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by: Harrison Reilly


Marketplace > University of Texas at Austin > PHIL-Philosophy > PHL 313Q > LOGIC AND SCIENTIFIC REASONING
Harrison Reilly
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This 49 page Class Notes was uploaded by Harrison Reilly on Monday September 7, 2015. The Class Notes belongs to PHL 313Q at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/181848/phl-313q-university-of-texas-at-austin in PHIL-Philosophy at University of Texas at Austin.

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Date Created: 09/07/15
Firstorder Predicate Logic Odysseusis hurting me Nobody is hurting me Predicates is hurting me is hurting 7 gives to Predicates represented by upperIcase letters If followed by no names complete statement If followed by one name quality or property If followed by two names binary relation etc Nonvariable constant name an expression used to name a particular individual thing person p1acetime Names are represented by lower case letters from a through s The symbol for identity 2 Socrates is Wise Socrates is Plato s teacher Sp Socrates is Plato s teacher You are not Socrates You are not Plato s teacher Socrates is human You are not Socrates You are not human sp a s 39a p Hs a s 39IHa Something is coming EIXCX Everything is coming VXCX Some swans are White EIXSX amp WX A11 swans are White VXSX gt WX No swans are white VXSX gt IWX IEXSX amp WX Translating English into Logical Symbols The language of predicate logic introduces two new logical symbols the universal quantifier V and the existential quantifier El These can be used to translate the words something and everything Something is coming EIXCX Everything is coming VXCX Every sentence in English contains a main verb These verbs come in several kinds Intransitive verbs sleeps walks thirsts These take a subject only Transitive verbs loves moves irritates These take a subject and a direct object Ditransitive verbs gives as in A gives B C These take a subject a direct object and an indirect object shall be translated by means of predicate letters We shall follow these letters by one two or three underscores indicating Whether the verb is intransitive transitive or ditransitive walks gt W likes gt L gives gt G As I described earlier shall be symbolized by lower case letters from a through 0 the individual constants We Will precede these constants by a question mark to indicate that the name must be combined With a predicate in order to produce a complete sentence An underscore shall stand for a missing name or variable and a question mark shall stand for a missing predicate John gt j Predicates consisting of to be plus an adjective or common noun are also translated by means of a single predicate letter followed by an underscore is White gt W Consider the sentence John walks We symbolize the name John as j and the predicate walks as W When these two are combined the constant j replaces the underscore in the predicate and the predicate repaces the s question mark in the subject The result is W Proper names are only one kind of term Another kind is a like a boy39 or 39all Circles in which a determiner a all is combined with a common noun boy circle or common noun phrase A common noun phrase can be created by starting with a common noun and adding a relative clause a clause headed by that who whom or which and adjective or a prepositional phrase The class of common noun phrases can be defined as follows 1 If 5Q is a common noun then 3Q is a common noun phrase 2 If ad is a noun phrase and 75 is a relative clause or a prepositional phrase then 3Q 75 is a noun phrase 3 If ad is a common noun phrase and 75 is an adjective then 75 3Q is a common noun phrase So the following are c noun phrases boy boy in Austin boy Who lives in Austin dark haired boy dark haired boy Who lives in Austin boy Who lives in a capital city A relative clause results from replacing the subject direct object or indirect object of a complete sentence by that Who Whom or Which Where Who must always replace the subject and Whom the direct object A prepositional phrase is the result of combining a preposition like 39in 39on39 through etc With a noun phrase First it is necessary to specify the translation of the of English The following table gives the translation of the most common determiners every 2 any 2 all 