SBORNÍK PRACÍ FILOZOFICKÉ FAKULTY BRNĚNSKÉ UNIVERZITY
STUDIA MINORA FACULTATIS PHILOSOPHICAE UNIVERSITATIS BRUNENSIS
B 47, 2000
Jiří Raclavský
Tichý's "Five modes of forming constructions" Pavlu Maternovi k 70. narozeninám
Tichý's (original and final) definitions of construction in Tichý
1988 are presented and commented. It is necessary to say that
these Tichý's definitions are embodied into the text of book
another way than standard explanation is given. Especially, it
is common to define first-order theory of types before exposing
constructions. The present article arose for the seminary "Gottlob
Frege's Foundation of Logic" as a presentation of respective
paragraph 15, Chapter Five: A Hierarchy of Entities (Tichý 1998).
Therefore the problem of variables (in Tichý's book paragraph
14) is mentioned rather briefly.
First, we should articulate one of several distinctions: we can
distinguish between simple and compound constructions. (Convention:
A construction containing a variable constructs one entity relative
to one valuation and another entity relative to another. A construction
constructs an entity relative to valuation v, we shall briefly
say that a construction v-constructs the entity assigned
to it by v.)
1) simple constructions (ätomic" construction - Materna
1998, p. 40; his tag is similar to l-calculi, where
x (if it is a variable) is an atomic term)
Variables are the only simple constructions. The variables xn
construct the n-th object from the given sequence of objects
yielded by the valuation, for any valuation v the variable v-constructs
what the valuation v assigns to it. A valuation is an objectual
valuation; valuations are total functions that associate each
variable with one object of the respective type; for every type a
there are denumerably many (a-)variables at our disposal.
The letters commonly used for variables (x, y,...) are conceived
to be names of variables here. Technically variables behave exactly
like letters, but notice that the approach is strictly objectual.
2) compound constructions (Tichý 1988 s. 64; "molecular" constructions)
Constructions other than variables have constituent parts, we
will discuss them after a while.
Further we will also distinguish between see, v-proper and v-improper
constructions, and complete constructions and incomplete ones
as well.
So we have to distinguish v-proper and v-improper
construction: a construction which v-constructs nothing at
all shall be called v-improper. Otherwise it is proper. There
are three constructions which can be v-improper: execution,
double execution and composition.
The latter distinction, complete and incomplete construction
(we can also find the terms closed and open constructions; in
Czech: úplné-neúplné, otevřené-uzavřené), is defined as follows:
the incomplete construction is a construction containing at least
one free variable. For example a variable (which is a construction)
is a simple case of an incomplete construction. The complete
constructions do not contain any free variable. Complete constructions
construct independently of valuations. The definition of free
or bound variables will be stated after explaining five modes
of forming constructions.
Briefly: assignings of objects (of respective types) to variables
are qualified by valuations.
Now, to define the class of constructions (which is infinite)
inductively, we must specify the modes of forming constructions,
i.e., of forming constructions from non-constructions ("mere
objects") and other constructions. In his book Tichý found useful
to state all in all five such modes.
1. Trivialization (trivializace)
Trivialization X, symbolized 0X, is a rudimentary construction.
Definition:
- Where X is any entity (any object or construction), we can
consider trivial construction whose starting point, as well as
outcome, is X itself. To realize, carry out, trivialization 0X,
we must start with X and leave it as it is. It constructs X without
any change. No matter how complex the construction X itself may
be, 0X is quite trivial. Every construction can be trivialized.
(If X is a first order object, 0X will be called a first order
trivialization. There are also higher order trivializations -
Materna 1998, p. 41.)
- Note that for no entity X and valuation v the trivialization 0X
is v-improper.
- Also note that what is v-constructed by 0X never depends
on v.
Examples: if X is a numerical construction, i.e., a construction
which v-constructs numbers, then for any v, X v-constructs
a number (if any), while 0X v-constructs X; hence if x
is a variable then 0x v-constructs x for any v.
Another examples: 03 v-constructs 3. Analogously - applied
to natural language - 0Bill Clinton constructs the individual
Bill Clinton (not the expression ,Bill Clinton', of course).
Further comments: Trivializations serve as ïmmediate" construction.
They can be seen as one-step procedures. Their counterparts in
the field of epistemology can be called ïmmediate identifications".
Thus the trivialization is more important than it possibly seems:
especially this construction enables us to distinguish between
objects and the way they are constructed (objects are not constructions).
The importance of trivialization will be obvious after introducing
a ramified hierarchy of types.
