Logical Connectives in Natural Languages
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Physical and Mathematical Sciences 2014, № 1, P. 3–6 Mathematics ON
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2014, № 1, p. 3–6 Mathematics ON ZIGZAG DE MORGAN FUNCTIONS V. A. ASLANYAN ∗ Oxford University UK, Chair of Algebra and Geometry YSU, Armenia There are five precomplete classes of De Morgan functions, four of them are defined as sets of functions preserving some finitary relations. However, the fifth class – the class of zigzag De Morgan functions, is not defined by relations. In this paper we announce the following result: zigzag De Morgan functions can be defined as functions preserving some finitary relation. MSC2010: 06D30; 06E30. Keywords: disjunctive (conjunctive) normal form of De Morgan function; closed and complete classes; quasimonotone and zigzag De Morgan functions. 1. Introduction. It is well known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the lattice of monotone Boolean functions of n variables. Analogous to these facts we have introduced the concept of De Morgan functions and proved that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables [1]. The Post’s functional completeness theorem for Boolean functions plays an important role in discrete mathematics [2]. In the paper [3] we have established a functional completeness criterion for De Morgan functions. In this paper we show that zigzag De Morgan functions, which are used in the formulation of the functional completeness theorem, can be defined by a finitary relation. -
Expressive Completeness
Truth-Functional Completeness 1. A set of truth-functional operators is said to be truth-functionally complete (or expressively adequate) just in case one can take any truth-function whatsoever, and construct a formula using only operators from that set, which represents that truth-function. In what follows, we will discuss how to establish the truth-functional completeness of various sets of truth-functional operators. 2. Let us suppose that we have an arbitrary n-place truth-function. Its truth table representation will have 2n rows, some true and some false. Here, for example, is the truth table representation of some 3- place truth function, which we shall call $: Φ ψ χ $ T T T T T T F T T F T F T F F T F T T F F T F T F F T F F F F T This truth-function $ is true on 5 rows (the first, second, fourth, sixth, and eighth), and false on the remaining 3 (the third, fifth, and seventh). 3. Now consider the following procedure: For every row on which this function is true, construct the conjunctive representation of that row – the extended conjunction consisting of all the atomic sentences that are true on that row and the negations of all the atomic sentences that are false on that row. In the example above, the conjunctive representations of the true rows are as follows (ignoring some extraneous parentheses): Row 1: (P&Q&R) Row 2: (P&Q&~R) Row 4: (P&~Q&~R) Row 6: (~P&Q&~R) Row 8: (~P&~Q&~R) And now think about the formula that is disjunction of all these extended conjunctions, a formula that basically is a disjunction of all the rows that are true, which in this case would be [(Row 1) v (Row 2) v (Row 4) v (Row6) v (Row 8)] Or, [(P&Q&R) v (P&Q&~R) v (P&~Q&~R) v (P&~Q&~R) v (~P&Q&~R) v (~P&~Q&~R)] 4. -
Completeness
Completeness The strange case of Dr. Skolem and Mr. G¨odel∗ Gabriele Lolli The completeness theorem has a history; such is the destiny of the impor- tant theorems, those for which for a long time one does not know (whether there is anything to prove and) what to prove. In its history, one can di- stinguish at least two main paths; the first one covers the slow and difficult comprehension of the problem in (what historians consider) the traditional development of mathematical logic canon, up to G¨odel'sproof in 1930; the second path follows the L¨owenheim-Skolem theorem. Although at certain points the two paths crossed each other, they started and continued with their own aims and problems. A classical topos of the history of mathema- tical logic concerns the how and the why L¨owenheim, Skolem and Herbrand did not discover the completeness theorem, though they proved it, or whe- ther they really proved, or perhaps they actually discovered, completeness. In following these two paths, we will not always respect strict chronology, keeping the two stories quite separate, until the crossing becomes decisive. In modern pre-mathematical logic, the notion of completeness does not appear. There are some interesting speculations in Kant which, by some stretching, could be realized as bearing some relation with the problem; Kant's remarks, however, are probably more related with incompleteness, in connection with his thoughts on the derivability of transcendental ideas (or concepts of reason) from categories (the intellect's concepts) through a pas- sage to the limit; thus, for instance, the causa prima, or the idea of causality, is the limit of implication, or God is the limit of disjunction, viz., the catego- ry of \comunance". -
