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Doubleplusungood double privation and multiply modified artefact properties Tutorial in two parts Deparment of Computer Science Technical University of Ostrava 26 February & 1 March 2013 Bjørn Jespersen TU Ostrava Dept. Computer Science

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relevant TIL literature A new logic of technical malfunction (with M. Carrara), Studia Logica, DOI /s , forthcoming Alleged(ly) in: The Logica Yearbook 2012, V. Punčochář, P. Švarný (eds.), College Publications, London, forthcoming Alleged assassins: realist and constructivist semantics for modal modifiers (with G. Primiero), LNCS 7758 (2013), Two kinds of procedural semantics for privative modification (with G. Primiero), LNAI 6284 (2010), Double privation and multiply modified properties (with M. Carrara), in submission Left subsectivity, in submission

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the problem If a property F has been multiply modified in this or that manner, is an individual a that has the so modified property an F? 0 M’ 0 M 0 F , ‘a happy bald child’ F/( ) ( ); M, M’/(( ) ( ) ) ( ) 0 M* 0 M 0 F , ‘a very happy child’ M* /((( ) ( ) ) /(( ) ( ) )) (( ) ( ))

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subsective, privative, modal 0 M s 0 F wt 0 a 0 M p 0 F wt 0 a 0 F wt 0 a 0 0 F wt 0 a A modal modifier, preliminarily speaking, is one that oscillates between being subsective and being privative. Subsection says what something is; privation, what something is not; and modal modification, what something may be.

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two main findings + main hypothesis + open question Problem: the received rule for single privative modification is too strong when extended to multiple privation. Solution: replace propositional (Boolean) negation by property negation in order to operate on the contraries of properties. Intuitive, since something that operates on properties (a modifier) is replaced by something else that also operates on properties (property negation). Result: a pair of privative modifiers is equivalent to one modal modifier. Hypothesis: the logic of multiple privation is a logic of contraries. Open question: where does logic end and semantics begin?

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double privation, 1 st and 2 nd order (TIL: degree): examples 0 Almost* 0 Finished 0 Meal 0 Almost* 0 Half 0 Pound 0 Former 0 Apparent 0 Heir 0 Former* 0 Apparent 0 Heir

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modifiers of propositions, of properties, of other modifiers DEFINITION 1 (first- and second-order modifier). A propositional modifier is of type ( ), forming a proposition from a proposition. A property modifier is of type ( ), forming a property from a property, and is thus a first- order (in TIL: first-degree) modifier. A modifier of property modifiers is of type (( ) ( )), i.e. a second-order (in TIL: second- degree) modifier.

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subsective modifier DEFINITION 2 (subsective property modifier). Let M/( ); let g s range over (( )); let x range over ; let F/ ; let /( (( ) ( ( )))): it is true or else false that a particular modifier M is an element of a particular set of modifiers. Then: M is subsective w.r.t. F iff M g [ 0 Req 0 F [g s 0 F]].

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double privation as double Boolean negation [[ 0 M p [ 0 M p 0 F]] wt 0 a] [[ 0 [ 0 0 F]] wt 0 a] [ 0 [ 0 [ 0 F wt 0 a]]] [ 0 F wt 0 a] [[[ 0 M p * 0 M p ] 0 F] wt 0 a] [[[ 0 0 ] 0 F] wt 0 a] [ 0 [ 0 [ 0 F wt 0 a]]] [ 0 F wt 0 a]

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what just went wrong? 0 Fake 0 Fake 0 F wt 0 a , [[[ 0 Fake* 0 Fake] 0 F] wt 0 a] ought obviously not to translate into 0 0 0 F wt 0 a there’s negation, and there’s negation: a is a non-F : property negation Not (a is an F) : Boolean/propositional/truth- value negation

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property negation (informally) The sentences “It is a not-white log” and “It is not a white log” do not imply one another’s truth. For if “It is a not-white log” is true, it must be a log: but that which is not a white log need not be a log at all. (Prior Analytics I, 46, 1) From the fact that John is not dishonest we cannot conclude that John is honest, but only that he is possibly so. (La Palma Reyes et al. 1999, p. 255.)

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non-Boolean negation [[ 0 M p ’ [ 0 M p 0 F]] wt 0 a] [[ 0 non [ 0 M p 0 F]] wt 0 a] [[ 0 non [ 0 non 0 F]] wt 0 a] ?