In the first phase of development of TIL (viz. Tichý 1986, "Constructions")
Tichý considered objects as trivial procedures (see Tichý 1996,
"Konstrukce", p. 120, above; or on page 133 he wrote: Every object
of type x is also x-construction; for any v it v-constructs
itself). In this article Tichý also did not define the trivialization
among constructions, the first occurrence of trivialization is
in his book (Tichý 1988).
(We can pose the question whether trivialization is a simple
construction. We cannot easily answer it. The trivialization
is useful quite independently of such answer.)
2. Execution (provedení) (cited from Tichý 1988)
For any entity X we shall also speak of the execution of
X and symbolize it as 1X. (Currently it is symbolized only X.)
Definition:
- If X is a construction, 1X is X. Construction consisting in
executing construction X is none other than X itself. It v-constructs
what is v-constructed by X.
- If, on the other hand, X is not a construction, then 1X is
the (abortive) construction whose starting point is X and which
yields nothing, i.e., a non-construction cannot be executed.
Thus if X is v-improper construction or not a construction
at all, 1X is v-improper.
Examples (x is a numerical variable):
13 is v-improper
1x v-constructs the number assigned to x by v.
3. Double execution (dvojité provedení) (cited from
Tichý 1988)
If what is constructed by X is itself a construction, one can
execute X and go on and execute the result. This two-stage construction
can be called double execution and symbolized as 2X.
Definition:
- 2X v-constructs what is v-constructed by what is v-constructed
by X.
- For any entity X the construction 2X is v-improper (i.e.,
yields, relative to v, nothing at all) if X is not itself
a construction, or if it does not v-construct a construction,
or if it v-constructs a v-improper construction.
Examples (x is a numerical variable):
a) 2x is v-improper
b) 2(0x) v-constructs the same as x, i.e., the
number assigned by v to x.
Note that 2X is not the same as 1(1X): if X is construction,
1X is the same as X, thus 1(1X) is the same as 1X which
in turn is X. Hence, if c is a variable ranging over numerical
constructions, 0c, 1c, a 2c are three distinct
constructions. 0c constructs c, quite independently
of v. 1c v-constructs numerical construction that
is assigned by v to c. 2c v-constructs whatever
number (if any) which is v-constructed by construction which v assigns
to c (note that what is v-constructed by 2c may depend
on what v assigns to variables others than c).
Remark: It is easy to see that inductively the whole class
of executions can be defined (Materna 1998, p. 39).
Another remark: The last two constructions, viz. execution
and double execution, can be found only in Tichý 1988. Other
research workers of TIL do not use these constructions - Materna
(in 1998) thinks that they are not principally necessary for
logical analysis of natural language (except special cases; execution
can be bypassed by the function from a construction to what it
constructs).
4. Composition (kompozice, složení)
Let F be a construction of a mapping and X a construction of
an argument of the mapping. F and X can be combined into a compound
construction which consists in i) executing F (remember foregoing
definition of execution), thus obtaining a mapping, then ii)
executing X, thus obtaining an argument of the mapping, and then
iii) applying the mapping to the argument, thus obtaining the
value (if any) of the former at the latter. We shall call this
compound construction the ,composition' F and X, or briefly [FX].
(Surely, the symbol ,[FX] , names the construction, not the number
constructed by it.) This kind of constructions is very similar
to the application of l-calculi, where the l-term [M
A] means application of the mapping M to the argument A. Now
to generalize the above we can put X0 to equal F and X1... Xm
to equal X (argument which can be, of course, m-tuple).
Definition:
- Let X0, X1, ... , Xm be arbitrary constructions. By
the composition [X0X1... Xm] of constructions
X0, X1,..., Xm (in this order) we shall understand the
construction consisting in: i) executing X0 to obtain an m-ary
mapping, then ii) executing X1, ... , Xm to obtain an m-tuple
of entities, and then iii) applying that mapping to the m-tuple.
- Thus for any valuation v, [X0X1... Xm] is v-improper,
if i) one of constructions X1, ... Xm is v-improper,
or if ii) X0 does not v-construct a mapping which is defined
at the m-tuple of entities v-constructed by X1, ... ,
Xm. (After introducing the type theory we can consider
the composition of type-incompatible entities also as improper.)
If X0 does construct such a mapping then [X0X1... Xm] v-constructs
the value which the mapping takes at the m-tuple.
Remember also that X0, X1 ... , Xm can
be complete or incomplete constructions.
Attention: The square brackets are not the same brackets
as the brackets used for the denoting of m-tuples (by the way,
Tichý disliked m-tuples, he construed them only as an aid for
an abbreviation of realizing m-ary functions (the functions applicable
to the m-tuples of arguments); see Cmorej&Tichý 1998).
