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INTRODUCTION to LOGIC Lecture 1 Validity Introduction to Sets And Outline INTRODUCTIONTOLOGIC (1) Introductory (2) Validity Lecture 1 (3) Course Overview Validity (4) Sets and Relations Introduction to Sets and Relations. Dr. James Studd Pure logic is the ruin of the spirit. Antoine de Saint-Exupéry Resources Why logic? The Logic Manual Logic is the scientific study of valid argument. logicmanual.philosophy.ox.ac.uk Exercises booklet Philosophy is all about arguments and reasoning. Lecture slides Logic allows us to rigorously test validity. Worked examples Past examination papers Modern philosophy assumes familiarity with logic. some with solutions Used in linguistics, mathematics, computer science,. Mark Sainsbury: Logical Forms: An Introduction to Helps us make fine-grained conceptual distinctions. Philosophical Logic, Blackwell, second edition, 2001, chs. 1–2. Logic is compulsory. 1.5 Arguments, Validity, and Contradiction 1.5 Arguments, Validity, and Contradiction Validity 1/3 Examples First approximation. Argument 1 Not valid When an argument is valid, the truth of the premisses Zeno is a tortoise. guarantees the truth of the conclusion. Therefore, Zeno is toothless. The truth of the premiss does not provide a sufficiently An argument is valid if it ‘can’t’ be the case that all of the strong guarantee of the truth of the conclusion premisses are true and the conclusion is false. Argument 2 Valid Validity does not depend on contingent facts. Zeno is a tortoise. Validity does not depend on laws of nature. All tortoises are toothless. Validity does not depend on the meanings of Therefore, Zeno is toothless. subject-specific expressions. Validity depends purely on the ‘form’ of the argument. 1.5 Arguments, Validity, and Contradiction 1.5 Arguments, Validity, and Contradiction Validity 2/3 Argument 1 revisited Characterisation (p. 19) Argument 1 Not valid An argument is logically valid if and only if: Zeno is a tortoise. there is no interpretation under which: Therefore, Zeno is toothless. (i) the premisses are all true, and Argument 1a Not valid (ii) the conclusion is false. Boris Johnson is a Conservative. Therefore, Boris Johnson is a Liberal Democrat. There is an interpretation under which: (i) the premiss is true, and (ii) the conclusion is false. 1.5 Arguments, Validity, and Contradiction 1.5 Arguments, Validity, and Contradiction Argument 2 revisited Validity 3/3. Argument 2 Valid Characterisation (p. 19) Zeno is a tortoise. An argument is logically valid if and only if: All tortoises are toothless. there is no [uniform] interpretation [of subject-specific Therefore, Zeno is toothless. expressions] under which: Argument 2a Valid (i) the premisses are all true, and Boris Johnson is a Conservative. (ii) the conclusion is false. All Conservatives are Liberal Democrats. Therefore, Boris Johnson is a Liberal Democrat. Each occurrence of an expression interpreted in the same way Argument 2b Valid Radon is a noble gas. Logical expression keep their usual English meanings. All noble gases are chemical elements. Therefore, Radon is a chemical element. Note: argument 2a is a valid argument with a false conclusion. 1.5 Arguments, Validity, and Contradiction 1.5 Arguments, Validity, and Contradiction Subject-specific versus logical expressions Argument 2 revisited again Examples: logical terms Argument 2 Valid all, every, some, no. Zeno is a tortoise. not, and, or, unless, if, only if, if and only if. All tortoises are toothless. Therefore, Zeno is toothless. Examples: subject-specific terms Argument 3 Not valid Zeno, Boris Johnson, France, The North Sea, Radon, soap, Boris Johnson is a Conservative. bread, GDP, logical positivism, . No Conservatives are Liberal Democrats. tortoise, toothless, Conservative, nobel gas, philosopher, Therefore, Boris Johnson is a Liberal Democrat. chemical element, . loves, owns, reacts with, voted for, . Argument 4 Not valid Radon is a noble gas. All noble gases are chemical elements. Therefore, air is a chemical element. Overview 1.1 Sets Course overview Sets 1/2 1: Validity; Introduction to Sets and Relations Characterisation A set is a collection of zero or more objects. 