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Foundations of TIL ; method of analysis

Explore the semantic schema of TIL ontology, basic notions, composition, execution, partial functions, and quantifiers. Understand the ramified hierarchy of types and constructions.

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Foundations of TIL ; method of analysis

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  1. Foundations of TIL; method of analysis Marie Duží http://www.cs.vsb.cz/duzi/

  2. Semantic schema Expression expresses denotesmeaning(construction) constructs object TIL Ontology ofobjects: ramified hierarchy of types

  3. Basic notions • Construction • Variables x, y, p, w, t, … v-construct • Trivialization 0X refers to the object X • Composition [F A1 … An] application of a function • Closure [x1…xn X] declaration of a function • Execution 1X, Double Execution 2X • Simple theory of types (non-procedural objects) • Base: {, , , } • Functional types: (1…n) • Partial functions (1 …  n)  

  4. Basic notions • The denoted object is always a function, possibly ofzero arity, i.e. an atomic object like individual, truth-value, number • The denoted object can be: • (PWS-)intension: (), frequently(()), orforshort • Extension: a function whose domain is not  • Construction (i.e. meaning of another embedded expression) • nothing (partiality) • Typicalextensions: • -sets are modelled by theircharacteristicfunctions: () • Relations(-in-extension) are of type () • Typicalintensions: • Propertiesofindividuals/(), individual offices (roles)/, propositions/, relations-in-intensionbetweenindividuals/(), attributes/() …

  5. Basic notions, notation • propositional-logic connectives (truth-value functions): implication (), conjunction (), disjunction () and equivalence () are functions of type (), negation () is of type (). • We can use infix notation without Trivialization • For instance, instead of ‘[0 [0 p q] [0q]]’ we can write ‘[[p  q]  q]’. • For relations of -identity, =/(), we also use infix notation without Trivialization and without the subscript . • For instance, let =/() be the identity of individuals, =(())/() the identity of propositions; a, b v , P v (). Then instead of • [0 [0=a b] [0=(()) [wt [Pwt a]] [wt [Pwt b]]]] we will simply write [[a = b]  [wt [Pwt a] = wt [Pwt b]]].

  6. Basic notions, notation • Quantifiers (total functions) ,  / (()). • Letx v , B v , hencex B(x)v (), then • [0x B(x)]v-constructsT, ifx B(x)v-constructs the whole type , otherwiseF, • [0x B(x)]v-constructsT, ifx B(x)v-constructsa non-empty set of elements of type , otherwiseF. • Notation: x B(x), x B(x) • Singularizers (partial functions) I / (()). • [0Ix B(x)]v-constructsthe only -elementof the set v-constructed byx B(x), if this set is a singleton, otherwiseundefined. • Notation: x B(x)reads „the onlyx, such thatB“

  7. Example • „the only man to run 100 m under 9 s“ • Man/(),Time/(()), Run/(), I/(()): the only…, the whole expression denotes . wt [0Ix [[0Manwt x]  [0Time t [0Runwt x 0100]] < 09]] ()  ()   (()) ()  () (())  ()  ()  

  8. Ramified hierarchy of types T1(types of order 1) – non-proceduralobjects, simple theory of types Cn (constructions of order n) • Letx be a variable that ranges over a type of ordern. Thenx is aconstruction of order n over B. • LetXis an object of a type of ordern. Then0X, 1X, 2Xareconstructions of order n over B. • LetX, X1, ..., Xm(m > 0) be constructions of ordern overB. Then [X X1... Xm] is aconstruction of order n over B. • Letx1, ..., xm, X (m > 0) be constructions of ordern overB. Then [x1...xmX] is aconstruction of order n over B. • Nothing else … Tn+1 (types of order n + 1) Letnis a collection of all constructions of order n overB. Then • nand every type of order naretypes of order n + 1 over B. • If, 1,...,m (m > 0) are types of ordern + 1 overB, then (1 ... m), is atype of order n + 1 over B. • Nothing else …

  9. examples, notation: C/ v  • 0+/1  (), x /1 v  • [0+ x 01]/1 v  • x [0+ x 01]/1 v () successorfunction • [x [0+ x 01] 05] /1 v  the number 6 • [0:x 00]/1 v  nothing • x [0:x 00]/1 v () degeneratefunction • LetImproper/(1)be the set of constructions of order 1 which are v-improper for every valuationv. HenceImproperis an extensional object belonging to (1), which is a type of order 2. • Then[0Improper0[0: x 00]] /2   is an element of2, which is a type of order 3, though it v-constructs the truth-valueT, the object of a type of order 1.

  10. examples • LetArithmeticbethe set of unary arithmetic functions defined on natural numbers;henceArithmetic/ (()); and letx v , where  is the type of natural numbers. • Then theComposition [0Aritmetic [x [0+ x 01]]] • belongs to 1, the type of order 2, and it constructsT, because the Closure [x [0+ x 01]] () • constructs the unary successor function, which is an arithmetic function.

