Definitions in
Empirical Sciences

In mathematics a definition of a function symbol presupposes a proof of an existence-and-uniqueness precondition. In empirical sciences we have only uncertain hypotheses instead of doubtless theorems of mathematics. The consequences are rather strange. A change in a theory of definitions can solve the problem. However the new concept of definition can be found interesting in itself.

1. SYMBOLISM

Object language formalized in the first order predicate calculus with identity and functions. Syntactic variables:

"u",...,"z" for individual variables, "a","a₁", "a₂",.. (in examples also "b","c","d"), "f","g","h" for function symbols (also 0-ary ones i.e. for individual constants),

"P","Q","R" for predicate symbols,

"U","V","Z" for words, possibly empty,

"A",...,"E" for formulas,

"t","s" for terms,

"M","N" for sets or sequences of formulas.

2. THE FIRST CONCEPT OF DEFINITION

For every formula B " "B" stands for the formula "a_i ... "a_j B, where a_i, ..., a_j is the list (possibly empty) of all variables free in B, i <...< j. (t/y)B is the result of replacing all free occurrences of the variable y in B by occurrences of t. $!yB is the formula $y "z (B « z = y).

A formula "D is a definition with respect to M iff formulas in M are closed and 2a or 2b holds true:

2a. 1. D has the form ( Px₁... x_n « A),

2. x₁,,.., x_n are distinct variables and exactly these variables are free in A,

3. P is an n place predicate symbol which does not occur in M Č {A},

4. all extralogical symbols which occur in A occur in M.

2b. 1. D has the form (fx₁ ...x_n= x « A),

2. x₁,.., x_n, x are distinct variables and exactly they are free in A,

3. f is an n-place function symbol which does not occur in M Č {A},

4. all extralogical symbols which occur in A occur in M,

5. M Ă " $!xA.

Remark. Suppose that 2b.2 - 2b.5 hold true. Then the definitions "D as described in 2b are equivalent to

"( fx₁ ...x_n/ x) A,

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if the substitution of fx₁ ...x_n for x is admissible in A. It is also equivalent to

" $ x( fx₁ ...x_n = x Ů A).

   Let Re be the first order fragment of the formalized theory of real numbers which does not contain the predicate symbol "Ł" and the function symbol "-". Then the formulas

2c. "a "b( a Ł b « ( a < b Ú a = b)),

2d. "a "b "c( a - b = c « a = b + c)

are examples of definitions in the sense of 2a and 2b. The definition of a predicate symbol is not "theory dependent", i.e. in the list of conditions 2a.1-4 under which they are definitions there is no condition of the form M Ă C, for any C. But the definitions of a function symbols are obviously "theory dependent", they have to meet the condition 2b.5. E.g. the formula 2d is a definition of "-" with respect to Re only if this holds true:

2e. Re Ă "a "b $!c a = b + c.

In mathematics definitions are axioms with some specific properties. These properties are expressed in some well known theorems. The most important of these theorems are the following.

2f. Conservative extension theorem. If "D is a definition as described in 2a (or 2b) and the formula B does not contain P (or f respectively), then

M Č { "D} Ă B iff M Ă B.

2g. Eliminability theorem.

(a) If "D is a definition as in 2a, UPt₁ ...t_nV is a formula and A^o is an alphabetic variant of A (the result of renaming bound variables) such that the simultaneous substitution t₁, ..., t_n for x₁, ..., x_n is admissible, then Ă ( "D ® ( UPt₁ ...t_nV « U(t₁ ...t_n/ x₁ ...x_n) A^oV)).

(b) If "D is a definition as in 2b Uft₁ ...t_n = yV is a formula, A^o is an alphabetic variant of A such that the simultaneous substitution of t₁, ..., t_n, y for x₁, .., x_n, x is admissible, then M Ă ( "D ® ( Uft₁ ...t_nV « U(t₁ ...t_n y/ x₁ ...x_n x)A^o V)).

Hence in both (a) and (b), if all extralogical symbols of formula B occur in M Č{D}, it is an effective task to find a formula B^o such that all extralogical symbols of B^o occur in M (and are distinct from P and f respectively), B and B^o have the same free variables, and M Č{ "D} Ă (B « B^o).

3. THE SECOND CONCEPT OF DEFINITION

There are authors who do not find the first concept of definition convenient, e.g. (G). According to them a formula "D is a definition with respect to M iff M is a set of closed formulas and 3a or 3b or 3c holds true:

3a. is identical with 2a,

3b. is identical with 2b,

3c. 1. D has the form ( C ® ( fx₁... x_n= x « A)),

2. x₁,... x_n, x are distinct variables, these and only these variables are free in A and at most the variables x₁, ..., x_n are free in C,

3. f is an n place function symbol which does not occur in M Č {C, A},

4. all extralogical symbols which occur in A occur in M,

5. M Ă "( C ® $!xA).

A typical example of a definition with respect to Re of the type 3c, called conditional definition (definicja warunkowa) is:

3d. "( ~b = 0 ® ( a : b = c « a = b . c)).