2 VX X gt X a 2 an 2 some 2 EIX X amp X no 2 none 2 EIX X amp X only VX x gt X everybody anybody VXPX gt X somebody EIXPX amp X The subscriped letters i and j indicate that two different predicates should be used to replace the question marks You should introduce new indices as you proceed through the English sentence The closest predicate to the determiner should be used to replace the question mark with the lower indeX Note that the crucial difference between every and only concerns the order of these indices when translating every put the first available predicate into the antecedent of the conditional while in translating only you should put the first predicate in the consequent instead Although the variable X is used throughout this table in fact one should introduce a new variable every time you translate a new deterrniner in an English sentence So if you ve already used X you should translate the next deterrniner using y instead Sorne swans are white 3X X amp X S W EIXSX amp X W EIXSX amp WX All swans are white VX X r X S W VXSX gt X W VXSX gt WX No swans are white 3X X amp X S W IEIXSX gt X W IEXSX amp WX consist of relative pronouns combined with predicates The relative pronouns that who whorn and which are all translated in the same way that 2 Who 2 whorn which 2 i amp i In this translation the shadow question marks stand for two different predicates but the underscores which share the same index stand for one individual attach to common noun phrases and produce new common noun phrases For example the adjective white can be translated as follows white 2 Wi amp i The two underscores are linked by a common index i and must both be replaced by the same term The shadowed question mark should be combined with the common noun or noun phrase to which the adjective is attached Thus the phrase a white swan would be translated as follows 3X X amp X Wk amp k l S EIX x amp X Wk amp Sk 3XWX amp SX amp X At this point we are ready to combine this noun phrase with a predicate replacing the remaining shadowed question mark producing a complete formula consist of prepositions plus noun phrases A preposition is translated by means of a two place predicate a predicate letter followed by two underscores This predicate represents the spatial or temporal relationship encoded by the preposition For example the translation of the predicate in includes the two place predicate Iij which represents the relationship of the thing corresponding to i s being inside the thing represented by j The full translation of a preposition is as follows in i amp 1 1 j The following translations illustrate the use of relative clauses and prepositional phrases All students Who attend Will do well VX X XS i amp i l A W VX X a X S 7185 i l A W VX X Xl 5 i amp A i W VXSX amp AX gt X W VXSX amp AX gt WX Some students Who attend Will do well 3X X amp X S7 i amp i l A W 3X X amp X S i amp i l A W 3X X amp X S i amp A i W EIXSX amp AX amp X W EIXSX amp CXC amp WX A boy in Austin likes Jane 3X X amp X l B 3 Lkm j 3X X amp X B7 1 amp L120 Lkj 3X X amp X Bi amp Ii aLk j EIXBX amp Ixa amp X Lkj VXBX amp Ixa amp LXj The following example illustrates the fact that non restrictive relative clauses those that are set off by commas are to be treated quite differently amp 1 1 j 1 All Plan 11 students who come to class will do well All Plan 11 students attend and all Plan 11 students will do well VXPX gt AX amp VyPy gt Wy It is possible to have more than one determiner and thus more than one quantifier in a single sentence When translating such sentences it can make a very material difference in what order we combine the elements In general when translating written text Consider the following simple example Every boy kisses a girl VX X r X B Kj k Ely yamp y Gm VX X r X B7 Kj k ElyGy amp y VX X XB3YGy amp K4 Y VxBX e x ElyGy amp K4 y VXBX gt EIyGy amp ny he she it him her should be translated by means of constants or variables In translating a pronoun in an English sentence one should try to discern which preceding noun phrase in the sentence is the antecedent of the pronoun Once you have identified this you can use the corresponding constant or variable as your translation of the pronoun For example Jane came and Mike greeted her Ci amp ij A visitor arrived and he asked for Sally EIXVX amp AX amp A39Xs Every student joined a club that accepted hirn VXSX gt