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# Introduction To Logic PHIL 110

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Date Created: 10/30/15

248 Hardegree Symbolic Logic 11 SUMMARY OF THE BASIC QUANTIFIER TRANSLATION PATTERNS SO FAR EXAMINED Before continuing it is a good idea to review the basic patterns of translation that we have examined so far These are given as follows 1 2 3 4 5 6 7 8 9 10 11 12 Simple Quantification Plus Negation everything is B VXBX something is B EIXBX nothing is B NEIXBX something is nonB EIXNBX everything is nonB VXNBX not everything is B NVXBX Syllogistic Forms Plus Negation every A is B VXAX gt BX some A is B EXAX amp BX no A is B EXAX amp BX some A is not B EXAX amp NBX every A is a nonB VXAX gt NBX not every A is B NVXAX gt BX In addition to these it is important to keep the following logical equivalences in mind when doing translations into predicate logic 1 2 3 4 Basic Logical Equivalences NEIXAX VXNAX NVXAX EIXNAX EXAX amp BX VXAX gt NBX NVXAX gt BX EXAX amp NBX In looking over the above patterns one might wonder why the following is not a correct translation Chapter 6 Translations in Monadic Predicate Logic 249 1 every A is B VXAX amp BX WRONG The correct translation is given as follows 2 every A is B VXAX gt BX RIGHT Remember there simply is no general symbolbysymbol translation between colloquial English and the language of predicate logic in the correct translation 2 no symbol in the formula corresponds to the is in the colloquial sentence and no symbol in the colloquial English sentence corresponds to gt in the formula The erroneous nature of 1 becomes apparent as soon as we translate the formula into English which goes as follows everything is such that it is A andit is B For example everything is such that it is an astronaut and it is brave In other words everything is an astronaut who is brave or equivalently everything is a brave astronaut This is also equivalent to everything is an astronaut and everything is brave Needless to say this does not say the same thing as every astronaut is brave OK arrow works when we have every A is B but ampersand does not work So why doesn39t arrow work just as well in the corresponding statement some A is B Why isn39t the following a correct translation 3 some A is B EXAX gt BX WRONG As noted above the correct translation is 4 some A is B EXAX amp BX RIGHT Once again please note that there is no symbolbysymbol translation between the colloquial English form and the predicate logic formula Let39s see what happens when we translate the formula of 3 into English the straight translation yields the following 250 Hardegree Symbolic Logic 3t there is at least one thing such that if it is A then it is B Does this say that some A is B No In fact it is not clear what it says If the conditional were subjunctive rather than truthfunctional then 3t might corre spond to the following colloquial subjunctive sentence there is someone who would be brave if he were an astronaut From this it surely does not follow that there is even a single brave astronaut or even a single astronaut To make this clear consider the following analogous sentence some Antarctican is brave Here let us understand Antarctican to mean a permanent citizen of Antarctica This sentence must be carefully distinguished from the following there is someone who would be brave if heshe were Antarctican To say that some Antarctican is brave to say that there is at least one An tarctican who is brave from which it obviously follows that there is at least one Antarctican The sentence some Antarctican is brave logically implies at least one Antarctican eXists By contrast the sentence there is someone who would be brave if heshe were Antarctican does not imply that any Antarctican exists Whether there is such a person who would be brave were heshe to become an Antarctican I really couldn39t say but I suspect it is probably true It takes a brave person to live in Antarctica When we take ifthen as a subjunctive conditional we see very quickly that EXAx gtBX simply does not say that some A is B What happens if we insist that ifthen is truthfunctional In that case the sentence EXAx gtBX is automatically true so long as we can find someone who is not Antarctican Suppose that Smith is not Antarctican Then the sentence Smith is Antarctican is false and hence the conditional sentence if Smith is Antarctican then Smith is brave is true Why Because of the truth table for ifthen But if Smith is such that if he is Antarctican then he is brave then at least one person is such that if he is An tarctican then he is brave Thus the following existential sentence is true there