We can come fairly close to an adequate English rendering of the material conditionalP→Qwith the sentenceIf P then Q. At any rate, it is clear that if. . .then
Material conditional symbol:→/ 179
this English conditional, like the material conditional, is false ifPis true and Qis false. Thus, we will translate, for example,If Max is home then Claire is at the library as:
Home(max)→Library(claire)
In this course we will always translate if. . . then. . . using →, but there are in fact many uses of the English expression that cannot be adequately expressed with the material conditional. Consider, for example, the sentence,
If Max had been at home, then Carl would have been there too.
This sentence can be false even if Max is not in fact at home. (Suppose the speaker mistakenly thought Carl was with Max, when in fact Claire had taken him to the vet.) But the first-order sentence,
Home(max)→Home(carl)
is automatically true if Max is not at home. A material conditional with a false antecedent is always true.
We have already seen that the connectivebecauseis not truth functional since it expresses a causal connection between its antecedent and consequent.
The English construction if. . . then. . . can also be used to express a sort of causal connection between antecedent and consequent. That’s what seems to be going on in the above example. As a result, many uses of if. . . then. . . in English just aren’t truth functional. The truth of the whole depends on something more than the truth values of the parts; it depends on there being some genuine connection between the subject matter of the antecedent and the consequent.
Notice that we started with the truth table for → and decided to read it asif. . . then. . .. What if we had started the other way around, looking for a truth-functional approximation of the English conditional? Could we have found a better truth table to go withif. . . then. . .? The answer is clearlyno.
While the material conditional is sometimes inadequate for capturing sub-tleties of English conditionals, it is the best we can do with a truth-functional connective. But these are controversial matters. We will take them up further in Section 7.3.
Necessary and sufficient conditions
Other English expressions that we will translate using the material conditional
P→Q include:P only if Q, Q provided P, and Q if P. Notice in particular only if, provided that P only if Q is translatedP→Q, while P if Qis translatedQ→P. To
Section 7.1
180 /Conditionals
understand why, we need to think carefully about the difference betweenonly if andif.
In English, the expression only if introduces what is called a necessary necessary condition
condition, a condition that must hold in order for something else to obtain.
For example, suppose your instructor announces at the beginning of the course that you will pass the course only if you turn in all the homework assignments.
Your instructor is telling you that turning in the homework is a necessary condition for passing: if you don’t do it, you won’t pass. But the instructor is not guaranteeing that you willpass if youdo turn in the homework: clearly, there are other ways to fail, such as skipping the tests and getting all the homework problems wrong.
The assertion that you will pass only if you turn in all the homework really excludes just one possibility: that you pass but did not turn in all the homework. In other words,P only if Qis false only when Pis true and Q is false, and this is just the case in which P→Qis false.
Contrast this with the assertion that you will pass the course if you turn in all the homework. Now this is a very different kettle of fish. An instructor who makes this promise is establishing a very lax grading policy: just turn in the homework and you’ll get a passing grade, regardless of how well you do on the homework or whether you even bother to take the tests!
In English, the expressionifintroduces what is called asufficient condition, sufficient condition
one that guarantees that something else (in this case, passing the course) will obtain. Because of this an English sentence P if Q must be translated as Q→P. The sentence rules out Q being true (turning in the homework) and P being false (failing the course).
Other uses of →
Infolwe also use→in combination with¬to translate sentences of the form Unless P, QorQ unless P. Consider, for example, the sentenceClaire is at the unless
library unless Max is home. Compare this with the sentence Claire is at the library if Max is not home. While the focus of these two sentences is slightly different, a moment’s thought shows that they are false in exactly the same circumstances, namely, if Claire is not at the library, yet Max is not home (say they are both at the movies). More generally,Unless P, Q or Q unless P are true in the same circumstances asQ if not P, and so are translated as
¬P→Q. A good way to remember this is to whisperif notwhenever you see unless. If you find this translation ofunlesscounterintuitive, be patient. We’ll say more about it in Section 7.3.