EIyCy amp AyX There are several special cases Where the rule given above for the translation of a an and sorne has to be broken In these exceptional cases we should translate these words as having the same meaning as every A friendship is forever A salesman is annoying Although these sentence employ the indefinite article a they are in fact ambiguous They could be used to assert that there is at least one thing of the specified kind or they could be used to assert that all or nearly all things mentioned in the subject have the characteristic expressed in the predicate In this second case they would be translated as VXFX gt EX VXSX gt AX 1 If you love someone you are fortunate 2 If you love someone you should be good to himher 3 Every salesman who greets a customer will receive a bonus 4 Every salesman who greets a customer should be polite to himher In sentences 1 and 3 the words some and a can be translated in the standard way employing existential quantifiers and conjunctions without difficulty However if we try to do the same thing with sentences 2 and 4 we run into a serious difficulty Simply applying our rules to these sentences would result in the following quottranslationsquot 20 2 EIyPy amp Lay gt Gay 439 VXSX amp EIyCy amp ny gt ny 2quot VyPy amp Lay a Gay 4quot VXVySX amp Cy amp ny e nygt If you love everyone you are fortunate VXPX gt Lax gt Fa If you love anyone you are fortunate VXPX gt Lax gt Fa I didn t catch every fish IVXFX gt CbX I didn t catch any fish VXFX gt CbX 21 I didn t catch a fish IEXFX amp CbX If you didn t catch every fish you should quit IVXFX gt Cbx gt Qb If you didn t catch any fish you should quit VXFX gt ICbX gt Qb Only mammals nurse their young VXNX gt MX Only Republicans are happy VXHX gt RX Some exercises for futher practice 1 Nobody knows all the trouble I ve seen 2 Nobody knows any of the trouble I ve seen 3 Some of the trouble I ve seen is not known by anybody 22 4 Everybody knows some of the trouble I ve seen 5 Some of the trouble I ve seen is known by everybody 6 Somebody knows all the trouble I ve seen 7 Each of the troubles I ve seen is known by somebody 8 Everybody knows all of the trouble I ve seen 9 Sornebody knows some of the trouble I ve seen Pa a is a person Ta a is trouble Sab a has seen b Kab a knows b i me 1 Nobody knows all the trouble I ve seen IEXPX amp X knows all the trouble I ve seen 23 IEXPX amp all the trouble I ve seen is known by X IEXPX amp VyTy amp Siy gt y is known MD IEXPX amp VyTy amp Siy gt ny 2 Nobody knows any of the trouble I ve seen VyTy amp Siy gt nobody knows y VyTy amp Siy gt IEX X knows y VyTy amp Siy gt IEX ny 3 Somebody knows all the trouble I ve seen EIXPX amp X knows all the trouble I ve seen EIXPX amp all the trouble I ve seen is known by X EIXPX amp VyTy amp Siy gt ny 4 Each of the troubles I ve seen is known by somebody VyTy amp Siy gt y is known by somebody 24 VyTy amp Siy amp somebody knows y VyTy amp Siy gt EIXPX amp X knows y VyTy amp Siy gt EIXPX amp KXy Lab 21 loves b Pa 21 is a person 21 Adam 5 Everyone loves someone Who loves himher VXPX gt X loves someone Who loves himher VXPX gt X loves someone Who loves X VXPX gt someone Who loves X is loved by X VXPX gt ElyPy amp LyX amp y is loved M VXPX gt ElyPy amp LyX amp LXy 6 Everyone loves someone Who loves himselfherself VXPX gt X loves someone Who loves himselfherself 25 VXPX gt someone Who loves himselfherself is loved by X VXPX gt EIyPy amp Lyy amp y is loved D31 VXPX gt EIyPy amp Lyy amp LXy 7 Only Adam loves Adam Everyone Who loves Adam is Adam VXPX amp LXa gt X is Adam VXPX amp an gt X 2a 8 m Adam loves himself Everyone Who loves himselfhersel f is Adam VXPX amp LXX gt X is Adam VXPX amp LXX gt X a 9 Adam loves LIIlX himself Everyone Who is loved by Adam is himself Everyone Who is loved by Adam is Adam VXPX amp LaX gt X a 26 10 Somebody loves only Adam EIXPX amp X loves LIIlX Adam EIXPX amp everything loved by X is Adam EIXPX amp Vyny gt y is Adam EIXPX amp Vyny gt y a Every man loves some woman VXEIyMX gt Wy amp ny VXMX gt ElyWy amp ny Some woman is loved by every man ElyVXWy amp MX gt ny ElyWy amp VXMX gt ny Some woman loves every man ElyWy amp VXMX gt LyX Every man is loved by some woman VXMX gt ElyWy amp LyX 27 In some cases when translated sentences whose main verb is a verb of action or change we must introduce an invisible term that stands for the event or action