is someone such that if he is Antarctican then he is brave Chapter 6 Translations in Monadic Predicate Logic 251 We conclude this section by presenting the following rule of thumb about how symbolizations usually go Of course in saying that it is a rule of thumb all one means is that it works quite often not that it works always Rule of Thumb not absolute If one has a universal formula then the connective immediately quotbeneathquot the universal quantifier is a conditional If one has an existential formula then the connective immediately quotbeneathquot the existential quantifier is a conjunction The slogan that goes with this reads as follows UNIVERSALCONDITIONAL EXISTENTIALCONJUNCTION Remember This is just a rule of thumb There are numerous exceptions which will be presented in subsequent sections 12 FURTHER TRANSLATIONS INVOLVING SINGLE QUANTIFIERS In the previous section we saw how one can formulate the statement forms of syllogistic logic in terms of predicate logic However the expressive power of predicate logic is significantly greater than syllogistic logic Syllogistic patterns are a very tiny fraction of the statement forms that can be formulated in predicate logic In the next three sections Sections 1214 we are going to explore numerous patterns of predicate logic that all have one thing in common with what we have so far examined Specifically they all involve exactly one quantifier More specifically still each one has one of the following forms f1 vm r2 EIxJZl r3 NVxJZl f4 Nam 252 Hardegree Symbolic Logic In particular either the main connective is a quanti er or the main connective is negation and the next connective is a quantifier We have already seen the simplest examples of these forms in Sections 10 and 11 Elex something is B VxBx everything is B NElex nothing is B NVxBx not everything is B ExBx something is nonB waBx everything is nonB We can also formulate sentences that have an overall form like one of the above but which have more complicated formulas in place of Bx The following are examples 1 everything is both A and B 2 everything is either A or B 3 everything is A but not B 4 something is both A and B 5 something is either A or B 6 something is A but not B 7 nothing is both A and B 8 nothing is either A or B 9 nothing is A but not B How do we translate these sorts of sentences into predicate logic One way is first to notice that the overall forms of these sentences may be written and symbolized respectively as follows 01 everything is J Vxe 02 everything is K VxKx 02 everything is L VxLx 03 something is J 3xe 04 something is K EIxKx 02 something is L Elex 05 nothing is J NElix o6 nothing is K NElxKx 02 nothing is L NElex Here the pseudoatomic formulas Jx Kx and Lx are respectively short for the more complex formulas given as follows Jx Ax amp Bx Kx Ax v Bx Lx Ax amp NBx Note the appearance of the outer parentheses Substituting in accordance with these equivalences we obtain the following translations of the above sentences Chapter 6 253 Translations in Monadic Predicate Logic t1 t2 t3 t4 t5 t6 t7 t8 t9 The VXAX amp BX VXAX v BX VXAX amp NBX EXAX amp BX EXAX v BX EXAX amp NBX EXAX amp BX EXAX v BX EXAX amp NBX following paraphrase chains may help to see how one might go about producing the symbolization 01 02 c3 everything is both A and B everything is such that it is both A and B everything is such that itisAanditisB VXAX amp BX everything is either A or B everything is such that it is either A or B everything is such that itisAoritisB VXAX v BX everything is A but not B everything is such that it is A but not B everything is such that it is A and it is not B VXAX amp NBX 254 Hardegree Symbolic Logic 04 something is both A and B there is at least one thing such that it is both A and B there is at least one thing such that itisAanditisB EXAX amp BX You will recall of course that something is both A and B is logically equivalent to some A is B as noted in the previous sections 05 c6 07 something is either A or B there is at least one thing such that it is either A or B there is at least one thing such that itisAoritisB EXAX v BX something is A but not B there is at least one thing such that it is A but not B there is at least one thing such that it is A and it is not B EXAX amp NBX nothing is both A and B it is not true that something is both A and B it is not true that there is at least one thing such that it is both A and B it is not true that there is at least one thing such that itisAanditisB EXAX amp BX Chapter 6 Translations in Monadic Predicate Logic 255 c8 nothing is either A or B it is not true that something is either A or B it is not true that there is at least one