1、Propositional LogicUSEM 40aSpring 2006James PustejovskyEvaluation 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 for all of
2、 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 notvote.(P2)Grover is dead.(C)Therefore,Grover does not vote.Formal Logic with many deductive arguments,validity is a matter simply
3、 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 alone,it is impossible for all of A
4、s 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 us a more robust understanding of validity in g
5、eneral It forms the building block for our model of meaning in language and for reasoning in generalIntroduction to Propositional(or“Sentential”or“Truth-Functional”)Logic deals with propositions whole statements;meaningful declarative sentences S is a simple proposition=df.S does not contain any oth
6、er proposition as a componentGrover is dead.S is a compound proposition=df.S contains at least one 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.Propositional Forms,Variables,Constants,and S
7、ubstitution 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 propositional constant is a capital letter that stands for a particular,definite proposition
8、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 replaced with different variables,but no two different propositions may be replaced by the
9、same one variablesome 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&Gp&p or p&q Grover and Frances are a couple now.CpPropositional Connectives(“Logical Operators”or Truth-Functional Connectives”)a definition
10、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 use a truth table sets out all of the possible truth value combinations for the simple component propositions and s
11、hows,for each combination,the value of the compound propositionConjunctionand,but,also,as well,pqp&qTTTTFFFTFFFFsome examplesGrover and Stevenson are dead.G&S Grover and Frances are a couple now.C All that I have left are photographs and memories.A?Grover and Frances are in love.?Disjunctionor,eithe
12、r orpqp v qTTTTFTFTTFFFInclusive 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 elderly or infirm people.E v I Either Kelly or Kerry is the best singer alive today.(B v P)&(B&P)Exclusive
13、Disjunction “either this or that,but not both”pqp vv q(p v q)&(p&q)TTFFTFTTFTTTFFFFNegationnot,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.KThe(Material)Conditionalif.,then.antece
14、dent consequentpqp qTTTTFFFTTFFT“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.“If p,the
15、n q.”=“Either q is the case or p is not the case.”=“It is not the case that p and not-q.”p qis equivalent toq v pis equivalent to (p&q)pqp qq v p(p&q)TTTTTTFFFFFTTTTFFTTT If Grover is decapitated,then Grover is dead.Some Other Constructions unless constructions can often be treated as conditionals e
16、.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 OSome Ifs that Are Not Conditionals uncertainty/iffy e.g.,Jen is not certain if Jack is
17、competent.“Bring a friend if you have one.”“I would appreciate tickets for the second performance,if there is one.”Parentheses(punctuation for propositional logic)allow us to specify the scope of an operator the truth value of a compound proposition is tied to the main operatorMary says John is beau
18、tiful.=“Mary,”says John,“is beautiful.”orMary says,“John is beautiful.”theres a big difference between (p v q)and p v qEquivalencesp qis equivalent to q v p two compound propositions p and q are logically equivalent if and only if p and q always have the same truth value two equivalent propositions“
19、have the same meaning”an example“Neither borrower nor lender be.”You should be neither a borrower nor a lender.You should not be a borrower and you should not be a lender.=(B v L)=B&LPropositional Arguments and Checking for Validitywe want a decision procedure for determining whether a propositional
20、 argument is valid:1.isolate the form of the argument(“translation”)2.do the truth table(for the entire argument)3.determine by inspection whether there are any cases in which all of the premises are true but the conclusion is false an argument form is a pattern for a whole bunch of particular argum
21、ents a substitution instance of an argument form is the argument that results from uniformly replacing the propositional variables with propositionsChecking for Validity:The Guiding Principles(GP1)an argument A is valid if A is a substitution instance of a valid argument form an argument can be a su
22、bstitution instance of a valid form and of an invalid form at the same time(P)Grover and Stevenson are dead.(C)Therefore,Grover is dead.(GP2)an argument form F is valid if and only if F has no substitution instances in which all of the premises are true and the conclusion is falseSome Common Argumen
23、t Forms:Conjunctionp&qtherefore,ppqtherefore,p&qPREMCONCpqp&qpTTTTTFFTFTFFFFFFDisjunctive Syllogismp v q ptherefore,qP1P2CONCpqp v q pqTTTFTTFTFFFTTTTFFFTFModus Ponensp qptherefore,qP1P2CONCpqp qpqTTTTTTFFTFFTTFTFFTFFModus Tollensp q qtherefore,pP1P2CONCpqp q q pTTTFFTFFTFFTTFTFFTTTHypothetical Syllogismp qq rtherefore,p rP1P2CONCpqrp qq rp rTTTTTTTTFTFFTFTFTTTFFFTFFTTTTTFTFTFTFFTTTTFFFTTT
