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1、Propositional Logic,USEM 40a Spring 2006 James Pustejovsky,Evaluation of Deductive Arguments,argument A is a deductive argument =df. A is an argument in which the conclusion is supposed to follow from the premises with necessity / with certainty deductive argument A is valid =df. it is not possible
2、for all of As premises to be true and its conclusion false deductive argument A is sound =df. (i) A is valid, and (ii) all of As premises are true,(P1) If Grover is dead, then Grover does not vote. (P2) Grover is dead. (C) Therefore, Grover does not vote.,Formal Logic,with many deductive arguments,
3、validity is a matter simply of form, of structure formal logic studies these cases in which validity depends solely on form not all valid arguments are formally valid: (P) Grover is a bachelor. (C) Therefore, Grover does not have a wife.,argument A is formally valid if, in virtue of As logical form
4、alone, it is impossible for all of As premises to be true and its conclusion false (P1) All 19th Cent. American presidents are dead people. (P2) All dead people are people who do not vote. (C) Therefore, all 19th Cent. American presidents are people who do not vote.,Why study formal logic?,It gives
5、us a more robust understanding of validity in general It forms the building block for our model of meaning in language and for reasoning in general,Introduction to Propositional (or “Sentential” or “Truth-Functional”) Logic,deals with propositions whole statements; meaningful declarative sentences S
6、 is a simple proposition =df. S does not contain any other proposition as a component Grover is dead. S is a compound proposition =df. S contains at least one simple proposition as a component Grover is dead and Stevenson is dead. It is not the case that Grover is beautiful. The woman who married Gr
7、over is beautiful.,Propositional Forms, Variables, Constants, and Substitution Instances,a propositional form is a pattern for a whole class of propositions (p & q) v p & ) p q ) a propositional variable is a lowercase letter (e.g., p, q, r, s) for which a proposition may be substituted,a propositio
8、nal constant is a capital letter that stands for a particular, definite proposition G = Grover is dead. S = Stevenson is dead. a substitution instance of a propositional form is the result of uniformly replacing the propositional variables in that form with propositions the same proposition may be r
9、eplaced with different variables, but no two different propositions may be replaced by the same one variable,some examples,Grover is dead and Stevenson is dead. G & S p & q Grover and Stevenson are beautiful men. B & M p & q Grover is dead and Grover is dead. G & G p & p or p & q Grover and Frances
10、are a couple now. C p,Propositional Connectives (“Logical Operators” or Truth-Functional Connectives”),a definition for each connective this simply specifies the truth conditions for any proposition in which the connective occurs this is a way of giving the meaning of the connective by specifying it
11、s use a truth table sets out all of the possible truth value combinations for the simple component propositions and shows, for each combination, the value of the compound proposition,Conjunction and, but, also, as well,some examples,Grover and Stevenson are dead. G & S Grover and Frances are a coupl
12、e now. C All that I have left are photographs and memories. A ? Grover and Frances are in love. ?,Disjunction or, either or,Inclusive Disjunction “either this or that, and perhaps both”,Some Examples,Either Zac wants to avoid you or hes out of town. W v O Special consideration is appropriate for eld
13、erly or infirm people. E v I Either Kelly or Kerry is the best singer alive today. (B v P) & (B & P),Exclusive Disjunction “either this or that, but not both”,Negation not, it is not the case that.,Grover is not alive. A It is not the case that Grover is alive. A Grover is not very attractive. V Fra
14、nces never knew about Grovers affair. K,The (Material) Conditional if., then.,antecedent consequent,“Why should we count the conditional claim as true when the antecedent is false and the consequent true or, especially, when both are false?” If you get an A on the final, then you get an A for the co
15、urse. If Shane is younger than 31, then Shane is younger than 33. “If p, then q.” = “Either q is the case or p is not the case.” = “It is not the case that p and not-q.”,p q is equivalent to q v p is equivalent to (p & q),If Grover is decapitated, then Grover is dead.,Some Other Constructions,unless
16、 constructions can often be treated as conditionals e.g., Otis remains quiet unless he is spoken to. S Q (also Q v S) provided that, given that, on condition that, and such like phrases only if constructions are different You get to be president only if you are over 34. P O,Some Ifs that Are Not Conditionals,uncertainty / iffy e.g., Jen is not certain if