  11. Examples • Composition[0Aritmetic2c]/3 v v-constructs thetruth-valueT, if the variablec/2 v 1 v-constructsfor instance the Closure[x [0+ x 01]]. • Double Execution2cv-constructs what is v-constructed by the Closure, which is an arithmetic successor function; • The Composition [0Aritmetic2c] is an object belonging to3, which is a type of order 4; • variablec v-constructs the Closure [x [0+ x 01]] belonging to 1, hencecbelongs to 2, which is a type of order 3; • Double Execution raises the order of a construction, hence2c belongs to 3, which is a type of order 4. Thus the whole Composition [0Aritmetic2c] belongs to3, a type of order 4.

  12. Method of analysis • Assingtypesto objects that are mentioned by the expression E, i.e. to the objects denoted by subexpressions of E • Compose constructionsof objects ad 1) to construct the object denoted by E Semantically simple expressions (including idioms) are furnished with Trivialization of the denoted object as their meaning • Type checking usually by drawing a derivation tree

  13. Example:‘The Mayor of Ostrava’ • Types: Mayor_of/((())) – abbr. (): attribute;Ostrava/, Mayor_of_Ostrava/(()) – abbr.  • Synthesis: wt[0Mayor_ofwt0Ostrava] • Type checking: w t[[[0Mayor_of w] t]0Ostrava] ((()))  (())  ()   () (()) abbreviated as (individual office)

  14. “The Mayor of Ostrava is rich” • Additional type: Rich/() • Synthesis: wt [0Richwt wt[0Mayor_ofwt0Ostrava]]wt] • Type checking(derivation tree; shortened): w t[[[0Richwtwt[0Mayor_ofwt0Ostrava]]wt] ()   () (()) abbr. (proposition)

  15. Paradoxes • The US President is the husband of Melania • Hillary wanted to become the US president –––––––––––––––––––––––––––––––––––––––– • Hillary wanted to become the husband of Melania wt [0= w’t’ [0Presw’t’0USA]wtw’t’ [0Husbandw’t’0Melania]wt] wt [0Wantwt0Hillary w’t’ [0Presw’t’0USA]] • =/(); Pres(-of), Husband(-of)/(); Usa, Melania/; w’t’ [0Presw’t’0USA], w’t’ [0Husbandw’t’0Melania]  ; Want(ed-to-become)/() • Substitution of another office for the presidential one is invalid; • the first premise establishes identity of individuals rather than offices; two different offices happen to be co-occupied by the same holder • In the second premise Hillary is related to the office rather than its value (individual)

  16. Paradoxes • Oidipus seeks the murderer of his father • Oidipus is the murderer of his father ––––––––––––––––––––––––––––––––– ??? • Oidipus seeksOidipus wt [0Seekwt0Oidipus w’t’ [0Murdererw’t’[0Fatherw’t’ 0Oidipus]]] wt [0=0Oidipus w’t’ [0Murdererw’t’[0Fatherw’t’ 0Oidipus]]wt] • =/(); Murderer(-of), Father(-of)/(); Oidipus/; [0Fatherw’t’0Oidipus]  ; [0Murdererw’t’ [0Fatherw’t’0Oidipus]]  ; w’t’ [0Murdererw’t’ [0Fatherw’t’0Oidipus]]  ; Seek/() • Substitution of the individualOidipus for the office (role) of the murderer is invalid; • 1st premise: Oidipus is related to the whole office (role); he wants to know who occupies the office (plays the role of the murderer) • 2nd premise: Oidipus happens to be the holder of the office

  17. Intensional vs. extensional occurence • Confusing intensional (de dicto) and extensional (de re) occurrences  paradoxes • Extensional: the valueof the function (office) is an object of predication wt [0= w’t’ [0Presw’t’0USA]wtw’t’ [0Husbandw’t’0Melania]wt] wt [0=0Oidipus w’t’ [0Murdererw’t’[0Fatherw’t’ 0Oidipus]]wt] de re • Intensional: the whole function (office) is an object of predication wt [0Wantwt0Hillary w’t’ [0Presw’t’0USA]] wt [0Seekwt0Oidipus w’t’ [0Murdererw’t’[0Fatherw’t’ 0Oidipus]]] de dicto

  18. paradoxes • John calculates 2 + 5 • 2 + 5 = 49 –––––––––––––––––??? • John calculates49 wt [0Calculatewt0John 0[0+ 02 05]] hyperint. [0= [0+ 02 05] [0 049]] Types. Calculate/(1); John/; +/(); /(); =/(); 0[0+ 02 05]/2 1; [0+ 02 05], [0 049]/1  ; 1st premise. John is related to the very construction that he wants to execute 2nd premise. Two different constructions produce the same value (number) Substitution is invalid

  19. Summary • Confusing different levels of abstraction yields paradoxes • Hyperintensional context; constructionis an object of predication • Intensional context; produced functionis an object of predication • Extensional context; valueof the produced functionis an object of predication

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