It turns out that the most important use of → in first-order logic is not in connection with the above expressions at all, but rather with universally
Biconditional symbol: ↔/ 181
quantifiedsentences, sentences of the form All A’s are B’sand Every A is a B. The analogous first-order sentences have the form:
For every object x (A(x)→B(x))
This says that any object you pick will either fail to be an A or else be a B.
We’ll learn about such sentences in Part II of this book.
There is one other thing we should say about the material conditional, which helps explain its importance in logic. The conditional allows us to reduce
the notion of logical consequence to that of logical truth, at least in cases reducing logical consequence to logical truth where we have only finitely many premises. We said that a sentence Q is a
consequence of premises P1, . . . ,Pn if and only if it is impossible for all the premises to be true while the conclusion is false. Another way of saying this is that it is impossible for the single sentence (P1∧. . . ∧Pn) to be true while Qis false.
Given the meaning of→, we see thatQ is a consequence ofP1, . . . ,Pn if and only if it is impossible for the single sentence
(P1∧. . . ∧Pn)→Q
to be false, that is, just in case this conditional sentence is a logical truth. Thus, one way to verify the tautological validity of an argument in propositional logic, at least in theory, is to construct a truth table for this sentence and see whether the final column contains onlytrue. This method is usually not very practical, however, since the truth tables quickly get too large to be manageable.
Remember
1. The following English constructions are all translated P→Q: If P then Q; Q if P; P only if Q;andProvided P, Q.
2. Unless P, QandQ unless Pare translated¬P→Q.
3. Q is a logical consequence of P1, . . . ,Pn if and only if the sentence (P1∧. . . ∧Pn)→Qis a logical truth.
Section 7.2
Biconditional symbol: ↔
Our final connective is called the material biconditional symbol. Given any sentences PandQ there is another sentence formed by connecting these by
Section 7.2
182 /Conditionals
means of the biconditional: P↔Q. A sentence of the form P↔Q is true if and only if PandQhave the same truth value, that is, either they are both true or both false. In English this is commonly expressed using the expression if and only if. So, for example, the sentenceMax is home if and only if Claire if and only if
is at the librarywould be translated as:
Home(max)↔Library(claire)
Mathematicians and logicians often write “iff” as an abbreviation for “if iff
and only if.” Upon encountering this, students and typesetters generally con-clude it’s a spelling mistake, to the consternation of the authors. But in fact it is shorthand for the biconditional. Mathematicians also use “just in case” as a just in case
way of expressing the biconditional. Thus the mathematical claims n is even iffn2 is even,andn is even just in casen2 is even,would both be translated as:
Even(n)↔Even(n2)
This use of “just in case” is, we admit, one of the more bizarre quirks of mathematicians, having nothing much to do with the ordinary meaning of this phrase. In this book, we use the phrase in the mathematician’s sense, just in case you were wondering.
An important fact about the biconditional symbol is that two sentences Pand Qare logically equivalent if and only if the biconditional formed from them, P↔Q, is a logical truth. Another way of putting this is to say that P ⇔ Qis true if and only if thefolsentenceP↔Q is logically necessary.
So, for example, we can express one of the DeMorgan laws by saying that the following sentence is a logical truth:
¬(P∨Q)↔(¬P∧ ¬Q)
This observation makes it tempting to confuse the symbols↔and⇔. This
↔vs.⇔
temptation must be resisted. The former is a truth-functional connective of fol, while the latter is an abbreviation of “is logically equivalent to.” It is not a truth-functional connective and is not an expression offol.
Semantics and the game rule for ↔
The semantics for the biconditional is given by the following truth table.
P Q P↔Q
t t T
t f F
f t F
f f T
truth table for↔
Biconditional symbol: ↔/ 183
Notice that the final column of this truth table is the same as that for (P→Q)∧(Q→P). (See Exercise 7.3 below.) For this reason, logicians often treat a sentence of the formP↔Qas an abbreviation of (P→Q)∧(Q→P).
Tarski’s World also uses this abbreviation in the game. Thus, the game rule game rule for↔ for P↔Q is simple. Whenever a sentence of this form is encountered, it is
replaced by (P→Q)∧(Q→P).
