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Logic primer Sentential logic Sentential (or propositional) logic i) Takes simple declarative sentences (i.e., sentences with no logical operators and that are either true or false) as its basic unit: For example, “Mary went to the store” but NOT “Mary did not go to the store”; the negation makes the latter a compound sentence. And NOT “Did Mary go the store?” as the latter is a question and does not have a truth value (i.e., is neither true nor false). ii) Uses Capital Roman Letters, A through Z with or without subscripts, to symbolize simple declarative sentences. That is, A is a sentence and so is A1, though we’ll never have reason to use the second. So, for example, we might symbolize “Mary went to the store” as M (or as S or any other letter we choose). Whatever you choose, remember it symbolizes the whole simple declarative sentence, not a word. iii) Uses P, Q and R (or p, q and r) as meta-variables that “range over” sentences (both simple and compound). That is, these letters “stand in” for sentences just as, in Algebra, x and y stand in for numbers. (iv) Assumes bivalence: that every declarative sentence is either true or false, including those whose truth value we don’t (or couldn’t) know: for example, “Julius Caesar had eggs for breakfast on the day he was killed” or “On July 8, 2008, an Amtrak train will derail on the way route to DC from Philadelphia.” Simple sentences are its basic unit and logical operators yield compound sentences. In sentential logic, the operators are: Conjunction ('and' or 'but' for example). The ampersand (&) is used to symbolize a conjunction so that conjunctions are of the form P & Q. Note that here and in all examples to come, P and Q might themselves be compound. For a conjunction to be true, both conjuncts must be true. Material conditional ("if then"), symbolized by the horseshoe : A sentence of the form P Q. The sentence following 'if' is called the antecedent; that following 'then' is called the consequent. A material conditional is false if and only if the antecedent is true and the consequent is false. Negation ("it is not the case that"), symbolized by the tilde ~: A sentence of the form ~P. ~P (a negation) is true if and only if P is false. 2 Material bi-conditional ("if and only if"), symbolized by a triple bar : A sentence of the form P Q. A material conditional is true if and only if the two halves have the same truth value (they are both true, or both false). The biconditional is shorthand for: (P Q) & (Q P) Disjunction ("either/or" in the inclusive sense), symbolized by the wedge v. A sentence of the form P v Q. In logic, "either/or" is understood as "at least one is true" rather than in the exclusive sense ("at most one is true"); to make a claim of the latter kind, we have to say more: specifically "Either P or Q, and not both." A disjunction is false if and only if neither the left disjunct nor the right is true. Definitions: Argument: A set of sentences, one of which is the conclusion and there is at least one premise. Note single, compound sentences can include arguments: for example: The U.S. will win the war on terror because the U.S. has god on its side. What follows ‘because’ is the premise; what precedes it is the conclusion. Deductive validity: an argument is deductively valid if and only if it is not possible for the premise(s) to be true and the conclusion false. The logical connection is tight enough to preclude that. Note: in everyday speech, we speak of statements being “valid”; valid is reserved in logic for whole arguments. An argument is deductively invalid if and only if it is not deductively valid. Validity is, in most instances, a matter of form not content. A valid argument: An invalid argument: All men are mortal Socrates is a man ---------------------Therefore Socrates is mortal. All men are mortal Socrates is mortal -----------------------Therefore Socrates is a man. The second argument is invalid precisely because it is possible for the premises to be true and the conclusion false (not the case in the first argument). Soundness: An argument is sound if and only if it is deductively valid and its premises are true. Special cases: Any argument with inconsistent premises (premises that contradict each other) is deductively valid. It will never be possible for the premises to be true and the conclusion false because it will never be possible for the premises to be true. However, no such argument will be sound. Why is that? Special cases continued: 3 There are three kinds of statements: Logically indeterminate: while true or false, they are so not because of their form but because of their content and how the world is. For example, “The Earth is spherical”. Logically true: a sentence that is always true, either because it is a definition or it is a tautology. Examples: “4 is a number that is larger than 3 and smaller than 131.” True on the basis of mathematical definitions. Another: “A rose is a rose.” A tautology, there is no content. Also: if A, then A. Couldn’t be false. Logically false: a sentence that is always false, because it involves a contradiction. For example: “A & ~A” or “If A, then ~A”. See if you can explain the following: Any argument with a logically false premise is deductively valid but not sound. Any argument with a logically true conclusion is deductively valid, and may or may not be sound. Inductive arguments: Inductive arguments are deductively invalid, but we use them all the time. Two common types: Empirical generalization: 1. I didn’t die today (it’s midnight). 2. I didn’t die yesterday. 3. I didn’t die the day before that. . . N. I’ve not died on any day I’ve lived. -----------------------------------------I’ll not die tomorrow (or I’ll never die…) Arguments by analogy: 1. Drug X does not harm rats. 2. Rats and humans are similar in just the right ways. ----------------------------------------------------------Drug X will not harm humans. Two schools of thought on induction: Hume: it is a matter of habit, it cannot be justified, and it is basically irrational. We cannot even move to say that the conclusions of empirical generalizations are 4 probable, as when we have a potentially infinite # of cases covered by the conclusion, and a finite (however large) # of observations, the probability of the conclusion is really zero. Some others: He’s right that it cannot be justified, but we can make sense of stronger and weaker arguments. The argument that I won’t die ever is weak; depending on the evidence supporting the second premise of the second argument, it could be a strong. We can make sense of some arguments making their conclusions more probable than other arguments. A weak inductive argument: 1. The 1st Australian I met was drunk. 2. The 2nd I met was drunk. 3. I’ve never met a sober Australian. -----------------------------------------All Australians are drunks. A strong one: 1. I’ve met every Australian currently alive. 2. All of them were drunk. -----------------------------------------------Probably all Australians currently alive are drunk. A logic puzzle for fun: In (another) mythical community, there are 3 kinds of persons: Knights who always tell the truth. Knaves who always lie. Normals who sometimes lie and sometimes tell the truth. There are 3 people – A, B, and C – one of whom is a Knight, one of whom is a Knave, and one of whom is a Normal but not necessarily in that order. They make 3 statements: A: I am normal. B: That is true. C: I am not normal. What are A, B, and C?