Dr. Ray is Associate Professor of Philosophy. His main interests lie in philosophical logic—which is a quite loosely identified domain whose common thread is the application of formal methods to philosophical problems. Topics in this area range from philosophy of logic and philosophy of language to some problems of metaphysics, and includes the analysis of philosophical paradoxes and special topics such as the concept of truth.

Show the LONG list Show the SHORT list     

Recent Work Recent Work WITH WORKING LIST Recent Work WITH EX LIST

  • "Tarski on Truth", in Oxford Handbook of Truth, Michael Glanzberg, ed. Oxford University Press. Forthcoming.
  • "Natural Natural Deduction for Modal Logics", with Fabio Lampert, Society for Exact Philosophy, Coral Gables, upcoming May 2016.
  • "Could Russell Have Been Right About Vagueness After All?", invited symposium commentator, American Philosophical Association, Chicago, March 2016.
  • "Teaching Reasoning with Technology", invited symposium, Florida Philosophical Association, St Augustine, FL, November 2016.
  • "The Truth About Vagueness", with Kirk Ludwig, invited lecture, Society for Exact Philosophy, Hamilton, Ontario, 2015.
  • "Meaning and Truth", Mind 123 (2014): 79-100.

    This paper concerns a key choice point in Donald Davidson's early work in philosophy of language—a fateful decision that set him and the discourse in the area on the path of truth-theoretic semantics. The decision of moment is the one Davidson makes when, in the face of a certain barrier, he gives up on the idea of spelling out a meaning theory proper along lines generally paralleling Tarski's way with truth theory. For Davidson, there was not much choice in the choice that he makes then. He tells us at that juncture that he just does not see how to do it any other way than in the radical way he then proposes.

    But there is a way to give such a meaning theory—a meaning theory proper which, using classical logic only, meets a meaning-theoretic analog of Convention T, satisfies Davidson's three key desiderata for a theory of meaning, reflects linguistic competence, and avoids quantifying over meanings. Just as Davidson initially sought, the meaning theory sketched here uses Tarskian strategies, but unlike Davidson's approach, does not go by way of a truth theory for the target language.

    (paper online)

  • Belief, Knowledge and Non-Existence, invited lecture series, Fudan University, Shanghai, China, May 2014.
    • "Logic & the Problem of Belief"
    • "Semantics & the Problem of Knowledge"
    • "Existence & the Problem of Fiction"
  • "The Problem of Negative Existentials, Inadvertently Solved", Empty Representations: Reference and Non-Existence, Manuel García-Carpintero & Genoveva Martí, eds., Oxford University Press, 2014, pp. 262-274.

    The problem of negative existentials is one of the classic problems in philosophy of language. Latter-day developments in semantics resolved this problem without our help, but due to accidents of history no one noticed.

  • Reasoning Toolbox. A suite of six interconnected, interactive web-based programs that constitute the primary working environment for students to learn seven fundamental processes for the analysis of argumentative texts, the representation of real arguments, the critique of such arguments, and for the construction of well-structured argumentative and critical text. This software is in use in my Reasoning class — which takes a novel approach to the teaching of reasoning skills.
  • "Analytic Satisfaction",
    • Society for Exact Philosophy, Columbus OH, 2012.
    • Florida Philosophical Association, Orlando FL, 2012.

Selected Publications Publications Selected Publications WITH WORKING LIST Selected Publications WITH EX LIST

Edited Volumes

Software

Selected Presentations Presentations Selected Presentations WITH WORKING LIST Selected Presentations WITH EX LIST

Unpublished, but Interesting Things Selected Unpublished WITH WORKING LIST Selected Unpublished WITH EX LIST

Software Software WITH WORKING LIST Software WITH EX LIST

Professional Associations

  • American Philosophical Association
    • Member: 1990-present.
  • Society for Exact Philosophy
    • Secretary: 1996-present.
    • Member: 1993-present.
  • Florida Philosophical Association
    • Technical Secretary: 2006-present
    • Acting Secretary-Treasurer: 2007-2008.
    • President: 2005-2006.
    • Vice President: 2004-2005.
    • Member: 1992-present.

