sign in sign up
<<
Home , Binary relation, Definition, False (logic), Interpretation (logic), Validity (logic)

Outline INTRODUCTIONTOLOGIC (1) Introductory (2) Lecture 1 (3) Course Overview Validity (4) Sets and Relations Introduction to Sets and Relations.

Dr. James Studd

Pure is the ruin of the spirit. Antoine de Saint-Exupéry

Resources Why logic?

The Logic Manual Logic is the scientific study of valid . logicmanual.philosophy.ox.ac.uk Exercises booklet Philosophy is all about 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, , ,. . . Mark Sainsbury: Logical Forms: An Introduction to Helps us make fine-grained conceptual distinctions. , Blackwell, second edition, 2001, chs. 1–2. Logic is compulsory. 1.5 Arguments, Validity, and 1.5 Arguments, Validity, and Contradiction Validity 1/3 Examples

First approximation. Argument 1 Not valid When an argument is valid, the 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 . 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 : Zeno is a tortoise. there is no 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 . 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 , . . . 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 is a collection of zero or more objects. 2: and of Propositional Logic 3: Formalization in Propositional Logic The objects are called elements of the set. 4: The Syntax of Logic a ∈ b is short for ‘a is an element of set b’. Examples 5: The Semantics of Predicate Logic The set of positive less than 4: 6: {1, 2, 3} or {n : n is an between 1 and 3} 7: Formalization in Predicate Logic The set of positive integers: {1, 2, 3, 4,...} or {n : n > 0} 8: Identity and Definite The : {} or {x : x is a round square} 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 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. {Lennon, McCartney, Harrison, Ringo} Example {Ringo, Lennon, Harrison, McCartney} hLondon, Munichi= 6 hMunich, Londoni {Ringo, Ringo, Ringo, Lennon, Harrison, McCartney} {London, Munich} = {Munich, London} {x : x is a Beatle} {x : x sang lead vocals on an Abbey Road track} 1.2 Binary relations 1.2 Binary relations Relations Worked example ' % Camilla Charles Diana Write down the following as a set of ordered pairs. Duchess of Cornwalle Prince of Walesd Princess of Wales Draw its diagram. The relation of 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 {hCharles, Dianai, hDiana, Charlesi, hCharles, Camillai, hCamilla, Charlesi, hKate, Williami, hWilliam, Katei, ... }

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 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 ∈ 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 The relation of having married. The relation of being the same height as {hKate, Williami, hCharles, Camillai, ···} {hd, ei : d married e}. Example Not reflexive on this set The empty set: ∅ The relation of being taller than Example Not reflexive on {1, 2, 3} {h1, 1i, h2, 2i, h1, 3i} Reflexive on {1, 2} 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 ∈ R then he, di ∈ 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 ∈ R and he, f i ∈ R, then also hd, f i ∈ 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 iff for all d, e, f: • ) • F ( • (Not transitive on S) if hd, ei ∈ F and hd, f i ∈ 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. {h1, 1i, h2, 4i, h3, 9i,...} 2 Õ ) Õ {hx, yi : y = x , for x a positive integer} • 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 = {hd, ei : d and e share exactly one of their parents} 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