he is not yours.ctessipus: To be sure he is.dionysodorus: Then he is a father, and he is yours; ergo,he is your father, and the puppies are your brothers.The argument seems silly, yet one with the same grammatical form seems unobjectionable: Fido is a dog. Fido isyours. Therefore Fido is your dog.An argument in a different vein is due to Raymond
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Smullyan, who is some logician:Some cars rattle.My car really is some car.So no wonder my car rattles.Getting back to dogs, why is one of the following arguments valid and the other not?A dog needs water to survive.Therefore my dog Linocera needs water to survive.A dog is barking in the backyard.Therefore my dog Ginger is barking in the backyard.
Perhaps surprisingly, the following two arguments arevalid.Everybody loves a lover.George doesn’t love himself.Therefore George doesn’t love Martha.Either everyone is a lover or some people are notlovers.If everyone is a lover, Waldo certainly is a lover.If everyone isn’t, then there is at least one nonlover.Call her Myrtle.
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Therefore if Myrtle is a lover, everyone is.The validity of the first argument depends on two facts:(1) A person is a lover if that person loves anyone, himselfincluded; (2) “If P, then Q” is true exactly when “If not Q,then not P” is. The validity of the second depends on theconditions under which “If P, then Q” is true.
36 inches = 1 yardSo 9 inches = 1⁄4 yardSo 9 inches = 1⁄4 yardTherefore 3 inches = 1⁄2 yard
The temperature is 93.The temperature will rise this afternoon.Therefore 93 will rise this afternoon.
Most common manipulations of statements involvinglogical constants such as “all,” “some,” and “not” are intuitively clear. The negation of “Everyone is bald” (symbolically, ∀xB(x)—for all x, x is bald) is not “No one is bald” butrather “Someone is not bald” (symbolically, ∃x ⬃B (x)—thereexists an x such that x is not bald) or, more longwindedly, “Itis not the case that everyone is bald.”
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Sometimes, though, the meanings of these constantsare not so clear. “You can fool some of the people all thetime” is, at second glance, ambiguous: it has two contraryinterpretations. It can mean that there are certain gulliblebumpkins, say George and Martha, who are bamboozledevery day of the week; or it can mean that on any given dayyou choose, you’ll always be able to fool some people—say,George on Monday, Martha and Waldo on Tuesday, Waldoand Myrtle on Wednesday, and so on. Using P(x) for “x is aperson,” T(y) for “y is a time,” and F (x,y) for “you can fool xat y,” the first interpretation can be rendered symbolicallyas ∃ x∀y (P(x) ⵩T(y)→F(x,y)), while the second is ∃ x∀y(P(x)⵩T(y)→F(x,y)).An example of a less common logical word subject tosimilar ambiguities is the word “most”:*: “Most of the people there had read most of the booksdiscussed.”Does * mean that more than half of the people therehad read more than half of the books discussed, or does itmean that more than half of the books discussed had beenread by more than half of the people there? That theseinterpretations differ can be seen if, for example, weassume that there were only three people there and onlythree books were discussed. The first interpretation corresponds to figure 1, and the second to figure 2.
GeorgeMarthaWaldo
Figure 1War and PeaceTropic of CancerPrincipiaMathematica
GeorgeMarthaWaldo
Figure 2War and PeaceTropic of CancerPrincipiaMathematica
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Even the simple logical word “is,” maybe even especiallythe word “is,” can be interpreted in everyday contexts inquite different ways. Consider the four statements below:(1) George is Mr. Kyriacopoulos.(2) George is anxious.(3) Man is anxious.(4) There is an anxious man here.The first “is” is the “is” of identity: g = k. The second “is” isthe “is” of predication: A(g)—g has the property A. The third“is” is the “is” of membership: ∀x [M(x) →A(x)]. The fourth“is” is existential: ∃x M(x) .
It’s odd that logical acuity, rather than helping one toclarify statements, often reveals hidden ambiguities withinthem. Instead of leading one to form more conclusions, itmakes clear that fewer conclusions are justified. BertrandRussell once observed (1956) that the keener one’s sense oflogical deduction, the less often one makes hard and fastinferences.However, only for deductive inferences is this the case.Most “common sense” inferences do not depend on explicitly formulated knowledge or on any precise rules, butrather on a kind of tacit and natural process that is hard todescribe. Thus mathematicians, accustomed as they are toclearly formulated principles and rigidly interpreted rules,sometimes have difficulty with “common sense” and often
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have a sense of humor characterized by the overly literalinterpretation of terms and phrases.
minister of war: That’s the last straw! I resign. I wash myhands of the whole business.firefly (groucho): A good idea. You can wash your necktoo.
george: In what two sports do face-offs occur?waldo: Ice hockey and . . . I give up. What’s the other one?george: Leper boxing.
