Bjørn Jespersen TU Ostrava Dept. Computer Science bjornjespersen@gmail.com

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Bjørn Jespersen TU Ostrava Dept. Computer Science bjornjespersen@gmail.com

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Doubleplusungooddouble privation and multiply modified artefact propertiesTutorial in two partsDeparment of Computer ScienceTechnical University of Ostrava26 February & 1 March 2013

Bjørn Jespersen

TU Ostrava

Dept. Computer Science

bjornjespersen@gmail.com

- A new logic of technical malfunction (with M. Carrara), StudiaLogica,
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

If a property F has been multiplymodified 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* /((() ()) /(() ())) (() ())

0Ms 0Fwt0a 0Mp 0Fwt0a

0Fwt0a0 0Fwt0a

A modal modifier, preliminarily speaking, is one that oscillates between being subsective and being privative.

Subsectionsays what something is; privation, what something isnot; and modal modification, what something may be.

- 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?

- 0Almost* 0Finished0Meal
- 0Almost* 0Half0Pound
- 0Former0Apparent 0Heir
- 0Former* 0Apparent0Heir

DEFINITION 1 (first- and second-order modifier).

Apropositional modifier is of type (), forming a proposition from a proposition.

A property modifieris of type (), forming a property from a property, and is thus a first-order (in TIL: first-degree) modifier.

A modifier of property modifiersis of type

(() ()), i.e. a second-order (in TIL: second-degree) modifier.

DEFINITION 2 (subsective property modifier).

Let M/(); let gsrange over (()); let x range over ; let F/; let /((() (()))): it is true or else false that a particular modifier Mis an element of a particular set of modifiers. Then:

M is subsective w.r.t. F

iff Mg [0Req0F [gs0F]].

[[0Mp [0Mp0F]]wt0a]

[[0[00F]]wt0a]

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

[0Fwt0a]

[[[0Mp* 0Mp] 0F]wt0a]

[[[00] 0F]wt0a]

[0[0[0Fwt0a]]]

[0Fwt0a]

- 0Fake0Fake 0Fwt0a,
[[[0Fake* 0Fake] 0F]wt0a] ought obviously not to translate into 0 00Fwt0a

there’s negation, and there’s negation:

- a is a non-F :property negation
- Not (a is an F) :Boolean/propositional/truth-value negation

- 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.)

[[0Mp’ [0Mp0F]]wt0a]

[[0non [0Mp0F]]wt0a]

[[0non [0non 0F]]wt0a]

?

DEFINITION 3 (privative property modifier).

Let M/(); let gprange over (()); let x range over ; let F/; let /((() (()))). Then:

M is privative w.r.t. F iff

Mgp[0Req [0non 0F] [gp0F]].

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

0Mpfwtx

0nonfwt x

DEFINITION 4 (modal property modifier).

Let M/(); let gmrange over (()); let x range over ; let F/; let /((() (())));

let /(()) and /(()). Then:

M is modal w.r.t. F iff

M gm0Reqwtx0w´0t´0Mm 0Fwt x

0Fw’t’ x0w´´0t´´0Mm 0Fwt x0non 0Fw´´t´´ xgm 0F.

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

0Mmfwt0a

w’ 0t’ 0Mmfwt0afw’t’0a

0w’’ 0t’’ 0Mmfwt0a0nonfw’’t’’0a

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

[[0Ms0F]wt0a]

[0Fwt0a]

[[0Mp0F]wt0a]

[[0non0F]wt0a]

[[0Ms’ [0Ms0F]]wt0a]

[[0Ms0F]wt0a]

(1)

[0Fwt0a]

[[0Ms’ [0Mp0F]]wt0a]

[[0Mp0F]wt0a]

(2)

0non 0Fwt0a

[[0Mp[0Ms0F]]wt0a]

[[0non [0Ms0F]]wt0a]

[[0Mp’ [0Mp0F]]wt0a]

[[0non [0Mp0F]]wt0a]/[[0Mp’ [0non 0F]]wt0a]

[[0non’ [0non 0F]]wt0a]

[[[0Ms* 0Ms] 0F]wt0a]

[[0Ms0F]wt0a]

(1)

[0Fwt0a]

[[[0Mp* 0Mp] 0F]wt0a]

[[[0non* 0Mp] 0F]wt0a]/ [[[0Mp* 0non] 0F]wt0a]

[[[0non* 0non] 0F]wt0a]

[[[0Ms* 0Mp] 0F]wt0a]

[[0Mp0F]wt0a]

(2)

[[0non0F]wt0a]

[[[0Mp* 0Ms] 0F]wt0a]

[[[0non* 0Ms] 0F]wt0a]

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 perspectiveas 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.

- The general rule of privation replaces the property constructed by 0Mp0F by the property constructed by 0non0F
- 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’

(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