misunderstanding here. We will take the 'if .... then ...' involved in the definition of admissible rules as a material implication. (There is a related notion of a constructively admissible rule, where from a proof of the premises we can construct a proof of the conclusion, the construction not necessarily taking place inside our system; cf. Anderson and Belnap (1975: w 54; w With this notion, 'if .... then ...' should better be taken as an intuitionistic, or even weaker implication. To prevent confusion, we shall not rely on this notion here.) On the other hand, we shall allow derived rules to make us perplexed. Suppose: is a derived rule. Is then: X also derived for any X? Classical and intuitionistic logic force us to say yes, because we have: ~(X~) X X ---~ 9~ But what if our logic lacks cp ---+ (X ~ ~), as relevant logic notoriously does? Should we then feel obliged to accept in the metalogic a principle corresponding to ~ ~ (X ~ ~P), which would make the second rule above derived? So it seems that depending on what logic we accept as our metalogic we may end up with different notions of derived rule. In Section 1 of this paper we explore various notions of derived rule. We first formulate a family of propositional logical systems, starting with a variant of the nonassociative Lambek calculus, and ending up with a system corresponding to intuitionistic logic. In between we have systems correspond- ing to the associative Lambek calculus, linear logic, relevant logics and BCK logic. These basic systems make a family because they may all be obtained by gradually adding structural rules in a suitable sequent formulation, whereas rules for the connectives are kept invariant as in Dogen (1988) (such logics are MODAL LOGIC AS METALOGIC 175 called substructural in Dogen (1992)). The nonassociative Lambek calculus, which is minimal in this family, is a good candidate for a minimal implicative logic. Inspired by Lambek and Scott (1986), we formulate these basic systems so that their theorems are sequents of the form A ~- t3. This we do because we have in mind possible extensions of our approach to categories, where A F-/3 will be a morphism from A to/3. In principle, we could have presented our basic systems as more orthodox Gentzen-type sequent systems, but the main things we want to say can also be said with our simpler sequents A ~- /3. When the turnstile t- in A ~-/3 is replaced by the connective of implication -% our basic systems look like ordinary Hilbert-type systems, but we should then bear in mind that the rules for these systems are rules for theorems, and that every implication A --+ B standing for A F- B is prefixed with F-. All primitive rules of our basic systems will be binary, i.e., with two premises. We require this because we then build, above our basic systems, sequent systems especially well suited to represent deductions with binary rules. These sequent systems have a binary comma between formulae on the left of their tumstile, rather than the n-ary commas of usual Gentzen formu- lations. (This type of sequent system was, so far as I know, first considered in Lambek (1961), and was further studied in Belnap (1982) and Dogen (1988).) We get various sequent systems by gradually adding structural rules, redu- plicating in this manner the family of the underlying basic systems. In these sequent systems we formalize various notions of derived rule for our basic systems. So, above a basic system we may assume, as its deductive metalogic, a sequent system weaker, equal in strength, stronger, or incomparable, with our basic system. The fact that there are various notions of derived rule has repercussions on the deduction theorem: with some notions we can prove it and with others we cannot. In Section 2 we then investigate what conditions are necessary and sufficient to prove the deduction theorem for our basic systems. These conditions also happen to be sufficient for translating the metalogic into the basic logic. Our deduction theorem is what is also in ordinary logical usage properly called a deduction theorem. It is a deduction theorem of higher level: one of its instances says that if from t- A we can derive k- B, then A t- B is provable in the basic system. On the other hand, an instance of a deduction theorem of lower level, better called introduction of implication, says that if we have A k-/3, then we have ~- A ~ B, which is already incorporated in our basic systems. These two types of deduction theorem are easy to confuse in ordinary versions of classical and intuitionistic propositional logic, where we have the higher-level deduction theorem and can translate the metalogic into the basic logic. But already in formulations of predicate logic with a rule of universal generalization, or in formulations of modal logic with a rule of 17 6 KOSTADO~EN necessitation, we can make the distinction, because the higher-level deduction theorem falls in full generality, whereas the lower-level one obtains with a proper understanding of F, like in Gentzen's sequent systems (cf. Kleene, 1952: w167 As usual, these sequent systems are interpreted as if they were about natural deduction. From F A we can derive F Vz A and F DA, but we shouldn't have A F Vx A and A F DA. We shall see with our weak basic systems that this distinction between the higher-level and lower-level deduction theorem should be drawn even in what amounts to implicational logic. Some specific problems for the deduction theorem arise if we allow our systems to be formulated with a primitive rule of substitution for variables. We shall not consider here this matter, which would get us involved in a discussion about the proper understanding of variables. For the propositional systems we shall introduce, substitution, though admissible, is not a derived rule. In Section 3 we extend all we have done previously to a richer language for our basic systems, with the usual connectives of propositional logic. Next, in Section 4, which is the central section of this paper, we show that though we don't always have the deduction theorem, we may always obtain a modal analogue of the deduction theorem for modal extensions of our basic systems.