It is true that

3e. Re Ă "a "b( ~b = 0 ® $!c a = b . c)

and so are 3c.1 - 3c.4.

Remark. "D as described in 3c is equivalent to

"( C ® ( fx₁ ...x_n/ x)A,

where the substitution is admissible in A, and to

"( C ® $x(fx₁ ...x_n= x Ů A).

The definition of the type 3c does not serve as a way of introducing symbols for non-total functions. It is a way of introducing symbols for total but not-totally-appointed functions (our term). "1:0" is a term of the language of Re enriched by 3d. 1:0 is in the standard interpretation of Re a real number but we never can appoint which one. The unique way how to eliminate the symbol ":" is given by 3d: we can derive from Re e.g. the formula ~1 = 0 and then the formula "a "c(a : 1 = c « c = 1 . c), which makes it possible to replace any formula of the form t : 1 =s by a ":" -less formula t = 1 . s. But nothing analogous can be done which should give

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us right for every term t and s to replace t : 0 = s by a ":" -less formula. It can be done only in special cases. E.g. the formula 1:0 = 1:0 is logically true and therefore it may be replaced e.g. by the ":" -less formula 1=1. The definitions of the type 3c are theory dependent too (look at 3c.5).

4. DEFINITIONS IN SCIENCE

In empirical sciences there are usually at least three sorts of individual objects to be distinguished: real numbers, time points and the proper objects of the science, e.g. celestial phenomena. Then the formalized language of the science can be a one-sorted or a many-sorted one. The definitions in a many-sorted language are defined by the same words as we have seen in sections 2 and 3 though the formulas are defined in a different way in comparison with the one-sorted language.

Example. Around the year 1760 J. Black began to use his concept of calory. According to him calory is the amount of heat required to increase the temperature of 1 kg of water from a^oC to (a+1)^oC, for every number a such that water can have the temperatures a^oC and (a+1)^oC. He believed that heat is a substance the amount of which within a body can be measured on a ratio scale. Assuming that our language is a many-sorted one and "a₁" and "a" are variables for real numbers, "b" for bodies and "c" for points of time, his explanation of the meaning of the word "calory" (more exactly of the expression "the heat of ... at time ... measured in calories") can be reproduced in the form 2b:

4a. " (the heat of b at time c measured in calories = a₁ « for every number a such that water can have both a^oC and (a+1)^oC, the amount of heat required to increase the temperature of 1 kg of water from a^oC to (a+1)^oC multiplied by a₁ equals the amount of heat of the body b at time c).

If our language is one-sorted one, his explanation of the meaning of "calory" has the form 4b:

4b. " ((a, a₁ are real numbers Ů b is a body Ů c a time point ) ® 4a).

Black believed that 4c was true:

4c. there exists exactly one a₁ such that for every a such that water can have both a^oC and (a+1)^oC, the amount of heat required to increase the temperature from 1 kg of water from a^oC to (a+1)^oC multiplied by a₁ equals the amount of heat of the body b at time c.

   He believed it without knowing it. The science Black's time did not provide sufficient evidence from which he could deduce 4c. It was doubtful whether 4c held true and therefore it was doubtful whether 4a or 4b was a definition. 4d has been found false later: the amount of heat

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needed to increase the temperature of kg water from 0^oC to 1^oC is not the same as that needed to increase it from 100^oC to 101^oC. Very important fact is that in empirical science you can almost never be sure whether an alleged definition of a function symbol is a definition indeed, because you can never be sure whether 2b.5 (or analogously 3c.5) holds true. Such alleged definitions are stipulations concerning ways of use of an expression and therefore they should be true ex definitione. At the same time two things are dependent on factual truths: whether it is a definition at all and if not whether its use in the role of a meaning postulate cannot make our future science inconsistent. We gave an example concerning calory. Many other examples of definitions concerning concepts of empirical functions and empirical singular objects based on doubtful or false factual conjectures could be given, e.g. the original the original definition of germanium.

5. THE THIRD CONCEPT OF DEFINITION

A formula " D is a definition with respect to M iff M is a set of closed formulas and 5a or 5b holds true:

5a. is the same as 2a,

5b. 1. D has the form ( $!xA ® ( fx₁...x_n = x « A)),

2. x₁, ...,x_n, x are distinct and exactly they are free in A,

3. f is an n place function symbol and it does not occur in M Č{A},

4. all extralogical symbols which occur in A occur in M.

Remark. The definition "D is equivalent e.g. to "(x!A ® fx₁...x_n=x), where x!A is the formula (A Ů "y(y/x)A). We call the formulas "D definitions but only theorems can assure us that they really satisfy the required claims. The definition of type 5a satisfies them; we can read it in any textbook. The conservative extension theorem for the definitions of type 5b has been proved in [G] (twierdzenie 19).

