Introductory Logic PHI 120 Presentation: "Intro to Formal Logic" Please turn off all cell phones!

Homework 1.Study Allen/Hand Logic Primer – "Well-formed Formula," pp. 6-7 – "Binary and Unary Connectives," p. 7 – "Parentheses Dropping Conventions," p. 9 – ("Denial,“ – logically opposite sentences, p. 7) 2.Handout on Class Web Page: – Truth Tables Handout Truth Tables Handout 3.Watch At Home: – “Basic Concepts Review” presentation Bring this handout to class from now on!

New Unit Formal (Symbolic) Logic Today: Basic Grammar of Sentences Sentential Logic

SYMBOLIC ELEMENTS OF THE LOGIC Part I

Expressions any sequence of symbols in the logic Sentences (WFFs) expressions that are well-formed The Well-Formed Formula An initial distinction

Sentences: two basic kinds i.atomic or simple i.cannot be broken into simpler sentences ii.no connectives ii.complex i.made up of simpler sentences ii.they always contain some connective

Symbolic Elements of the Logic 1.Atomic sentences 2.Connectives (or Logical Operators) 3.Parentheses ( … )

Symbolic Elements of the Logic 1.Atomic or Simple Sentences Sentence variables – Examples: » Pe.g., “John dances on the table.” » Q e.g., “The table will be broken.” » R e.g., "James is the man next to the wall over there.

Symbolic Elements of the Logic 2.Connectives (or Logical Operators) ~ the tilde “it is not the case that …” or simply "not" & the ampersand “ … and … ” v the wedge “either … or … ” -> the arrow “if … then … ” the double arrow “ … if and only if … ” Examples: ~P P & Q P v Q P -> Q P Q

Symbolic Elements of the Logic 3.Parentheses – Examples: 1.( P & (Q -> R )) 2.P & (Q -> R) 3.P & Q -> R 4.P & (Q & R) P and (if Q then R) If P and Q then R See page 9: “parentheses dropping conventions” P and (Q and R) Outermost parentheses unnecessary Inner Parentheses When necessary?

Parenthesis Dropping 1.Drop parentheses surrounding sentence. 2.Drop embedded parentheses only if unambiguous. Parenthesis Dropping 1.Drop parentheses surrounding sentence. 2.Drop embedded parentheses only if unambiguous.

KINDS OF VARIABLES Excursus

Kinds of Variables Sentence Variable: P, Q, R, S, T,... – an element of the formal language – stands for any simple (atomic) sentence in natural language Metavariable: Φ (Phi) or Ψ (Psi) – not an element of the formal language – stands for the any WFF – used to represent logical form

The 6 Sentences (WFFs) (pages 6-7) 1) Atomic Sentence (P, Q, R, S, …) 2) Negation~Φ 3) ConjunctionΦ & Ψ 4) Disjunction Φ v Ψ 5) Conditional Φ -> Ψ 6) BiconditionalΦ Ψ 7) and nothing else Unary Binary

READING SYMBOLIC LOGIC Part III (Order of Operations)

The Key to Recognizing Sentences Binding Strength See page 9 Strongest ~ & and/or v -> Weakest

P = We are studying symbolic logic. Q = It is interesting. P = We are studying symbolic logic. ~P = We are not studying symbolic logic. ~~P =It is false that we are not studying symbolic logic. Recognizing Negations The ~ attaches to the symbol directly to the right of it. Examples: ~P~P ~~P ~(P & Q) ~P & ~Q ~(~P & ~Q) NB: the middle statement is not a negation (Note the parentheses) Strongest ~ & and/or v -> Weakest ~Φ~Φ ~Φ~Φ ~Φ~Φ ~Φ~Φ

P = You study hard Q = You will do well on the exams R = Your GPA will go up Conjunctions and Disjunctions The & or v connects two WFFs. Examples: P & Q P v Q P & (Q v R) (P & Q) v R P & (Q -> R) (P -> Q) v R (Note the parentheses) Strongest ~ & and/or v -> Weakest Φ & Ψ and Φ v Ψ Φ & Ψ and Φ v Ψ Φ & Ψ and Φ v Ψ Φ & Ψ and Φ v Ψ P = You study hard Q = You will do well on the exams

P = You study hard Q = You will do well on the exams R = Your GPA will go up Conditional Statements The -> connects two WFFs. Examples: P -> Q P -> ~Q P -> (Q -> R) (P -> Q) -> R P -> Q v R P & Q -> R (Note the parentheses) Strongest ~ & and/or v -> Weakest Φ -> Ψ

P = You study hard Q = You will do well on the exams R = Your GPA will go up Biconditionals The connects two WFFs. Examples: P Q P ~Q P Q & R P v Q R P -> Q R P (Q R) (Note the parentheses) Strongest ~ & and/or v -> Weakest Φ Ψ

Parentheses and Ambiguity What kind of statement is this? P v (Q & R) P v Q & R (unambiguous) (ambiguous) Strongest ~ & and/or v -> Weakest

Summary 1.Elements of Symbolic Logic – (i) Variables, (ii) Connectives, (iii) Parentheses 2.Sentences (or WFFs) – Atomic – Complex 3.Key to Reading Symbolic Logic – Binding Strength of Connective

Homework 1.Study Allen/Hand Logic Primer – "Well-formed Formula," pp. 6-7 – "Binary and Unary Connectives," p. 7 – "Parentheses Dropping Conventions," p. 9 – ("Denial,“ – logically opposite sentences, p. 7) 2.Handout on Class Web Page: – Truth Tables Handout Truth Tables Handout 3.Watch At Home: – “Basic Concepts Review” presentation Bring this handout to class from now on!