Remember
1. IfPandQare sentences of fol, then so isP↔Q.
2. The sentenceP↔Qis true if and only ifPandQhave the same truth value.
Exercises
For the following exercises, use Boole to determine whether the indicated pairs of sentences are tauto-logically equivalent. Feel free to have Boole build your reference columns and fill them out for you. Don’t forget to indicate your assessment.
7.1
➶
A→Band¬A∨B.
7.2
➶
¬(A→B) andA∧ ¬B.
7.3
➶
A↔Band (A→B)∧(B→A).
7.4
➶
A↔Band (A∧B)∨(¬A∧ ¬B).
7.5
➶
(A∧B)→CandA→(B∨C).
7.6
➶
(A∧B)→CandA→(B→C).
7.7
➶
A→(B→(C→D)) and ((A→B)→C)→D.
7.8
➶
A↔(B↔(C↔D)) and ((A↔B)↔C)↔D.
7.9
✎
(Just in case) Prove that the ordinary (nonmathematical) use ofjust in casedoes not express a truth-functional connective. Use as your example the sentenceMax went home just in case Carl was hungry.
7.10
➶
(Evaluating sentences in a world) Using Tarski’s World, run throughAbelard’s Sentences, eval-uating them inWittgenstein’s World. If you make a mistake, play the game to see where you have gone wrong. Once you have gone through all the sentences, go back and make all the false ones true by changing one or more names used in the sentence. Submit your edited sentences asSentences 7.10.
Section 7.2
184 /Conditionals
7.11
➶
(Describing a world) Launch Tarski’s World and chooseHide Labelsfrom theDisplaymenu.
Then, with the labels hidden, openMontague’s World. In this world, each object has a name, and no object has more than one name. Start a new sentence file where you will describe some features of this world. Check each of your sentences to see that it is indeed a sentence and that it is true in this world.
1. Notice that ifcis a tetrahedron, thenais not a tetrahedron. (Remember, in this world each object has exactly one name.) Use your first sentence to express this fact.
2. However, note that the same is true of b andd. That is, ifb is a tetrahedron, thend isn’t. Use your second sentence to express this.
3. Finally, observe that if bis a tetrahedron, thencisn’t. Express this.
4. Notice that if ais a cube andb is a dodecahedron, thenais to the left ofb. Use your next sentence to express this fact.
5. Use your next sentence to express the fact that if b and c are both cubes, then they are in the same row but not in the same column.
6. Use your next sentence to express the fact that b is a tetrahedron only if it is small.
[Check this sentence carefully. If your sentence evaluates as false, then you’ve got the arrow pointing in the wrong direction.]
7. Next, express the fact that if a and d are both cubes, then one is to the left of the other. [Note: You will need to use a disjunction to express the fact that one is to the left of the other.]
8. Notice thatdis a cube if and only if it is either medium or large. Express this.
9. Observe that ifbis neither to the right nor left ofd, then one of them is a tetrahedron.
Express this observation.
10. Finally, express the fact thatbandcare the same size if and only if one is a tetrahedron and the other is a dodecahedron.
Save your sentences as Sentences 7.11. Now choose Show Labels from the Display menu.
Verify that all of your sentences are indeed true. When verifying the first three, pay particular attention to the truth values of the various constituents. Notice that sometimes the conditional has a false antecedent and sometimes a true consequent. What it never has is a true antecedent and a false consequent. In each of these three cases, play the game committed to true. Make sure you understand why the game proceeds as it does.
7.12
➶
(Translation) Translate the following English sentences intofol. Your translations will use all of the propositional connectives.
1. Ifais a tetrahedron then it is in front of d.
2. ais to the left of or right ofdonly if it’s a cube.
3. cis between eitheraandeor aandd.
4. cis to the right ofa, provided it (i.e.,c) is small.
Biconditional symbol: ↔/ 185
5. cis to the right ofdonly ifb is to the right ofcand left ofe.