2: Syntax and Semantics of Propositional Logic 3: Formalization in Propositional Logic The objects are called elements of the set. 4: The Syntax of Predicate Logic a 2 b is short for ‘a is an element of set b’. Examples 5: The Semantics of Predicate Logic The set of positive integers less than 4: 6: Natural Deduction f1; 2; 3g or fn : n is an integer between 1 and 3g 7: Formalization in Predicate Logic The set of positive integers: f1; 2; 3; 4;:::g or fn : n > 0g 8: Identity and Definite Descriptions The empty set: f g or fx : x is a round squareg or ; 1.1 Sets 1.2 Binary relations Sets 2/2 Ordered pairs Fact about sets Characterisation Sets are identical if and only if they have the same elements. An ordered pair comprises two components in a given order. Example hd; ei is the ordered pair whose first component is d and The following sets are all identical: whose second component is e, in that order. fLennon, McCartney, Harrison, Ringog Example fRingo, Lennon, Harrison, McCartneyg hLondon, Munichi 6= hMunich, Londoni fRingo, Ringo, Ringo, Lennon, Harrison, McCartneyg fLondon, Munichg = fMunich, Londong fx : x is a Beatleg fx : x sang lead vocals on an Abbey Road trackg 1.2 Binary relations 1.2 Binary relations Relations Worked example ' % Camilla Charles Diana Write down the following relation as a set of ordered pairs. Duchess of Cornwalle Prince of Walesd Princess of Wales Draw its arrow diagram. The relation of being countries in GB sharing a border Catherine Prince& William Prince George Duchess of Cambridge Duke of Cambridge of Cambridge f The relation of having married fhCharles, Dianai; hDiana, Charlesi; hCharles, Camillai; hCamilla, Charlesi; hKate, Williami; hWilliam, Katei; ::: g 1.2 Binary relations 1.2 Binary relations Relations Properties of relations 1/3 Definition (p. 8) Definition (p. 9) A set R is a binary relation if and only if it contains only A binary relation R is reflexive on a set S iff: ordered pairs. for all d in S: the pair hd; di is an element of R. Informally: hd; ei 2 R indicates that d stands in R to e. Informally: every member of S bears R to itself. Example Example Reflexive on the set of human beings The relation of having married. The relation of being the same height as fhKate, Williami; hCharles, Camillai; · · · g fhd; ei : d married eg. Example Not reflexive on this set The empty set: ; The relation of being taller than Example Not reflexive on f1; 2; 3g fh1; 1i; h2; 2i; h1; 3ig Reflexive on f1; 2g 1.2 Binary relations 1.2 Binary relations Reflexivity on S Properties of relations 2/3 Every point in S has a “loop”. Definition (p. 9) A binary relation R is symmetric on set S iff: Õ Õ Õ • • F • (Reflexive on S) for all d; e in S: if hd; ei 2 R then he; di 2 R. Õ Õ Õ Informally: any member of S bears R to a second only if the • • F • (Reflexive on S) second bears R back to the first. Õ Õ Õ Example Symmetric on the set of human beings • / • / • (Reflexive on S) F The relation of being a sibling of Õ Õ •• F • (Not Reflexive on S) Example Not symmetric on this set Key: Member of S: • The relation of being a brother of Non-member of S: F 1.2 Binary relations 1.2 Binary relations Symmetry on S Properties of relations 3/3 Every “outward route” between points in S has a “return route”. Definition A binary relation R is transitive on S iff: ) for all d; e; f in S: • i • F • (Symmetric on S) if hd; ei 2 R and he; f i 2 R, then also hd; f i 2 R • ) • ( • (Symmetric on S) i h F Informally: if any member of S bears R to a second, and the second also bears R to a third, the first bears R to the third. • ) • F • (Not symmetric on S) Example Transitive on the set of human beings ) + • i • F • (Symmetric on S) The relation of being taller than Example Not transitive on this set The relation of not having the same height (±1cm) 1.2 Binary relations 1.3 Functions Transitivity on S Functions Every “double-step” between points in S has a “one-step shortcut”. Definition (p. 14) A binary relation F is a function iff for all d; e; f: • ) • F ( • (Not transitive on S) if hd; ei 2 F and hd; f i 2 F then e = f. ) ( Informally, everything stands in F to at most one thing. • • F 4 • (Transitive on S) Example ) • i • F • (Not transitive on S) The function that squares positive integers. fh1; 1i; h2; 4i; h3; 9i;:::g 2 Õ ) Õ fhx; yi : y = x ; for x a positive integerg • i • F • (Transitive on S) 1.3 Functions Example F is a function A “straightforward and elementary” example Everything stands in F to at most one thing (“many-one” or “one-one”) + + + (a) What is a binary relation? 1 1 1 4< 1 1 1 (b) Consider the relation R of sharing exactly one parent: 2 4 2 2 2 2 2 R = fhd; ei : d and e share exactly one of their parentsg Determine whether R is: 3 3 3 3 3 + 3 (i) reflexive on the set of human beings (ii) symmetric on the set of human beings (iii) transitive on the set of human beings 4 4 4 + 4 4 + 4 Explain your answers. “one-one” “many-one” “one-many” function function not a function.
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