described by the verb For example consider the following two sentences John walks John walks slowly EIXWalkingX amp ByJohn X or EIXWX amp BjX EIXWalkingX amp ByJohnX amp SlowX or EIXWX amp B jX amp SX For example we can translate John came late to the party as EIXComingX amp ToXthe party amp ByJohnX amp LateX 28 We can also represent the information contained by the tenses of English statements in a similar fashion We can introduce a special constant n for the present moment and add to any sentence in the past tense a clause of the form Pxn specifying that the event X is temporally prior to the present An example of using function symbols in translation In a hierarchy every employee tends to rise to his own level of incompetency Hx X is a hierarchy Exy X is an employee in y ny X tends to rise to y lX X s level of incompetency VXVyHX amp EyX gt RXlX 29 In order to translate mathematical texts we need to be able to represent functional expressions such as addition subtraction and multiplication We will do this by means of function symbols followed by a series of terms enclosed in parentheses and set off by commas For example to represent the complex term 2 3 we could use a two place function symbol representing addition and two indiVidual constants representing 2 and 3 abc where b stands for 2 c for 3 and a for the sum function Function symbols can also be used to translate certain definite descriptions in English such as the capital of Texas or J ohn s mother These could be translated as ce or as m0 where e stands for Texas j for John c for the 30 capital of function and m for the mother of function Suppose f is a l place g a 2 place h a 3 place function symbol Then the following are terms of our formal language fa gaa habc fX gXa hXyZ ffafgab gfx haxb hfahxabc fghabc b Notice that terms can be of any complexity There are two kinds of terms open and closed Open terms contain variables and closed terms do not lAny upper case letter is a sentence 31 2 Any upper case letter followed by 11 names is a sentence 3 Ifc and d are names c 2 dis a sentence 3 If 3Q is a sentence so is Q 4 If 3Q and 75 are sentences With no variables in common then 3Q gt 75 3st amp 75 3st v 75 and 3Q H 75 are sentences 5 If 3Q is a sentence c is a name and z is a variable that does not appear in 3Q then 3Z3Slzc and VV3Slzc are sentences 6 Every sentence can be constructed by a finite number of applications of these rules Well formed sentences EIXFX EIXA EIXFX gt A VXEInyy VXVyFy 32 VXFX gt Elnyy VXEIyAxyX amp VZEIWGZW gt HWWZ VXEInyy 3X FX gt Vnyy EIXFX HXFX EIX I FX Fax Fax amp EIXGaX VyGy ny HXFX amp VXGX VXGX amp EIXFX EIXFX amp VyGy FVXa Translations of mathematical statements Hilbert s Geometry 33 If two planes oz 5 have a point A in common then they have at least one more point in common VXVszAX amp Ay amp Pz amp X 72 y amp CXZ amp Cyz gt EIWPW amp CXW amp CyW amp W 72 z There exist at least four points that do not lie in a plane X y z W do not lie in a plane EIXEIyEIZEIWPXampPyampPZampPWamp X yampX ZampX Wampy Z ampy7 39WampZ7 39Wamp IEuAu ampCuXampCuyampCuzamp CuW 34 Set Theory If two sets have the same members they are identical VXVySX amp Sy amp Vzz E X H z E y e X y For any set X there is a set Whose members are exactly the subsets of X VXSX gt Ely Sy amp Vzz E y H z E X There is no set to Which every set belongs There is no set X every set belongs to X VXSX gt IVySy gt y E X IEXSX amp VySy gt y E X 35 Every set is a subset of itself VXSX gt X E X No set belongs to itself VXSX gt 1 E X Instantiate Universal Quantifier Instantiate Existential Quantifier Alter Denied Quantification From VXFX VXFX gt GaX VXEInyyb We can get Fa Fa gt Gaa EIyHayb Fb Fb Gab ElyHbyb 36 The universally quantified variable can be replaced by any constant narne whatsoever Exactly the same process With the restriction The name introduced by instantiating the variable on a path must be to the sub tables to which the path belongs Consider the argument John is bald Something is thin John is thin 37 BJ 3XTX IT Tj X John is bald Everything is material John is material 1VXMX 2 Bj 3 IMj 4 Mj from 1 X Something is thin Everything is material Something is thin and material 38 Alter Denied Quantifier From IEX3S we get VxIy l From VX we get ElxIs l If it s false that something is red then everything is not red If it s false that everything is red then something is