thing such that it is either A or B it is not true that there is at least one thing such that itisAoritisB EXAX v BX c9 nothing is A but not B it is not true that something A but not B it is not true that there is at least one thing such that it is A but not B it is not true that there is at least one thing such that it is A and it is not B EXAX amp NBX In the next section we will further examine these kinds of sentences but will introduce a further complication 13 CONJUNCTIVE COMBINATIONS OF PREDICATES So far we have concentrated on formulas that have at most two predicates In the present section we drop that restriction and discuss formulas with three or more predicates However for the most part we will concentrate on conjunctive combinations of predicates Consider the following sentences which pertain to a ctional group of people called Bozonians who inhabit the ctional country of Bozonia E e1 every Adult Bozonian is a Criminal e2 every Adult Criminal is a Bozonian e3 every Criminal Bozonian is an Adult e4 every Adult is a Criminal Bozonian e5 every Bozonian is an Adult Criminal e6 every Criminal is an Adult Bozonian 256 Hardegree Symbolic Logic S sl some Adult Bozonian is a Criminal s2 some Adult Criminal is a Bozonian s3 some Criminal Bozonian is an Adult s4 some Adult is a Criminal Bozonian s5 some Bozonian is an Adult Criminal s6 some Criminal is an Adult Bozonian N nl no Adult Bozonian is a Criminal n2 no Adult Criminal is a Bozonian n3 no Criminal Bozonian is an Adult n4 no Adult is a Criminal Bozonian n5 no Bozonian is an Adult Criminal n6 no Criminal is an Adult Bozonian The predicate terms have been capitalized for easy spotting The official predicates are as follows A is an adult B is a Bozonian C is a criminal You will notice that every sentence above involves at least one of the following predicate combinations AB adult Bozonian Bozonian adult AC adult criminal criminal adult BC Bozonian criminal criminal Bozonian In these particular cases the predicates combine in the simplest manner possible ie conjunctively In other words the following are equivalences for the complex predicates X is an Adult Bozonian X is an Adult and X is a Bozonian X is an Adult Criminal X is an Adult and X is a Criminal X is a Bozonian Criminal X is a Bozonian and X is a Criminal The above predicates combine conjunctively this is not a universal feature of English as evidenced by the following eXamples X is an alleged criminal X is a putative solution X is imitation leather X is an eXpectant mother X is an eXperienced sailor X is an eXperienced hunter X is a large whale X is a small whale X is a large shrimp X is a small shrimp X is a deer hunter X is a shrimp fisherman Chapter 6 Translations in Monadic Predicate Logic 257 For example an alleged criminal is not a criminal who is alleged indeed an alleged criminal need not be a criminal at all Similarly an expectant mother need not be a mother at all By contrast an experienced sailor is a sailor but not a sailor who is generally experienced Similarly an experienced hunter is a hunter but not a hunter who is generally experienced In each case the person is not experienced in general but rather is experienced at a particular thing sailing hunting Along the same lines a large whale is a whale and a large shrimp is a shrimp but neither is generally large neither is nearly as large as a small ocean let alone a small planet or a small galaxy Finally a deer hunter is not a deer who hunts but someone or something that hunts deer and a shrimp sherman is not a shrimp who shes but someone who shes for shrimp 1 am sure that the reader can come up with numerous other examples of predicates that don39t combine conjunctively Sometimes a predicate combination is ambiguous between a conjunctive and a nonconjunctive reading The following is an example x is a Bostonian Cabdriver This has a conjunctive reading x is a Bostonian who drives a cab perhaps in Boston perhaps elsewhere But it also has a nonconjunctive reading x is a person who drives a cab in Boston who lives perhaps in Boston perhaps elsewhere Another example which seems to engender confusion is the following x is a male chauvinist This has a conjunctive reading x is a male and x is a chauvinist which means x is a male who is excessively and blindly patriotic loyal However this is not what is usually meant by the phrase male chauvinist As originally intended by the author of this phrase a male chauvinist need not be male and a male chauvinist need not be a chauvinist Rather a male chauvinist is a person male or female who is excessively and blindly loyal in respect to the alleged superiority of men to women 258 Hardegree