Elementary Logic

Elementary Logic PHIL 105-302 Intersession 2013 MTWHF 10:00 – 12:00 ASA0118C Steven A. Miller Day 3

Argument evaluation What makes an argument good? 1) true premises 2) conclusion at least probable, given 1 3) premises are related to the conclusion 4) the conclusion can’t be undermined

Argument evaluation 3) premises are related to the conclusion 1) Scientists have not yet found any sign of aliens. 2) The universe would be boring without any aliens. 3) Therefore, there must be undiscovered aliens. Premise 1 is relevant to the opposite of the conclusion; premise 2 is irrelevant to either the conclusion or its opposite.

Argument evaluation 4) the conclusion can’t be undermined A deductive argument is valid when it is impossible (in the strongest sense) for the premises to all be true and the conclusion to be false.

Argument evaluation it is impossible for … the conclusion to be false Any argument where the conclusion cannot be false is always valid. The prime case of this is found with logically necessary conclusions. 1) All squares are rectangles. 2) 10 is greater than 6. 3) If it is red, then it is red.

Argument evaluation 1) Lincoln is the capital of Nebraska. 2) Barack Obama is the president. 3) Therefore, all bachelors are unmarried. Valid, because it is impossible for the premises to all be true and the conclusion false. The conclusion cannot be false.

Argument evaluation it is impossible for … the premises to all be true Any argument where the premises cannot all be true is always valid. The prime case of this is found with logically inconsistent premises. These are premises where the truth of one guarantees the falsehood of another.

Argument evaluation 1) All dogs are brown. 2) No dogs are brown. 3) Therefore, Mark McGwire is overrated. Valid, because it is impossible for the premises to all be true and the conclusion false. The premises cannot both be true (truth of one implies the falsity of the other).

Argument evaluation Inductive arguments do not share this standard of validity. Instead, they rely on complex standards of evidence, relevance, and probability… (which we’re skipping in the interest of time, see S, pp. 31-43, etc.).

Formalization Arguments in natural language (e.g. English, Spanish, Arabic, etc.) can be very complicated to follow. Formalization offers a way to simplify and to make explicit the underlying structure (that is, logic) of arguments.

Argument form Consider the following arguments: • The ice cream is either chocolate or vanilla. • The ice cream isn’t chocolate. • So, the ice cream is vanilla.

Argument form Consider the following arguments: • John is either taller or shorter than Mary. • John isn’t taller than Mary. • So, John is shorter than Mary.

Argument form Consider the following arguments: • The sauce is either ketchup or mustard. • It isn’t ketchup. • So, the sauce is mustard.

Argument form Commonality between the three: 1) This or that. 2) NOT THIS. 3) Therefore, THAT. This is called a “disjunctive syllogism.” “Disjunctive syllogism” is the argument’s form.

Argument form All arguments have forms, though some are easier to find than others. All (deductive) argument forms are either valid or invalid. Disjunctive syllogism, for instance, is valid.

Seventh Inning Stretch (“…Take Me Out to the Crowd, …”)

Formalizing Symbolization: - simplifies arguments - quicker than natural language - helps in finding form

Formalizing Symbolizing Propositional Logic Basic unit: the proposition (the statement, for us) Statement symbolization: any capital letter

Formalizing Statement symbolization: • The ice cream is either chocolate or the cream is vanilla. • The ice cream isn’t chocolate. • So, the ice cream is vanilla. “The ice cream is chocolate” = P “The ice cream is vanilla” = Q “The ice cream isn’t chocolate” = Z “It is not the case that P”

Formalizing Statement symbolization: • The ice cream is either chocolate or the cream is vanilla. • The ice cream isn’t chocolate. • So, the ice cream is vanilla. • Either P or Q. 2) It is not the case that P. 3) So, Q. Symbol for conclusions: ∴ 3) ∴

Formalizing Negation symbolization: “The ice cream isn’t chocolate.” “It is not the case that P.” Symbol for “it is not the case that”: ~ “It is not the case that P” = ~P (You may also see ¬P, −P, NP, !P)

Formalizing Conjunction symbolization: “The ice cream is chocolate and cold.” “The ice cream is chocolate and cold” = P “The ice cream is chocolate” = P “Cold” = Q “The ice cream is cold” = Q

Formalizing Conjunction symbolization: “The ice cream is chocolate and cold.” P and Q Symbol for “and” = & “The ice cream is chocolate and cold” = P&Q (You may also see ∧, ·)

Formalizing Conjunction symbolization: Symbol for “and” = & Other words that take “&” as their symbol: but yet although nevertheless moreover

Formalizing Disjunction symbolization: “The ice cream is chocolate or vanilla.” P or Q Symbol for “or” = v “The ice cream is chocolate or vanilla” = PvQ

Formalizing Material conditional symbolization: “If the ice cream is chocolate, then it is cold.” If P, then Q Symbol for “if … then” = → “If the ice cream is chocolate, then it’s cold” = P → Q (You may also see ⊃, >)

Formalizing Material conditional symbolization: Symbol for “if … then” = → If I get a puppy, then I’ll be so happy. Antecedent = the part after the ‘if’ Consequent = the part after the ‘then’

Formalizing Biconditional symbolization: “The ice cream is chocolate if and only if it’s cold.” P if and only if Q Symbol for “if and only if” = ↔ “The ice cream is chocolate if and only if it’s cold” = P ↔ Q (You may also see ≡.)

Formalizing Biconditional symbolization: Symbol for “if and only if” = ↔ “The ice cream is chocolate if and only if it is cold.” is equivalent to “If the ice cream is chocolate, then it is cold and if the ice cream is cold, then it is chocolate.”

Formalizing Symbolization chart: It is not the case = ~ And = & Or = v If … then = → If and only if = ↔ Therefore = ∴

Formalization tips 1) Negation applies to whatever is to its immediate right ~P & Q is different than ~(P & Q) (they’re also different than ~P & ~Q) 2) All other operators apply both to their left and right

Formalization tips 3) Use parentheses, and then brackets, to avoid confusion. ~P v Q & Z = confusing (~P v Q) & Z, ~(PvQ) & Z, ~[P v (Q & Z)], etc. = less confusing

Formalization tips All formulas must be well-formed This means they must be syntactically correct (Exactly what the ‘English sentence’ “jo.!hn wa2s He@#rD” isn’t.) 1) All statement letters are well-formed formula. 2) All negations of statement letters are wffs. 3) All operators between two statement letters, and their negations, are well-formed formula. (A slightly less complex version of S, p. 52-4.)

For next time… Read ahead, S, pp. 55-68 Not difficult, but lots to memorize / apply

Elementary Logic