No Parking signs often indicate “Violators will betowed,” yet I’ve never yet seen a tow truck dragging anyonedown the street. Trash cans that warn “Keep litter in itsplace” are also amusing if taken literally (litterly?): if something is litter, its place by definition, it would seem, is theground.
Finally, consider this last sillygism, which leads into thetopic of the next section:
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This argument is helopita.Nothing that’s helopita is steedibeep.Everything that’s valid is steedibeep.Therefore this argument is not valid.Yet it is.
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The Titl of This SectionContains Three ErorsThe study of self-referential statements dates back to Stoiclogicians of the fourth and fifth century b.c. The oldest,best-known such paradox concerned Epimenides theCretan, who stated: “All Cretans are liars.” The crux of thisso-called liar paradox is clearer if we simplify his statementto “I am lying” or, better yet, “This sentence is false.”(Douglas Hofstadter writes of a more recent version, that ofNixonides the Cretin, who in 1974 stated: “This sentence isinoperative” [Hofstadter].)Let us give the label Q to “This sentence is false.” Nowwe notice that, if Q is true, then by what it says it must befalse. On the other hand, if Q is false, then what it says istrue, and Q must then be true. Hence, Q is true if and onlyif it’s false.The statement Q and variants of it are intimately connected with some of the deepest and most important ideasin logic and philosophy, and possibly even with consciousness itself. Despite this they are dismissed by most people assilly diversions, suitable only for amusing logicians andother useless people. I must admit that on Tuesdays I feelthe same way, and since this is Tuesday, I will demonstratehow a relative of Q can be used to prove the existence ofGod. Consider the box below, which contains two sentences:
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1. God exists.2. Both of these sentences are false.
The second sentence is either true or false. If it’s true,then both sentences are false. In particular, the second sentence is false; but the only way the second sentence can befalse is for the first sentence to be true. Thus in this caseGod exists. On the other hand, if we assume directly thatthe second sentence is false, then again we must note thatthe only way for this to be is for the first sentence to be true.Thus in this case also God exists. Hence God exists.Of course, in a similar way we can demonstrate thatGod had a hangnail, or that he doesn’t exist, or that LudwigWittgenstein was in love with Mae West.A related trick can be pulled with the following statement:*: “If this statement is true, then God exists.” If * istrue, then it’s true. On the other hand, if * is false, then theantecedent (the if clause) of * is false, which ensures that *itself is true. (Remember that a conditional if-then statement is true if the antecedent is false.) Thus in both cases *is true. Thus * is true, and hence the antecedent of * is true.Together these two facts ensure that the consequent (thethen clause of *) is true. Hence God exists.Even nonparadoxical statements can, if the situation isappropriate, be combined to yield a paradox. If Socrateswere to say “What Plato just said is false,” there would usu-
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ally be nothing strange about the utterance. But if Platowere previously to have said “What Socrates says next istrue,” we would have a paradox.
There is an ancient story about the Sophist philosopherProtagoras, who agreed to instruct Euathlus in rhetoric sothe latter could practice law. Euathlus in turn agreed to payProtagoras his fee only after winning his first case.However, Euathlus chose not to practice law upon completing his training, and so Protagoras sued him for his fee.Protagoras maintained that he should be paid no matterwhat: he argued that if he won the case, he should be paidby order of the court; while if he lost, he should be paid bythe terms of his agreement with Euathlus. Euathlus, whohad learned something from his study with Protagoras,maintained that he should not pay no matter what: he arguedthat if he won the case, he should not pay by order of thecourt; while if he lost, he should not pay by the terms of hisagreement with Protagoras.
B r ea k fas t S ceneBig, brawny woman, hair in pincurls, wearing a tornbathrobe, to her scrawny, bald husband sitting in hisunderwear: “I want you to dominate me, to make me feellike a real woman.”