Symbolic Logic It is important to realize that many predicates don39t combine conjunctively Nonetheless we are going to concentrate exclusively on ones that do for the sake of simplicity When there are two readings of a predicate combination we will opt for the conjunctive reading and ignore the nonconjunctive reading Now let39s go back to the original problem of paraphrasing the various sen tences concerning adults Bozonians and criminals We do two examples from each group in each case by presenting a paraphrase chain e1 every Adult Bozonian is a Criminal every AB is C everything is such that if it is AB then it is C everything is such that if it is A and it is B then it is C VXAX amp BX gt CX e4 every Adult is a Bozonian Criminal every A is BC everything is such that if it is A then it is BC everything is such that ifit is A then it is B and it is C VXAX gt BX amp CX s3 some Criminal Bozonian is an Adult some CB is A there is at least one thing such that it is CB and it is A there is at least one thing such that it is C and it is B and it is A EXCX amp BX amp AX Chapter 6 Translations in Monadic Predicate Logic 259 s5 some Bozonian is an Adult Criminal some B is AC there is at least one thing such that it is B and it is AC there is at least one thing such that it is B and it is A and it is C EXBX amp AX amp CX n3 no Criminal Bozonian is an Adult no CB is A it is not true that some CB is A it is not true that there is at least one thing such that it is CB and it is A it is not true that there is at least one thing such that it is C and it is B and it is A EXCX amp BX amp AX n6 no Criminal is an Adult Bozonian no C is AB it is not true that some C is AB it is not true that there is at least one thing such that it is C and it is AB it is not true that there is at least one thing such that it is C and it is A and it is B EXCX amp AX amp BX The reader is invited to symbolize the remaining sentences from the above groups We can further complicate matters by adding an additional predicate letter say D which symbolizes say is deranged Consider the following two examples 260 Hardegree Symbolic Logic el every Deranged Adult is a Criminal Bozonian e2 no Adult Bozonian is a Deranged Criminal The symbolizations go as follows s1 VXDX amp AX gt CX amp BX s2 EXAX amp BX amp DX amp CX Another possible complication concerns internal negations in the sentences The following are examples together with their stepwise paraphrases 1 every Adult who is not Bozonian is a Criminal every A who is not B is C everything is such that if it is an A who is not B then it is C everything is such that if it is A and it is not B then it is C VXAX amp NBX gt CX 2 some Adult Bozonian is not a Criminal some AB is not C there is at least one thing such that it is AB and it is not C there is at least one thing such that it is A and it is B and it is not C EXAX amp BX amp NCX 3 some Bozonian is an Adult who is not a Criminal some B is an A who is not a C there is at least one thing such that it is B and it is an A who is not C there is at least one thing such that it is B and it is A and it is not C EXBX amp AX amp CX Chapter 6 Translations in Monadic Predicate Logic 261 4 no Adult who is not a Bozonian is a Criminal no A who is not B is C it is not true that some A who is not B is C it is not true that there is at least one thing such that it is an A who is not B and it is C it is not true that there is at least one thing such that it is A and it is not B and it is C EXAX amp NBX amp CX 262 Hardegree Symbolic Logic 14 SUMMARY OF BASIC TRANSLATION PATTERNS FROM SECTIONS 12 AND 13 Forms With Only Two Predicates 1 2 3 1 2 3 1 2 3 everything is both A and B VXAX amp BX everything is A but not B VXAX amp NBX everything is either A or B VXAX v BX something is both A and B EXAX amp BX something is A but not B EXAX amp NBX something is either A or B EXAX v BX nothing is both A and B EXAX amp BX nothing is A but not B EXAX amp NBX nothing is either A or B EXAX v BX Simple Conjunctive Combinations 1 2 3 4 5 6 7 8 every AB is C VXAX amp Bx gt CX some AB is C EXAX amp BX amp CX some AB is not C EXAX amp BX amp NCX no AB is C EXAX amp BX amp CX every A is BC VXAX gt BX amp CX some A is BC EXAX amp BX amp CX some A is not BC EXAX amp BX amp CX no A is BC EXAX amp BX amp CX Conjunctive Combinations Involving Negations 1 2 3 4 5 6 7 every A that is not B is C VXAX amp NBX gt CX some A that is not B is C EXAX amp NBX amp CX some A that is not B is not C EXAX amp NBX amp NCX no A that is not B is C EXAX amp NBX amp CX every A is B but not C VXAX gt BX amp CX some A is B but not C EXAX amp BX amp CX no A is B but not C EXAX amp BX amp CX Chapter 6 Translations