Introduction to Deductive 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 ”. 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 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 ~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 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 ”. 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. Introduction to Deductive Logic Summer 2004 44

Main Connective and Sentential Components (p. 65/72)

1. If P is an atomic sentence, P contains no connectives and hence does not have a main connective. P has no immediate sentential components.

2. If P is of the form ~Q, where Q is a sentence, then the main connective of P is the that occurs before Q, and Q is the immediate sentential component of P.

3. If P is of the form Q & R, Q  R, Q  R, or Q  R, where Q and R are sentences, then the main connective of P is the connective that occurs between Q and R. Q and R are the immediate sentential components of P. 4. The sentential components of P are P itself, the immediate sentential components of P and the sentential components of the immediate sentential components of P. 5. The atomic sentential components of P are all of the sentential components of P that are atomic sentences. Introduction to Deductive Logic Summer 2004 45

P Q P & Q P Q P  Q T T T T T T T F F T F T F T F F T T F F F F F F P Q P  Q P Q P  Q T T T T T T T F F T F F F T T F T F F F T F F T P ~P T F F T

Truth-Value Assignment (p. 68/76) A truth-value assignment is an assignment of truth-values (T’s or F’s) to [all of] the atomic sentences of SL. Introduction to Deductive Logic Summer 2004 46

B F89 J D217 (B  (( D217  J)  ~F89 )) & ~J T T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F F T T T F T T F F T F T F T F F F F T T F F T F F F F T F F F F Introduction to Deductive Logic Summer 2004 47

True on a Truth-Value Assignment (p. 74/82) A sentence is true on a truth-value assignment if and only if it has the truth-value T on that truth-value assignment.

False on a Truth-Value Assignment (p. 74/82) A sentence is false on a truth-value assignment if and only if it has the truth-value F on that truth-value assignment. We shall often refer to a “truth-value assignment” as a “tva” and thus say of SL-sentences that they are true or false on some tva.

NB We have defined “true on a truth-value assignment” and “false on a truth-value assignment”. We have not defined “true” and “false” as applied to sentences of SL. “True” and “false” are the names of the two truth-values, and we shall never apply them to sentences directly. I guarantee that some of you will loose points on assignments or tests by referring to SL-sentences as “true” simpliciter. Strictly speaking “Sentence P of SL is true” makes no sense at all. (It is like saying “2 is red”.) Introduction to Deductive Logic Summer 2004 48

Truth-Functionally True Sentence (p. 76/84)

A sentence P of SL is truth-functionally true if and only if P is true on every truth-value assignment. Truth-functionally true sentences are also called tautologies.

O L (O  O)  (L & O) T T T F F T F F Introduction to Deductive Logic Summer 2004 49

Truth-Functionally False Sentence (p. 76/84)

A sentence P of SL is truth-functionally false if and only if P is false on every truth-value assignment. Truth-functionally false sentences are also called .

K C ~ (K  ~K) & C T T T F F T F F Introduction to Deductive Logic Summer 2004 50

Truth-Functionally Indeterminate Sentence (p. 78/86) A sentence P of SL is truth-functionally indeterminate if and only if P is neither truth-functionally true nor truth- functionally false. Truth-functionally indeterminate sentences are also called contingent.

D D L14 ~ ( L14 & L14) T T T F T F F F T F Introduction to Deductive Logic Summer 2004 51

Truth-Functionally Equivalent Sentences (p. 80/93)

Sentences P and Q of SL are truth-functionally equivalent if and only if there is no truth-value assignment on which P and Q have different truth-values.

G V Y G  ~ (V & Y) ~ (~ ( ~V  ~Y)  ~G) T T T T T F T F T T F F F T T F T F F F T F F F