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As the last story might suggest, there is a close connection between these paradoxes and certain “double-bind”situations. The command “Be spontaneous” is anothersimple example, as is the injunction “Whatever you do,don’t let the image of a sweet red watermelon covered withthick yellow mustard enter your mind.” Unfortunately (orfortunately), most such situations that require contradictory behavior are more complex and somewhat more disguised—so disguised, in fact, that most people, especiallythose who consider self-referential paradoxes frivolous, donot realize that these situations pervade their lives.Situations, incidents, conversations are often so complexlytextured that it’s not uncommon for some aspect of thesecommunications to “say” of itself that it’s not true.Consider the act of telling a joke. W. F. Fry andGregory Bateson have shown that when someone tells ajoke, there is usually some kind of behavioral cue—a different voice inflection, an arched eyebrow or wink, the use ofa dialect, a mock serious tone, even the explicit clause,“Have you heard the one about. . . .” These cues, ormetacues, qualify what is being said and function as a kindof nonverbal liar paradox. They say, in effect, “This wholebusiness is false, unreal, not to be taken seriously; it’s onlyjoking.” In fact, all dramatic performance—all art, even—has these two aspects: the content, and the frame or setting,which sets it apart from non-art and which says of itself,“This is not an everyday sort of communication. This isunreal.”
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A man, smiling and holding a small tree branch abovehis head, informs a bank teller, “This is a stick-up.”
Russel l ’s D r. Gold berg a ndD r. Rubi nConsider a country whose laws allow nonresident mayors—that is, some mayors live in the cities they govern, while others do not. A reform-minded dictator comes to power andorders that all the nonresident mayors, and only the nonresident mayors, live in one place—call it city C. City C nowrequires a mayor. Where shall the mayor of city C reside?This story is a popularization of Russell’s paradox,whose derivation is similar. We first note that some setscontain themselves as members. The set of all things mentioned on this page is mentioned on this page, and thuscontains itself. Likewise, the set of all those sets with morethan seven members itself contains more than seven members, and is thus a member of itself. Most naturally occurring sets do not contain themselves as members. The set ofhairs on my head is not itself a hair on my head, and thus isnot a member of itself. Similarly, the set of odd numbers isnot itself an odd number, and thus does not contain itselfas a member.Dividing the set of all sets into two non-overlappingsets, let us denote by M the set of all those sets that do contain themselves as members, and by N the set of all those setsthat do not contain themselves as members. In other words,for any set x, if x is a member of M, then x is a member ofitself; and if x is a member of itself, then x is a member of
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M. On the other hand, for any set x, if x is a member of N,then x is not a member of itself; and if x is not a member ofitself, then x is a member of N.Now we may ask whether N is a member of itself or not.(Compare this question with “Where does the mayor of cityC live?”) If N is a member of itself, then by definition of N,N is not a member of itself. But if N is not a member ofitself, then by definition of N, N is a member of itself. ThusN is a member of itself if and only if it is not a member ofitself. This contradiction constitutes Russell’s paradox.
Robert Benchley once remarked: “There may be said tobe two classes of people in the world: those who constantlydivide the people of the world into two classes and thosewho do not.” He should have added paradoxically that hebelongs to the latter class.
Recall Groucho Marx’s quip that he would never joinany club that would be willing to have him as a member.
The following is a true story: A well-known philosopher was delivering a talk on linguistics and had just statedthat the double negative construction has a positive mean-
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ing in some natural languages and a (very) negative meaning in others. He went on to observe, however, that in nolanguage was it the case that a double positive constructionhas a negative meaning. To this, Sidney Morgenbesser,another well-known philosopher, who was sitting in therear of the lecture room, responded with a jeering “Yeah,yeah.”
V: “This sentence is true.” V is a little odd, but notparadoxical. A variant of it, due to David Moser, reads:“This sentence is a !!! premature punctuator.” “This sentence is graduellement changeant en français” is another.
