Doubleplusungood

double privation and multiply modified artefact properties

Tutorial in two partsDeparment of Computer ScienceTechnical University of Ostrava26 February & 1 March 2013

Bjørn JespersenTU Ostrava

Dept. Computer Sciencebjornjespersen@gmail.com

relevant TIL literature

• A new logic of technical malfunction (with M. Carrara), Studia Logica, DOI 10.1007/s11225-012-9397-8, 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), 94-114

• Two kinds of procedural semantics for privative modification (with G. Primiero), LNAI 6284 (2010), 252-71

• Double privation and multiply modified properties (with M. Carrara), in submission

• Left subsectivity, in submission

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?

0M’ 0M 0F , ‘a happy bald child’F/() (); M, M’/(() ()) ()

0M* 0M 0F, ‘a very happy child’M* /((() ()) /(() ())) (() ())

subsective, privative, modal

0Ms 0Fwt 0a 0Mp

0Fwt 0a

0Fwt

0a 0 0Fwt 0a

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.

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?

double privation, 1st and 2nd order (TIL: degree): examples

• 0Almost* 0Finished 0Meal

• 0Almost* 0Half 0Pound

• 0Former 0Apparent 0Heir

• 0Former* 0Apparent 0Heir

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.

subsective modifier

DEFINITION 2 (subsective property modifier). Let M/(); let gs 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 Mg [0Req 0F [gs

0F]].

double privation as double Boolean negation

[[0Mp [0Mp 0F]]wt 0a]

[[0[0 0F]]wt 0a]

[0 [0[0Fwt

0a]]]

[0F wt 0a]

[[[0Mp* 0Mp] 0F]wt 0a]

[[[0 0] 0F]wt 0a]

[0 [0[0Fwt 0a]]]

[0Fwt 0a]

what just went wrong?

• 0Fake 0Fake 0Fwt 0a,

[[[0Fake* 0Fake] 0F]wt 0a] ought obviously not to translate into 0 00Fwt 0a

there’s negation, and there’s negation:• a is a non-F : property negation• Not (a is an F) : Boolean/propositional/truth-

value negation

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.)

non-Boolean negation

[[0Mp’ [0Mp 0F]]wt 0a]

[[0non [0Mp 0F]]wt 0a]

[[0non [0non 0F]]wt 0a]

?

privative modifier

DEFINITION 3 (privative property modifier). Let M/(); let gp range over (()); let x range over ; let F/; let

/((() (()))). Then:

M is privative w.r.t. F iff Mgp [0Req [0non 0F] [gp

0F]].

From Def. 3 we obtain the following elimination rule for privative modifiers Mp:

0Mp fwt x

0non fwt x

modal modifier

DEFINITION 4 (modal property modifier). Let M/(); let gm range over (()); let x range over ; let F/; let /((() (())));

let /(()) and /(()). Then:M is modal w.r.t. F iff

M gm 0Req wt x 0 w´ 0 t´0Mm 0Fw t x

0Fw ’t ’ x 0 w´´0 t´´0Mm 0Fw t x 0non 0Fw ´´t ´ ´ x gm

0F.

From Def. 4 we obtain the following conditional elimination rule for Mm:

0Mm fw t 0a

w’ 0 t’ 0Mm fw t 0a fw ’t ’ 0a

0 w’’ 0 t ’’ 0Mm fw t 0a 0non fw ’’ t ’ ’ 0a

Gloss: “From a being an 0Mm f at w, t, infer that there is a w´, t´ such that if a is an 0Mm 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 0Mm f at w, t then a is a 0non f at w´´, t´´.”

rule 1

[[0Ms 0F]wt 0a]

[0Fwt 0a]

rule 2

[[0Mp 0F]wt 0a]

[[0non 0F]wt 0a]

rule 3

[[0Ms’ [0Ms 0F]]wt 0a]

[[0Ms 0F]wt 0a]

(1)[0F wt 0a]

rule 4

[[0Ms’ [0Mp 0F]]wt 0a]

[[0Mp 0F]wt 0a]

(2)0non 0Fwt 0a

rule 5

[[0Mp [0Ms 0F]]wt 0a]

[[0non [0Ms 0F]]wt 0a]

rule 6

[[0Mp’ [0Mp 0F]]wt 0a]

[[0non [0Mp 0F]]wt 0a] / [[0Mp’ [0non 0F]]wt 0a]

[[0non’ [0non 0F]]wt 0a]

rule 7

[[[0Ms* 0Ms] 0F]wt 0a]

[[0Ms 0F]wt 0a]

(1)[0Fwt 0a]

rule 8

[[[0Mp* 0Mp] 0F]wt 0a]

[[[0non* 0Mp] 0F]wt 0a] / [[[0Mp* 0non] 0F]wt 0a]

[[[0non* 0non] 0F]wt 0a]

rule 9

[[[0Ms* 0Mp] 0F]wt 0a]

[[0Mp 0F]wt 0a]

(2)[[0non 0F]wt 0a]

rule 10

[[[0Mp* 0Ms] 0F]wt 0a]

[[[0non* 0Ms] 0F]wt 0a]

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.

conclusions

• The general rule of privation replaces the property constructed by 0Mp 0F by the property constructed by 0non 0F

• 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’

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