Load more

Recommended publications
  • PHIL 677 Lec 01 Metalogic Fall Term 2019 MW 14:00–15:15, Math Sciences 319
    PHIL 677 Lec 01 Metalogic Fall Term 2019 MW 14:00–15:15, Math Sciences 319 Course Outline Instructor: Richard Zach (he/him) Email: [email protected] Oce: Social Sciences 1230 Oce Hours: MW 13:00–13:50, or by appointment Phone: (403) 220–3170 Teaching Assistant: Weidong Sun (he/him) Email: [email protected] Oce: Social Sciences 1242 Oce Hours: T 15:40-16:40, R 10:00–11:00 Phone: (403) 220–6464 Email is preferred over phone. However please keep the following in mind: 1. Please ensure that “Phil 677” or some other clearly identifying term occurs in the subject line. Otherwise there is a strong possibility that your message will be deleted unread as spam. 2. If you email to make an appointment please indicate the times when you are available. 3. Please make sure your first and last names are clearly included in the body of any email message. 4. We will not respond to email after 7pm or on weekends. 1 Course Information Formal logic has many applications both within philosophy and outside (espe- cially in mathematics, computer science, and linguistics). This second course will introduce you to the concepts, results, and methods of formal logic neces- sary to understand and appreciate these applications as well as the limitations of formal logic. It will be mathematical in that you will be required to master abstract formal concepts and to prove theorems about logic; but it does not presuppose any (advanced) knowledge of mathematics. We will start from the basics. We will begin by studying some basic formal concepts: sets, relations, and functions, and the sizes of infinite sets.
    [Show full text]
  • Notes on Incompleteness Theorems
    The Incompleteness Theorems ● Here are some fundamental philosophical questions with mathematical answers: ○ (1) Is there a (recursive) algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is true? ○ (2) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of Peano or Robinson Arithmetic? ○ (3) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of pure (first-order) logic? ○ (4) Is there a complete (even if not recursive) recursively axiomatizable theory in the language of first-order arithmetic? ○ (5) Is there a recursively axiomatizable sub-theory of Peano Arithmetic that proves the consistency of Peano Arithmetic (even if it leaves other questions undecided)? ○ (6) Is there a formula of arithmetic that defines arithmetic truth in the standard model, N (even if it does not represent it)? ○ (7) Is the (non-recursively enumerable) set of truths in the language of first-order arithmetic categorical? If not, is it ω-categorical (i.e., categorical in models of cardinality ω)? ● Questions (1) -- (7) turn out to be linked. Their philosophical interest depends partly on the following philosophical thesis, of which we will make frequent, but inessential, use. ○ Church-Turing Thesis:A function is (intuitively) computable if/f it is recursive. ■ Church: “[T]he notion of an effectively calculable function of positive integers should be identified with that of recursive function (quoted in Epstein & Carnielli, 223).” ○ Note: Since a function is recursive if/f it is Turing computable, the Church-Turing Thesis also implies that a function is computable if/f it is Turing computable.
    [Show full text]
  • On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem
    On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem Yannick Forster Dominik Kirst Gert Smolka Saarland University Saarland University Saarland University Saarbrücken, Germany Saarbrücken, Germany Saarbrücken, Germany [email protected] [email protected] [email protected] Abstract like decidability, enumerability, and reductions are avail- We formalise the computational undecidability of validity, able without reference to a concrete model of computation satisfiability, and provability of first-order formulas follow- such as Turing machines, general recursive functions, or ing a synthetic approach based on the computation native the λ-calculus. For instance, representing a given decision to Coq’s constructive type theory. Concretely, we consider problem by a predicate p on a type X, a function f : X ! B Tarski and Kripke semantics as well as classical and intu- with 8x: p x $ f x = tt is a decision procedure, a function itionistic natural deduction systems and provide compact д : N ! X with 8x: p x $ ¹9n: д n = xº is an enumer- many-one reductions from the Post correspondence prob- ation, and a function h : X ! Y with 8x: p x $ q ¹h xº lem (PCP). Moreover, developing a basic framework for syn- for a predicate q on a type Y is a many-one reduction from thetic computability theory in Coq, we formalise standard p to q. Working formally with concrete models instead is results concerning decidability, enumerability, and reducibil- cumbersome, given that every defined procedure needs to ity without reference to a concrete model of computation. be shown representable by a concrete entity of the model.
