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An Introduction to Propositional Logic Translations: Ordinary Language to Propositional Form 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. Simple Propositions Fast foods tend to be unhealthy. Parakeets are colorful birds. (p. 290) Simple propositions are grammatically independent expressions of information. 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. 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. 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) Propositional Symbols Symbolizing propositions allows us to focus on the relations between propositions (logic) rather than the content of those propositions (truth). P1: UCLA will either raise fees or P1: R or C cut back on services. P2: UCLA will not raise fees. P2: Not R Con: UCLA will cut back on Con: C services. 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. 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. 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. 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. Basics of Propositional Logic All arguments are reducible to symbols, which represent either elements of an argument or ways these elements are put together. 1. All arguments contain statements, by definition. Each statement is represented by a “propositional variable” – p, q, r, s 2. 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. Propositional Operators ~ (“tilde,” negation) • (“dot,” conjunction) v (“wedge,” either-or) Not, it is false that, conjunctions like “don’t” And, also, but, in addition, moreover Or, unless > (“implication” or “conditional,” if,then). Is a sufficient (or necessary) condition of, ifthen, 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 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.) . 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) 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. Well-formulated formula Not a well-formulated (wff) formula (non-wff) pv~q pv~q>r 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.). Well-formulated formula Not a well-formulated (wff) formula (non-wff) ~ (p ≡ q) ● r (p ~ ≡ q) ● r Working with Propositional Symbols - 1 Expression: If I get 80 points on the test, I’ll get a B on the test. Partial symbolization of the expression: If P, then B (material implication; conditional statement) Statement form: P>B p > q (in variables) Working with Propositional Symbols - 2 Expression: p. 290, #5 Psychologists and psychiatrists do not both prescribe prescription drugs. Partial symbolization of the expression: Not both G and T → conjunction Statement form: ~(G●T) ~(p●q) Working with Propositional Symbols - 3 Expression: p. 290, #11 If Internet use continues to grow, then more people will become cyberaddicts and normal human relations will deteriorate. Partial symbolization of the expression: If I, then both C and D → mixed forms Statement form: I>(C●D) p > (q●r) 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: ~