DOCSLIB.ORG

The Surprise Examination Paradox and the Second Incompleteness Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Surprise Examination Paradox and the Second Incompleteness Theorem The Surprise Examination Paradox and the Second Incompleteness Theorem Shira Kritchman and Ran Raz Few theorems in the history of mathematics which the derivation is done; which is impossible have inspired mathematicians and philosophers by the second incompleteness theorem. as much as G¨odel’s incompleteness theorems. The first incompleteness theorem states that, for The First Incompleteness Theorem 1 2 any rich enough consistent mathematical theory, G¨odel’s original proof for the first incompleteness there exists a statement that cannot be proved or theorem [Gödel31] is based on the liar paradox. disproved within the theory. The second incom- The liar paradox: consider the pleteness theorem states that for any rich enough statement “this statement is false.” consistent mathematical theory, the consistency The statement can be neither true of the theory itself cannot be proved (or disproved) nor false. within the theory. G¨odel considered the related statement “this state- In this paper we give a new proof for G¨odel’s second incompleteness theorem, based on Kol- ment has no proof.” He showed that this statement mogorov complexity, Chaitin’s incompleteness can be expressed in any theory that is capable of theorem, and an argument that resembles the expressing elementary arithmetic. If the statement surprise examination paradox. has a proof, then it is false; but since in a consistent We then go the other way around and suggest theory any statement that has a proof must be that the second incompleteness theorem gives a true, we conclude that if the theory is consistent, possible resolution of the surprise examination the statement has no proof. Since the statement N paradox. Roughly speaking, we argue that the has no proof, it is true (over ). Thus, if the the- flaw in the derivation of the paradox is that it ory is consistent, we have an example for a true N contains a hidden assumption that one can prove statement (over ) that has no proof. the consistency of the mathematical theory in The main conceptual difficulty in G¨odel’s orig- inal proof is the self-reference of the statement Shira Kritchman did her M.Sc in applied mathematics at “this statement has no proof.” A conceptually the Weizmann Institute of Science and is currently work- simpler proof of the first incompleteness the- ing in logic design at ECI Telecom. Her email address is orem, based on Berry’s paradox, was given by [email protected]. Chaitin [Chaitin71]. Ran Raz is professor on the faculty of mathematics and Berry’s paradox: consider the ex- computer science at the Weizmann Institute of Science. pression “the smallest positive in- His email address is [email protected]. teger not definable in under eleven Research supported by the Israel Science Foundation (ISF), words.” This expression defines the Binational Science Foundation (BSF), and the Minerva that integer in under eleven words. Foundation. Article©2010 Shira Kritchman, Ran Raz. To formalize Berry’s paradox, Chaitin uses the 1We require that the theory can express and prove ba- sic arithmetical truths. In particular, ZFC and Peano notion of Kolmogorov complexity. The Kolmogorov arithmetic (PA) are rich enough. complexity K(x) of an integer x is defined to be the 2Here and below, we consider only first-order theories length (in bits) of the shortest computer program with recursively enumerable sets of axioms. For simplic- that outputs x (and stops). Formally, to define K(x) ity, let us assume that the set of axioms is computable. one has to fix a programming language, such as 1454 Notices of the AMS Volume 57, Number 11 LISP, Pascal or C++. Alternatively, one can define proofs only show the existence of a true statement K(x) by considering any universal Turing machine. (over N) that has no proof, without giving an Chaitin’s incompleteness theorem states that explicit example of such a statement. for any rich enough consistent mathematical A different proof for the second incomplete- theory, there exists a (large enough) integer L ness theorem, based on Berry’s paradox, was (depending on the theory and on the programming given by Kikuchi [Kikuchi97]. This proof is model- language that is used to define Kolmogorov com- theoretic and seems to us somewhat less intuitive plexity) such that, for any integer x, the statement for people who are less familiar with model theory. “K(x) > L” cannot be proved within the theory. For previous model-theoretic proofs for the