    [Show full text]
  • Classical Metalogic an Introduction to Classical Model Theory & Computability
    Classical Metalogic An Introduction to Classical Model Theory & Computability Notes by R.J. Buehler January 5, 2015 2 Preface What follows are my personal notes created in preparation for the UC-Berkeley Group in Logic preliminary exam. I am not a computability theorist, nor a model theorist; I am a graduate student with some knowledge who is–alas–quite fallible. Accordingly, this text is made available as a convenient reference, set of notes, and summary, but without even the slight hint of a guarantee that everything contained within is factual and correct (indeed, some areas are entirely unchanged from the moment I copied them off the blackboard). This said, if you find a serious error, I would greatly appreciate it if you would let me know so that it can be corrected. The material for these notes derives from a wide variety of sources: Lectures by Wes Holliday Lectures by Antonio Montalban Lectures by John Steel Kevin Kelly’s computability theory notes David Marker’s “Model Theory: An Introduction” Wilfrid Hodge’s “A Shorter Model Theory” Robert Soare’s “Recursively Enumerable Sets and Degrees" Richard Kaye’s “Models of Peano Arithmetic” Chang and Keisler’s “Model Theory" While I certainly hope my notes are beneficial, if you’re attempting to learn the contained material for the first time, I would highly suggest picking up (at least) a copy of Marker and Soare’s texts in addition. To those Group in Logic students who may be using these notes to help themselves prepare for their preliminary exam, chapters 1-5, 7, and 9-18 contain relevant material, as well as chapter 8, section 3.
    [Show full text]
  • METALOGIC METALOGIC an Introduction to the Metatheory of Standard First Order Logic
    METALOGIC METALOGIC An Introduction to the Metatheory of Standard First Order Logic Geoffrey Hunter Senior Lecturer in the Department of Logic and Metaphysics University of St Andrews PALGRA VE MACMILLAN © Geoffrey Hunter 1971 Softcover reprint of the hardcover 1st edition 1971 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. First published 1971 by MACMILLAN AND CO LTD London and Basingstoke Associated companies in New York Toronto Dublin Melbourne Johannesburg and Madras SBN 333 11589 9 (hard cover) 333 11590 2 (paper cover) ISBN 978-0-333-11590-9 ISBN 978-1-349-15428-9 (eBook) DOI 10.1007/978-1-349-15428-9 The Papermac edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent, in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. To my mother and to the memory of my father, Joseph Walter Hunter Contents Preface xi Part One: Introduction: General Notions 1 Formal languages 4 2 Interpretations of formal languages. Model theory 6 3 Deductive apparatuses. Formal systems. Proof theory 7 4 'Syntactic', 'Semantic' 9 5 Metatheory. The metatheory of logic 10 6 Using and mentioning. Object language and metalang- uage. Proofs in a formal system and proofs about a formal system. Theorem and metatheorem 10 7 The notion of effective method in logic and mathematics 13 8 Decidable sets 16 9 1-1 correspondence.
    [Show full text]
  • MAT 440: Mathematical Logic
    Program Cover Sheet --- MAT 440: Mathematical Logic I. Basic Course Information MAT 440: Mathematical Logic is an upper level course scheduled for two 80-minute lecture periods. Sophomore standing or above is a prerequisite. Every upper level math course is designed in part to increase a student’s ability to reason logically, and this 400-level Mathematical Logic course is no different. Despite its name, however, it is not particularly designed for this goal. Instead, this course is designed to take a mathematical and philosophical view at the mechanisms of logical reasoning. There will be three major themes: the process, characteristics and limits of logical reasoning. Much of the mathematical content and processes that will be used in this course will already have been introduced somewhere else in the curriculum, most directly in our Discrete Mathematics which serves as an introduction to techniques of proof. The most notable exception to this is that we will require some knowledge of set theory, in particular, infinite sets and cardinalities. II. Learning Goals The learning goals of this course reflect its major themes. Students will learn the process of logic by exploring propositional and first-order logic languages. Students will learn how to work inside these languages to derive proofs of statements. Secondly, students will study these languages from outside, in the language of metalogic. Here, students will learn the characteristics of these languages. For example, the different notions of ‘truth’ and ‘proof’ will be compared and contrasted. Lastly, students will learn the limits of mathematics in the context of completeness and consistency.
