An Introduction to Propositional Logic

1. An Introduction to Propositional Logic Translations: Ordinary Language to Propositional Form

2. What is a Proposition? • Propositions are the meanings of statements. • I have no money • Ich habe kein geld. • Meanings are the thoughts, concepts, ideas we are trying to convey through speech and writing.

3. Simple Propositions • Fast foods tend to be unhealthy. • Parakeets are colorful birds. (p. 290) • Simple propositions are grammatically independent expressions of information.

4. Compound Propositions • If fast foods tend to be unhealthy, then you shouldn’t eat them. • Parakeets are colorful birds, and colorful birds are good to have at home. • People are free, if and only if they can choose their actions and there are no forces compelling those actions.

5. The Focus of Propositional Logic • Propositional truth is determined by consulting typical sources of information. • Propositional logic is determined by examining how various propositions are related.

6. Types of relations between propositions • 1 proposition is offered in support of another (simple argument) • 1 proposition expressed the condition under which a 2nd proposition is true (conditional statement) • 1 sentence offers two proposed alternatives, and a 2nd proposition negates one of these alternatives (disjunctive syllogism)

7. Propositional Symbols • Symbolizing propositions allows us to focus on the relations between propositions (logic) rather than the content of those propositions (truth).

8. Module Objectives - 1 • Learn how to symbolize complex propositions • Simple propositions, which express grammatically independent units of information, are easy to symbolize: “It is cold” = “C” • Complex propositions are sentences which contain 2 or more simple propositions. These can be more difficult to symbolize.

9. Module Objectives - 2 • Learn how to determine the truth-value of complex propositions Remember, an argument can only establish the truth of it’s conclusion if all its premises are true (and its reasoning is valid) • “Fees are rising at UCLA” is either true or false • “Either fees are rising or services are being cut back” could be true or false – depending on the actual situation regarding fees and services at UCLA.

10. Module Objectives - 3 • Create and interpret truth tables for both propositions and arguments (series of propositions). Truth tables allow us to see all possible conditions under which a statement could be true and could be false.

11. Module Objectives – 4 & 5 • Learn to recognize common argument forms, and know when an argument form is valid or invalid • Prove that an argument is valid or invalid when it doesn’t fit a common argument form.

12. Basics of Propositional Logic All arguments are reducible to symbols, which represent either elements of an argument orways these elements are put together. • All arguments contain statements, by definition. Each statement is represented by a “propositional variable” – p, q, r, s • All arguments also contain connections, or ways in which individual propositions are related. Each of these connections are represented by one of five “operators”: Putting propositional variables together with operators creates a “statement form,” or a symbolic blueprint identifying typical structures of English expressions.

13. Propositional Operators ~ (“tilde,” negation) Not, it is false that, conjunctions like “don’t” • (“dot,” conjunction) And, also, but, in addition, moreover v (“wedge,” either-or) Or, unless > (“implication” or “conditional,” if,then). Is a sufficient (or necessary) condition of, if-then, implies, given that, only if Ξ (“biconditional,” if and only if) If and only if, is equivalent to, is a sufficient and necessary condition of

14. Note on the “Tilde” 1. All operators except the tilde must relate at least two propositions. 2. The tilde negates either a proposition directly, or an operator relating to propositions (by standing directly before a parentheses/bracket/etc.) .

15. Examples of “Tilde” Functions ~ p = not p; p is not true, etc ~ p ● ~ q = p is false and q is false; p and q are both false ~ ( p ● q ) = not both p and q (maybe one is true and one false)

16. Rules for Operator Types - 1 If there is more than one operator (excluding tildes), then some portion of the statement must be included in parentheses/brackets/etc.

17. Rules for Operator Types - 2 The tilde negates either a proposition directly, or an operator relating to propositions (by standing directly before a parentheses/bracket/etc.).

18. Working with Propositional Symbols - 1

19. Working with Propositional Symbols - 2

20. Working with Propositional Symbols - 3

21. Tips for Translation • Use “clue words”: • “If, then”; “on the condition that”: > • Both; and; also; etc: ● • Either, or; or maybe both:v • If one, then the other; if and only if; always occur together: • Negation; it is not true that, not: ~