sec- The proof given by Chaitin is as follows. Let L be ond incompleteness theorem see [Kreisel50] (see a large enough integer. Assume for a contradiction also [Smory´nski77]). that, for some integer x, there is a proof for the statement “K(x) > L”. Let w be the first proof Our Approach (say, according to the lexicographic order) for a We give a new proof for the second incomplete- statement of the form “K(x) > L”. Let z be the ness theorem, based on Chaitin’s incompleteness integer x such that w proves “K(x) > L”. It is easy theorem and an argument that resembles the to give a computer program that outputs z: the surprise examination paradox (also known as the program enumerates all possible proofs w, one by unexpected hanging paradox). one, and for the first w that proves a statement of The surprise examination para- the form “K(x) > L”, the program outputs x and dox: the teacher announces in stops. The length of this program is a constant + log L. Thus, if L is large enough, the Kolmogorov class: “next week you are going complexity of z is less than L. Since w is a proof to have an exam, but you will not for “K(z) > L” (which is a false statement), we be able to know on which day of conclude that the theory is inconsistent. the week the exam is held until Note that the number of computer programs of that day.” The exam cannot be length L bits is at most 2L+1. Hence, for any integer held on Friday, because otherwise, L, there exists an integer 0 ≤ x ≤ 2L+1, such that the night before the students will K(x) > L. Thus, for some integer x, the statement know that the exam is going to be “K(x) > L” is a true statement (over N) that has no held the next day. Hence, in the proof. same way, the exam cannot be held A different proof for G¨odel’s first incomplete- on Thursday. In the same way, the ness theorem, also based on Berry’s paradox, was exam cannot be held on any of the given by Boolos [Boolos89] (see also [Vopenka66, days of the week. Kikuchi94]). Other proofs for the first incomplete- Let T be a (rich enough) mathematical theory, ness theorem are also known (for a recent survey, such as PA or ZFC. For simplicity, the reader can see [Kotlarski04]). assume that T is ZFC, the theory of all mathematics; thus any mathematical proof, and in particular any The Second Incompleteness Theorem proof in this paper, is obtained within T . The second incompleteness theorem follows di- Let L be the integer guaranteed by Chaitin’s rectly from G¨odel’s original proof for the first in- incompleteness theorem. Thus, for any integer x, completeness theorem. As described above, G¨odel the statement “K(x) > L” cannot be proved (in the expressed the statement “this statement has no theory T ) unless the theory is inconsistent. Note, proof” and showed that, if the theory is consistent, however, that for any integer x such that K(x) ≤ L, this is a true statement (over N) that has no proof. there is a proof (in T ) for the statement “K(x) ≤ L”, Informally, because the proof that this is a true simply by giving the computer program of length at statement can be obtained within any rich enough most L that outputs x and stops and by describing theory, such as Peano arithmetic (PA) or ZFC, if the running of that computer program until it the consistency of the theory itself can also be stops. proved within the theory, then the statement can Let m be the number of integers 0 ≤ x ≤ 2L+1 be proved within the theory, which is a contra- such that K(x) > L. (The number m is analogous diction. Hence, if the theory is rich enough, the to the day of the week on which the exam is consistency of the theory cannot be proved within held in the surprise examination paradox.) Recall the theory. that because the number of computer programs Thus the second incompleteness theorem fol- of length L bits is at most 2L+1, there exists at lows directly from G¨odel’s original proof for the least one integer 0 ≤ x ≤ 2L+1 such that K(x) > L. first incompleteness theorem. However, the sec- Hence, m ≥ 1. ond incompleteness theorem doesn’t follow from Assume that m = 1. Thus there exists a sin- Chaitin’s and Boolos’s simpler proofs for the first gle integer x ∈ {0,..., 2L+1} such that K(x) > L, incompleteness theorem. The problem is that these and every other integer y ∈ {0,..., 2L+1} satisfies December 2010 Notices of the AMS 1455 K(y) ≤ L. In this case, one can prove that x satis- see, for example, [Kikuchi97], Theorem 1.2 and fies K(x) > L by proving that every other integer Section 2). + ∈ { L 1} ≤ + y 0,..., 2 satisfies K(y) L (and recall that (2) T ⊢ ∀y ∈ {0,..., 2L 1} ((K(y) ≤ L) there is a proof for every such statement).
