DOCSLIB.ORG

INTRODUCTION to LOGIC Lecture 2 Syntax and Semantics Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
INTRODUCTION to LOGIC Lecture 2 Syntax and Semantics Of Outline INTRODUCTIONTOLOGIC 1 Syntax vs Semantics. 2 Syntax of L1. Lecture 2 3 Semantics of L1. Syntax and Semantics of 4 Truth-table methods. Propositional Logic. Dr. James Studd Logic is the beginning of wisdom. Thomas Aquinas Syntax vs. Semantics Syntax vs. Semantics Syntax Semantics Syntax is all about expressions: words and sentences. Semantics is all about meanings of expressions. Examples of semantic claims Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘Bertrand Russell’ refers to a British philosopher. ‘Bertrand Russell’ refers to Bertrand Russell. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. ‘likes logic’ expresses a property Russell has. ‘Bertrand Russell likes logic’ is true. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics 2.2 The Syntax of the Language of Propositional Use vs Mention Syntax: English vs. L1. Note our use of quotes to talk about expressions. English has many different sorts of expression. ‘Bertrand Russell’ refers to Bertrand Russell. Some expressions of English (1) Sentences: ‘Bertrand Russell likes logic’, ‘Philosophers like Mention conceptual analysis’, etc.. The first occurrence of ‘Bertrand Russell’ is an example (2) Connectives: ‘it is not the case that’, ‘and’, etc.. of mention. (3) Noun phrases: ‘Bertrand Russell’, ‘Philosophers’, etc.. This occurrence (with quotes) mentions—refers to—an (4) Verb phrases: ‘likes logic’, ‘like conceptual analysis’, etc.. expression. (5) Also: nouns, verbs, pronouns, etc., etc., etc.. Use L1 has just two sorts of basic expression. The second occurrence of ‘Bertrand Russell’ is an example of use. Some basic expressions of L1 This occurrence (without quotes) uses the expression to (1) Sentence letters: e.g. ‘P’, ‘Q’. refer to a man. (2) Connectives: e.g. ‘:’, ‘^’. 2.2 The Syntax of the Language of Propositional 2.2 The Syntax of the Language of Propositional Combining sentences and connectives makes new sentences. Connectives Some complex sentences Here’s the full list of L1-connectives. ‘It is not the case that’ and ‘Bertrand Russell likes logic’ make: ‘It is not the case that Bertrand Russell likes logic’. name in English symbol ‘:’ and ‘P’ make: ‘:P ’. conjunction and ^ ‘Bertrand Russell likes logic’ and ‘and’ and ‘Philosophers like disjunction or _ conceptual analysis’ make: negation it is not the : ‘Bertrand Russell likes logic and philosophers like conceptual case that analysis’. arrow if . then ! ‘P ’, ‘^’ and ‘Q’ make: ‘(P ^ Q)’. double arrow if and only if $ Logic convention: no quotes around L1-expressions. P , ^ and Q make: (P ^ Q). 2.2 The Syntax of the Language of Propositional 2.2 The Syntax of the Language of Propositional The syntax of L1 How to build a sentence of L1 Here’s the official definition of L1-sentence. Example Definition The following is a sentence of L1: (i) All sentence letters are sentences of L1: P; Q; R; P1;Q1;R1;P2;Q2;R2;P3;::: ::(((P ^ Q) ! (P _ :R45)) $ :((P3 _ R) _ R)) (ii) If φ and are sentences of L1, then so are: :φ (φ ^ ) (φ _ ) Definition of L1-sentences (repeated from previous page) (φ ! ) (φ $ ) (i) All sentence letters are sentences of L1. (ii) If φ and are sentences of L1, then :φ, (φ ^ ), (iii) Nothing else is a sentence of L1. (φ _ ), (φ ! ) and (φ $ ) are sentences of L1. Greek letters: φ (‘PHI’) and (‘PSI’): not part of L . 1 (iii) Nothing else is a sentence of L1. 2.2 The Syntax of the Language of Propositional 2.3 Rules for Dropping Brackets Object vs. Metalanguage Bracketing conventions I mentioned that φ and are not part of L1. There are conventions for dropping brackets in L1. :P is a L1-sentence. Some are similar to rules used for + and × in arithmetic. :φ describes many L1-sentences (but is not one itself). Example in arithmetic e.g. :P , :(Q _ R), :(P $ (Q _ R))... 