FORMAL-LOGIC-I-PROPOSITIONAL-LOGIC我的命題邏輯形式邏輯課件

1、Propositional LogicUSEM 40aSpring 2006James PustejovskyPropositional LogicUSEM 40aEvaluation 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 certaintydeductive argument A is valid =df

2、.it is not possible for all of As premises to be true and its conclusion falsedeductive argument A is sound =df.(i) A is valid, and (ii) all of As premises are true Evaluation of Deductive Argume(P1)If Grover is dead, then Grover does notvote.(P2)Grover is dead. (C)Therefore, Grover does not vote. (

3、P1)If Grover is dead, then GFormal Logicwith many deductive arguments, validity is a matter simply of form, of structureformal logic studies these cases in which validity depends solely on formnot all valid arguments are formally valid:(P)Grover is a bachelor.(C)Therefore, Grover does not have a wif

4、e.Formal Logicwith many deductivargument A is formally valid if, in virtue of As logical form 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

5、19th Cent. American presidents are people who do not vote.argument A is formally valid iWhy study formal logic?It gives us a more robust understanding of validity in generalIt forms the building block for our model of meaning in language and for reasoning in generalWhy study formal logic?It giveIntr

6、oduction to Propositional(or “Sentential” or “Truth-Functional”) Logicdeals with propositions whole statements; meaningful declarative sentencesS is a simple proposition =df. S does not contain any other proposition as a componentGrover is dead.S is a compound proposition =df. S contains at least on

7、e simple proposition as a componentGrover is dead and Stevenson is dead.It is not the case that Grover is beautiful. The woman who married Grover is beautiful.Introduction to PropositionalPropositional Forms, Variables, Constants, and Substitution Instancesa propositional form is a pattern for a who

8、le 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 substitutedPropositional Forms, Variablesa propositional constant is a capital letter that stands for a particular, definite propositionG = Grover is dead. S

9、= Stevenson is dead.a substitution instance of a propositional form is the result of uniformly replacing the propositional variables in that form with propositionsthe same proposition may be replaced with different variables, but no two different propositions may be replaced by the same one variable

10、a propositional constant is a some examplesGrover is dead and Stevenson is dead.G & S p & qGrover and Stevenson are beautiful men. B & M p & qGrover is dead and Grover is dead. G & Gp & p or p & qGrover and Frances are a couple now. Cpsome examplesGrover is dead anPropositional Connectives(“Logical

11、Operators” or Truth-Functional Connectives”)a definition for each connective this simply specifies the truth conditions for any proposition in which the connective occursthis is a way of giving the meaning of the connective by specifying its usea truth table sets out all of the possible truth value

12、combinations for the simple component propositions and shows, for each combination, the value of the compound propositionPropositional Connectives(“LoConjunctionand, but, also, as well,pqp & qTTTTFFFTFFFFConjunctionand, but, alssome examplesGrover and Stevenson are dead.G & S Grover and Frances are

13、a couple now. C All that I have left are photographs and memories. A? Grover and Frances are in love. ?some examplesGrover and StevenDisjunctionor, either orpqp v qTTTTFTFTTFFFInclusive Disjunction “either this or that, and perhaps both”Disjunctionor, either orSome ExamplesEither Zac wants to avoid

14、you or hes out of town.W v OSpecial consideration is appropriate for elderly or infirm people.E v IEither Kelly or Kerry is the best singer alive today.(B v P) & (B & P)Some ExamplesEither Zac wants Exclusive Disjunction “either this or that, but not both”pqp vv q(p v q) & (p & q)TTFFTFTTFTTTFFFFExc

15、lusive Disjunction “eitheNegationnot, it is not the case that.ppTFFTGrover is not alive. AIt is not the case that Grover is alive. A Grover is not very attractive. V Frances never knew about Grovers affair. KNegationnot, it is not theThe (Material) Conditionalif., then.antecedent consequentpqp qTTTT

16、FFFTTFFTThe (Material) Conditionalif“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 course.If Shane is younger than 31, then Shane is younger than 33.“

17、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.” “Why should we count the condip qis equivalent toq v pis equivalent to (p & q) pqp qq v p (p & q)TTTTTTFFFFFTTTTFFTTT If Grover is decapitated, then Grover is dead.p qis equivalent topqp Some Other C

18、onstructionsunless constructions can often be treated as conditionalse.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 phrasesonly if constructions are differentYou get to be president only if you are over 34.P OSome Other Con

19、structionsunlesSome Ifs that Are Not Conditionalsuncertainty / iffye.g., Jen is not certain if Jack is competent.“Bring a friend if you have one.”“I would appreciate tickets for the second performance, if there is one.”Some Ifs that Are Not ConditParentheses(punctuation for propositional logic)allow

20、 us to specify the scope of an operatorthe truth value of a compound proposition is tied to the main operatorMary says John is beautiful. =“Mary,” says John, “is beautiful.”orMary says, “John is beautiful.”theres a big difference between (p v q)and p v qParentheses(punctuation for pEquivalencesp qis

21、 equivalent to q v ptwo compound propositions p and q are logically equivalent if and only if p and q always have the same truth valuetwo equivalent propositions “have the same meaning”Equivalencesp qis equivalean example“Neither borrower nor lender be.”You should be neither a borrower nor a lender.

22、You should not be a borrower and you should not be a lender.= (B v L)= B & Lan example“Neither borrower noPropositional Arguments and Checking for Validitywe want a decision procedure for determining whether a propositional argument is valid:isolate the form of the argument (“translation”)do the tru

23、th table (for the entire argument)determine by inspection whether there are any cases in which all of the premises are true but the conclusion is falsePropositional Arguments and Chan argument form is a pattern for a whole bunch of particular argumentsa substitution instance of an argument form is t

24、he argument that results from uniformly replacing the propositional variables with propositionsan argument form is a pattern Checking for Validity:The Guiding Principles(GP1) an argument A is valid if A is a substitution instance of a valid argument forman argument can be a substitution instance of a valid form and of an invalid form at the same time(P) Grover and Stev