Awards

  • NEH Summer Institute, "20th Century American Philosophy: Quine and Davidson", Princeton, Summer 2011.
  • NEH Summer Institute, "Meaning", Rutgers, Summer 1994.
Its Ce r t a I n m aKE M O M ENTS I N F OR THE P HI l o S o ph e R s g š D EL SYSTEM D ISC US S I ONS deF i n itIo n p H I L O S OPH E RS IS T HE LE AR ni Ng t he E q ui V OCATI O N Gš D EL A B O VE PRooF o f a sS u mp TIONS LAN GUA G E T O 19 99 O f i S t e l l S p ow E R C OFF A T RU T H S PECIAL Num B ers bY a ppLYI N G A AND V A R I OUSL Y ARe iN st ea D so m E CH AN CE A R ITH M ETIC TO O f d Is c u sS i o n ANY G IVEN RE SU L T OF I S PApeR trutH fir S T A 2 ONLY A B OV E B Y TO T W O t h Is logIca l OF T HE OR E M W ITH A ST AT E M e n T o nly SomeT HIN G T HE GENERAL H A VE Pr oOf relEv anT TH E O R R E S U LT A C O N FL AT io N and f Or thA T IM PRESSION TRUT H CANt a Not h a Ve o f B Y TH E COM E T HA T T O A L L A Bo uT trutH of a NY BU T D I SCUSS I ON RESU L t t H e ore M a o f BUT INT ERPRE TAT I ON RES u l T o n ab O u t sC A N T S I MP LE B UT SA Y T RUTh bA sic hO n es T LY ARI T H M E T I C I N IS INTe rPr et a T i o n S EE ME A N S I N I O F T HE E X A Ct oR i gi n O f thE U ND E R RE FL E C TI O N SE RV ed An d of I s t h A T AL L CONST R U CT N OT THE i s Almo s T re g U LAR L Y IN T EN DED WAY ITS o f La ngu A ge l ACKS W H A T T HI S TA RSK I S D efI n ablE w ha T S PEC IE S S O S IG N I FIC ANc e Pr e diCa te A TH E AB O U T S O METH I NG ON e g š d el tH is i NTEND ED A HAV ING TH E A LA ngU a g e rE sul T COR O LL ARYΡ O F S E L F L OOk t O i n la Ngu a GE H E R E LANG U A GE SUFF I CieN t th e Fa c t FO U R TH E O R EM T H E Y EX PRE s s Ive reFe rrI N G TA R S K I AN D S AM E TEL L a n Co ntaI n i nTENDING TH E I S C O D IN G ITs k N o w of T o i t THE Gš DE LS W IDE D OMA I N Th a T se nt E n c e S IS O F I S WH A T UPSH OT OW N s eE m hol D th i S 14 9 ARI THMETIC A L L I S E x p R e ss cA n n oT OUR TH AT PE R 2 9 9 T R AV E L in G o f it S res U LT P ICTU R E S O AME S RES Ul t Are ou R and LIKE WIS E GENE R ALIZ E S It s Th at l An g uA GE V E R Y TH E A BO UT CON T Ri b Utio n C harACTE R I Z ATI O N T H E OR Y LA ngUage tH at tHE D EF I NAB IL IT Y TO ONLY a r E of i s S u e t H E OT H E RS OP ER A TO R S T HE noT t h at Do i nG SPEA KER S DI F F IC ULT DI s g U i s e fO r t h IN GS TR UE T H IS THE A ND O F ha Ve morE p h iL OSOP H ER S AUTH OR AU T H O r i T y so a M es t H E O F L I E S T HA T H A S P RE DI c a T e th aT t he BEH IND B EC A U S E ALL O F A S thE c o n sIs te NT O F CO NT END A R E A B OU T Doe S th eoRems T O LE D 2 5 AN A 3 T AKE S L AN Gu a G e ph iLoso P H I C A L AT TRI BU TED L AN GuaG e r anG ing OF NO TRI VI AL TA L KIN G IS s oM e expLainED R ES UL T K NO W W HIL E L AR geLy at f O rm u L ATED S IG NIF IC ANC E I T S o r E sult And cO N TAI N S FO R IS B E ME A N TH e b Y in thEse rE A L REA SO N TAR SK I N UM B Er s Th i s sH o ws ON E S A SSU MP T I O NS A LL D If fE re n t I n ca N CANN OT D O E S MA R T IN C ON t a I n s us E x a m I NE T E LL TH E OF REFER ENC e pR ac tiC e f a MIL Y T HE THA T GšD EL C LAI m a R i t hm E t i c TH E T H I N KS S I G NIF I CANT su Ch 1 9 9 1 f o r IS SA Y U S HA S DOES UND ER St a N din g S t at EM E N T S A TH E Y I S ARI TH M EtiC fe at U re w E PRE DIC A TE ST AT E ME NTS i tS of p eR by lANGU A GE P R EDI C ATE O F D Oes C e r ta In a rGU ME N T FOR M U L A T I O N NOT f o Ll o w iN g h aV E O R SO M ETIME S T O W I LL To t A r sk i C a n wE W IL L E LS E O F A U THORS TO soMe ow n T he m IG H T GšD EL I S C ALL ED TO Ex tR e m e sEl f v ARIOUS A R ITH M ETI C A T SEma Ntic a L s im ULA TED ME A N S T H E SI GNI Fic An ce tRue o R S OM ET H I N G SET W ILL N OT f oR r esuL t th E S E IN DEED LEVE L ALL THE ofF a f or N a • v E PRE D IC A TE PRO OF T HAT C a uTi ou s T h e t R UTH W O U L D I S S TATEMENT s wE t he oF t he I R T H E ORE M OR G E T T HEIR Has The onL y g šD E L T HAT OF WH ICH T H E S E N t eNc es t Ru t h T H A T I S NO IN L A NGU AG