Introductory Logic and It's Rabbit Holes Ch.1 Part 2
Introductory Logic and It's Rabbit Holes Ch.1 Part 2 PHI 120
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This 4 page Class Notes was uploaded by Maddi Caudill on Tuesday January 26, 2016. The Class Notes belongs to PHI 120 at University of Kentucky taught by Daniel B. Cole in Spring 2016. Since its upload, it has received 13 views.
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Date Created: 01/26/16
PART TWO The Artiﬁcial Language: Propositional Symbols and Connectives • Artiﬁcial language contains: propositional symbols, connectives, and parenthesis • Propositional symbols are uppercase letters from the English Alphabet (A, B, C, etc) and they are divisible into two sets: 1. Propositional constants ----> propositional symbol that designates (or represents or stands for) a speciﬁc (i.e., exactly one) proposition 2. Propositional variables ----> designates (or represents or stands for) any proposition whatsoever Thus A, B, C could be the propositional constants designating the proposition "Ardbeg drowns," "Bobo drowns" and "Coco drowns," respectively. * Think of it like numbers and variables in math! Where as the constants = numerals and the variables = variables/letters in math such as "x" * The number represents only a speciﬁc number, and a variable can represent ANY number! Connectives: the words that are more equal than others- such as "not" "and" "or" "if- then" and "if and only if" • Its with these words that we construct proportions of increasingly greater complexity • IT'S NOT SOMETHING THAT CONNECTS TWO OR MORE SENTENCES • It’s a device (or operator) that operates on one or more sentences in order to generate a new sentence Consider these sentences, BOTH containing connectives: 1. Arbeg howls and Bobo howls 2. It is not the case that Ardbeg howls * THE WORDS IN BOLD ARE BOTH CONNECTIVES!!!! * Sentences come in 2 forms: 1. Atomic: (simple) sentence is a sentence WITHOUT any connectives 2. Molecular: (compound) sentence is a sentence containing at least ONE connective So, lets look at some of that artiﬁcial language now…. A = Ardbeg howls B = Bobo howls The dash _ "It is not the case that…." or "it is false that…" operates on a sentence to create the negation of that sentence The caret ^ "and" creates conjunction The wedge v "or" / "either" creates disjunction The arrow ---> "if-then" creates the conditional (consequent) The double arrow <----> "If and only if" (iff) creates bi-conditional • In the English language, "or" is ambiguous • It has the exclusive sense and the inclusive sense • "You maybe have either the cake or the ice cream" ---> one or the other, but not both (exclusive) •P exclusive- or Q means P or Q but not both P and Q •P inclusive - or Q means P or Q and possibly both P and Q • You do not need a connective for exclusive or inclusive FROM THIS POINT ON…. You must treat the English language uses of "or" (or "either- or") as inclusive-or!!! Regardless of how strained it might seem. Truth Tables: Deﬁnitions 1. The sentence "-P" is true iff P is false 2. The sentence "P ^ Q" is true iff P is true and Q is true 3. The sentence "P v Q" is true iff P is false and Q is false 4. The sentence "P --> Q" is false iff P is true and Q is false 5. The sentence "P <--> Q" is true iff either P and Q are both true OR P and Q are both false • A truth table shows all possible combinations of truth value that a given sentence can have • With a single sentence there are two possibilities: true or false • Aristotles Law of the Excluded Middle: there is no third possibility/value • Artistotles Law of Contradiction: assume that no sentence can be both true and false • A sentence that appears to be both true and false is AMBIGUOUS • An ambiguous sentence is not a sentence strictly speaking, but a form of a sentence but lacks in content or meaning • A string of words if ambiguous iff it possesses more than one meaning, such that its an instance of more than one proposition The magic formula for ﬁnding the number of possibilities of a sentence is 2 raised to the nth power where n = the number of atomic sentences you are considering In sum: given one atomic, a truth table will have two rows; given two atomics, four rows; with three atomics, eight possibilities, etc. Rows = horizontal arrays of T's (truths) and F's (falses) Columns = vertical * truth values are highlighted * The Truth Table Deﬁnition of the Dash P -P _ __ What does this mean? ---> -P is F if P is T (in the ﬁrst row) T FT -P is T if P is F (second row) F TF Truth Table Deﬁnition of the Caret PQ P ^ Q ___ _____ TT TTT P ^ Q is T iff P is T and Q is T ----> the conjunction is true iff both of its conuncts are truth TF TFF P ^ Q is F iff either P is F or Q is F or oth P and Q are F FT FFT a conjunction is F iff at least one of its conjuncts is false FF FFF The Truth Table Deﬁnition of the Wedge PQ P v Q ___ _____ TT TTT P v Q is T iff either P is T or Q is T or both P and Q are T TF TTF A disjunction is true iff at least one of its disjuncts is true FT FTT P v Q is F iff P is F and Q is F ---> disjunction is F iff both FF FFF of its disjuncts are false * note the curious symmetry between the truth table def of caret and the truth table def of the wedge: a conjunction is true iff each conjunct is true, whereas a disjunction if false iff each disjunct is false * The Truth Table Deﬁnition of the Arrow PQ P ---> Q ___ _______ TT TTT P --> Q IS T except when P is T and Q is F ---> kind of counter intuitive!! TF TFT P --> Q is F iff P is T and Q is F FT FTT FF FTF The Truth Table Deﬁnition of the Double Arrow PQ (P --> Q) ^ (Q --> P) ___ ________________ TT TTT T TTT TF TFF F FTT FT FTT F TFF FF FTF T FTF OR (TABLES ARE IDENTICAL) -----> They both say the same thing: P <--> Q is T iff either P and Q are both true or P and Q are both false. Iff P and Q have the same truth value. PQ P <--> Q ___ _______ TT TTT TF TFT FT FFT FF FTF