Reasoning with Ambiguity
Noname manuscript No. (will be inserted by the editor) Reasoning with Ambiguity Christian Wurm the date of receipt and acceptance should be inserted later Abstract We treat the problem of reasoning with ambiguous propositions. Even though ambiguity is obviously problematic for reasoning, it is no less ob- vious that ambiguous propositions entail other propositions (both ambiguous and unambiguous), and are entailed by other propositions. This article gives a formal analysis of the underlying mechanisms, both from an algebraic and a logical point of view. The main result can be summarized as follows: sound (and complete) reasoning with ambiguity requires a distinction between equivalence on the one and congruence on the other side: the fact that α entails β does not imply β can be substituted for α in all contexts preserving truth. With- out this distinction, we will always run into paradoxical results. We present the (cut-free) sequent calculus ALcf , which we conjecture implements sound and complete propositional reasoning with ambiguity, and provide it with a language-theoretic semantics, where letters represent unambiguous meanings and concatenation represents ambiguity. Keywords Ambiguity, reasoning, proof theory, algebra, Computational Linguistics Acknowledgements Thanks to Timm Lichte, Roland Eibers, Roussanka Loukanova and Gerhard Schurz for helpful discussions; also I want to thank two anonymous reviewers for their many excellent remarks! 1 Introduction This article gives an extensive treatment of reasoning with ambiguity, more precisely with ambiguous propositions. We approach the problem from an Universit¨atD¨usseldorf,Germany Universit¨atsstr.1 40225 D¨usseldorf Germany E-mail: [email protected] 2 Christian Wurm algebraic and a logical perspective and show some interesting surprising results on both ends, which lead up to some interesting philosophical questions, which we address in a preliminary fashion. -
Some Analogues of the Sheffer Stroke Function in N-Valued Logic
MA THEMA TICS SOME ANALOGUES OF THE SHEFFER STROKE FUNCTION IN n-VALUED LOGIe BY NORMAN M. MARTIN (Communicated by Prof. A. HEYTING at the meeting of June 24, 1950) The interpretation of the ordinary (two-valued) propositional logic in terms of a truth-table system with values "true" and "false" or, more abstractly, the numbers "I" and "2" has become customary. This has been useful in giving an algorithm for the concept "analytic" , in giving an adequacy criterion for the definability of one function by another and it has made possible the proof of adequacy of a list of primitive terms for the definition of all truth-functions (functional completeness). If we generalize the concept of truth-function so as to allow for systems of functions of 3, 4, etc. values (preserving the "extensionality" requirement on functions examined) we obtain systems of functions of more than 2 values analogous to the truth-table interpretation of the usual pro positional calculus. The problem of functional completeness (the term is due to TURQUETTE) arises in each ofthe resulting systems. Strictly speaking, this problem is notclosely connected with problems of deducibility but is rather a combinatorial question. It will be the purpose of this paper to examine the problem of functional completeness of functions in n-valued logic where by n-valued logic we mean that system of functions such that each function of the system determines, by substitution of an arbitrary numeral a for the symbol n in the definition of the n-valued function, a function in the truth table system of a values. -
A Computer-Verified Monadic Functional Implementation of the Integral
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 411 (2010) 3386–3402 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs A computer-verified monadic functional implementation of the integral Russell O'Connor, Bas Spitters ∗ Radboud University Nijmegen, Netherlands article info a b s t r a c t Article history: We provide a computer-verified exact monadic functional implementation of the Riemann Received 10 September 2008 integral in type theory. Together with previous work by O'Connor, this may be seen as Received in revised form 13 January 2010 the beginning of the realization of Bishop's vision to use constructive mathematics as a Accepted 23 May 2010 programming language for exact analysis. Communicated by G.D. Plotkin ' 2010 Elsevier B.V. All rights reserved. Keywords: Type theory Functional programming Exact real analysis Monads 1. Introduction Integration is one of the fundamental techniques in numerical computation. However, its implementation using floating- point numbers requires continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the end-user by providing a library of exact analysis in which the computer handles the error estimates. For high assurance we use computer-verified proofs that the implementation is actually correct; see [21] for an overview. It has long been suggested that, by using constructive mathematics, exact analysis and provable correctness can be unified [7,8]. Constructive mathematics provides a high-level framework for specifying computations (Section 2.1). -