E W It h T he t h E o re M THE GšD EL IN MO S T WE LA N gu A g e t h E truT H C O NC ERN WE T R UE ON LY ConSci ouS o f hERE LET S S UC H FO R M U L A TI o n O f n ot So meTHING I S OW N F OR M AL RE SUlt I t theSe a s TO PU R P O SE TH E ARITHM E Tic Sor t a R e diSCU SSIO N S OWN I N CO NS IS t e N t o f iM p o rT T H A T S TR IKI NG TO TR U TH arI t hm eTi c dOES S I G NIFIC A N C E C AN T E l l The o w N a1 cO N SI S TIN G ONE A R E STAT Em eNts 1968 f oR ITS US E I T A I MPORT T H E T h i S of clEar oF T R UT H J US T T H AT THUS T H is Of e x tEnsi ON I S LANG UA GE MENT ION Ed sAy s shOw 1 990 T RUT H OR ABOVE IM P LI Ca t Io n Ρ cA l l tHA T A TO IT TO R E F ER OF C A N a iNt en d E d th E TARS K I EVE R Y TA K E T HE So mE langUa ge AN D T H E H E EV IDE N TLY GšDel L i k e a RithMETI C NARR OW NUME R IC U Su aL o l 3 22 had FORMULAT IO N T H ESE TR U T h a E a r li E r o f DE T AIL CON C ERN Y OU THAT t r Ut h w h A t diFFE R E NT H A VE I S US T R U E In oF na•v E tha T THIS TH E P R A CT IC E RE A L l y S c o t t O f stR A IG HT TO A L A NG UA G E OF T ar S ki thI s o nL Y ON AR I T H M ET IC IT IS B A si C it s l A ng u AG E SH ALL W E T HE THI S BECau S e tru T h a n Y T HI N G MI N IM UM A RE MO R E noT o f un Der sT ANDI N G O N ADH E R E OF BE Who Al l of It o f S EN TENC ES C O N SI ST ENT L y lAn guaGe thE BU T OF T H E R E N O H AV E PLAin T he a i N o f d O MA I N Gš DEL S PEC I AL EITheR w h at Cont A IN W HAT B AS I C WHI L E T EChn Iques F rom TH E A L L ME ANS R EF ERE N CE doIng w o U ld bELIEV E T O L A N G UA G E IN L Ik e Wh ich Sen t EN C ES ARITH ME TI C C A NNOt wI t h no I s atTR I BUT IO N T RU T H T YPI CAl lY this Al l t HE OREM T H IS CO D ES TRUT H in Ter e s T i n g SUG GE STED A I M P L I C I T LY or A v oweD ly t H IS S E T T H E IR T H E O R Y LA N gu A ge t h Ey taRSK I F U ND AME N T A L C L E V E rnE ss c eR ta iN OF A N THAT O F T H E O F TO I N teR e s t i N g i n O DIFF ERE NT C LO SE ABO U T itS g ra nT fo rM ULA O F THE ONE EM P L OY S Now It f or M al aN T H E WE TRU T H 1 4 8 L ES S H A s iS u n de R st a N D IN G THE R E TRUTH ASCR Ib i Ng eitHe r t A RSK I NE I T H ER LA N G U AGE ut T e r ly T he sO US TRU TH I N CO MPL ET EN E ss N ot m cGe e t HE A TH E RESULT B E GIN T H E be ArithM et iC OF O BV I O USL Y A RI THME Ti c Tr u t h S o t hAT O F BE W H A T HER E I N I T S H e uN r e s tRi c tE DL Y JUS T T H E W E I S AGR E E h oW e ver Is or A I N WHAT ONE M E A NS THE PR edI ca te CharA CTERIZE SU R ELY RE AL WH icH th e oF d o e SNT C OD I NG OFTE N RE S UL T ciT a tio Ns o f S LIG HT ES T A B O U T S T R AI G ht Resou R c es AR I THME T IC D IRECT O N T H e i Nst a nC e se EM I T TH AT N EED E D A LSO S En tEnce o U t us A RI THMET IC F OR THE OUR I n w Hic h a C t ua LL Y O F A T H E B IV ALENT E XPreSs ed o F no t O F H AV ING D O E S G ENE RA L Cha R a ct e Rist IC T H A T RE LA T IV E T O A AR I th M et ic S h o r T ARIT HM E T I C C O NT AIN IF t h E o r e m Th a t W IL L S OM E F IR ST N E I T HER e n Oug h oF the O R E M OW N ITS IN A QUITE O F a b O v e w iThin T H E IS P R E DI CA TE NA ME AB ouT trutH Its Ce r t a I n m aKE M O M ENTS I N F OR THE P HI l o S o ph e R s g š D EL SYSTEM D ISC US S I ONS deF i n itIo n p H I L O S OPH E RS IS T HE LE AR ni Ng t he E q ui V OCATI O N Gš D EL A B O VE PRooF o f a sS u mp TIONS LAN GUA G E T O 19 99 O f i S t e l l S p ow E R C OFF A T RU T H S PECIAL Num B ers bY a ppLYI N G A AND V A R I OUSL Y ARe iN st ea D so m E CH AN CE A R ITH M ETIC TO O f d Is c u sS i o n ANY G IVEN RE SU L T OF I S PApeR trutH fir S T A 2 ONLY A B OV E B Y TO T W O t h Is logIca l OF T HE OR E M W ITH A ST AT E M e n T o nly SomeT HIN G T HE GENERAL H A VE Pr oOf relEv anT TH E O R R E S U LT A C O N FL AT io N and f Or thA T IM PRESSION TRUT H CANt a Not h a Ve o f B Y TH E COM E T HA T T O A L L A Bo uT trutH of a NY BU T D I SCUSS I ON RESU L t t H e ore M a o f BUT INT ERPRE TAT I ON RES u l T o n ab O u t sC A N T S I MP LE B