4 + 5 × 3 does not abbreviate (4 + 5) × 3. φ and are part of the metalanguage, not the object one. × ‘binds more strongly’ than +. Object language 4 + 5 × 3 abbreviates 4 + (5 × 3). The object language is the one we’re theorising about. Examples in L1 The object language is L . 1 ^ and _ bind more strongly than ! and $. Metalanguage (P ! Q ^ R) abbreviates (P ! (Q ^ R)). The metalanguage is the one we’re theorising in. One may drop outer brackets. The metalanguage is (augmented) English. P ^ (Q !:P4) abbreviates (P ^ (Q !:P4)). One may drop brackets on strings of ^s or _s. φ and are used as variables in the metalanguage: (P ^ Q ^ R) abbreviates ((P ^ Q) ^ R). in order to generalise about sentences of the object language. 2.4 The Semantics of Propositional Logic 2.4 The Semantics of Propositional Logic Semantics L1-structures Recall the characterisation of validity from week 1. We interpret sentence letters by assigning them truth-values: either T for True or F for False. Characterisation An argument is logically valid if and only if there is no Definition interpretation of subject-specific expressions under which: An L1-structure is an assignment of exactly one (i) the premisses are all true, and truth-value (T or F) to every sentence letter of L1. (ii) the conclusion is false. Examples We’ll adapt this characterisation to L1. One may think of an L1-structure as an infinite list that Logical expressions: :; ^; _; ! and $. provides a value T or F for every sentence letter. Subject specific expressions: P; Q; R; ... PQRP1 Q1 R1 P2 Q2 R2 ··· Interpretation: L1-structure. A : TFFFTFTTF ··· B : FFFFFFFFF ··· We use A, B, etc. to stand for L1-structures. 2.4 The Semantics of Propositional Logic 2.4 The Semantics of Propositional Logic Truth-values of complex sentences 1/3 Worked example 1 φ :φ L1-structures only directly specify truth-values for P , Q, R,... jφj is the truth-value of φ under A. The logical connectives have fixed meanings. A T F F T These determine the truth-values of complex sentences. Notation: jφjA is the truth-value of φ under A. Compute the following truth-values. Let the structure A be partially specified as follows. Truth-conditions for : The meaning of : is summarised in its truth table. PQRP1 Q1 R1 P2 Q2 R2 ··· TFFFTFTTF ··· φ :φ T F Compute: F T jP jA = jQjA = jR1jA = j:P jA = j:QjA = j:R1jA = j::P j = j::Qj = j::R j = In words: j:φjA = T if and only if jφjA = F. A A 1 A 2.4 The Semantics of Propositional Logic 2.4 The Semantics of Propositional Logic Truth-values of complex sentences 2/3 Truth-values of complex sentences 3/3 Truth-conditions for ^ and _ Truth-conditions for ! and $ The meanings of ^ and _ are given by the truth tables: The meanings of ! and $ are given by the truth tables: φ (φ ^ ) φ (φ _ ) φ (φ ! ) φ (φ $ ) T T T T T T T T T T T T T F F T F T T F F T F F F T F F T T F T T F T F F F F F F F F F T F F T j(φ ^ )jA = T if and only if jφjA = T and j jA = T. j(φ ! )jA = T if and only if jφjA = F or j jA = T. j(φ _ )jA = T if and only if jφjA = T or j jA = T (or both). j(φ $ )jA = T if and only if jφjA = j jA. 2.4 The Semantics of Propositional Logic 2.4 The Semantics of Propositional Logic Worked example 2 For actual calculations it’s usually better to use tables. Suppose jP jB = T and jQjB = F. Let jP jB = T and jQjB = F. Compute j:(P ! Q) ! (P ^ Q)jB Compute j:(P ! Q) ! (P ^ Q)jB What is the truth value of :(P ! Q) ! (P ^ Q) under B? P Q : (P ! Q) ! (P ^ Q) 1 j(P ! Q)jB = F and j(P ^ Q)jB = F 2 j:(P ! Q)jB = T φ (φ ^ ) (φ ! ) 3 j:(P ! Q) ! (P ^ Q)jB = F φ :φ T T T T T F T F F F φ (φ ^ ) (φ ! ) F T F T F T φ :φ T T T T F F F T T F T F F F F T F T F T F F F T 2.4 The Semantics of Propositional Logic 2.4 The Semantics of Propositional Logic Using the same technique we can fill out the full truth table Validity for :(P ! Q) ! (P ^ Q) Let Γ be a set of sentences of L1 and φ a sentence of L1. P Q : (P ! Q) ! (P ^ Q) Definition T T FTTT T TTT The argument with all sentences in Γ as premisses and φ as T F TTFF F TFF conclusion is valid if and only if there is no L1-structure F T FFTT T FFT under which: F F FFTF T FFF (i) all sentences in Γ are true; and The main column (underlined) gives the truth-value of the (ii) φ is false. whole sentence. Notation: when this argument is valid we write Γ φ. fP !:Q; Qg j= :P means that the argument whose premises are P !:Q and Q, and whose conclusion is :P is valid. Also written: P !:Q; Q j= :P 2.4 The Semantics of Propositional Logic 2.4 The Semantics of Propositional Logic Worked example 3 Other logical notions We can use truth-tables to show that L1-arguments are valid. Definition Example A sentence φ of L1 is logically true (a tautology) iff: Show that fP !:Q; Qg j= :P .