Boolean Logic
Boolean logic Lecture 12 Contents . Propositions . Logical connectives and truth tables . Compound propositions . Disjunctive normal form (DNF) . Logical equivalence . Laws of logic . Predicate logic . Post's Functional Completeness Theorem Propositions . A proposition is a statement that is either true or false. Whichever of these (true or false) is the case is called the truth value of the proposition. ‘Canberra is the capital of Australia’ ‘There are 8 day in a week.’ . The first and third of these propositions are true, and the second and fourth are false. The following sentences are not propositions: ‘Where are you going?’ ‘Come here.’ ‘This sentence is false.’ Propositions . Propositions are conventionally symbolized using the letters Any of these may be used to symbolize specific propositions, e.g. :, Manchester, , … . is in Scotland, : Mammoths are extinct. The previous propositions are simple propositions since they make only a single statement. Logical connectives and truth tables . Simple propositions can be combined to form more complicated propositions called compound propositions. .The devices which are used to link pairs of propositions are called logical connectives and the truth value of any compound proposition is completely determined by the truth values of its component simple propositions, and the particular connective, or connectives, used to link them. ‘If Brian and Angela are not both happy, then either Brian is not happy or Angela is not happy.’ .The sentence about Brian and Angela is an example of a compound proposition. It is built up from the atomic propositions ‘Brian is happy’ and ‘Angela is happy’ using the words and, or, not and if-then. -
Edinburgh Research Explorer
Edinburgh Research Explorer Propositions as Types Citation for published version: Wadler, P 2015, 'Propositions as Types', Communications of the ACM, vol. 58, no. 12, pp. 75-84. https://doi.org/10.1145/2699407 Digital Object Identifier (DOI): 10.1145/2699407 Link: Link to publication record in Edinburgh Research Explorer Document Version: Peer reviewed version Published In: Communications of the ACM General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 28. Sep. 2021 Propositions as Types ∗ Philip Wadler University of Edinburgh [email protected] 1. Introduction cluding Agda, Automath, Coq, Epigram, F#,F?, Haskell, LF, ML, Powerful insights arise from linking two fields of study previously NuPRL, Scala, Singularity, and Trellys. thought separate. Examples include Descartes’s coordinates, which Propositions as Types is a notion with mystery. Why should it links geometry to algebra, Planck’s Quantum Theory, which links be the case that intuitionistic natural deduction, as developed by particles to waves, and Shannon’s Information Theory, which links Gentzen in the 1930s, and simply-typed lambda calculus, as devel- thermodynamics to communication. -
A Multi-Modal Analysis of Anaphora and Ellipsis
University of Pennsylvania Working Papers in Linguistics Volume 5 Issue 2 Current Work in Linguistics Article 2 1998 A Multi-Modal Analysis of Anaphora and Ellipsis Gerhard Jaeger Follow this and additional works at: https://repository.upenn.edu/pwpl Recommended Citation Jaeger, Gerhard (1998) "A Multi-Modal Analysis of Anaphora and Ellipsis," University of Pennsylvania Working Papers in Linguistics: Vol. 5 : Iss. 2 , Article 2. Available at: https://repository.upenn.edu/pwpl/vol5/iss2/2 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/pwpl/vol5/iss2/2 For more information, please contact [email protected]. A Multi-Modal Analysis of Anaphora and Ellipsis This working paper is available in University of Pennsylvania Working Papers in Linguistics: https://repository.upenn.edu/pwpl/vol5/iss2/2 A Multi-Modal Analysis of Anaphora and Ellipsis Gerhard J¨ager 1. Introduction The aim of the present paper is to outline a unified account of anaphora and ellipsis phenomena within the framework of Type Logical Categorial Gram- mar.1 There is at least one conceptual and one empirical reason to pursue such a goal. Firstly, both phenomena are characterized by the fact that they re-use semantic resources that are also used elsewhere. This issue is discussed in detail in section 2. Secondly, they show a striking similarity in displaying the characteristic ambiguity between strict and sloppy readings. This supports the assumption that in fact the same mechanisms are at work in both cases. (1) a. John washed his car, and Bill did, too. b. John washed his car, and Bill waxed it. -
