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

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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!

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

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SYMBOLIC ELEMENTS OF THE LOGIC Part I

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

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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

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

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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.

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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

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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?

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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.

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KINDS OF VARIABLES Excursus

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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

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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

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READING SYMBOLIC LOGIC Part III (Order of Operations)

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

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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

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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

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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

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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

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

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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

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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!

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