Russell’s resolution of the paradox is to restrict thenotion of set to a well-defined collection of already-existing sets. In his famous theory of types, he classifed setsaccording to their type or level, thereby creating a set-theoretic hierarchy. On the lowest level, type 1, are individualobjects. On the next level, type 2, are sets of type 1 objects.On the next level, type 3, are sets of type 1 or type 2 objectsor sets; and so on. The elements of type n sets are sets oftype (n  1) or lower. In this way Russell’s paradox is avoided, since a set can be a member only of a set of a higher typeand not of itself. Informal uses of this idea are commonplace: characters in cartoons, television, and movies are
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forever saying things like they’re anxious about their anxieties, bored with their boredom, or tired of being tired.The liar paradox can be avoided in a similar way.Applying a hierarchical solution to Epimenides’ utterancerequires that “All Cretans are liars” be assigned a highertype than other statements made by Cretans. We make a distinction between type 1 statements (usually called firstorder statements), which do not refer to other statements;second-order statements, which refer to first-order statements; third-order statements, which refer to first- andsecond-order statements; and so on. Thus if Epimenidesthe Cretan states that all statements made by him are false,he is to be understood as making a statement that does notapply to itself; he is making a statement of an order higherthan that of his other statements. In this way the self-negation of the liar paradox is prevented.The following three statements are first-, second-, andthird-order statements, respectively:(1) Wittgenstein was bald.(2) Statement (1) is false.(3) Statement (2) is true.This hierarchical notion of truth for statements hasbeen extensively developed by the logician Alfred Tarski andothers. It is not, however, the only way to handle thesequestions. An alternative approach, due to the philosopherSaul Kripke, takes statements like Q—“This sentence isfalse”—to be “ungrounded,” there being no ground, nofirst-order statement, from which to build up to the truth
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or falsity of Q. Statements in the Kripke formulation arenot (as in the Tarski account) assigned fixed levels ororders, but attain their level naturally depending on whatother statements have been made and the facts of the situation. (Cf. the Plato-Socrates story above.) The truth or falsity of these statements is determined in a gradual, inductive manner, with not every sentence receiving a truth value(e.g., Q). Self-referential statements are allowed, though,and can, under appropriate circumstances, receive a truthvalue.
A literal interpretation of “self-addressed stampedenvelope” is a stamped envelope addressed to itself. Idealfor instant delivery.
“You always overreact. You never ever respond moderately. If you shrug one more time, I’m going to scream.”
Language and Metalanguage:D o Yo u G e t I t ?Implicit in the discussion of language levels in Russell’s theory of types is a very general and extremely important distinction in logic and philosophy, that between object language and metalanguage. Object-level statements are statements within a (usually) formal system that is the object ofstudy. Examples are:
(1) A B→⬃C B(If A and B, then either not C or B)(2)∀x∃yP (x,y)(For all x there is a y such that x bears relation P to y)(3) p(x 2 − 1)→px(If p divides x 2 − 1, then p divides x)Metalevel statements are statements about the formalsystem, or about the object-level statements within it.Examples are:(1) Statement Q has two different interpretations.(2) Statement S is true but not provable.(3) P, S, and Q are inconsistent.If one is studying Japanese grammar, Japanese is theobject language and English is the metalanguage.
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Groucho says to an acquaintance, “Did you hear theone about the organization for people whose IQs are in thebottom 2 percent? I was just looking at its newsletter calledDENSA.” Groucho stops, bends closer, and says casually,“Do you get it? Do you get it?” The acquaintance, thinkinghe’s being asked if he understands the joke, answers “Yes,”to which Groucho rejoins, “I’m surprised. I thought youwere a little brighter than that.”Risking a charge of pedantry, I’ll note that Groucho’sacquaintance misinterprets “Do you get it?” as a metalevelquestion about the joke, and not an object-level questionthat is part of the joke.The tortoise in Lewis Carroll’s “What the Tortoise Saidto Achilles” (excerpted below) makes the opposite mistake.He confuses the metalevel rule C with the object-level statements A, B, and Z. C needs no further rule to explain whenand how it applies, yet the tortoise insists on a meta-metalevel rule D to do just that, and then a meta-metametalevel rule E to explain when and how D applies, and so on. Thetortoise just isn’t playing the logic game.“That beautiful First Proposition by Euclid!” theTortoise murmured dreamily. “You admire Euclid?”“Passionately! So far, at least, as one can admire atreatise that won’t be published for some centuries tocome!”