The Logic of Internal Rational Agent 1 Introduction
Australasian Journal of Logic Yaroslav Petrukhin The Logic of Internal Rational Agent Abstract: In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev’s Logic of Rational Agent LRA [16]. We call our logic LIRA (Logic of Internal Rational Agency). In contrast to LRA, it has three designated values instead of one and a different interpretation of truth values, the same as in Zaitsev and Shramko’s bi-facial truth logic [42]. This logic may be useful in a situation when according to an agent’s point of view (i.e. internal point of view) her/his reasoning is rational, while from the external one it might be not the case. One may use LIRA, if one wants to reconstruct an agent’s way of thinking, compare it with respect to the real state of affairs, and understand why an agent thought in this or that way. Moreover, we discuss Kubyshkina and Zaitsev’s necessity and possibility operators for LRA definable by means of four-valued Kripke-style semantics and show that, due to two negations (as well as their combination) of LRA, two more possibility operators for LRA can be defined. Then we slightly modify all these modalities to be appropriate for LIRA. Finally, we formalize all the truth-functional n-ary extensions of the negation fragment of LIRA (including LIRA itself) as well as their basic modal extension via linear-type natural deduction systems. Keywords: Logic of rational agent, logic of internal rational agency, four-valued logic, logic of generalized truth values, modal logic, natural deduction, correspondence analysis. -
Topics in Philosophical Logic
Topics in Philosophical Logic The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Litland, Jon. 2012. Topics in Philosophical Logic. Doctoral dissertation, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:9527318 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA © Jon Litland All rights reserved. Warren Goldfarb Jon Litland Topics in Philosophical Logic Abstract In “Proof-Theoretic Justification of Logic”, building on work by Dummett and Prawitz, I show how to construct use-based meaning-theories for the logical constants. The assertability-conditional meaning-theory takes the meaning of the logical constants to be given by their introduction rules; the consequence-conditional meaning-theory takes the meaning of the log- ical constants to be given by their elimination rules. I then consider the question: given a set of introduction (elimination) rules , what are the R strongest elimination (introduction) rules that are validated by an assertabil- ity (consequence) conditional meaning-theory based on ? I prove that the R intuitionistic introduction (elimination) rules are the strongest rules that are validated by the intuitionistic elimination (introduction) rules. I then prove that intuitionistic logic is the strongest logic that can be given either an assertability-conditional or consequence-conditional meaning-theory. In “Grounding Grounding” I discuss the notion of grounding. My discus- sion revolves around the problem of iterated grounding-claims. -
6C Lecture 2: April 3, 2014
6c Lecture 2: April 3, 2014 2.1 Functional completeness, normal forms, and struc- tural induction Before we begin, lets give a formal definition of a truth table. Definition 2.1. A truth table for a set of propositional variables, is a function which assigns each valuation of these variables either the value true or false. Given a formula φ, the truth table of φ is the truth table assigning each valuation of the variables of v the corresponding truth value of φ. We are ready to begin: Definition 2.2. We say that a set S of logical connective is functionally com- plete if for every finite set of propositional variables p1; : : : ; pn and every truth table for the variables p1; : : : ; pn, there exists a propositional formula φ using the variables p1; : : : ; pn and connectives only from S so that φ has the given truth table. We will soon show that the set f:; ^; _g is functionally complete. Before this, lets do a quick example. Here is an example of a truth table for the variables p1; p2; p3: p1 p2 p3 T T T T T T F F T F T T T F F T F T T F F T F F F F T F F F F F If f:; ^; _g is functionally complete, then we must be able to find a formula using only f:; ^; _g which has this given truth table (indeed, we must be able to do this for every truth able). In this case, one such a formula is p1 ^ (:p2 _ p3).