THEOREM 1 (the eliminability theorem for the definitions of the type 5b). Assume that "D is a definition as in 5b, Uft₁...t_n = yV is a formula, y₁,...y_h is the list of variables u such that u is free in ft₁ ...t_n and the occurrence of this term within Uft₁...t_n = yV between U and = yV lies within the scope of "u or $u. Assume also that A^o is an alphabetic variant of A such that the simultaneous substitution of t₁,..., t_n for x₁,..., x_n is admissible. If

(*) M Ă "y₁.. "y_h $!y( t₁ ...t_n y/ x₁ ...x_nx)A^o,

then

M Ă ( Uft₁...t_n=yV « U( t₁...t_ny/x₁...x_nx)A^o).

The proof is easy. Almost everything we need are rules for equivalent replacements of formulas and distribution of quantifiers. We see that the elimination of f is possible only if the condition (*) holds true. Therefore in general the elimination is an inductively definable procedure.

   Definitions of the type 5a, 5b are theory-independent: in order to know that a formula is a definition with respect to M we need not know anything about the set of consequences of M.

   Disciples of contemporary mathematical textbooks, i.e. adherents of the first concept of definition may refuse this third concept, because the definitions of the type 5b make it possible to introduce non-totally-appointed functions. On the other side the definition of type 5b is in an important sense neither weaker nor stronger than the definition of type 2b. We can use the definitions of type 5a and 5b to any purpose to which a definition of another type can serve. The following theorem can express it more distinctly.

THEOREM 2. Suppose that "D is a definition with respect to M described in 2b. Then there exists a definition "D^o with respect to M of the type 5b such that

M Ă " ( D « D^o)

and therefore for all B

M Č{ "D} Ă B iff M Č{ "D^o} Ă B.

Proof. Let D^o be the formula ( $!xA ® D). Obviously

Ă ( D ® D^o).

M Ă $!xA,

M Č{D^o} Ă ( $!xA ® ( fx₁ ...x_n = x « A)),

M Ă "( D^o ® D).

THEOREM 3. If "D is a definition described in 3c and D^o is the formula

( $!x(A Ů C) ® ( fx₁ ...x_n = x « ( A Ů C)))

(i.e. a definition with respect to M of the type 5b), then

M Ă "( D « D^o)

and consequently for all B

M Č { "D} Ă B iff M Č{ " D^o} Ă B.

Proof. The variable x is not free in C and M Ă (C ® $!xA). Therefore M Ă (C « $!x(C Ů A)). Hence M Č{ C, D^o} Ă ( fx₁ ...x_n= x « ( C Ů A)) and

M Č{ C, D^o} Ă ( fx₁ ...x_n= x « A),

M Č{ D^o} Ă D. And also

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M Č{ D, $! x(C Ů A)} Ă C,

M Č{ D, $! x(C Ů A)} Ă ( fx₁ ...x_n= x « A),

M Č{ D, $! x(C Ů A)} Ă ( fx₁ ...x_n= x « (C Ů A)),

M Č{ D} Ă D^o.

6. FACTUAL KNOWLEDGE AND LANGUAGE CONVENTIONS IN EMPIRICAL SCIENCE

   The third concept of definition seems to be the most convenient concept for empirical science.

(a) These definitions are theory-independent. Therefore there is no doubt that they are definitions. No factual knowledge can change their status as definitions, i.e. their status of a meaning postulate, a convention concerning the use of symbols.

(b) Let us transform the definitions as described in 2b and 3c in order to obtain a theory-independent definition. The results are

( " $!xA ® "(fx₁ ...x_n=x « A))

which is equivalent to

6a. "( " $!xA ® (fx₁ ...x_n=x « A)),

and

( "(C ® $!xA) ® "(C ® (fx₁ ...x_n=x « A)))

which is equivalent to:

6b. "(( "(C ® $!xA) Ů C) ® (fx₁ ...x_n=x « A)).

The definition as conceived in 5b has the form

6c. "( $!xA ® (fx₁ ...x_n=x « A)).

The formulas 6a, 6b, 6c have the form

" (B ® (fx₁ ...x_n=x « A)).

B will be labelled here as precondition. The precondition of 6c can be weaker than those of 6a and 6b. This fact is not important in mathematics. But in empirical sciences unnecessarily strong preconditions are often found later as false. Then the false instances of B cannot be detached and the definition cannot be applied. Empirical sciences need week preconditions, but at the same time strong enough to make it possible to deduce the eliminability and conservativeness theorems.

   We believe that the unique strange outcome of employing the third concept of definition is this: we can use nowhere appointed symbols. E.g. an individual constant which is semantically a variable, because it is impossible to find any constant extension of this symbol. This outcome seems to be harmless.

7. DEFINITIONS AND DESCRIPTIONS

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   If we have a suitable logical calculus with descriptions, definitions are superfluous. Assume that formulas of the form

"( $!xA ® ( iz (z/x)A = x « A)),

where z is x or a variable which is not free in A and such that the substitution of z for x is admissible in A, are axioms of our calculus. Then the definitions of the form 5b are redundant. Special instances of this axiom scheme play the same role as definitions and their instances. The scheme (restricted to the language of set theory only) can be found in [R].

References

(G) A. Grzegorczyk, Zarys logiki matematycznej, Warszawa, PWN, 1969

[R] J.B. Rosser, On the consistency of Quine's New Foundations for Mathematical Logic, JSL, vol.4, No 1, 1939

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