Another remark: In Tichý 1986 Tichý used Compm (F, X1,....,
Xm) for composition, he also argued that it should be called
composition, not application, because the parts are not lost
here.
Example: let us start with 2+3, 2 and 3 are objects-numbers,
+ is the addition mapping; all objects must be trivialized, the
resulting construction consists in applying the addition mapping
to (the couple of) two and three: [0+ 02 03]. This construction
constructs number 5 similarly like 05 but it is a distinct construction,
these constructions are not identical only ,congruent,.
Another example: where × and - are the multiplication
and the subtraction mappings, [0- [0× xx] 03]
is the incomplete construction of multiplying an unspecified
number by itself and subtracting three from the result, the unspecified
number will be given by a valuation.
Let us add the definition of congruency: Two constructions will
be called v-congruent if they v-construct one
and the same object or are both v-improper. Moreover, they
will be called congruent if they are v-congruent
for any v.
5. Closure (uzávěr)
A certain incomplete construction X (with an unspecified value
of the variable included in X) can be turned into the complete
construction of some mapping. We shall call this complete construction
,a-closure of [X] on x' and symbolized it [lax
[X] ]. Thus if composition "computes" the value of some function
at certain argument, the closure "generates" a function. The
function arises as follows: we let run (every) variable through
all valuations and hence the dependence on valuation is omitted.
Note that notation ,[lax Y] , names construction,
not the mapping constructed by it. In closure we can recognize
the third kind of term of l-calculi: l-abstraction
(briefly abstraction; for a sign of l we use the name l-abstractor,
briefly abstractor). Bottom index a shows the type of
the values of the resulting mapping.
Definition:
- Let t be a (certain) collection, x1, ... , xm
distinct variables ranging over the respective collections x1,
... , xm and v a valuation. Any construction Y
can be used in constructing a mapping from x1, ... , xm into t
; we shall call this latter construction the t-closure
Y on x1,... , xm , or briefly [ltx1... xmY].
For any valuation v, [ltx1... xmY] v-constructs
the mapping which takes any X1, ... , Xm of the
respective types x1, ... , xm into that
member (if any) of t which is v(X1/x1, ... , Xm/xm)-constructed
by Y, where v(X1/x1, ... , Xm/xm) is
like v except for assigning X1 to x1, ... , and Xm to
xm.
- Consequently, for any t , Y, x1, ... , xm and v,
construction [ltx1... xmY] is v-proper.
Remark: The valuation v(X1/x1, ... , Xm/xm),
which could be called v', is such a valuation which is quite
similar to valuation v, except assigning X1, ... , Xm
to bound variables x1, ... , xm. In other words, the given
valuation v is accepted only for variables (possibly occurring
in Y) which are distinct from x1, ... , xm. Thus v does
not concern x1, ... , xm (v(xi) 1 v'(xi) but v(yi)
= v'(yi) ).
Another remark: Now when Y is v'-improper and we consequently
cannot find the value for given argument, the function constructed
by [lt x1... xmY] is undefined for this
argument (the construction [ltx1... xmY]
is, in spite of that, proper). Example of this: [0> x 00]
is improper but lx[0> x 00] constructs a function
undefined at all arguments (in every line of the table).
Further comments: The construction called closure is the
same as the functional instructions, prescriptions, (used commonly
from the beginning of the 17th century) like 2x2+3. In the
case of 2×x2+3 and y2×y2+3, they are two
distinct constructions of the same mapping. 2x2+3 can be naturally
turned into the term of typed l -calculi denoting the
respective constructions. Whereas functions as mappings are ,flat'
(Materna's term) - you cannot recognize the parts of Y, you can
,see' only a table with the m-tuples of arguments on the left
side and the values on the right side, closures are structured:
they might be construed as instructions how to create a function
- you can see every step, every partial instruction.
Remark: All the time Tichý used the term collection because
he dismissed the term set which could be possibly problematized
by an alternative approach to the theory of set.
(Remark: in Tichý 1986 the closure is signed Closx1,... xm(Y).)
Remember also carefully the difference between construction and
expression. The term [lx1x2 [0+ x1 x2]] contains
four occurrences of variables but the construction only two occurences
of variables. And, moreover: no construction contains l.
Examples (x is a numerical variable):
a) the mathematical expression f: (x1+x2) can be transcribed
as follows: [l x1x2 [0+ x1 x2]]
b) incomplete construction [0+ x1 03] can be turned into
the complete construction [lx1 [0+ x1 03] ] (commonly
written x1+3)
Convention: we can omit the outermost brackets: so [lx1x2
[0+ x1 x2] can be written lx1x2[0+ x1 x2] .