in Monadic Predicate Logic 263 15 ONLY The standard quantifiers of predicate logic are every and at least one We have already seen how to paraphrase various nonstandard quantifiers into standard form In particular we paraphrase all as every some as at least one and no as not at least one In the present section we examine another nonstandard quanti er only in particular we show how it can be paraphrased using the standard quantifiers In a later section we examine a subtle variant the only But for the moment let us concentrate on only by itself The basic quantificational form for only is only A are 83 Examples include 1 only Men are NFL football players 2 only Citizens are Voters Occasionally signs use only as in employees only members only passenger cars only These can often be paraphrased as follows 3 only Employees are Allowed 4 only Members are Allowed 5 only Passenger cars are Allowed What is in fact allowed or disallowed depends on the context Generally signs employing only are intended to exclude certain things specifically things that fail to have a certain property being an employee being a member being a passenger car etc Before dealing with the quantifier only let us recall a similar expression in sentential logic namely only if In particular recall that A only if B may be paraphrased as not A if not B which in standard form is written if not B then not A MB gt MA 264 Hardegree Symbolic Logic In other words only modi es i by introducing two negations The word i always introduces the antecedent and the word only modifies i by adding two negations in the appropriate places When combined with the connective i the word only behaves as a special sort of doublenegative modi er When only acts as a quanti er it behaves in a similar doublenegative manner Recall the signs involving only they are intended to exclude persons who fail to have a certain property Indeed we can paraphrase only JZl are 13 in at least two very different ways involving doublenegatives First we can paraphrase only JZl are 13 using the negative quanti er no as follows 0 only A are B p ma non A are B Strictly speaking non is not an English word but simply a pre x properly speaking we should write the following p n0 nonA are B However the hyphen will generally be dropped simply to avoid clutter in our intermediate symbolizations Thus the following is the quotskeletalquot paraphrase only no non only no non However in various colloquial examples the following more quotmeatyquot paraphrase is more suitable p no one who is not A is B so for example 14 may be paraphrased as follows pl no one who isn39t a Man is an NFL football player p2 no one who isn39t a Citizen is a Voter p3 no one who isn39t an Employee is Allowed p4 no one who isn39t a Member is Allowed Next we turn to symbolization First the general form is 0 only A are B which is paraphrased p no non A are B no non A is B This is symbolized as follows s ExAxamp Bx Chapter 6 Translations in Monadic Predicate Logic 265 Similarly ol05 are symbolized as follows s1 EXMXamp NX s2 EXCX amp VX s3 EXEX amp AX s4 EXMX amp AX s5 EXPX amp AX The quickest way to paraphrase only is using the equivalence ONLY NO NON An alternative paraphrase technique uses all every plus two occurrences of non not as follows 0 only A are B p1 all non A are non B p2 every non A is non B p3 everyone who is not A is not B These are symbolized as follows s VXAX gt BX So for example 15 may be paraphrased as follows p1 everyone who isn39t a Man isn39t an NFL football player p2 everyone who isn39t a Citizen isn39t a Voter p3 everyone who isn39t an Employee isn39t Allowed p4 everyone who isn39t a Member isn39t Allowed p5 everyone who isn39t driving a Passenger car isn39t Allowed These in turn are symbolized as follows s1 VXMX gt NNX s2 VXCX gt NVX s3 VXEX gt MAX s4 VXMX gt MAX s5 VXPX gt MAX The two approaches above are equivalent since the following is an equivalence of predicate logic EXAXamp BX VXAX gt BX To see this equivalence first recall the following quantificational equivalence NEIXf39 VXNf And recall the following sentential equivalence 266 Hardegree Symbolic Logic NJZl amp B MA gt NB Accordingly EXAXamp BX VXAXamp BX And Axamp BX AX gt BX So EXAXamp BX VXAX gt BX There is still another sentential equivalence NJZl gt NB B gt 1 So Bx gt MAX BX gt AX So EXAXampBX VXBX gtAX This equivalence enables us to provide yet another paraphrase and symbolization of only A are B as follows 0 only A are 3 p all B are A s VX BX gt AX The latter symbolization is admitted in intro logic just as P gtQ is admitted as a symbolization of P only if Q in addition to the official Q gtP The problem is that the nonnegative construals