Load more

Recommended publications
  • Chapter 2 Introduction to Classical Propositional
    CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called, go back to antiquity and are due to Stoic school of philosophy (3rd century B.C.), whose most eminent representative was Chryssipus. But the real development of this calculus began only in the mid-19th century and was initiated by the research done by the English math- ematician G. Boole, who is sometimes regarded as the founder of mathematical logic. The classical propositional calculus was ¯rst formulated as a formal ax- iomatic system by the eminent German logician G. Frege in 1879. The assumption underlying the formalization of classical propositional calculus are the following. Logical sentences We deal only with sentences that can always be evaluated as true or false. Such sentences are called logical sentences or proposi- tions. Hence the name propositional logic. A statement: 2 + 2 = 4 is a proposition as we assume that it is a well known and agreed upon truth. A statement: 2 + 2 = 5 is also a proposition (false. A statement:] I am pretty is modeled, if needed as a logical sentence (proposi- tion). We assume that it is false, or true. A statement: 2 + n = 5 is not a proposition; it might be true for some n, for example n=3, false for other n, for example n= 2, and moreover, we don't know what n is. Sentences of this kind are called propositional functions. We model propositional functions within propositional logic by treating propositional functions as propositions.
    [Show full text]
  • COMPSCI 501: Formal Language Theory Insights on Computability Turing Machines Are a Model of Computation Two (No Longer) Surpris
    Insights on Computability Turing machines are a model of computation COMPSCI 501: Formal Language Theory Lecture 11: Turing Machines Two (no longer) surprising facts: Marius Minea Although simple, can describe everything [email protected] a (real) computer can do. University of Massachusetts Amherst Although computers are powerful, not everything is computable! Plus: “play” / program with Turing machines! 13 February 2019 Why should we formally define computation? Must indeed an algorithm exist? Back to 1900: David Hilbert’s 23 open problems Increasingly a realization that sometimes this may not be the case. Tenth problem: “Occasionally it happens that we seek the solution under insufficient Given a Diophantine equation with any number of un- hypotheses or in an incorrect sense, and for this reason do not succeed. known quantities and with rational integral numerical The problem then arises: to show the impossibility of the solution under coefficients: To devise a process according to which the given hypotheses or in the sense contemplated.” it can be determined in a finite number of operations Hilbert, 1900 whether the equation is solvable in rational integers. This asks, in effect, for an algorithm. Hilbert’s Entscheidungsproblem (1928): Is there an algorithm that And “to devise” suggests there should be one. decides whether a statement in first-order logic is valid? Church and Turing A Turing machine, informally Church and Turing both showed in 1936 that a solution to the Entscheidungsproblem is impossible for the theory of arithmetic. control To make and prove such a statement, one needs to define computability. In a recent paper Alonzo Church has introduced an idea of “effective calculability”, read/write head which is equivalent to my “computability”, but is very differently defined.