    [Show full text]
  • Mental Metalogic: a New Paradigm in Psychology of Reasoning
    Mental MetaLogic: A New Paradigm in Psychology of Reasoning Yingrui Yang & Selmer Bringsjord [email protected] [email protected] Dept. of Philosophy, Psychology &• Cognitive Science (S.B. & Y.Y.) The Minds & Machines Laboratory (S.B. & Y.Y.) Department of Computer Science (S.B.) Rensselaer Polytechnic Institute Troy NY USA Abstract mental logic theory claims that logically untrained hu- mans reason using domain-independent inference rules We introduce a new theory of reasoning, Mental Meta- or schemas (e.g., that from p q and p one can infer to q; Logic (MML), which uni®es the mental logic and men- proponents of mental logic would_ say: that something like tal models theories currently deadlocked in opposition to each other. MML, as we explain, is based on the ªpsychol- this particular rule is operative in the case of our hypo- ogizingº of all of symbolic logic Ð which includes a syn- thetical Smith). On the other hand, mental model theory tactic component (proof theory) with which mental logic holds that ordinary humans reason using not abstract syn- is associated, a semantic component (model theory and tactic rules, but semantic possibilities they in some sense its relatives) with which mental models is associated, and a ªmetaº component (metatheory or metalogic), hitherto imagine. Scientists on both sides are incontestably sem- largely ignored in the psychology of reasoning. We set inal and proli®c, and the content offered by each is pow- out the seven research questions that MML is designed to erful. But the two sides stand vehemently apart, and as a enable answers to, present the research agenda that arises result the science of reasoning is, in a real sense, funda- from the goal of answering three of these questions, and mentally paralyzed.2 MML will end this paralysis.
    [Show full text]
  • PL 3014 - Metalogic
    PL 3014 - Metalogic Important topics in Chapter 5 (Mathematical Systems) Let S be a first order system. 1. Logical axioms of S are axioms of S that are logically valid (i.e., (K1)-(K6)). 2. Proper axioms of S are axioms of S that are not logically valid. Def. 5.3. A first order system with equality is an extension of K obtained by including the following proper axioms (axioms of equality): 2 (E7) A1 (x1, x1) 2 2 n n n (E8) A1 (tk, u) → A1 (fi (t1, ..., tk, ..., tn), fi (t1, ..., u, ..., tn)), where t1, ..., tn, u are any terms, and fi is any function letter of L. 2 n n n (E9) (A1 (tk, u) → (Ai (t1, ..., tk, ..., tn) → Ai (t1, ..., u, ..., tn))), where t1, ..., tn, u are any terms, and Ai is any predicate letter of L. 2 2 A1 is a 2-place predicate letter representing equality. Convention: We abbreviate A1 (x, x) by x = x. The first order system N of arithmetic . The language LN of N contains: individual variables: x1, x2, ... (below abbreviated x, y, z, ... ) individual constant: a1 (for 0) 1 2 2 function letters: f1 , f1 , f2 (successor ', sum +, product, ×) 2 predicate letter: A1 (equality =) punctuation: (, ), , connectives: ∼, → quantifier: ∀ N is the extension of K obtained by including as axioms (E7), (E8), (E9), and the following: (N1) (∀x)∼(x') = 0 (N2) (∀x)(∀y)(x' = y' → x = y) (N3) (∀x)(x + 0 = x) (N4) (∀x)(∀y)(x + y' = (x + y)') (N5) (∀x)(x × 0 = 0) (N6) (∀x)(∀y)(x × y' = (x × y) + x) (N7) A(0) → ((∀x)(A(x) → A(x') → (∀x)A(x)), for each wf A(x) of L N in which x occurs free.