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privative modifier DEFINITION 3 (privative property modifier). Let M/( ); let g p range over (( )); let x range over ; let F/ ; let /( (( ) ( ( )))). Then: M is privative w.r.t. F iff M g p [ 0 Req [ 0 non 0 F] [g p 0 F]]. From Def. 3 we obtain the following elimination rule for privative modifiers M p : 0 M p f wt x 0 non f wt x

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modal modifier DEFINITION 4 (modal property modifier). Let M/( ); let g m range over (( )); let x range over ; let F/ ; let /( (( ) ( ( )))); let /( ( )) and /( ( )). Then: M is modal w.r.t. F iff M g m 0 Req w t x 0 w´ 0 t´ 0 M m 0 F wt x 0 F w’t’ x 0 w´´ 0 t´´ 0 M m 0 F wt x 0 non 0 F w´´t´´ x g m 0 F . From Def. 4 we obtain the following conditional elimination rule for M m : 0 M m f wt 0 a w’ 0 t’ 0 M m f wt 0 a f w’t’ 0 a 0 w’’ 0 t’’ 0 M m f wt 0 a 0 non f w’’t’’ 0 a Gloss: “From a being an 0 M m f at w, t , infer that there is a w´, t´ such that if a is an 0 M m f at w, t then a is an f at w´, t´ and that there is an alternative w´´, t´´ such that if a is an 0 M m f at w, t then a is a 0 non f at w´´, t´´ .”

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rule 1 [[ 0 M s 0 F] wt 0 a] [ 0 F wt 0 a]

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rule 2 [[ 0 M p 0 F] wt 0 a] [[ 0 non 0 F] wt 0 a]

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rule 3 [[ 0 M s ’ [ 0 M s 0 F]] wt 0 a] [[ 0 M s 0 F] wt 0 a] (1) [ 0 F wt 0 a]

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rule 4 [[ 0 M s ’ [ 0 M p 0 F]] wt 0 a] [[ 0 M p 0 F] wt 0 a] (2) 0 non 0 F wt 0 a

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rule 5 [[ 0 M p [ 0 M s 0 F]] wt 0 a] [[ 0 non [ 0 M s 0 F]] wt 0 a]

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rule 6 [[ 0 M p ’ [ 0 M p 0 F]] wt 0 a] [[ 0 non [ 0 M p 0 F]] wt 0 a] / [[ 0 M p ’ [ 0 non 0 F]] wt 0 a] [[ 0 non’ [ 0 non 0 F]] wt 0 a]

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rule 7 [[[ 0 M s * 0 M s ] 0 F] wt 0 a] [[ 0 M s 0 F] wt 0 a] (1) [ 0 F wt 0 a]

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rule 8 [[[ 0 M p * 0 M p ] 0 F] wt 0 a] [[[ 0 non* 0 M p ] 0 F] wt 0 a] / [[[ 0 M p * 0 non] 0 F] wt 0 a] [[[ 0 non* 0 non] 0 F] wt 0 a]

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rule 9 [[[ 0 M s * 0 M p ] 0 F] wt 0 a] [[ 0 M p 0 F] wt 0 a] (2) [[ 0 non 0 F] wt 0 a]

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rule 10 [[[ 0 M p * 0 M s ] 0 F] wt 0 a] [[[ 0 non* 0 M s ] 0 F] wt 0 a]

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the logic of non (intuitive sketch) Formally, non takes a (modified or basic) property to one of its contraries, leaving it open which particular contrary. Imagine a residing in the capital of some country. When a leaves the capital, a moves to a town in the province. When a leaves that town, a has the choice between returning to the capital or going to some other town in the province. From the point of view of the first town a goes to, its complement includes both the capital and all the other towns in the province. So each new privation introduces a shift in perspective as to what the complement is. It is crucial not to confuse non, which operates on properties, with the complement function \, which operates on sets. The complement of a complement is the original set, thereby reinstalling the problem with Boolean negation.

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conclusions The general rule of privation replaces the property constructed by 0 M p 0 F by the property constructed by 0 non 0 F A pair of privative modifiers is equivalent to one modal modifier The present framework serves an extensional, set- theoretic purpose: is a in or out? Further research will be hyperintensional, semantic: ‘is an almost finished meal’ versus ‘is almost half a pound’

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exercise (1) What are the various ways of carving up the scopes of the adjective ‘doubleplusungood’? (Orwell, 1984, 1949) (2) Is any one analysis superior? doubleplusungood

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