of only statements sound funny even wrong to some people in English In short our official paraphrasesymbolization goes as follows 0 only A are B pl ma non A is B s1 EXAXamp BX p2 every non A is non B s2 VXAX gt BX Note carefully however for the sake of having a single form the former para phrasesymbolization will be used exclusively in the answers to the exercises Chapter 6 Translations in Monadic Predicate Logic 267 16 AMBIGUITIES INVOLVING ONLY Having discussed the basic only statement forms we now move to examples involving more than two predicates As it turns out adding a third predicate can complicate matters Consider the following eXample el only Poisonous Snakes are Dangerous Let us assume that poisonous combines conjunctively so that a poisonous snake is simply a snake that is poisonous even though a poisonous snake is quite different from a poisonous mushroom a mushroom39s bite is not very deadly Granting this simplifying assumption we have the following paraphrase X is a Poisonous Snake X is Poisonous and X is a Snake Now if we follow the pattern of paraphrase suggested in the previous section we obtain the following paraphrase pl no non Poisonous Snakes are Dangerous no non Poisonous Snake is Dangerous Unfortunately the scope of non is ambiguous For the sentence 1 X is a non poisonous snake has two different readings and hence two different symbolizations r1 X is a nonpoisonous snake r2 X is a nonpoisonous snake X is not a poisonous snake s1 NPX amp SX s2 PX amp SX On one reading to be a non poisonous snake is to be a snake that is not poisonous On the other reading to be a non poisonous snake is simply to be anything but a poisonous snake Our original sentence and its paraphrase only poisonous snakes are dangerous no non poisonous snakes are dangerous are correspondingly ambiguous between the following readings EXPX amp SX amp DX 268 Hardegree Symbolic Logic there is no thing X such that X is not a Poisonous Snake but X is Dangerous EXPX amp SX amp DX there is no thing X such that X is a nonPoisonous Snake but X is Dangerous To see that the original sentence really is ambiguous consider the following four very short paragraphs 1 Few snakes are dangerous In fact only poisonous snakes are dangerous 2 Few reptiles are dangerous In fact only poisonous snakes are dangerous 3 Few animals are dangerous In fact only poisonous snakes are dangerous 4 Few things are dangerous In fact only poisonous snakes are dangerous In each paragraph when we get to the second sentence it is clear what the topic is snakes reptiles animals or things in general What the topic is helps to determine the meaning of the second sentence For eXample in the first paragraph by the time we get to the second sentence it is clear that we are talking exclusively about snakes and not things in general In particular the sentence does not say whether there are any dangerous tigers or dangerous mushrooms By contrast in the fourth paragraph the first sentence makes it clear that we are talking about things in general so the second sentence is intended to eXclude from the class of dangerous things anything that is not a poisonous snake An alternative method of clarifying the topic of the sentence is to rewrite the four sentences as follows 1 only Poisonous Snakes are Dangerous snakes 2 only Poisonous Snakes are Dangerous reptiles 3 only Poisonous Snakes are Dangerous animals 4 only Poisonous Snakes are Dangerous things These may be straightforwardly paraphrased and symbolized as follows 0 only A are B no non A is B EXAX amp BX Chapter 6 269 Translations in Monadic Predicate Logic 1 2 3 4 only PS are DS no nonPS is DS EXPX amp SX amp DX amp SX only PS are DR no nonPS is DR EXPX amp SX amp DX amp RX only PS are DA no nonPS is DA EXPX amp SX amp DX amp AX only PS are D no nonPS is D EXPX amp SX amp DX If we prefer to use the every paraphrase of only then the paraphrase and sym bolization goes as follows 0 1 2 3 4 only A are B every non A is non B VXAX gt NBX only PS are DS every nonPS is nonDS VXPX amp SX gt DX amp SX only PS are DR every nonPS is nonDR VXPX amp SX gt DX amp RX only PS are DA every nonPS is nonDA VXPX amp SX gt DX amp AX only PS are D every nonPS is nonD VXPX amp SX gt NDX 17 THE ONLY The word the subtleties of only are further complicated by combining it with the to produce the only Still more complications arise when the is combined with only all to produce only the all the however we are only going to deal with the only The nice thing about the only is that it enables us to make only statements without the kind of ambiguity seen in the previous section Recall that

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