    [Show full text]
  • The Decision Problem and the Development of Metalogic
    Historical Background Behmann’s Solution for the Monadic Class Decision Problems after Behmann Historical Discussion The Decision Problem and the Development of Metalogic Richard Zach University of Calgary, Canada Department of Philosophy and Calgary Institute for the Humanities www.ucalgary.ca/∼rzach/ ASL Annual Meeting March 12, 2007 1/38 Richard Zach Decision Problem and Development of Metalogic Historical Background Behmann’s Solution for the Monadic Class Decision Problems after Behmann Historical Discussion Outline 1 Historical Background 2 Behmann’s Solution for the Monadic Class 3 Decision Problems after Behmann 4 Historical Discussion 2/38 Richard Zach Decision Problem and Development of Metalogic Historical Background Introduction Behmann’s Solution for the Monadic Class Dramatis personae Decision Problems after Behmann Early Work on Decidability Historical Discussion The Decision Problem Introduction Decision Problem: find a procedure which decides after finitely many steps if a given formula is valid or not One of the fundamental problems for the development of modern logic in the 1920s and 1930s Intimately tied up with Hilbert’s philosophical project development of model theory development of notion of decision procedure/computation 3/38 Richard Zach Decision Problem and Development of Metalogic Historical Background Introduction Behmann’s Solution for the Monadic Class Dramatis personae Decision Problems after Behmann Early Work on Decidability Historical Discussion The Decision Problem David Hilbert, 1862–1943 Studied mathematics
    [Show full text]
  • Modal Metalogic: Completeness
    Philosophy 244: #7|Modal Metalogic: Completeness Soundness for a system S says that its theorems are S-valid, valid in all S-frames. Com- pleteness says that every S-valid wff is provable in S. The two together show system S is adequate. You might think this is no great accomplishment; why would it be called S-validity if it wasn't going to line up with provability in S? For provability in S to agree with S-validity is about as informative as being told that Lou Gehrig died of Lou Gehrig's disease. What we Well, but that is somewhat informative; learn is that there's a distinctive form of validity corresponding to system T. Given how va- we learn that he had a disease and it was that disease that killed him; he lidity was defined we learn that there's a distinctive sort of frame. The surprise isn't that the didn't die of boredom or whatever. relevant frames are called T-frames; it's that there's that kind of frame in the first place. Adequacy results are informative in a similar way. Systems can be defined either in terms of an axiomatic basis, or in terms of theorems; the Not all modal systems have associated first approach is more fine-grained since there can be two bases for the same set of theorems. frames. KH = K + 2(2p≡p)⊃p For the most part we'll be using the second notion. A system is a set of wffs. Not any old set, doesn't (160-1) e.g., not the wffs of length 51! The set has to be closed under some kind of implication.
    [Show full text]
  • Formal Semantics and Logic.Pdf
    Formal Semantics and Logic Bas C. van Fraassen Copyright c 1971, Bas C. van Fraassen Originally published by The Macmillan Company, New York This eBook was published by Nousoul Digital Publishers. Its formatting is optimized for eReaders and other electronic reading devices. For information, the publisher may be con- tacted by email at: [email protected] 2 To my parents 3 Preface to the .PDF Edition With a view to the increasing academic importance of digital media this electronic edition was created by Nousoul Digital Publishers. Thanks to the diligent work and expertise of Brandon P. Hopkins this edition has fea- tures that no book could have in the year of its original publication: searchable text and hyperlinked notes. The text itself remains essentially unchanged, but in addition to typographical corrections there are also some substantive corrections. Apart from the change in the solution to exercise 5.4 of Chapter 3, none comprise more than a few words or symbols. However, as different as digital media are from print media, so too is digital for- matting different from print formatting. Thus there are significant formatting differences from the earlier edition. The font and page dimensions differ, as well as the page numbering, which is made to accord with the pagina- tion automatically assigned to multi-paged documents by most standard document-readers. Bas van Fraassen 2016 4 Contents Preface (1971)9 Introduction: Aim and Structure of Logical Theory 12 1 Mathematical Preliminaries 19 1.1 Intuitive Logic and Set Theory....... 19 1.2 Mathematical Structures.......... 23 1.3 Partial Order and Trees.........
    [Show full text]
  • 21 Metatheory of Logics and the Characterization Problem
    21 Metatheory of Logics and the Characterization Problem JAN WOLENSKI´ 1 Introduction The word ‘metatheory’ denotes or perhaps suggests a theory of theories. Metascientific studies in the twentieth century used the term ‘metatheory’ to refer to investigations of theories in a variety of disciplines, for example, logic, sociology, psychology, history, etc. However, the philosophers of the Vienna Circle who made metatheoretical studies of science the main concern of their philosophy restricted metatheory to the logic of science modeled on developments in the foundations of mathematics. More specifically, the logic of science was intended to play a role similar to metamathematics in Hilbert’s sense; that is, it was projected as formal analysis of scientific theories understood as well-defined linguistic items. The word ‘metamathematics’ was used before Hilbert, but with a different meaning from his (see Ritter et al. 1980: 1175–8). In the early nine- teenth century, mathematicians, like Gauss, spoke about metamathematics in an explicitly pejorative sense. It was for them a speculative way of looking at mathematics – a sort of metaphysics of mathematics. A negative attitude to metaphysics was at that time inherited from Kant and early positivists. The only one serious use of ‘metamath- ematics’ was restricted to metageometry, and that was due to the fact that the invention of different geometries in the nineteenth century stimulated comparative studies. For example, investigations were undertaken of particular axiomatizations, their mutual relations, models of various geometrical systems, and attempts to prove their consistency. The prefix ‘meta’ presently suggests two things. First, it indicates that metatheoretical considerations appear ‘after’ (in the genetic sense) theories are formulated.
    [Show full text]