Load more

Recommended publications
  • 6.5 the Recursion Theorem
    6.5. THE RECURSION THEOREM 417 6.5 The Recursion Theorem The recursion Theorem, due to Kleene, is a fundamental result in recursion theory. Theorem 6.5.1 (Recursion Theorem, Version 1 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. For every total recursive function f, there is some n such that ϕn = ϕf(n). The recursion Theorem can be strengthened as follows. Theorem 6.5.2 (Recursion Theorem, Version 2 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. There is a total recursive function h such that for all x ∈ N,ifϕx is total, then ϕϕx(h(x)) = ϕh(x). 418 CHAPTER 6. ELEMENTARY RECURSIVE FUNCTION THEORY A third version of the recursion Theorem is given below. Theorem 6.5.3 (Recursion Theorem, Version 3 ) For all n ≥ 1, there is a total recursive function h of n +1 arguments, such that for all x ∈ N,ifϕx is a total recursive function of n +1arguments, then ϕϕx(h(x,x1,...,xn),x1,...,xn) = ϕh(x,x1,...,xn), for all x1,...,xn ∈ N. As a first application of the recursion theorem, we can show that there is an index n such that ϕn is the constant function with output n. Loosely speaking, ϕn prints its own name. Let f be the recursive function such that f(x, y)=x for all x, y ∈ N. 6.5. THE RECURSION THEOREM 419 By the s-m-n Theorem, there is a recursive function g such that ϕg(x)(y)=f(x, y)=x for all x, y ∈ N.
    [Show full text]
  • April 22 7.1 Recursion Theorem
    CSE 431 Theory of Computation Spring 2014 Lecture 7: April 22 Lecturer: James R. Lee Scribe: Eric Lei Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. An interesting question about Turing machines is whether they can reproduce themselves. A Turing machine cannot be be defined in terms of itself, but can it still somehow print its own source code? The answer to this question is yes, as we will see in the recursion theorem. Afterward we will see some applications of this result. 7.1 Recursion Theorem Our end goal is to have a Turing machine that prints its own source code and operates on it. Last lecture we proved the existence of a Turing machine called SELF that ignores its input and prints its source code. We construct a similar proof for the recursion theorem. We will also need the following lemma proved last lecture. ∗ ∗ Lemma 7.1 There exists a computable function q :Σ ! Σ such that q(w) = hPwi, where Pw is a Turing machine that prints w and hats. Theorem 7.2 (Recursion theorem) Let T be a Turing machine that computes a function t :Σ∗ × Σ∗ ! Σ∗. There exists a Turing machine R that computes a function r :Σ∗ ! Σ∗, where for every w, r(w) = t(hRi; w): The theorem says that for an arbitrary computable function t, there is a Turing machine R that computes t on hRi and some input. Proof: We construct a Turing Machine R in three parts, A, B, and T , where T is given by the statement of the theorem.
    [Show full text]
  • Intuitionism on Gödel's First Incompleteness Theorem
    The Yale Undergraduate Research Journal Volume 2 Issue 1 Spring 2021 Article 31 2021 Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem Quinn Crawford Yale University Follow this and additional works at: https://elischolar.library.yale.edu/yurj Part of the Mathematics Commons, and the Philosophy Commons Recommended Citation Crawford, Quinn (2021) "Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem," The Yale Undergraduate Research Journal: Vol. 2 : Iss. 1 , Article 31. Available at: https://elischolar.library.yale.edu/yurj/vol2/iss1/31 This Article is brought to you for free and open access by EliScholar – A Digital Platform for Scholarly Publishing at Yale. It has been accepted for inclusion in The Yale Undergraduate Research Journal by an authorized editor of EliScholar – A Digital Platform for Scholarly Publishing at Yale. For more information, please contact [email protected]. Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem Cover Page Footnote Written for Professor Sun-Joo Shin’s course PHIL 437: Philosophy of Mathematics. This article is available in The Yale Undergraduate Research Journal: https://elischolar.library.yale.edu/yurj/vol2/iss1/ 31 Crawford: Intuitionism on Gödel’s First Incompleteness Theorem Crawford | Philosophy Incomplete? Or Indefinite? Intuitionism on Gödel’s First Incompleteness Theorem By Quinn Crawford1 1Department of Philosophy, Yale University ABSTRACT This paper analyzes two natural-looking arguments that seek to leverage Gödel’s first incompleteness theo- rem for and against intuitionism, concluding in both cases that the argument is unsound because it equivo- cates on the meaning of “proof,” which differs between formalism and intuitionism.