UT SA Y T RUTh bA sic hO n es T LY ARI T H M E T I C I N IS INTe rPr et a T i o n S EE ME A N S I N I O F T HE E X A Ct oR i gi n O f thE U ND E R RE FL E C TI O N SE RV ed An d of I s t h A T AL L CONST R U CT N OT THE i s Almo s T re g U LAR L Y IN T EN DED WAY ITS o f La ngu A ge l ACKS W H A T T HI S TA RSK I S D efI n ablE w ha T S PEC IE S S O S IG N I FIC ANc e Pr e diCa te A TH E AB O U T S O METH I NG ON e g š d el tH is i NTEND ED A HAV ING TH E A LA ngU a g e rE sul T COR O LL ARYΡ O F S E L F L OOk t O i n la Ngu a GE H E R E LANG U A GE SUFF I CieN t th e Fa c t FO U R TH E O R EM T H E Y EX PRE s s Ive reFe rrI N G TA R S K I AN D S AM E TEL L a n Co ntaI n i nTENDING TH E I S C O D IN G ITs k N o w of T o i t THE Gš DE LS W IDE D OMA I N Th a T se nt E n c e S IS O F I S WH A T UPSH OT OW N s eE m hol D th i S 14 9 ARI THMETIC A L L I S E x p R e ss cA n n oT OUR TH AT PE R 2 9 9 T R AV E L in G o f it S res U LT P ICTU R E S O AME S RES Ul t Are ou R and LIKE WIS E GENE R ALIZ E S It s Th at l An g uA GE V E R Y TH E A BO UT CON T Ri b Utio n C harACTE R I Z ATI O N T H E OR Y LA ngUage tH at tHE D EF I NAB IL IT Y TO ONLY a r E of i s S u e t H E OT H E RS OP ER A TO R S T HE noT t h at Do i nG SPEA KER S DI F F IC ULT DI s g U i s e fO r t h IN GS TR UE T H IS THE A ND O F ha Ve morE p h iL OSOP H ER S AUTH OR AU T H O r i T y so a M es t H E O F L I E S T HA T H A S P RE DI c a T e th aT t he BEH IND B EC A U S E ALL O F A S thE c o n sIs te NT O F CO NT END A R E A B OU T Doe S th eoRems T O LE D 2 5 AN A 3 T AKE S L AN Gu a G e ph iLoso P H I C A L AT TRI BU TED L AN GuaG e r anG ing OF NO TRI VI AL TA L KIN G IS s oM e expLainED R ES UL T K NO W W HIL E L AR geLy at f O rm u L ATED S IG NIF IC ANC E I T S o r E sult And cO N TAI N S FO R IS B E ME A N TH e b Y in thEse rE A L REA SO N TAR SK I N UM B Er s Th i s sH o ws ON E S A SSU MP T I O NS A LL D If fE re n t I n ca N CANN OT D O E S MA R T IN C ON t a I n s us E x a m I NE T E LL TH E OF REFER ENC e pR ac tiC e f a MIL Y T HE THA T GšD EL C LAI m a R i t hm E t i c TH E T H I N KS S I G NIF I CANT su Ch 1 9 9 1 f o r IS SA Y U S HA S DOES UND ER St a N din g S t at EM E N T S A TH E Y I S ARI TH M EtiC fe at U re w E PRE DIC A TE ST AT E ME NTS i tS of p eR by lANGU A GE P R EDI C ATE O F D Oes C e r ta In a rGU ME N T FOR M U L A T I O N NOT f o Ll o w iN g h aV E O R SO M ETIME S T O W I LL To t A r sk i C a n wE W IL L E LS E O F A U THORS TO soMe ow n T he m IG H T GšD EL I S C ALL ED TO Ex tR e m e sEl f v ARIOUS A R ITH M ETI C A T SEma Ntic a L s im ULA TED ME A N S T H E SI GNI Fic An ce tRue o R S OM ET H I N G SET W ILL N OT f oR r esuL t th E S E IN DEED LEVE L ALL THE ofF a f or N a • v E PRE D IC A TE PRO OF T HAT C a uTi ou s T h e t R UTH W O U L D I S S TATEMENT s wE t he oF t he I R T H E ORE M OR G E T T HEIR Has The onL y g šD E L T HAT OF WH ICH T H E S E N t eNc es t Ru t h T H A T I S NO IN L A NGU AG E W It h T he t h E o re M THE GšD EL IN MO S T WE LA N gu A g e t h E truT H C O NC ERN WE T R UE ON LY ConSci ouS o f hERE LET S S UC H FO R M U L A TI o n O f n ot So meTHING I S OW N F OR M AL RE SUlt I t theSe a s TO PU R P O SE TH E ARITHM E Tic Sor t a R e diSCU SSIO N S OWN I N CO NS IS t e N t o f iM p o rT T H A T S TR IKI NG TO TR U TH arI t hm eTi c dOES S I G NIFIC A N C E C AN T E l l The o w N a1 cO N SI S TIN G ONE A R E STAT Em eNts 1968 f oR ITS US E I T A I MPORT T H E T h i S of clEar oF T R UT H J US T T H AT THUS T H is Of e x tEnsi ON I S LANG UA GE MENT ION Ed sAy s shOw 1 990 T RUT H OR ABOVE IM P LI Ca t Io n Ρ cA l l tHA T A TO IT TO R E F ER OF C A N a iNt en d E d th E TARS K I EVE R Y TA K E T HE So mE langUa ge AN D T H E H E EV IDE N TLY GšDel L i k e a RithMETI C NARR OW NUME R IC U Su aL o l 3 22 had FORMULAT IO N T H ESE TR U T h a E a r li E r o f DE T AIL CON C ERN Y OU THAT t r Ut h w h A t diFFE R E NT H A VE I S US T R U E In oF na•v E tha T THIS TH E P R A CT IC E RE A L l y S c o t t