not red Numerical statements There is at least one F EIXFX There are at least two F s EIXFXampFyampIXy There are at least three Fs EIXEIyEzFXampFyampFzampIXy ampIyz amp IXZ There is at most one F VXVyFXampFy gt Xzy 39 There are at most two Fs VXVszFXampFyampFz gtXyvyZVXz There is exactly one F EIXFX amp VyFy gt Xy To say a is the rnost VX IXa gt Max To say a is the least VXIXa gt an Enthyrnernes in English An enthymerne is an argument in which one or more premises consisting of universal generalizations have been left unstated 40 Persons of genius are by definition more individual than any other people less capable consequently of fitting themselves Without hurtful compression into any of the small number of molds which society provides JSMill Goc oc is a genius loaf oc is more individual than 5 Focfiy oc is more capable of fitting himself Without hurtful compression into any of the small number of molds 5 provides than is y a society VXVyGXampIGy gtIxy 39 VXVyGXampIGy gtFan Missing premise 41 Sir when a man is tired of London he is tired of life for in London is to be found all that life affords Samuel Johnson 1 London b life Tocfi o is tired of 5 Aocfi oc affords 5 or 5 is to be found in o VXAbX gt AlX VXTXl gtTXb Missing premises VXVyVzAyz gt TXZ gt TXy For all X and y X is tired of y only if X is tired of everything y affords VXVyPX gt TXy gt VzAyz gt TXZ For all X and y X is tired of y if X is tired of everything y affords 42 If individuals live only 70 years then a state or a nation or a civilization which may last for a thousand years is more important than an individual But if Christianity is true then the individual is not only more important but incomparably more important for he is everlasting and the life of a state or a civilization compared with his is only a moment C S Lewis Ioc oc is an individual human Soc oc is a state Coc oc is a civilization Moe oc lives only 70 years or so Loc oc lives for centuries or millenia Eoc oc lives forever DOLE oc lives longer than 5 Vocfi oc is more important than 5 A Christianity is true 43 Premises A gt VXHx gtEX VXHx gtEX gt VszHyampSZVCz gtDyz Conclusion A gt VXVyHXampSyva gtny Unstated premise VXVyny gtny Paralogisrn frorn Plato s Euthydemos Learners do not know What they are learning Those Who do not know something are ignorant Learners are ignorant VXVyny gt ny VXVyIny gt Ixy VXVyny gt Ixy 44 Learners come to know what they are learning Those who come to know something are not ignorant Learners are not ignorant VXXVyny gt K Xy VXVyK Xy gt ley VXVyny gt ley Inconsistency results if we add EIXEIy ny A poem is nothing but letters Those who learn a poem already know all the letters Those who learn a poem learn what they already know 45 VXPX gt Ay VXEyny amp Py gt VZAZ gt KXZ VXEyny amp Py gt EIZLXZ amp KXZ or VXVyny amp Py gt ny Distinguish between is a letter and is a collection of letters Ay y is a letters like alpha beta gamma A y y is composed by letters like poems histories etc You want Cleanthes to be a Wise person The only Wise person is Socrates You want Cleanthes to be Socrates You want Cleanthes not to be Cleanthes any more You want Cleanthes to die W wants to be 46 EIXSX amp WaCX VXSX gt X s Wacs C t s 39 Wacc Wacd Verbs of attitude takes states of affairs as objects Hocfiy in state o it holds that 5 has property y s property of Wisdom squot the property of being identical to Socrates e the actual world EIXWaX amp chs VXHexs39 gt s s or VXHexs39 gt Hexsquot EIXWaX amp chsquot Your dog is a father Your dog is yours Your dog is your father 47 When someone tells a lie he speaks of something Whatever someone speaks of is real When someone speaks of something real he speaks truly When someone tells a lie he speaks truly The following are equivalent VXVyFX gt ny VXFX gt Vnyy So are these VXEIyFX gt ny VXFX gt Elnyy And these VXVyny gt FX These too VXEIyny gt FX VXEnyy gt FX VXVnyy gt FX The following pairs are equivalent VXVyny gt FX VXVnyy gt FX VXEIyny gt FX VXEnyy gt FX Moving a quantifer through a negative context changes its polarity V to El and vice versa 48 49 VXEnyy gt IIX equiv to VXVyny gt IIX VXVzsz gt IX equiv to VXEIZ LXZ gt IX Moving through a positive context no change All students and all teachers have a conference VxSx gt VyTy gt ElzszyD VXVyEIZSX gt Ty gt CZXYD VXVyEIZSXamp Ty gt szy


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