    [Show full text]
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • Theorem Proving in Classical Logic
    MEng Individual Project Imperial College London Department of Computing Theorem Proving in Classical Logic Supervisor: Dr. Steffen van Bakel Author: David Davies Second Marker: Dr. Nicolas Wu June 16, 2021 Abstract It is well known that functional programming and logic are deeply intertwined. This has led to many systems capable of expressing both propositional and first order logic, that also operate as well-typed programs. What currently ties popular theorem provers together is their basis in intuitionistic logic, where one cannot prove the law of the excluded middle, ‘A A’ – that any proposition is either true or false. In classical logic this notion is provable, and the_: corresponding programs turn out to be those with control operators. In this report, we explore and expand upon the research about calculi that correspond with classical logic; and the problems that occur for those relating to first order logic. To see how these calculi behave in practice, we develop and implement functional languages for propositional and first order logic, expressing classical calculi in the setting of a theorem prover, much like Agda and Coq. In the first order language, users are able to define inductive data and record types; importantly, they are able to write computable programs that have a correspondence with classical propositions. Acknowledgements I would like to thank Steffen van Bakel, my supervisor, for his support throughout this project and helping find a topic of study based on my interests, for which I am incredibly grateful. His insight and advice have been invaluable. I would also like to thank my second marker, Nicolas Wu, for introducing me to the world of dependent types, and suggesting useful resources that have aided me greatly during this report.
    [Show full text]
  • On the Mathematical Foundations of Learning
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 1, Pages 1{49 S 0273-0979(01)00923-5 Article electronically published on October 5, 2001 ON THE MATHEMATICAL FOUNDATIONS OF LEARNING FELIPE CUCKER AND STEVE SMALE The problem of learning is arguably at the very core of the problem of intelligence, both biological and artificial. T. Poggio and C.R. Shelton Introduction (1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the mainstream of mathematics. In particular, there are large roles for probability theory, for algorithms such as least squares,andfor tools and ideas from linear algebra and linear analysis. An advantage of doing this is that communication is facilitated and the power of core mathematics is more easily brought to bear. We illustrate what we mean by learning theory by giving some instances. (a) The understanding of language acquisition by children or the emergence of languages in early human cultures. (b) In Manufacturing Engineering, the design of a new wave of machines is an- ticipated which uses sensors to sample properties of objects before, during, and after treatment. The information gathered from these samples is to be analyzed by the machine to decide how to better deal with new input objects (see [43]). (c) Pattern recognition of objects ranging from handwritten letters of the alpha- bet to pictures of animals, to the human voice. Understanding the laws of learning plays a large role in disciplines such as (Cog- nitive) Psychology, Animal Behavior, Economic Decision Making, all branches of Engineering, Computer Science, and especially the study of human thought pro- cesses (how the brain works).
    [Show full text]
  • Plato on the Foundations of Modern Theorem Provers
    The Mathematics Enthusiast Volume 13 Number 3 Number 3 Article 8 8-2016 Plato on the foundations of Modern Theorem Provers Ines Hipolito Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Hipolito, Ines (2016) "Plato on the foundations of Modern Theorem Provers," The Mathematics Enthusiast: Vol. 13 : No. 3 , Article 8. Available at: https://scholarworks.umt.edu/tme/vol13/iss3/8 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol. 13, no.3, p.303 Plato on the foundations of Modern Theorem Provers Inês Hipolito1 Nova University of Lisbon Abstract: Is it possible to achieve such a proof that is independent of both acts and dispositions of the human mind? Plato is one of the great contributors to the foundations of mathematics. He discussed, 2400 years ago, the importance of clear and precise definitions as fundamental entities in mathematics, independent of the human mind. In the seventh book of his masterpiece, The Republic, Plato states “arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument” (525c). In the light of this thought, I will discuss the status of mathematical entities in the twentieth first century, an era when it is already possible to demonstrate theorems and construct formal axiomatic derivations of remarkable complexity with artificial intelligent agents the modern theorem provers.
    [Show full text]
  • Axioms and Theorems for Plane Geometry (Short Version)
    Axioms and theorems for plane geometry (Short Version) Basic axioms and theorems Axiom 1. If A; B are distinct points, then there is exactly one line containing both A and B. Axiom 2. AB = BA. Axiom 3. AB = 0 iff A = B. Axiom 4. If point C is between points A and B, then AC + BC = AB. Axiom 5. (The triangle inequality) If C is not between A and B, then AC + BC > AB. ◦ Axiom 6. Part (a): m(\BAC) = 0 iff B; A; C are collinear and A is not between B and C. Part (b): ◦ m(\BAC) = 180 iff B; A; C are collinear and A is between B and C. Axiom 7. Whenever two lines meet to make four angles, the measures of those four angles add up to 360◦. Axiom 8. Suppose that A; B; C are collinear points, with B between A and C, and that X is not collinear with A, B and C. Then m(\AXB) + m(\BXC) = m(\AXC). Moreover, m(\ABX) + m(\XBC) = m(\ABC). Axiom 9. Equals can be substituted for equals. Axiom 10. Given a point P and a line `, there is exactly one line through P parallel to `. Axiom 11. If ` and `0 are parallel lines and m is a line that meets them both, then alternate interior angles have equal measure, as do corresponding angles. Axiom 12. For any positive whole number n, and distinct points A; B, there is some C between A; B such that n · AC = AB. Axiom 13. For any positive whole number n and angle \ABC, there is a point D between A and C such that n · m(\ABD) = m(\ABC).
