Introduction to Deductive Logic Summer 2004 31 Material Conditional (p. 37/41) Take the sentence If that monkey is friendly, then it is safe to play with it. This is a compound of That monkey is friendly and That monkey is safe to play with. Let us symbolize these as ‘F’ and ‘S’ respectively. We introduce into SL the symbol ‘’ to represent truth-functional uses of ‘if . then . ’ in English. We thus symbolize our sentence as F S. Sentences of SL of the form P Q are called material conditionals. P (the sentence on the left of the connective) is called the antecedent and Q (the sentence on the right) is called the consequent of P Q. ‘’ is called “the hook” or “the horseshoe”. Introduction to Deductive Logic Summer 2004 32 Material Conditional Continued Material conditionals have the following truth-table: P Q P Q T T T T F F F T T F F T They are false when their antecedent is true and their consequent is false, and true otherwise. The conditions for a material conditional being false are clear enough. “If Brian v. is in this room, then he is a student in Introduction to Deductive Logic” is pretty clearly false, just because I am in the room, but am not an Intro Logic student. What someone who says “If Brian v. is in this room, then he is a student in Introduction to Deductive Logic” is trying to rule out is just that I am in the room and not a student. If what they are trying to rule out does not obtain, then what they have said is true. (Given the assumptions of truth-functionality and exactly two truth-values, every sentence having exactly one.) Introduction to Deductive Logic Summer 2004 33 Material Conditional Continued The text does not introduce sentences of the form P Q directly. Instead, it suggests that we paraphrase our original sentence as: Either it is not the case that that monkey is friendly or that monkey is safe to play with. And then symbolize it as: ~F S. Only then does the text introduce 'F S' as an abbreviation of '~F S'. Thus, the text introduces P Q as an abbreviation for ~P Q as a way of helping to motivate its truth-table: ~P Q is false if and only if ~P is false and Q is false. So ~P Q is false if and only if P is true and Q is false. And, you will note, these are exactly the truth-conditions we gave for P Q. Introduction to Deductive Logic Summer 2004 34 Material Conditional Continued So, why does the text take the detour? Consider the “Paradoxes of the Material Conditional”: If 2+2=9 then my name is Brian. If baboons are reptiles then the president of the USA is a potato. If grass is green then sky is blue. If we symbolize these as T B, etc., then the each come out as true sentences of SL, given the state of the world and the truth-table for ''. (We've not yet really introduced our full semantics, so this is a bit sloppy, but fine for now.) Many people feel uncomfortable with the claim that these sentences are true. (After all, logically speaking, the colour of grass and the sky have nothing to do with each other.) The presentation in the text tries to hide this, by first offering the translation in terms of negation and disjunction. The hope is that you will find the translation plausible and thus accept the somewhat counter-intuitive consequences as following from the translation. Introduction to Deductive Logic Summer 2004 35 Material Conditional Continued This seems dishonest. If you are bothered by the “paradoxes,” you could just as easily react by rejecting the translation of English “If . then . .”'s into the negation/disjunction form. Fortunately we can do better. Recall that we are doing truth- functional logic. We thus must read “If . then . .” as a truth- function. Say I promise you: If you don't take the final exam, then you will fail the course. (We can symbolize this as D F.) When have I broken my promise? Well, say you don't take the final and yet you pass. Pretty clearly, my promise has been broken. What if you don't take the final and you fail the course? Well, I've kept my promise! And what if you do take the final? Well, whether you pass or fail the course, I haven't broken my promise, have I? So, if we look at P Q as a promise that “if P is true, then Q is true”, then if P is false, the promise is kept, no matter the truth-value of Q. Introduction to Deductive Logic Summer 2004 36 Material Conditional Continued Here is another approach. What would prove the English sentence: If Jane is in the library, then she is studying to be false? Clearly Jane's being in the library and not studying. And, if it is not false, it is true. (We are assuming that every sentences is either true or false.) If Jane is there and studying, it has to be true —it is no good to say that maybe she is sometimes there without studying, etc. because we are doing truth-functional logic. But if the only circumstances where our sentence (we can symbolize it as L S) is false is when she is in the library and not studying, then it has to be true, no matter what else obtains (under our assumptions of bivalenent truth and truth- functionality.) Introduction to Deductive Logic Summer 2004 37 Material Conditional Continued: One more approach: with our Jane in the library example, we have given clear justification of some of the table as follows: P Q P Q T T T T F F F T T F F T What can we say to justify the other two values? They cannot both get false, otherwise we would have conjunction. But if either of the false antecedent cases get true, it would seem that they both should. (Given that the antecedent fails to obtain, what difference to the truth of the truth- functional conditional should the truth-value of the consequent make?) Introduction to Deductive Logic Summer 2004 38 Material Conditional Continued: Not all English conditionals are plausibly seen as truth-functional. Consider If these crystals are salt then they will dissolve in water. This sentence is plausibly construed as making a causal claim — it is claiming that being crystals of salt is connected (beyond a merely truth-functional connection) to dissolving in water. If we symbolize it as S D we are giving it a symbolization that is true if the crystals in question are not salt. But then S ~D would also be true. But only one of the two English conditionals: If these crystals are salt then they will dissolve in water and If these crystals are salt then they will not dissolve in water is true. Something has been lost. Introduction to Deductive Logic Summer 2004 39 Material Conditional Continued: Other Conditionals Forms in English: Mary is in this room only if Mary is interested in logic. We can paraphrase this as If Mary is in this room then Mary is interested in logic. Ferrets are docile unless they are frightened. We can paraphrase this as If it is not the case that ferrets are frightened then ferrets are docile. Introduction to Deductive Logic Summer 2004 40 Material Biconditional (p. 42/45) Consider Koalas are quadrupeds if and only if they have four legs. We can paraphrase this as Both (if koalas are quadrupeds then they have four legs) and (if koalas have four legs then they are quadrupeds). We can symbolize this as (Q F) & (F Q). We introduce the symbol ‘’ into SL to capture this truth- functional relationship. ‘’ is called “the triple bar”. We can thus symbolize our sentence as Q F. Sentences of SL of the form P Q are called material biconditionals. Introduction to Deductive Logic Summer 2004 41 Material Biconditional Continued: A material biconditional is true if and only both sentences on either side of the ‘’ have the same truth-value. Thus, material biconditionals have the following truth-table: P Q P Q T T T T F F F T F F F T Introduction to Deductive Logic Summer 2004 42 The Vocabulary of SL (p. 62/69) We have now introduced all of the elements of our language SL. The formal syntax of our language is fairly simple. The “lexicon” is as follows: Sentence letters of SL: { A, B, . , Z, A1, B1, . , Z1, A2, B2, . , Z2, . } Connectives of SL: { ~, &, , , } The punctuation marks of SL: { (, ) } Next, we define Sentence of SL. Introduction to Deductive Logic Summer 2004 43 Definition of Sentence of SL (p. 63/70) 1. Every sentence letter is a sentence. 2. If P is a sentence then ~P is a sentence. 3. If P and Q are sentences, then (P & Q ) is a sentence. 4. If P and Q are sentences, then (P Q ) is a sentence. 5. If P and Q are sentences, then (P Q ) is a sentence. 6. If P and Q are sentences, then (P Q ) is a sentence. 7. Nothing else is a sentence. We can refer to sentences of SL as SL-sentences and, more generally, to sentences of some logical language L as L- sentences. A Convention If R is a sentence by one of clauses 3-6 we may drop the outer pair of parentheses.