    [Show full text]
  • CAS LX 522 Syntax I
    It is likely… CAS LX 522 IP This satisfies the EPP in Syntax I both clauses. The main DPj I′ clause has Mary in SpecIP. Mary The embedded clause has Vi+I VP is the trace in SpecIP. V AP Week 14b. PRO and control ti This specific instance of A- A IP movement, where we move a likely subject from an embedded DP I′ clause to a higher clause is tj generally called subject raising. I VP to leave Reluctance to leave Reluctance to leave Now, consider: Reluctant has two θ-roles to assign. Mary is reluctant to leave. One to the one feeling the reluctance (Experiencer) One to the proposition about which the reluctance holds (Proposition) This looks very similar to Mary is likely to leave. Can we draw the same kind of tree for it? Leave has one θ-role to assign. To the one doing the leaving (Agent). How many θ-roles does reluctant assign? In Mary is reluctant to leave, what θ-role does Mary get? IP Reluctance to leave Reluctance… DPi I′ Mary Vj+I VP In Mary is reluctant to leave, is V AP Mary is doing the leaving, gets Agent t Mary is reluctant to leave. j t from leave. i A′ Reluctant assigns its θ- Mary is showing the reluctance, gets θ roles within AP as A θ IP Experiencer from reluctant. required, Mary moves reluctant up to SpecIP in the main I′ clause by Spellout. ? And we have a problem: I vP But what gets the θ-role to Mary appears to be getting two θ-roles, from leave, and what v′ in violation of the θ-criterion.
    [Show full text]
  • The Validities of Pep and Some Characteristic Formulas in Modal Logic
    THE VALIDITIES OF PEP AND SOME CHARACTERISTIC FORMULAS IN MODAL LOGIC Hong Zhang, Huacan He School of Computer Science, Northwestern Polytechnical University,Xi'an, Shaanxi 710032, P.R. China Abstract: In this paper, we discuss the relationships between some characteristic formulas in normal modal logic with their frames. We find that the validities of the modal formulas are conditional even though some of them are intuitively valid. Finally, we prove the validities of two formulas of Position- Exchange-Principle (PEP) proposed by papers 1 to 3 by using of modal logic and Kripke's semantics of possible worlds. Key words: Characteristic Formulas, Validity, PEP, Frame 1. INTRODUCTION Reasoning about knowledge and belief is an important issue in the research of multi-agent systems, and reasoning, which is set up from the main ideas of situational calculus, about other agents' states and actions is one of the most important directions in Al^^'^\ Papers [1-3] put forward an axiom scheme in reasoning about others(RAO) in multi-agent systems, and a rule called Position-Exchange-Principle (PEP), which is shown as following, was abstracted . C{(p^y/)^{C(p->Cy/) (Ce{5f',...,5f-}) When the length of C is at least 2, it reflects the mechanism of an agent reasoning about knowledge of others. For example, if C is replaced with Bf B^/, then it should be B':'B]'{(p^y/)-^iBf'B]'(p^Bf'B]'y/) 872 Hong Zhang, Huacan He The axiom schema plays a useful role as that of modus ponens and axiom K when simplifying proof. Though examples were demonstrated as proofs in papers [1-3], from the perspectives both in semantics and in syntax, the rule is not so compellent.
    [Show full text]
  • The Metamathematics of Putnam's Model-Theoretic Arguments
    The Metamathematics of Putnam's Model-Theoretic Arguments Tim Button Abstract. Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted mod- els. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I exam- ine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to meta- mathematical challenges. Copyright notice. This paper is due to appear in Erkenntnis. This is a pre-print, and may be subject to minor changes. The authoritative version should be obtained from Erkenntnis, once it has been published. Hilary Putnam famously attempted to use model theory to draw metaphys- ical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: [T]he world consists of a fixed totality of mind-independent objects. (Putnam 1981, p. 49; cf. 1978, p. 125). Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things. (1981, p. 49; cf. 1989, p. 214) [W]hat is epistemically most justifiable to believe may nonetheless be false. (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an \externalist perspective" whose \favorite point of view is a God's Eye point of view" (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction.