    [Show full text]
  • Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
    Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite.
    [Show full text]
  • Logic and Proof Release 3.18.4
    Logic and Proof Release 3.18.4 Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn Sep 10, 2021 CONTENTS 1 Introduction 1 1.1 Mathematical Proof ............................................ 1 1.2 Symbolic Logic .............................................. 2 1.3 Interactive Theorem Proving ....................................... 4 1.4 The Semantic Point of View ....................................... 5 1.5 Goals Summarized ............................................ 6 1.6 About this Textbook ........................................... 6 2 Propositional Logic 7 2.1 A Puzzle ................................................. 7 2.2 A Solution ................................................ 7 2.3 Rules of Inference ............................................ 8 2.4 The Language of Propositional Logic ................................... 15 2.5 Exercises ................................................. 16 3 Natural Deduction for Propositional Logic 17 3.1 Derivations in Natural Deduction ..................................... 17 3.2 Examples ................................................. 19 3.3 Forward and Backward Reasoning .................................... 20 3.4 Reasoning by Cases ............................................ 22 3.5 Some Logical Identities .......................................... 23 3.6 Exercises ................................................. 24 4 Propositional Logic in Lean 25 4.1 Expressions for Propositions and Proofs ................................. 25 4.2 More commands ............................................
    [Show full text]
  • CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction Theorem
    CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction Theorem The Hilbert proof systems are systems based on a language with implication and contain a Modus Ponens rule as a rule of inference. They are usually called Hilbert style formalizations. We will call them here Hilbert style proof systems, or Hilbert systems, for short. Modus Ponens is probably the oldest of all known rules of inference as it was already known to the Stoics (3rd century B.C.). It is also considered as the most "natural" to our intuitive thinking and the proof systems containing it as the inference rule play a special role in logic. The Hilbert proof systems put major emphasis on logical axioms, keeping the rules of inference to minimum, often in propositional case, admitting only Modus Ponens, as the sole inference rule. 1 Hilbert System H1 Hilbert proof system H1 is a simple proof system based on a language with implication as the only connective, with two axioms (axiom schemas) which characterize the implication, and with Modus Ponens as a sole rule of inference. We de¯ne H1 as follows. H1 = ( Lf)g; F fA1;A2g MP ) (1) where A1;A2 are axioms of the system, MP is its rule of inference, called Modus Ponens, de¯ned as follows: A1 (A ) (B ) A)); A2 ((A ) (B ) C)) ) ((A ) B) ) (A ) C))); MP A ;(A ) B) (MP ) ; B 1 and A; B; C are any formulas of the propositional language Lf)g. Finding formal proofs in this system requires some ingenuity. Let's construct, as an example, the formal proof of such a simple formula as A ) A.
    [Show full text]
  • Symbolic Logic: Grammar, Semantics, Syntax
    Symbolic Logic: Grammar, Semantics, Syntax Logic aims to give a precise method or recipe for determining what follows from a given set of sentences. So we need precise definitions of `sentence' and `follows from.' 1. Grammar: What's a sentence? What are the rules that determine whether a string of symbols is a sentence, and when it is not? These rules are known as the grammar of a particular language. We have actually been operating with two different but very closely related grammars this term: a grammar for propositional logic and a grammar for first-order logic. The grammar for propositional logic (thus far) is simple: 1. There are an indefinite number of propositional variables, which we have been symbolizing as A; B; ::: and P; Q; :::; each of these is a sen- tence. ? is also a sentence. 2. If A; B are sentences, then: •:A is a sentence, • A ^ B is a sentence, and • A _ B is a sentence. 3. Nothing else is a sentence: anything that is neither a member of 1, nor constructable via successive applications of 2 to the members of 1, is not a sentence. The grammar for first-order logic (thus far) is more complex. 2 and 3 remain exactly the same as above, but 1 must be replaced by some- thing more complex. First, a list of all predicates (e.g. Tet or SameShape) and proper names (e.g. max; b) of a particular language L must be specified. Then we can define: 1F OL. A series of symbols s is an atomic sentence = s is an n-place predicate followed by an ordered sequence of n proper names.
    [Show full text]