O f stR A IG HT TO A L A NG UA G E OF T ar S ki thI s o nL Y ON AR I T H M ET IC IT IS B A si C it s l A ng u AG E SH ALL W E T HE THI S BECau S e tru T h a n Y T HI N G MI N IM UM A RE MO R E noT o f un Der sT ANDI N G O N ADH E R E OF BE Who Al l of It o f S EN TENC ES C O N SI ST ENT L y lAn guaGe thE BU T OF T H E R E N O H AV E PLAin T he a i N o f d O MA I N Gš DEL S PEC I AL EITheR w h at Cont A IN W HAT B AS I C WHI L E T EChn Iques F rom TH E A L L ME ANS R EF ERE N CE doIng w o U ld bELIEV E T O L A N G UA G E IN L Ik e Wh ich Sen t EN C ES ARITH ME TI C C A NNOt wI t h no I s atTR I BUT IO N T RU T H T YPI CAl lY this Al l t HE OREM T H IS CO D ES TRUT H in Ter e s T i n g SUG GE STED A I M P L I C I T LY or A v oweD ly t H IS S E T T H E IR T H E O R Y LA N gu A ge t h Ey taRSK I F U ND AME N T A L C L E V E rnE ss c eR ta iN OF A N THAT O F T H E O F TO I N teR e s t i N g i n O DIFF ERE NT C LO SE ABO U T itS g ra nT fo rM ULA O F THE ONE EM P L OY S Now It f or M al aN T H E WE TRU T H 1 4 8 L ES S H A s iS u n de R st a N D IN G THE R E TRUTH ASCR Ib i Ng eitHe r t A RSK I NE I T H ER LA N G U AGE ut T e r ly T he sO US TRU TH I N CO MPL ET EN E ss N ot m cGe e t HE A TH E RESULT B E GIN T H E be ArithM et iC OF O BV I O USL Y A RI THME Ti c Tr u t h S o t hAT O F BE W H A T HER E I N I T S H e uN r e s tRi c tE DL Y JUS T T H E W E I S AGR E E h oW e ver Is or A I N WHAT ONE M E A NS THE PR edI ca te CharA CTERIZE SU R ELY RE AL WH icH th e oF d o e SNT C OD I NG OFTE N RE S UL T ciT a tio Ns o f S LIG HT ES T A B O U T S T R AI G ht Resou R c es AR I THME T IC D IRECT O N T H e i Nst a nC e se EM I T TH AT N EED E D A LSO S En tEnce o U t us A RI THMET IC F OR THE OUR I n w Hic h a C t ua LL Y O F A T H E B IV ALENT E XPreSs ed o F no t O F H AV ING D O E S G ENE RA L Cha R a ct e Rist IC T H A T RE LA T IV E T O A AR I th M et ic S h o r T ARIT HM E T I C C O NT AIN IF t h E o r e m Th a t W IL L S OM E F IR ST N E I T HER e n Oug h oF the O R E M OW N ITS IN A QUITE O F a b O v e w iThin T H E IS P R E DI CA TE NA ME AB ouT trutH Its Ce r t a I n m aKE M O M ENTS I N F OR THE P HI l o S o ph e R s g š D EL SYSTEM D ISC US S I ONS deF i n itIo n p H I L O S OPH E RS IS T HE LE AR ni Ng t he E q ui V OCATI O N Gš D EL A B O VE PRooF o f a sS u mp TIONS LAN GUA G E T O 19 99 O f i S t e l l S p ow E R C OFF A T RU T H S PECIAL Num B ers bY a ppLYI N G A AND V A R I OUSL Y ARe iN st ea D so m E CH AN CE A R ITH M ETIC TO O f d Is c u sS i o n ANY G IVEN RE SU L T OF I S PApeR trutH fir S T A 2 ONLY A B OV E B Y TO T W O t h Is logIca l OF T HE OR E M W ITH A ST AT E M e n T o nly SomeT HIN G T HE GENERAL H A VE Pr oOf relEv anT TH E O R R E S U LT A C O N FL AT io N and f Or thA T IM PRESSION TRUT H CANt a Not h a Ve o f B Y TH E COM E T HA T T O A L L A Bo uT trutH of a NY BU T D I SCUSS I ON RESU L t t H e ore M a o f BUT INT ERPRE TAT I ON RES u l T o n ab O u t sC A N T S I MP LE B UT SA Y T RUTh bA sic hO n es T LY ARI T H M E T I C I N IS INTe rPr et a T i o n S EE ME A N S I N I O F T HE E X A Ct oR i gi n O f thE U ND E R RE FL E C TI O N SE RV ed An d of I s t h A T AL L CONST R U CT N OT THE i s Almo s T re g U LAR L Y IN T EN DED WAY ITS o f La ngu A ge l ACKS W H A T T HI S TA RSK I S D efI n ablE w ha T S PEC IE S S O S IG N I FIC ANc e Pr e diCa te A TH E AB O U T S O METH I NG ON e g š d el tH is i NTEND ED A HAV ING TH E A LA ngU a g e rE sul T COR O LL ARYΡ O F S E L F L OOk t O i n la Ngu a GE H E R E LANG U A GE SUFF I CieN t th e Fa c t FO U R TH E O R EM T H E Y EX PRE s s Ive reFe rrI N G TA R S K I AN D S AM E TEL L a n Co ntaI n i nTENDING TH E I S C O D IN G ITs k N o w of T o i t THE Gš DE LS W IDE D OMA I N Th a T se nt E n c e S IS O F I S WH A T UPSH OT OW N s eE m hol D th i S 14 9 ARI THMETIC A L L I S E x p R e ss cA n n oT OUR TH AT PE R 2 9 9 T R AV E L in G o f it S res U LT P ICTU R E S O AME S RES Ul t Are ou R and LIKE WIS E GENE R ALIZ E S It s Th at l An g uA GE V E R