Chapter 4 INTRO LOGIC. Sentential Logic DAY 03. Schedule for Unit 1. Review. An argument is valid or invalid purely in virtue of its form.

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1 INO LOGIC DAY 03 Chapter 2 entential Logic 1 3 chedule for Unit 1 eview Day 1 Day 2 Day 3 Day 4 Day 5 Intro Chapter 1 Chapter 2 Chapter 3 Chapter 4 warm-up 40% of Exam 1 An argument is valid or invalid purely in virtue of its form. orm is a function of the arrangement of the terms in the argument, where the LOGICAL EM play a primary role. Day 6 Day 7 Chapter 4 Chapter 4 60% of Exam 1 Day 8 EXAM #

2 Classical yllogistic Logic What is a tatement Connective? Logical terms all some no are not Example Arguments all X are Y all Y are Z / all X are Z all X are Y no Y are Z / no X are Z A statement connective (or simply, a connective) is an "incomplete" expression i.e., an expression with one or more blanks such that, whenever the blanks are filled by statements, the resulting expression is also a statement. statement 1 connective statement 2 all X are Y some X are not Z / some Y are not Z statement entential Logic Example 1 In sentential logic the logical terms are AND 2 statement connectives snow is white AND grass is green it is raining AND it is sleeting 2+2 = 4 AND 3+3 =

3 1-Place, 2-Place, a 1-place connective has 1 blank a 2-place connective has 2 blanks a 3-place connective has 3 blanks etc. Examples 2-place AND 2 O 2 I 2 ONLY I 2 UNLE 2 I HEN Examples 1-place Examples 3-place I I ALE HA I 2 OHEWIE 3 I I POIBLE HA Jay BELIEVE HA UNLE 2 IN WHICH CAE 3 Kay HOPE HA

4 Atoms and Molecules ruth-values A compound (molecular) statement is one that is constructed from one or more smaller statements by the application of a statement connective. the truth-value of a true statement is the truth-value of a false statement is A simple (atomic) statement is one that is not constructed out of smaller statements by the application of a statement connective A implification Intro Logic is not concerned with all connectives, but only special ones namely truth-functional connectives ruth-unctional o say that a connective is truth-functional is to say that the truth-value of any compound statement produced by that connective is a function of the truth-values of its immediate parts. the whole is merely the sum of its parts

5 Abbreviation cheme 1. atomic sentences are abbreviated by upper-case letters (of the oman alphabet) 2. connectives are abbreviated by special symbols (logograms) 3. compound sentences are abbreviated by algebraic-combinations of 1 and 2 erminology he symbol & is called ampersand, which is a stylized way of writing the Latin word et, which means and. & & & & & & Example 1 Conjunction erminology (cont) expression it is raining it is sleeting and it is raining and it is sleeting abbreviation & ( & ) the word ampersand is a children s pronunciation of the original word and per se and & is called the conjunction of and. and are individually called conjuncts

6 Conjunction is truth-functional Example 2 Disjunction ( or ) & expression abbreviation case 1 it is raining case 2 it is sleeting case 3 or case 4 it is raining or it is sleeting ( ) logan A conjunction A&B is true if and only if both conjuncts A and B are true. A conjunction A&B is true if both conjuncts A and B are true; otherwise, it is false. erminology he symbol is called wedge, which is a stylized way of writing the letter v, which initializes the Latin word vel, which means or. is called the disjunction of and. and are individually called disjuncts

7 Exclusive ense vs. Inclusive ense Disjunction is truth-functional case 1 case 2 would you like a baked potato, O rench fries? case 3 would you like coffee or dessert? case 4 would you like soup, O salad? would you like cream or sugar? inclusive or 25 Exclusive or vs. Inclusive or 27 logan exclusive or soup O salad inclusive or cream or sugar A disjunction d e is true if and only if at least one disjunct d or e is true. Latin has two words: aut is exclusive or vel is inclusive or A disjunction d e is false if both disjuncts d and e are false; otherwise, it is true. Legalistic English has the word and/or Logic concentrates on inclusive or

8 a Connective that is not ruth-unctional because??? because??? erminology he symbol is called tilde (as in matilda ); which is a highly stylized way of writing the letter N, which is short for not. merely knowing that and are both true tells us nothing about whether one is responsible for the other Example 3 Negation ( not ) Negation is truth-functional expression it is raining not it is not raining abbreviation ~ ~ if is true, then is false if is false, then is true and have opposite truth-values

9 Example 4 if...then... my car runs out of gas my car stops Aside the prefix ante means before other words that contain ante if then if my car runs out of gas, then my car stops if my car stops, then my car runs out of gas ( ) ( ) ante antechamber antediluvian antebellum ante meridian (a.m.) antipasto (Italian form) is not equivalent to erminology Non-ruth-unctional If-hen A C is called a conditional (of A and C). A is called the antecedent. C is called the consequent. if antecedent, then consequent I live in Los Angeles I live in New York City I live in California if I lived in L.A., then I would live in CAL if I lived in NYC, then I would live in CAL L N C L C N C

10 NO UH-UNCIONAL! ruth-unctional version of if-then I live in LA I live in Cal L C case 1 I live in NYC I live in Cal N C in one case "adding" and produces in one case "adding" and produces case 2 case 3 case 4 true by default ruth-unctional If-hen he Oddness of Cases 3 and 4 it rains I shut the windows if it rains, then I (will) shut the windows If you promise to shut the windows I it rains, then only one scenario (case) constitutes breaking your promise the scenario in which it rains but you don t shut the windows. In case 3 and case 4, you keep your promise "by default"