Also [lx1[lx2... [lxm Y]...]]
can be abbreviated by l x1lx2... lxm Y.
c) l x [0> x 00] constructs the function which
assigns true to every number greater than 0, i.e., the class
of positive numbers. (Remark concerning relations vs. functions
(citing Materna 1998): Relations and functions are mutually convertible,
i.e., any n-place relation can be viewed as an n-adic function,
and any n-adic function is an n- or (n+1)-relation. Well, beginning
with the notion of function enables us to define a most important
operation - the application of a function to its arguments. In
contrast to predicate logic which is based on relations, transparent
intensional logic is based on functional approach.)
d) lx1[0> x1 x2] v-constructs the function
which assigns the truth value True to every number (x1) if
this number is greater than number assigned to x2 by valuation.
So when the valuation assigns x2 number 0, the construction l
x1[0> x1 x2] v-constructs the class of positive
numbers (without respect to any valuation for x1). When the
valuation assigns to x2 number 6 then the construction lx1[0> x1 x2] v-constructs
the class of numbers greater than 6 (and again: we take into
account all valuations v').
e) Now let be both variables from the previous example bound: lx1x2 [0> x1 x2] ; this
construction constructs the function which associates every couple
of numbers to a truth value quite independently of any valuation.
This mean that l x1x2 [0> x1 x2] v-constructs
the same as 0> , viz. the relation "greater than".
f) lx1x2 [0> x1 x2] constructs the function
from couples of numbers to a truth value, the function that assigns
true when the first number is greater than the second, whereas lx1x2 [0> x2 x1]
constructs the relation < , because the relation is first
applied to the second number.
Subconstructions (podkonstrukce)
It is also possible to add the definition of subconstructions which
is not a part of Tichý's Chapter Five. (In l-calculi
the definition of subterms approximately corresponds to it.)
For the definition see Materna 1998, p. 91 (or Materna 1995,
p. 70).
Literature:
CMOREJ, PAVEL & TICHý, PAVEL (1998): KOMPLEXY [COMPLEXES]. ORGANON
F, NO. 2-3 MATERNA, PAVEL (1989): LOGICKá ANALýZA PřIROZENéHO
JAZYKA [LOGICAL ANALYSIS OF NATURAL LANGUAGE]. PRAHA: ACADEMIA
MATERNA, PAVEL (1995): SVěT POJMů A LOGIKA [THE WORLD
OF CONCEPTS AND LOGIC]. PRAHA: FILOSOFIA AV ČR MATERNA, PAVEL (1998): CONCEPTS AND OBJECTS. HELSINKI:
ACTA PHILOSOPHICA FENNICA TICHý, PAVEL (1986): CONSTRUCTIONS. PHILOSOPHY OF SCIENCE
53, PP. 514-534; (IN CZECH: (1996): KONSTRUKCE, IN: O čEM MLUVíME?
VYBRANé STATI K LOGICE A SéMANTICE, PRAHA: FILOSOFIA) TICHý, PAVEL (1988): THE FOUNDATIONS OF FREGE'S LOGIC.
BERLIN-NEW YORK: WALTER DE GRUYTER TICHý, PAVEL (1995): CONSTRUCTION AS THE SUBJECT MATTER OF
MATHEMATICS, IN: W. DEPAULI-SCHIMANOVICH, E. KőHLER, FR. STADLER,
EDS.: THE FOUNDATIONAL DEBATE (COMPLEXITY AND CONSTRUCTIVITY
IN MATHEMATICS AND PHYSICS), DODRECHT-BOSTON-LONDON: KLUWER ACADEMIC
PUBLISHERS, PP. 175-185; (IN CZECH: (1998) KONSTRUKCE JAKO PřEDMěT
MATEMATIKY, FILOSOFICKý čASOPIS 2)
Tichého "Pět způsobů formování konstrukcí"
Stať Tichého "Pět způsobů formování konstrukcí" podává,
vysvětluje a komentuje Tichého definice z [Tichý 1988]. Konstrukce
je v (Tichým vybudované) Transparentní intenzionální logice (TIL)
explikací fregovského smyslu, toho, díky čemu výrazům rozumíme.
Konstrukce konstruují intenze (resp. extenze), jsou to přitom
nemnožinové entity. Tím je zachycen fakt, že např. pro jednu
funkci existuje n funkčních předpisů (konstrukce jsou modifikací
lambda termů). Identifikací (a rozčleněním druhů) konstrukcí
se TIL zásadně odlišuje od jiných intenzionálních logik. Konstrukce
nám umožňují lépe logicky analyzovat věty, a tudíž lépe podchytit
vyplývání.
File translated from
TEX
by
TTH,
version 3.00.
On 22 Feb 2002, 20:17.