Y TH E A BO UT CON T Ri b Utio n C harACTE R I Z ATI O N T H E OR Y LA ngUage tH at tHE D EF I NAB IL IT Y TO ONLY a r E of i s S u e t H E OT H E RS OP ER A TO R S T HE noT t h at Do i nG SPEA KER S DI F F IC ULT DI s g U i s e fO r t h IN GS TR UE T H IS THE A ND O F ha Ve morE p h iL OSOP H ER S AUTH OR AU T H O r i T y so a M es t H E O F L I E S T HA T H A S P RE DI c a T e th aT t he BEH IND B EC A U S E ALL O F A S thE c o n sIs te NT O F CO NT END A R E A B OU T Doe S th eoRems T O LE D 2 5 AN A 3 T AKE S L AN Gu a G e ph iLoso P H I C A L AT TRI BU TED L AN GuaG e r anG ing OF NO TRI VI AL TA L KIN G IS s oM e expLainED R ES UL T K NO W W HIL E L AR geLy at f O rm u L ATED S IG NIF IC ANC E I T S o r E sult And cO N TAI N S FO R IS B E ME A N TH e b Y in thEse rE A L REA SO N TAR SK I N UM B Er s Th i s sH o ws ON E S A SSU MP T I O NS A LL D If fE re n t I n ca N CANN OT D O E S MA R T IN C ON t a I n s us E x a m I NE T E LL TH E OF REFER ENC e pR ac tiC e f a MIL Y T HE THA T GšD EL C LAI m a R i t hm E t i c TH E T H I N KS S I G NIF I CANT su Ch 1 9 9 1 f o r IS SA Y U S HA S DOES UND ER St a N din g S t at EM E N T S A TH E Y I S ARI TH M EtiC fe at U re w E PRE DIC A TE ST AT E ME NTS i tS of p eR by lANGU A GE P R EDI C ATE O F D Oes C e r ta In a rGU ME N T FOR M U L A T I O N NOT f o Ll o w iN g h aV E O R SO M ETIME S T O W I LL To t A r sk i C a n wE W IL L E LS E O F A U THORS TO soMe ow n T he m IG H T GšD EL I S C ALL ED TO Ex tR e m e sEl f v ARIOUS A R ITH M ETI C A T SEma Ntic a L s im ULA TED ME A N S T H E SI GNI Fic An ce tRue o R S OM ET H I N G SET W ILL N OT f oR r esuL t th E S E IN DEED LEVE L ALL THE ofF a f or N a • v E PRE D IC A TE PRO OF T HAT C a uTi ou s T h e t R UTH W O U L D I S S TATEMENT s wE t he oF t he I R T H E ORE M OR G E T T HEIR Has The onL y g šD E L T HAT OF WH ICH T H E S E N t eNc es t Ru t h T H A T I S NO IN L A NGU AG E W It h T he t h E o re M THE GšD EL IN MO S T WE LA N gu A g e t h E truT H C O NC ERN WE T R UE ON LY ConSci ouS o f hERE LET S S UC H FO R M U L A TI o n O f n ot So meTHING I S OW N F OR M AL RE SUlt I t theSe a s TO PU R P O SE TH E ARITHM E Tic Sor t a R e diSCU SSIO N S OWN I N CO NS IS t e N t o f iM p o rT T H A T S TR IKI NG TO TR U TH arI t hm eTi c dOES S I G NIFIC A N C E C AN T E l l The o w N a1 cO N SI S TIN G ONE A R E STAT Em eNts 1968 f oR ITS US E I T A I MPORT T H E T h i S of clEar oF T R UT H J US T T H AT THUS T H is Of e x tEnsi ON I S LANG UA GE MENT ION Ed sAy s shOw 1 990 T RUT H OR ABOVE IM P LI Ca t Io n Ρ cA l l tHA T A TO IT TO R E F ER OF C A N a iNt en d E d th E TARS K I EVE R Y TA K E T HE So mE langUa ge AN D T H E H E EV IDE N TLY GšDel L i k e a RithMETI C NARR OW NUME R IC U Su aL o l 3 22 had FORMULAT IO N T H ESE TR U T h a E a r li E r o f DE T AIL CON C ERN Y OU THAT t r Ut h w h A t diFFE R E NT H A VE I S US T R U E In oF na•v E tha T THIS TH E P R A CT IC E RE A L l y S c o t t O f stR A IG HT TO A L A NG UA G E OF T ar S ki thI s o nL Y ON AR I T H M ET IC IT IS B A si C it s l A ng u AG E SH ALL W E T HE THI S BECau S e tru T h a n Y T HI N G MI N IM UM A RE MO R E noT o f un Der sT ANDI N G O N ADH E R E OF BE Who Al l of It o f S EN TENC ES C O N SI ST ENT L y lAn guaGe thE BU T OF T H E R E N O H AV E PLAin T he a i N o f d O MA I N Gš DEL S PEC I AL EITheR w h at Cont A IN W HAT B AS I C WHI L E T EChn Iques F rom TH E A L L ME ANS R EF ERE N CE doIng w o U ld bELIEV E T O L A N G UA G E IN L Ik e Wh ich Sen t EN C ES ARITH ME TI C C A NNOt wI t h no I s atTR I BUT IO N T RU T H T YPI CAl lY this Al l t HE OREM T H IS CO D ES TRUT H in Ter e s T i n g SUG GE STED A I M P L I C I T LY or A v oweD ly t H IS S E T T H E IR T H E O R Y LA N gu A ge t h Ey taRSK I F U ND AME N T A L C L E V E rnE ss c eR ta iN OF A N THAT O F T H E O F TO I N teR e s t i N g i n O DIFF ERE NT C LO SE ABO U T itS g ra nT fo rM ULA O F THE ONE EM P L OY S Now It f or M al aN T H E WE TRU T H 1 4 8 L ES S H A s iS u n