Propositional Logic USEM 40a Spring 2006 James Pustejovsky.
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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 for all of A’s premises to be true and its conclusion false deductive argument A is sound =df. (i) A is valid, and (ii) all of A’s 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, 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 A’s logical form alone, it is impossible for all of A’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 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 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 Grover 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 propositional 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 replaced 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 & Gp & p or p & q Grover and Frances are a couple now. Cp
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 its 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’,… p q p & q TTT TFF FTF FFF
some examples Grover 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. ??
Disjunction ‘or’, ‘either… or…’ p q p v q TTT TFT FTT FFF Inclusive Disjunction “either this or that, and perhaps both”
Some Examples Either Zac wants to avoid you or he’s 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 Disjunction “either this or that, but not both” p q p vv q(p v q) & ~ (p & q) TTFF TF TT FTTT FFFF
Negation ‘not’, ‘it is not the case that...’ p~p TF FT Grover is not alive.~ A It is not the case that Grover is alive.~ A Grover is not very attractive. ~ V Frances never knew about Grover’s affair.~ K
The (Material) Conditional ‘if..., then...’ [antecedent] [consequent] pq p q TTT TFF FTT FFT
“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, then 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 to q v ~ pis equivalent to ~ (p & ~ q) pq p q q v ~p~ (p & ~q) TTTTT TFFFF FTTTT FFTTT If Grover is decapitated, then Grover is dead.
Some Other Constructions ‘unless’ 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 ‘If’s that Are Not Conditionals uncertainty / ‘iffy’ –e.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.”
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 operator Mary says John is beautiful. = “Mary,” says John, “is beautiful.” or Mary says, “John is beautiful.” there’s a big difference between ‘~ (p v q)’ and ‘~p v q’
Equivalences p 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 “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 & ~ L
Propositional Arguments and Checking for Validity we want a decision procedure for determining whether a propositional 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 arguments a substitution instance of an argument form is the argument that results from uniformly replacing the propositional variables with propositions
Checking 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 substitution 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 false
Some Common Argument Forms: Conjunction p & q therefore, p p q therefore, p & q PREMCONC pqp & qp TTTT TFFT FTFF FFFF
Disjunctive Syllogism p v q ~ p therefore, q P1P2CONC pqp v q~ pq TTTFT TFTFF FTTTT FFFTF
Modus Ponens p q p therefore, q P1P2CONC pq p q pq TTTTT TFFTF FTTFT FFTFF
Modus Tollens p q ~ q therefore, ~ p P1P2CONC pq p q ~ q~ p TTTFF TFFTF FTTFT FFTTT
Hypothetical Syllogism p q q r therefore, p r
P1P2CONC pqr p qq rp r TTTTTT TTFTFF TFTFTT TFFFTF FTTTTT FTFTFT FFTTTT FFFTTT