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Well, now, let’s take a little bit of the argument inthat First Proposition—just two steps, and the conclusion drawn from them. Kindly enter them inyour note-book. And in order to refer to them conveniently, let’s call them A, B, and Z:A) Things that are equal to the same are equal toeach other.B) The two sides of this Triangle are things thatare equal to the same.Z) The two sides of this Triangle are equal to eachother. . . .“I’m to force you to accept Z, am I?” Achilles saidmusingly. “And your present position is that youaccept A and B, but you don’t accept theHypothetical—”“Let’s call it C,” said the Tortoise.“—but you don’t accept C) If A and B are true, Zmust be true.”“That is my present position,” said the Tortoise.“Then I must ask you to accept C.”“I’ll do so,” said the Tortoise, “as soon as you’veentered it in that note-book of yours. What else haveyou got in it?”“Only a few memoranda,” said Achilles, nervouslyfluttering the leaves: “A few memoranda of—of thebattles in which I have distinguished myself!”“Plenty of blank leaves, I see!” the Tortoise cheerily remarked. “We shall need them all!” (Achillesshuddered.) “Now write as I dictate:—
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A) Things that are equal to the same are equal toeach other.B) The two sides of this Triangle are things thatare equal to the same.C) If A and B are true, Z must be true.Z) The two sides of this Triangle are equal to eachother.”“You should call it D, not Z,” said Achilles. “Itcomes next to the other three. If you accept A and Band C, you must accept Z.”“And why must I?”“Because it follows logically from them. If A and Band C are true, Z must be true. You don’t dispute that,I imagine?”“If A and B and C are true, Z must be true,” theTortoise thoughtfully repeated. “That’s anotherHypothetical, isn’t it? And, if I failed to see its truth,I might accept A and B and C, and still not accept Z,mightn’t I?”“You might,” the candid hero admitted; “thoughsuch obtuseness would certainly be phenomenal.Still, the event is possible. So I must ask you to grantone more Hypothetical.”“Very good. I’m quite willing to grant it, as soonas you’ve written it down. We will call it D) If A and Band C are true, Z must be true. Have you entered thatin your note-book?”“I have!” Achilles joyfully exclaimed, as he ran thepencil into its sheath. “And at last we’ve got to the
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end of this ideal racecourse! Now that you accept Aand B and C and D, of course you accept Z.”“Do I?” said the Tortoise innocently. “Let’s makethat quite clear. I accept A and B and C and D.Suppose I still refused to accept Z?”“Then Logic would take you by the throat, andforce you to do it!” Achilles triumphantly replied.“Logic would tell you ‘You can’t help yourself. Nowthat you’ve accepted A and B and C and D, you mustaccept Z!’ So you’ve no choice, you see.”“Whatever Logic is good enough to tell me is worthwriting down,” said the Tortoise. “So enter it in yourbook, please. We will call it E) If A and B and C and Dare true, Z must be true. Until I’ve granted that, ofcourse I needn’t grant Z. So it’s quite a necessary step,you see?”“I see,” said Achilles; and there was a touch ofsadness in his tone.(Carroll)
Godel’s famous incompleteness (meta)theorem alsodepends crucially on the object-metalevel distinction.Godel considered a simple formal system containing thebasic axioms of the arithmetic of whole numbers. Hemethodically assigned each object-level statement a uniquecode number; he also assigned a code number to each proofof an object-level statement. By means of this coding,object-level statements about numbers can also be under-
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stood as expressing metalevel statements about the system,or about individual object-level statements. If one is careful and clever, one can find a statement G that is true if andonly if it is unprovable.Loosely speaking, we note that this object-level statement about whole numbers says of itself, via the numericalcoding, that it is not provable. If the axioms are all true andthe system is consistent, it is possible to conclude that sucha statement G (about whole numbers, remember) is neitherprovable nor disprovable from the axioms—that it is independent of them. The same idea can be extended to showthat any formal system is incapable of proving some truthsand thus that no formal system can prove all truths.
The following old joke has something of the same flavor: A new prisoner was puzzled because his fellow inmateslaughed whenever one of them called out a number. He wastold that the numbers were a code for certain jokes, whichthus did not need to be repeated verbatim. Intrigued, thenew prisoner called out “63” and was greeted by totalsilence. Later his cellmate explained that everythingdepends on how the joke is told.This metajoke can, I suppose, itself be assigned a codenumber and. . . .
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Comedians, after a joke has failed, often follow it witha self-deprecating comment on the joke, thereby salvagingat least a metajoke. So do authors.
This sentance has three erors. As in the titl of a previous section, one of these erors is of a quite different type(level) than the other two.