6. Ife is a tetrahedron, then it’s to the right of bif and only if it is also in front ofb.
7. Ifb is a dodecahedron, then if it isn’t in front ofdthen it isn’t in back of deither.
8. cis in back of abut in front of e.
9. eis in front of dunless it (i.e.,e) is a large tetrahedron.
10. At least one of a, c, ande is a cube.
11. ais a tetrahedron only if it is in front of b.
12. bis larger than bothaande.
13. aandeare both larger than c, but neither is large.
14. dis the same shape as bonly if they are the same size.
15. ais large if and only if it’s a cube.
16. bis a cube unlesscis a tetrahedron.
17. Ife isn’t a cube, either bordis large.
18. bord is a cube if eitheraorc is a tetrahedron.
19. ais large just in case dis small.
20. ais large just in case eis.
Save your list of sentences asSentences 7.12. Before submitting the file, you should complete Exercise 7.13.
7.13
➶
(Checking your translations) OpenBolzano’s World. Notice that all the English sentences from Exercise 7.12 are true in this world. Thus, if your translations are accurate, they will also be true in this world. Check to see that they are. If you made any mistakes, go back and fix them.
Remember that even if one of your sentences comes out true inBolzano’s World, it does not mean that it is a proper translation of the corresponding English sentence. If the translation is correct, it will have the same truth value as the English sentence ineveryworld. So let’s check your translations in some other worlds.
Open Wittgenstein’s World. Here we see that the English sentences 3, 5, 9, 11, 12, 13, 14, and 20 are false, while the rest are true. Check to see that the same holds of your translations.
If not, correct your translations (and make sure they are still true inBolzano’s World).
Next openLeibniz’s World. Here half the English sentences are true (1, 2, 4, 6, 7, 10, 11, 14, 18, and 20) and half false (3, 5, 8, 9, 12, 13, 15, 16, 17, and 19). Check to see that the same holds of your translations. If not, correct your translations.
Finally, open Venn’s World. In this world, all of the English sentences are false. Check to see that the same holds of your translations and correct them if necessary.
There is no need to submit any files for this exercise, but don’t forget to submit Sentences 7.12.
Section 7.2
186 /Conditionals
7.14
➶
(Figuring out sizes and shapes) OpenEuler’s Sentences. The nine sentences in this file uniquely determine the shapes and sizes of blocksa,b, andc. See if you can figure out the solution just by thinking about what the sentences mean and using the informal methods of proof you’ve already studied. When you’ve figured it out, submit a world in which all of the sentences are true.
7.15
➶
(More sizes and shapes) Start a new sentence file and use it to translate the following English sentences.
1. Ifais a tetrahedron, then bis also a tetrahedron.
2. cis a tetrahedron ifbis.
3. aandcare both tetrahedra only if at least one of them is large.
4. ais a tetrahedron butcisn’t large.
5. Ifc is small anddis a dodecahedron, then dis neither large nor small.
6. cis medium only if none ofd,e, andfare cubes.
7. dis a small dodecahedron unlessa is small.
8. eis large just in case it is a fact thatdis large if and only if fis.
9. dandeare the same size.
10. dandeare the same shape.
11. fis either a cube or a dodecahedron, if it is large.
12. cis larger thane only ifbis larger thanc.
Save these sentences asSentences 7.15. Then see if you can figure out the sizes and shapes of a, b, c, d,e, andf. You will find it helpful to approach this problem systematically, filling in the following table as you reason about the sentences:
a b c d e f
Shape:
Size:
When you have filled in the table, use it to guide you in building a world in which the twelve English sentences are true. Verify that your translations are true in this world as well. Submit both your sentence file and your world file.
7.16
➶
(Name that object) OpenSherlock’s WorldandSherlock’s Sentences. You will notice that none of the objects in this world has a name. Your task is to assign the namesa,b, andcin such a way that all the sentences in the list come out true. Submit the modified world asWorld 7.16.
7.17
➶
(Building a world) OpenBoolos’ Sentences. Submit a world in which all five sentences in this file are true.