1. Foundations of TIL; method of analysis
Marie Duží

2. Semantic schema Expression expresses constructs object
denotes meaning (construction)
constructs
object
TIL Ontology of objects: 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 can be: Typical extensions:
The denoted object is always a function, possibly of zero arity, i.e. an atomic object like individual, truth-value, number
The denoted object can be:
(PWS-)intension: (), frequently (()), or  for short
Extension: a function whose domain is not 
Construction (i.e. meaning of another embedded expression)
nothing (partiality)
Typical extensions:
-sets are modelled by their characteristic functions: ()
Relations(-in-extension) are of type ()
Typical intensions:
Properties of individuals/(), individual offices (roles)/, propositions/, relations-in-intension between individuals/(), 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) ,  / (()).
Let x v , B v , hence x B(x) v (), then
[0 x B(x)] v-constructs T, if x B(x) v-constructs the whole type , otherwise F,
[0 x B(x)] v-constructs T, if x B(x) v-constructs a non-empty set of elements of type , otherwise F.
Notation: x B(x), x B(x)
Singularizers (partial functions) I / (()).
[0I x B(x)] v-constructs the only -element of the set v-constructed by x B(x), if this set is a singleton, otherwise undefined.
Notation: x B(x) reads „the only x, such that B“

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-procedural objects, simple theory of types
Cn (constructions of order n)
Let x be a variable that ranges over a type of order n. Then x is a construction of order n over B.
Let X is an object of a type of order n. Then 0X, 1X, 2X are constructions of order n over B.
Let X, X1, ..., Xm (m > 0) be constructions of order n over B. Then [X X1... Xm] is a construction of order n over B.
Let x1, ..., xm, X (m > 0) be constructions of order n over B. Then [x1...xm X] is a construction of order n over B.
Nothing else …
Tn+1 (types of order n + 1)
Let n is a collection of all constructions of order n over B. Then
n and every type of order n are types of order n + 1 over B.
If , 1,...,m (m > 0) are types of order n + 1 over B, then ( 1 ... m), is a type of order n + 1 over B.

9. examples, notation: C/ v 
0+/1  (), x /1 v 
[0+ x 01]/1 v 
x [0+ x 01]/1 v () successor function
[x [0+ x 01] 05] /1 v  the number 6
[0: x 00]/1 v  nothing
x [0: x 00]/1 v () degenerate function
Let Improper/(1) be the set of constructions of order 1 which are v-improper for every valuation v. Hence Improper is an extensional object belonging to (1), which is a type of order 2.
Then [0Improper 0[0: x 00]] /2   is an element of 2, which is a type of order 3, though it v- constructs the truth-value T, the object of a type of order 1.

10. examples
Let Arithmetic be the set of unary arithmetic functions defined on natural numbers; hence Arithmetic / (()); and let x v , where  is the type of natural numbers.
Then the Composition
[0Aritmetic [x [0+ x 01]]]  
belongs to 1, the type of order 2, and it constructs T, because the Closure
[x [0+ x 01]]  ()
constructs the unary successor function, which is an arithmetic function.

11. Examples
Composition [0Aritmetic 2c]/3 v  v-constructs the truth-value T, if the variable c/2 v 1 v-constructs for instance the Closure [x [0+ x 01]].
Double Execution 2c v-constructs what is v-constructed by the Closure, which is an arithmetic successor function;
The Composition [0Aritmetic 2c] is an object belonging to 3, which is a type of order 4;
variable c v-constructs the Closure [x [0+ x 01]] belonging to 1, hence c belongs to 2, which is a type of order 3;
Double Execution raises the order of a construction, hence 2c belongs to 3, which is a type of order 4. Thus the whole Composition [0Aritmetic 2c] belongs to 3, a type of order 4.

12. Method of analysis
Assing types to objects that are mentioned by the expression E, i.e. to the objects denoted by subexpressions of E
Compose constructions of 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_ofwt 0Ostrava]
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_ofwt 0Ostrava]]wt]
Type checking (derivation tree; shortened):
w t [[[0Richwt wt [0Mayor_ofwt 0Ostrava]]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 [0Wantwt 0Hillary 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 seeks Oidipus
wt [0Seekwt 0Oidipus 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 individual Oidipus 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 value of 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 [0Wantwt 0Hillary w’t’ [0Presw’t’ 0USA]]
wt [0Seekwt 0Oidipus w’t’ [0Murdererw’t’ [0Fatherw’t’ 0Oidipus]]]
de dicto

18. paradoxes John calculates 2 + 5 2 + 5 = 49 ––––––––––––––––– ???
––––––––––––––––– ???
John calculates 49
wt [0Calculatewt 0John 0[ ]]
hyperint.
[0= [ ] [0 049]]
Types. Calculate/(1); John/; +/(); /(); =/();
0[ ]/2  1; [ ], [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; construction is an object of predication
Intensional context; produced function is an object of predication
Extensional context; value of the produced function is an object of predication