Related to the language-metalanguage distinction—infact, a special case of it—is the use-mention distinction. Toillustrate, the first pair of sentences below uses the words“laughing” and “Daniel,” while the second pair merelymentions them:(1) Leah was laughing at the Smurf cartoons.(2) Daniel loves lawnmowers.(1’) “Laughing” contains eight letters.(2’) “Daniel” is a boy’s name.That the distinction can be grasped by a five-year-old isdemonstrated by the following true story: A colleague ofmine, working on a paper at home in his study, was disturbed repeatedly by his five-year-old son’s use of theexpletive “shit” in the next room whenever his blocks collapsed. My colleague firmly warned him not to use the wordanymore. Returning to his study he heard his son say it
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again, whereupon he whirled around and barged into hisson’s room only to hear him continue defensively “. . . is abad word. ‘Shit’ is a bad word. Right, Daddy?”Failure to distinguish between using and mentioningcan also lead to arguments like the following concerningformer-President and Mrs. Ford:Betty loves Ford.Ford is a four-letter word.Therefore Betty loves a four-letter word.
i should begin with a capital letter.
Meaning, Reference, andDora Black’s First HusbandMeaning, reference, names, and descriptions: thesenotions are at the heart of many disputes in philosophicallogic. Whatever the resolution of these, some well-knownpuzzles will have to be solved (or dissolved).The meaning of “meaning” is difficult if not impossible to formulate in full generality. One clear thoughwrongheadedly narrow account was provided by the earlylogical positivists, who identified the meaning of a proposition with its method of verification—that is, with whateverobservations point to its truth. Using this so-called verifiability principle, they thought they had managed to do awaywith metaphysics, theology, and ethics, whose propositionsare seemingly unverifiable. Hence we have the boring sterility of the logical positivists’ program. One major problemwith this theory of meaning, of course, is that the verifiability principle itself is embarrassingly not verifiable.“Reference” too is a difficult notion to pin down. Still,whatever these terms mean we can say something about theconnection between them.Two terms or expressions can refer to the same entity(or set of entities) though they differ in meaning. To use aclassical example, due to the logician Gottlob Frege, wenote that “the morning star” and “the evening star” certainly differ in meaning, or, as is sometimes said, in inten-
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sion (with an s). It required an empirical discovery to realize that both those expressions have the same referent (i.e.,both refer to the same entity)—the planet Venus. Similarly,“the younger coauthor of Principia Mathematica” and the “firsthusband of Dora Black,” though they do not mean the samething, both refer to Bertrand Russell. The term “renates”means “animals with kidneys,” while “cordates” means“animals with hearts”; yet they both refer to the same collection of animals, since it so happens that all animals withhearts have kidneys and vice versa.A term or expression is said to occur extensionally in asentence if replacing it with any other term or expressionwith the same referent does not change the truth or falsityof the sentence. In many contexts, certainly in all purelymathematical ones, this substituting of equals for equals isextensional and causes no problem. Its use in fact was soobvious to Euclid that he included it in his development ofplane geometry as “a common notion” not in need of anyfurther elucidation or proof. If a is greater than x2 <ms> 1,and a = b, then b is greater than x2 <ms> 1. Still, outside ofmathematics this substitution principle can fail—as the following arguments show:The president thought that the city of Copenhagen wasin Norway.The city of Copenhagen is the capital of Denmark.Therefore the president thought the capital ofDenmark was in Norway.
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It is a mathematical theorem that the number six isgreater than the number three.The number six is the number of men who were husbands of Elizabeth Taylor.Therefore it is a mathematical theorem that the number of men who were husbands of Elizabeth Taylor is greaterthan the number three.
Statements or expressions can have a meaning yet lack areferent (at least according to many philosophers’accounts). Russell’s celebrated “The present King of Franceis bald” is an example. Russell takes it to mean “There is asingle person who is King of France, and that person isbald”; on this analysis it is meaningful but false.george: Peter Pan doesn’t exist.martha: You mean the boy who flies through the air, whodoes battle with Captain Hook, and whom little childrenlove.george: Yes, he doesn’t exist.martha: Who doesn’t exist?george: Peter Pan.waldo: When are you going to pay the balance of this bill,George?george: Don’t worry. I’ll get it to you by the secondTuesday of next week.Sometimes even more serious difficulties can arise, aswhen George announces, “My brother is an only child.”
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Just as two terms can differ in meaning yet have thesame referent, a term can have many referents but only onemeaning.