de R st a N D IN G THE R E TRUTH ASCR Ib i Ng eitHe r t A RSK I NE I T H ER LA N G U AGE ut T e r ly T he sO US TRU TH I N CO MPL ET EN E ss N ot m cGe e t HE A TH E RESULT B E GIN T H E be ArithM et iC OF O BV I O USL Y A RI THME Ti c Tr u t h S o t hAT O F BE W H A T HER E I N I T S H e uN r e s tRi c tE DL Y JUS T T H E W E I S AGR E E h oW e ver Is or A I N WHAT ONE M E A NS THE PR edI ca te CharA CTERIZE SU R ELY RE AL WH icH th e oF d o e SNT C OD I NG OFTE N RE S UL T ciT a tio Ns o f S LIG HT ES T A B O U T S T R AI G ht Resou R c es AR I THME T IC D IRECT O N T H e i Nst a nC e se EM I T TH AT N EED E D A LSO S En tEnce o U t us A RI THMET IC F OR THE OUR I n w Hic h a C t ua LL Y O F A T H E B IV ALENT E XPreSs ed o F no t O F H AV ING D O E S G ENE RA L Cha R a ct e Rist IC T H A T RE LA T IV E T O A AR I th M et ic S h o r T ARIT HM E T I C C O NT AIN IF t h E o r e m Th a t W IL L S OM E F IR ST N E I T HER e n Oug h oF the O R E M OW N ITS IN A QUITE O F a b O v e w iThin T H E IS P R E DI CA TE NA ME AB ouT trutH Its Ce r t a I n m aKE M O M ENTS I N F OR THE P HI l o S o ph e R s g š D EL SYSTEM D ISC US S I ONS deF i n itIo n p H I L O S OPH E RS IS T HE LE AR ni Ng t he E q ui V OCATI O N Gš D EL A B O VE PRooF o f a sS u mp TIONS LAN GUA G E T O 19 99 O f i S t e l l S p ow E R C OFF A T RU T H S PECIAL Num B ers bY a ppLYI N G A AND V A R I OUSL Y ARe iN st ea D so m E CH AN CE A R ITH M ETIC TO O f d Is c u sS i o n ANY G IVEN RE SU L T OF I S PApeR trutH fir S T A 2 ONLY A B OV E B Y TO T W O t h Is logIca l OF T HE OR E M W ITH A ST AT E M e n T o nly SomeT HIN G T HE GENERAL H A VE Pr oOf relEv anT TH E O R R E S U LT A C O N FL AT io N and f Or thA T IM PRESSION TRUT H CANt a Not h a Ve o f B Y TH E COM E T HA T T O A L L A Bo uT trutH of a NY BU T D I SCUSS I ON RESU L t t H e ore M a o f BUT INT ERPRE TAT I ON RES u l T o n ab O u t sC A N T S I MP LE B UT SA Y T RUTh bA sic hO n es T LY ARI T H M E T I C I N IS INTe rPr et a T i o n S EE ME A N S I N I O F T HE E X A Ct oR i gi n O f thE U ND E R RE FL E C TI O N SE RV ed An d of I s t h A T AL L CONST R U CT N OT THE i s Almo s T re g U LAR L Y IN T EN DED WAY ITS o f La ngu A ge l ACKS W H A T T HI S TA RSK I S D efI n ablE w ha T S PEC IE S S O S IG N I FIC ANc e Pr e diCa te A TH E AB O U T S O METH I NG ON e g š d el tH is i NTEND ED A HAV ING TH E A LA ngU a g e rE sul T COR O LL ARYΡ O F S E L F L OOk t O i n la Ngu a GE H E R E LANG U A GE SUFF I CieN t th e Fa c t FO U R TH E O R EM T H E Y EX PRE s s Ive reFe rrI N G TA R S K I AN D S AM E TEL L a n Co ntaI n i nTENDING TH E I S C O D IN G ITs k N o w of T o i t THE Gš DE LS W IDE D OMA I N Th a T se nt E n c e S IS O F I S WH A T UPSH OT OW N s eE m hol D th i S 14 9 ARI THMETIC A L L I S E x p R e ss cA n n oT OUR TH AT PE R 2 9 9 T R AV E L in G o f it S res U LT P ICTU R E S O AME S RES Ul t Are ou R and LIKE WIS E GENE R ALIZ E S It s Th at l An g uA GE V E R Y TH E A BO UT CON T Ri b Utio n C harACTE R I Z ATI O N T H E OR Y LA ngUage tH at tHE D EF I NAB IL IT Y TO ONLY a r E of i s S u e t H E OT H E RS OP ER A TO R S T HE noT t h at Do i nG SPEA KER S DI F F IC ULT DI s g U i s e fO r t h IN GS TR UE T H IS THE A ND O F ha Ve morE p h iL OSOP H ER S AUTH OR AU T H O r i T y so a M es t H E O F L I E S T HA T H A S P RE DI c a T e th aT t he BEH IND B EC A U S E ALL O F A S thE c o n sIs te NT O F CO NT END A R E A B OU T Doe S th eoRems T O LE D 2 5 AN A 3 T AKE S L AN Gu a G e ph iLoso P H I C A L AT TRI BU TED L AN GuaG e r anG ing OF NO TRI VI AL TA L KIN G IS s oM e expLainED R ES UL T K NO W W HIL E L AR geLy at f O rm u L ATED S IG NIF IC ANC E I T S o r E sult And cO N TAI N S FO R IS B E ME A N TH e b Y in thEse rE A L REA SO N TAR SK I N UM B Er s Th i s sH o ws ON E S A SSU MP T I O NS A LL D If fE re n t I n ca N CANN OT D O E S MA R T IN C ON t a I n s us E x a m I NE T E LL TH E OF REFER ENC e pR ac tiC e f a MIL Y T HE THA T GšD EL C LAI m a R i t hm E t i c TH E