INTRO LOGIC DAY 03
Schedule for Unit 1 warm-up 40% of Exam 1 60% of Exam 1
Chapter 2 Sentential Logic
Review • An argument is valid or invalidpurely in virtue of its form. Form is a function of the arrangement of the terms in the argument, where theLOGICAL TERMS play a primary role.
Classical Syllogistic Logic • Logical terms Example Arguments all some no are not all X are Yall Y are Z/ all X are Z all X are Yno Y are Z/ no X are Z all X are Ysome X arenot Z/ some Y arenot Z
Sentential Logic • In sentential logic • the logical terms are statement connectives
What is a Statement Connective? • 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. statement1 statement2 statement3
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.
Atoms and Molecules • A compound (molecular)statement isone that is constructed from one or more smaller statements by the application of a statement connective. A simple (atomic)statement isone that is not constructed out of smaller statements by the application of a statement connective.
A Simplification • Intro Logic is not concerned • with all connectives, • but only special ones – namely… truth-functional connectives
Truth-Values • the truth-value of a true statement is T the truth-value of a false statement is F
Truth-Functional • To say that a connective istruth-functional is to say that • the truth-value of any compound statementproduced by that connective • is afunction of the truth-values • of its immediate parts. the whole is merely the sum of its parts
Abbreviation Scheme • 1. atomic sentences are abbreviated by upper-case letters (of the Roman alphabet) 2. connectives are abbreviated by special symbols (logograms) 3. compound sentences are abbreviated by algebraic-combinations of 1 and 2
Example 1 – Conjunction expression abbreviation it is raining R it is sleeting S and & it is raining and it is sleeting ( R & S )
Terminology • The symbol ‘&’ is called ampersand, • which is a stylized way of writing • the Latin word ‘et’, • which means “and”. & &&&& &
Terminology (cont) the word ‘ampersand’ is a children’s pronunciation of the original word and per se and R&S is called the conjunction of R and S. R and S are individually called conjuncts.
Conjunction is truth-functional R S R&S case 1 T T T case 2 T F F case 3 F T F case 4 F F F
Slogan • A conjunction & is trueif and only if both conjuncts and are true. A conjunction & is true if both conjuncts and are true; otherwise, it is false.
Example 2 – Disjunction (‘or’) expression abbreviation it is raining R it is sleeting S or it is raining or it is sleeting ( R S )
Terminology • The symbol ‘’ is called wedge, • which is a stylized way of writing the letter ‘v’, • which initializes the Latin word ‘vel’, • which means “or”. RS is called the disjunction of R and S. R and S are individually called disjuncts.
Exclusive Sense vs. Inclusive Sense • would you like soup,OR salad? would you like a baked potato,OR French fries? would you like coffee or dessert? would you like cream or sugar?
Exclusive ‘or’ vs. Inclusive ‘or’ • exclusive ‘or’ soup OR salad • inclusive ‘or’ cream or sugar • Latin has two words: • ‘aut’ is exclusive ‘or’ • ‘vel’ is inclusive ‘or’ Legalistic English has the word ‘and/or’ Logic concentrates on inclusive ‘or’.
Disjunction is truth-functional R S RS case 1 T T T case 2 T F T case 3 F T T case 4 F F F inclusive ‘or’
Slogan • A disjunction is trueif and only if at least one disjunct or is true. A disjunction is false if both disjuncts and are false; otherwise, it is true.
a Connective that is not Truth-Functional R S R because S S because R T T ??? ??? T F F F F T F F F F F F merely knowing that R and S are both true tells us nothing about whether one is responsible for the other
Example 3 – Negation (‘not’) expression abbreviation it is raining R not it is notraining R
Terminology • The symbol ‘’ is called “tilde” • (as in ‘matilda’); • which is a highly stylized way of writing the letter ‘N’, • which is short for ‘not’.
Negation is truth-functional • if R is true, then R is false • if R is false, then R is true R and R have opposite truth-values
Example 4 – ‘if...then...’ my car runs out of gas R my car stops S if… then… ifmy car runs out of gas, then my car stops ( R S ) ifmy car stops,then my car runs out of gas ( S R ) RS is not equivalent to SR.
Terminology AC is called a conditional(of A and C). A is called the antecedent. C is called the consequent. ifantecedent, thenconsequent
Aside • the prefix ‘ante’ means ‘before’ other words that contain ‘ante’ ante antechamber antediluvian antebellum ante meridian (a.m.) antipasto (Italian form)
Truth-Functional version of ‘if-then’ R S RS case 1 T T T case 2 T F F case 3 F T T case 4 F F T true by “default”
The Oddness of Cases 3 and 4 • If you promise to shut the windowsIF 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".