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

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 2 nd proposition is true (conditional statement) 1 sentence offers two proposed alternatives, and a 2 nd 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 cut back on services. P2: UCLA will not raise fees. Con: UCLA will cut back on services. P1: R or C P2: Not R Con: C

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 argumentor 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 (p. 281) ~ (“tilde,” negation) v (“wedge,” either-or) Ξ (“biconditional,” if and only if) (“dot,” conjunction) > (“implication” or “conditional,” if,then). Not, it is false that, conjunctions like “don’t” And, also, but, in addition, moreover Or, unless Is a sufficient (or necessary) condition of, if- then, implies, given that, 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 ) = not both p and q (maybe one is true and one false) ~ p ● ~ q = p is false and q is false; p and q are both 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 (wff) Not a well-formulated formula (non-wff) p v ~ qp v ~ 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 (wff) Not a well-formulated 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 - 1 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: ~

Tips for Translation - 2 Develop a system to mark every important part of a complex proposition before trying to symbolize it. DMX abandons explicit lyrics if and only if neither Columbia nor BMG stops advertising.