    [Show full text]
  • Thinking Recursively Part Four
    Thinking Recursively Part Four Announcements ● Assignment 2 due right now. ● Assignment 3 out, due next Monday, April 30th at 10:00AM. ● Solve cool problems recursively! ● Sharpen your recursive skillset! A Little Word Puzzle “What nine-letter word can be reduced to a single-letter word one letter at a time by removing letters, leaving it a legal word at each step?” Shrinkable Words ● Let's call a word with this property a shrinkable word. ● Anything that isn't a word isn't a shrinkable word. ● Any single-letter word is shrinkable ● A, I, O ● Any multi-letter word is shrinkable if you can remove a letter to form a word, and that word itself is shrinkable. ● So how many shrinkable words are there? Recursive Backtracking ● The function we wrote last time is an example of recursive backtracking. ● At each step, we try one of many possible options. ● If any option succeeds, that's great! We're done. ● If none of the options succeed, then this particular problem can't be solved. Recursive Backtracking if (problem is sufficiently simple) { return whether or not the problem is solvable } else { for (each choice) { try out that choice. if it succeeds, return success. } return failure } Failure in Backtracking S T A R T L I N G Failure in Backtracking S T A R T L I N G S T A R T L I G Failure in Backtracking S T A R T L I N G S T A R T L I G Failure in Backtracking S T A R T L I N G Failure in Backtracking S T A R T L I N G S T A R T I N G Failure in Backtracking S T A R T L I N G S T A R T I N G S T R T I N G Failure in Backtracking S T A R T L I N G S T A R T I N G S T R T I N G Failure in Backtracking S T A R T L I N G S T A R T I N G Failure in Backtracking S T A R T L I N G S T A R T I N G S T A R I N G Failure in Backtracking ● Returning false in recursive backtracking does not mean that the entire problem is unsolvable! ● Instead, it just means that the current subproblem is unsolvable.
    [Show full text]
  • Boolean Satisfiability Solvers: Techniques and Extensions
    Boolean Satisfiability Solvers: Techniques and Extensions Georg WEISSENBACHER a and Sharad MALIK a a Princeton University Abstract. Contemporary satisfiability solvers are the corner-stone of many suc- cessful applications in domains such as automated verification and artificial intelli- gence. The impressive advances of SAT solvers, achieved by clever engineering and sophisticated algorithms, enable us to tackle Boolean Satisfiability (SAT) problem instances with millions of variables – which was previously conceived as a hope- less problem. We provide an introduction to contemporary SAT-solving algorithms, covering the fundamental techniques that made this revolution possible. Further, we present a number of extensions of the SAT problem, such as the enumeration of all satisfying assignments (ALL-SAT) and determining the maximum number of clauses that can be satisfied by an assignment (MAX-SAT). We demonstrate how SAT solvers can be leveraged to solve these problems. We conclude the chapter with an overview of applications of SAT solvers and their extensions in automated verification. Keywords. Satisfiability solving, Propositional logic, Automated decision procedures 1. Introduction Boolean Satisfibility (SAT) is the problem of checking if a propositional logic formula can ever evaluate to true. This problem has long enjoyed a special status in computer science. On the theoretical side, it was the first problem to be classified as being NP- complete. NP-complete problems are notorious for being hard to solve; in particular, in the worst case, the computation time of any known solution for a problem in this class increases exponentially with the size of the problem instance. On the practical side, SAT manifests itself in several important application domains such as the design and verification of hardware and software systems, as well as applications in artificial intelligence.
    [Show full text]
  • Syntax and Semantics of Propositional Logic
    INTRODUCTIONTOLOGIC Lecture 2 Syntax and Semantics of Propositional Logic. Dr. James Studd Logic is the beginning of wisdom. Thomas Aquinas Outline 1 Syntax vs Semantics. 2 Syntax of L1. 3 Semantics of L1. 4 Truth-table methods. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. Examples of syntactic claims ‘likes logic’ is a verb phrase. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘Bertrand Russell likes logic’ is a sentence. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun. ‘likes logic’ is a verb phrase. Combining a proper noun and a verb phrase in this way makes a sentence. Syntax vs. Semantics Syntax Syntax is all about expressions: words and sentences. Examples of syntactic claims ‘Bertrand Russell’ is a proper noun.
