Declarative Statements and Logic: Translation, Truth Tables, and Validity

2. Chapter 17 The Logic of Declarative Statements

3. Learning Outcomes Identify negations, conjunctions, disjunctions, and conditional declarative statements Translate simple and compound natural language declarative statements to and from symbolic notation

4. Learning Outcomes Identify tautologies, inconsistent statements, and contingent declarative statements using truth tables Test declarative statements for implication and equivalence using truth tables Test arguments composed of declarative statements for validity using truth tables

5. Chapter Opening Video

6. Declarative Statements Simple statements Negations Statement compounds: And, or, if, then

7. Simple Statements • Grammatically correct construction in a given language used to assert that an idea is true • Janet is going to the beach • Sally and Broderick met online • Evaluation is limited to only two values • True • False

8. Negations • Grammatically correct construction used to assert that a statement is false • I do not like frozen yogurt • It is not true that the abacus is obsolete • Expressed by the symbol ~ (tilde)

9. Statement Compounds: And, Or, If, Then • Conjunction: Grammatically correct construction used to assert that two statements are both true • Expressed by ampersand (&) • (p & q)

10. Statement Compounds: And, Or, If, Then • Disjunction: Grammatically correct construction used to assert that one or both of two statements are true • Expressed by a v-shaped wedge • (p v q)

11. Statement Compounds: And, Or, If, Then • Conditional • Grammatically correct construction used to assert that if an antecedent statement is true, then a consequent statement is true • Expressed by an arrow (→) with opening and closing parentheses

12. Translating Between Symbolic Logic and a Natural Language Grammatically correct expressions Translation to English

13. Grammatically Correct Expressions • Rules to form grammatically correct expressions in the language of symbolic logic • Statement letter is a grammatically correct expression • Placing ~ in front of any grammatically correct expression generates another grammatically correct expression

14. Grammatically Correct Expressions • Placing &, v, or → between any two grammatically correct expressions and enclosing it with a pair of parentheses generates another grammatically correct expression

15. Translating to English • Rules • Render ~A as “It is not the case that A” • Render (A & B) as “A and B” • Render (A v B) as “Either A or B” • Render (A → B) as “If A, then B”

16. Translating to Symbolic Logic • Translating a telephone tree • Telephone tree instruction is an exercise in the Logic of Statements • Symbolic representation • ((((p → q) v (r → s)) v (p1 → q1))v (r1 & s1))

17. Translating to Symbolic Logic • Translation knowledge from telephone tree • Determine whether the sentence is a declarative statement • Sentences used to make assertions can be translated into symbolic logic • Declarative statements handle negations, conjunctions, disjunctions, conditionals, and simple assertions

18. Detecting the Logical Characteristics of Statements Building truth tables Tautologies, inconsistent statements, and contingent statements Testing for implication and equivalence

19. Building Truth Tables • Truth table for a grammatically correct expression in symbolic notation, A • Count how many different statement letters are used in A • Columns on the left-hand side of a truth table • Rows of a truth table are organized in a predictable order

20. Truth Table of ((p & q) → ~q)

21. Contingent Statement Grammatically correct expression True under at least one possible assignment of truth values to its component simple statements False under another possible assignment of truth values to its component simple statements

22. Inconsistent Statement Grammatically correct expression False under every possible assignment of truth values to its component simple statements Referred as self-contradictory

23. Tautology Grammatically correct expression True under every possible assignment of truth values to its component simple statements Logic of Statements does not contain all possible tautologies

24. Testing for Implication and Equivalence • Implication • A implies B • If there is no interpretation of the statement letters of A and B such that A is true and B is false • If the grammatical expression generated by the structure (A → B) is a tautology • Equivalence • A and B, are equivalent: If, and only if, the biconditional (A ≡ B) is a tautology

25. Evaluating Arguments for Validity Testing symbolic arguments for validity Testing natural language arguments for validity

26. Testing Symbolic Arguments for Validity • Consider the example • Premise #1 (q v r) • Premise #2 ~r • Conclusion q • Form the conditional (((q v r) & ~r) → q) • Build the truth table • Conditional is a tautology then the argument is valid at this level of logic

27. Testing Natural Language Arguments for Validity Translate natural language premises and conclusion into symbolic logic notation Form the conditional ((conjunction of the premises) → conclusion) Build the conditional’s truth table If the conditional is a tautology, then the argument is valid

28. Discussion Question • What are the advantages that the careful analysis of language provides when trying to interpret exactly what is being said? • Answer by giving an example from your own experience and explain your example

29. Sketchnote Video