Introduction to Formal Logic: Understanding Symbolic Elements and Well-Formed Formulas

1. Presentation: "Intro to Formal Logic" Introductory LogicPHI 120 Please take out your book. Please turn off all cell phones!

2. Homework • 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) • Handout on Class Web Page: • Truth Tables Handout • Watch At Home: • “Basic Concepts Review” presentation Bring this handout to class from now on!

3. New UnitFormal (Symbolic) Logic Sentential Logic Today: Basic Grammar of Sentences

4. Part I Symbolic Elements of the Logic

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

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

7. Symbolic Elements of the Logic • Atomic sentences • Connectives (or Logical Operators) • Parentheses ( … )

8. Symbolic Elements of the Logic • Atomic or Simple Sentences • Sentence variables • Examples: • P e.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.

9. Symbolic Elements of the Logic • 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

10. Symbolic Elements of the Logic • Parentheses • Examples: • ( P & (Q -> R )) • P & (Q -> R) • P & Q -> R • P & (Q & R) Outermost parentheses unnecessary P and (if Q then R) See page 9: “parentheses dropping conventions” P and (if Q then R) If P and Q thenR P and (Q and R) Inner ParenthesesWhen necessary?

11. Parenthesis Dropping Drop parentheses surrounding sentence. Drop embedded parentheses only if unambiguous.

12. Excursus Kinds of variables

13. 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

14. The 6 Sentences (WFFs)(pages 6-7) • Atomic Sentence (P, Q, R, S, …) • Negation ~Φ • Conjunction Φ&Ψ • Disjunction ΦvΨ • Conditional Φ->Ψ • BiconditionalΦ<->Ψ • and nothing else Unary Binary

15. Part III Reading Symbolic Logic (Order of Operations)

16. The Key to Recognizing Sentences Binding Strength Strongest ~ &and/orv -> <-> Weakest See page 9

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

18. 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 Φ & Ψ and Φ v Ψ Φ & Ψ and Φ v Ψ (Note the parentheses) Strongest ~ &and/orv -> <-> Weakest P = You study hard Q = You will do well on the exams R = Your GPA will go up P = You study hard Q = You will do well on the exams

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

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

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

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

23. Homework • 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) • Handout on Class Web Page: • Truth Tables Handout • Watch At Home: • “Basic Concepts Review” presentation Bring this handout to class from now on!