    [Show full text]
  • Data Types & Arithmetic Expressions
    Data Types & Arithmetic Expressions 1. Objective .............................................................. 2 2. Data Types ........................................................... 2 3. Integers ................................................................ 3 4. Real numbers ....................................................... 3 5. Characters and strings .......................................... 5 6. Basic Data Types Sizes: In Class ......................... 7 7. Constant Variables ............................................... 9 8. Questions/Practice ............................................. 11 9. Input Statement .................................................. 12 10. Arithmetic Expressions .................................... 16 11. Questions/Practice ........................................... 21 12. Shorthand operators ......................................... 22 13. Questions/Practice ........................................... 26 14. Type conversions ............................................. 28 Abdelghani Bellaachia, CSCI 1121 Page: 1 1. Objective To be able to list, describe, and use the C basic data types. To be able to create and use variables and constants. To be able to use simple input and output statements. Learn about type conversion. 2. Data Types A type defines by the following: o A set of values o A set of operations C offers three basic data types: o Integers defined with the keyword int o Characters defined with the keyword char o Real or floating point numbers defined with the keywords
    [Show full text]
  • The Substitutional Analysis of Logical Consequence
    THE SUBSTITUTIONAL ANALYSIS OF LOGICAL CONSEQUENCE Volker Halbach∗ dra version ⋅ please don’t quote ónd June óþÕä Consequentia ‘formalis’ vocatur quae in omnibus terminis valet retenta forma consimili. Vel si vis expresse loqui de vi sermonis, consequentia formalis est cui omnis propositio similis in forma quae formaretur esset bona consequentia [...] Iohannes Buridanus, Tractatus de Consequentiis (Hubien ÕÉßä, .ì, p.óóf) Zf«±§Zh± A substitutional account of logical truth and consequence is developed and defended. Roughly, a substitution instance of a sentence is dened to be the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category. In particular atomic formulae can be replaced with any formulae containing. e denition of logical truth is then as follows: A sentence is logically true i all its substitution instances are always satised. Logical consequence is dened analogously. e substitutional denition of validity is put forward as a conceptual analysis of logical validity at least for suciently rich rst-order settings. In Kreisel’s squeezing argument the formal notion of substitutional validity naturally slots in to the place of informal intuitive validity. ∗I am grateful to Beau Mount, Albert Visser, and Timothy Williamson for discussions about the themes of this paper. Õ §Z êZo±í At the origin of logic is the observation that arguments sharing certain forms never have true premisses and a false conclusion. Similarly, all sentences of certain forms are always true. Arguments and sentences of this kind are for- mally valid. From the outset logicians have been concerned with the study and systematization of these arguments, sentences and their forms.
    [Show full text]
  • Deduction (I) Tautologies, Contradictions And
    D (I) T, & L L October , Tautologies, contradictions and contingencies Consider the truth table of the following formula: p (p ∨ p) () If you look at the final column, you will notice that the truth value of the whole formula depends on the way a truth value is assigned to p: the whole formula is true if p is true and false if p is false. Contrast the truth table of (p ∨ p) in () with the truth table of (p ∨ ¬p) below: p ¬p (p ∨ ¬p) () If you look at the final column, you will notice that the truth value of the whole formula does not depend on the way a truth value is assigned to p. The formula is always true because of the meaning of the connectives. Finally, consider the truth table table of (p ∧ ¬p): p ¬p (p ∧ ¬p) () This time the formula is always false no matter what truth value p has. Tautology A statement is called a tautology if the final column in its truth table contains only ’s. Contradiction A statement is called a contradiction if the final column in its truth table contains only ’s. Contingency A statement is called a contingency or contingent if the final column in its truth table contains both ’s and ’s. Let’s consider some examples from the book. Can you figure out which of the following sentences are tautologies, which are contradictions and which contingencies? Hint: the answer is the same for all the formulas with a single row. () a. (p ∨ ¬p), (p → p), (p → (q → p)), ¬(p ∧ ¬p) b.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]