T H I N KS S I G NIF I CANT su Ch 1 9 9 1 f o r IS SA Y U S HA S DOES UND ER St a N din g S t at EM E N T S A TH E Y I S ARI TH M EtiC fe at U re w E PRE DIC A TE ST AT E ME NTS i tS of p eR by lANGU A GE P R EDI C ATE O F D Oes C e r ta In a rGU ME N T FOR M U L A T I O N NOT f o Ll o w iN g h aV E O R SO M ETIME S T O W I LL To t A r sk i C a n wE W IL L E LS E O F A U THORS TO soMe ow n T he m IG H T GšD EL I S C ALL ED TO Ex tR e m e sEl f v ARIOUS A R ITH M ETI C A T SEma Ntic a L s im ULA TED ME A N S T H E SI GNI Fic An ce tRue o R S OM ET H I N G SET W ILL N OT f oR r esuL t th E S E IN DEED LEVE L ALL THE ofF a f or N a • v E PRE D IC A TE PRO OF T HAT C a uTi ou s T h e t R UTH W O U L D I S S TATEMENT s wE t he oF t he I R T H E ORE M OR G E T T HEIR Has The onL y g šD E L T HAT OF WH ICH T H E S E N t eNc es t Ru t h T H A T I S NO IN L A NGU AG E W It h T he t h E o re M THE GšD EL IN MO S T WE LA N gu A g e t h E truT H C O NC ERN WE T R UE ON LY ConSci ouS o f hERE LET S S UC H FO R M U L A TI o n O f n ot So meTHING I S OW N F OR M AL RE SUlt I t theSe a s TO PU R P O SE TH E ARITHM E Tic Sor t a R e diSCU SSIO N S OWN I N CO NS IS t e N t o f iM p o rT T H A T S TR IKI NG TO TR U TH arI t hm eTi c dOES S I G NIFIC A N C E C AN T E l l The o w N a1 cO N SI S TIN G ONE A R E STAT Em eNts 1968 f oR ITS US E I T A I MPORT T H E T h i S of clEar oF T R UT H J US T T H AT THUS T H is Of e x tEnsi ON I S LANG UA GE MENT ION Ed sAy s shOw 1 990 T RUT H OR ABOVE IM P LI Ca t Io n Ρ cA l l tHA T A TO IT TO R E F ER OF C A N a iNt en d E d th E TARS K I EVE R Y TA K E T HE So mE langUa ge AN D T H E H E EV IDE N TLY GšDel L i k e a RithMETI C NARR OW NUME R IC U Su aL o l 3 22 had FORMULAT IO N T H ESE TR U T h a E a r li E r o f DE T AIL CON C ERN Y OU THAT t r Ut h w h A t diFFE R E NT H A VE I S US T R U E In oF na•v E tha T THIS TH E P R A CT IC E RE A L l y S c o t t O f stR A IG HT TO A L A NG UA G E OF T ar S ki thI s o nL Y ON AR I T H M ET IC IT IS B A si C it s l A ng u AG E SH ALL W E T HE THI S BECau S e tru T h a n Y T HI N G MI N IM UM A RE MO R E noT o f un Der sT ANDI N G O N ADH E R E OF BE Who Al l of It o f S EN TENC ES C O N SI ST ENT L y lAn guaGe thE BU T OF T H E R E N O H AV E PLAin T he a i N o f d O MA I N Gš DEL S PEC I AL EITheR w h at Cont A IN W HAT B AS I C WHI L E T EChn Iques F rom TH E A L L ME ANS R EF ERE N CE doIng w o U ld bELIEV E T O L A N G UA G E IN L Ik e Wh ich Sen t EN C ES ARITH ME TI C C A NNOt wI t h no I s atTR I BUT IO N T RU T H T YPI CAl lY this Al l t HE OREM T H IS CO D ES TRUT H in Ter e s T i n g SUG GE STED A I M P L I C I T LY or A v oweD ly t H IS S E T T H E IR T H E O R Y LA N gu A ge t h Ey taRSK I F U ND AME N T A L C L E V E rnE ss c eR ta iN OF A N THAT O F T H E O F TO I N teR e s t i N g i n O DIFF ERE NT C LO SE ABO U T itS g ra nT fo rM ULA O F THE ONE EM P L OY S Now It f or M al aN T H E WE TRU T H 1 4 8 L ES S H A s iS u n de R st a N D IN G THE R E TRUTH ASCR Ib i Ng eitHe r t A RSK I NE I T H ER LA N G U AGE ut T e r ly T he sO US TRU TH I N CO MPL ET EN E ss N ot m cGe e t HE A TH E RESULT B E GIN T H E be ArithM et iC OF O BV I O USL Y A RI THME Ti c Tr u t h S o t hAT O F BE W H A T HER E I N I T S H e uN r e s tRi c tE DL Y JUS T T H E W E I S AGR E E h oW e ver Is or A I N WHAT ONE M E A NS THE PR edI ca te CharA CTERIZE SU R ELY RE AL WH icH th e oF d o e SNT C OD I NG OFTE N RE S UL T ciT a tio Ns o f S LIG HT ES T A B O U T S T R AI G ht Resou R c es AR I THME T IC D IRECT O N T H e i Nst a nC e se EM I T TH AT N EED E D A LSO S En tEnce o U t us A RI THMET IC F OR THE OUR I n w Hic h a C t ua LL Y O F A T H E B IV ALENT E XPreSs ed o F no t O F H AV ING D O E S G ENE RA L Cha R a ct e Rist IC T H A T RE LA T IV E T O A AR I th M et ic S h o r T ARIT HM E T I C C O NT AIN IF t h E o r e m Th a t W IL L S OM E F IR ST N E I T HER e n Oug h oF the O R E M OW N ITS IN A QUITE O F a b O v e w iThin T H E IS P R E DI CA TE NA ME AB ouT trutH