Teo GRÜNBERG
1. Interpretation of Arbitrary Languages (General Semantics)
General (Formal) Semantics studies the interpretation of arbitrary for-malized languages. An arbitrary language is a system of the form
where is the alphabet (or set of primitive symbols), and is t h e set of (closed) sentences of (We shall use t h e word ''sentence'' always in t h e sense of closed sentence.) The other members of (if any) are intermediary ones such as t h e set of (singular) terms and the set of well-formed formulas.
We assume t h a t consists of two disjoint subsets, viz. t h e set of constants and t h e set of variables of We shall consider here exclusively the set of constants Hence we can denote any formalized language as follows:
Expressions containing defined symbols will be considered as mere ab-breviations of expressions which contain only primitive symbols. We shall assume t h a t all expressions of in particular all sentences, consist exclusively of primitive symbols.
An interpretation of is a function I which assigns to every constant c the meaning of c. (The meaning of c, i.e. I(c), may be either t h e extension or the intension of c; but it is irrelevant for our purpose to state which of these is t h e case.)
It is assumed t h a t for each sentence of t h e form
there is a unique sentence in the metalanguage of which is uniquely and effectively determined by the sequence of meanings
I(c1), ... , I(cn)
and which is true in iff (i.e. if and only if) the sentence S is true in the object language The sentence in t h e metalanguage which corresponds uniquely to S is called the truth condition of S. The metalanguage is as-sumed to be an intuitively interpreted language.
An interpreted language is a (formalized) language with an intended interpretation I+, i.e. a system of the form
A formalized language may have besides its intended interpretation I+ an infinity of other arbitrary interpretations. We shall assume t h a t for
each formalized language there is a well determined set of (possible) in-terpretations of Of course the intended interpretation I+ must be also an
element of this set, i.e.
I+
2. C-Preserving Interpretations and C-determinacy
Consider an intuitively interpreted formalized language In such a lan-guage the set of constants consist of two disjoint sets, viz. the set L of logical constants and the set of nonlogical constants of On the other hand the set
of sentences of can be divided into the set of analytic sentences and the set of synthetic (or empirical) sentences. The analytic sentences of are those sentences of whose truth-value is determined uniquely by the intended meaning of their constituent constants, whereas the truth-value of the synthetic sentences of are not uniquely determined by the meaning of their constituent constants. Let us remark t h a t it follows from this definition t h a t analyticity is relative to a given interpretation of t h e language. Thus a sentence S which is analytic in a certain interpretation of may be synthetic in another one. Unless stated otherwise, we shall use the term 'analytic' when unqualified only in the sense of ''analytic in the intended interpretation J+" . Any analytic
Now consider t h e logically true sentences of an intuitively interpreted language £. All these sentences are obviously analytically true, and the negation of any logically true sentence is analytically false. For example
(1) the sky is blue then the sky is blue
is logically true and therefore analytically t r u e , whereas
(2) (the sky is blue the sky is blue)
is logically false and therefore analytically false.
A logically true or logically false sentence is called logically determinate, while an analytically true or analytically false sentence is an analytic one. Hence we can say t h a t all logically determinate sentences are also analytic. But t h e converse does not hold, there are analytic sentences which are not logically determinate. For example:
(3) (This surface is wholly red) (this surface is wholly green)
is analytically t r u e , though it is by no means logically true. Similarly,
(4) (This surface is wholly red) (this surface is wholly green)
is analytically false b u t not logically false.
In ordinary logic, a logically true sentence is defined as a sentence which is true in all interpretations which preserves the meaning of t h e logical constants. For instance, the sentence (1) is logically true because it remains true for all re-interpretations of its constitutive terms ' t h e sky' and 'is blue', provided t h e logical constant ' ' is assigned its usual meaning. But we can re-interpret the non-logical constants of sentence (3) in such a way t h a t (3) becomes false. Hence we see t h a t sentence (3) is not logically t r u e , though it is obviously analytic since its being true depends only of t h e meaning of its constituent constants.
We see t h a t in ordinary logic we define the concept of logical truth in terms of t h e concept of a logical constant. But logical constants are merely enumerated without being defined in any way. My aim in this paper is to give a general definition of "logical constant" in terms of the concept of analyticity. Analytic-i t y Analytic-is not defAnalytic-inable Analytic-in terms of logAnalytic-ical t r u t h . It Analytic-is rather t h e converse whAnalytic-ich is possible, i.e., we have to define logical t r u t h in terms of analytic t r u t h .
Let now be an arbitrary formalized language of the form
(5)
where is t h e set of (primitive) constants, t h e set of analytically t r u e sen-tences, t h e set of analytically false sensen-tences, and t h e set of synthetic sentences of the language The set of all (closed) sentences of is t h e n :
(6)
The three sets and are mutually disjoint.
Obviously and must satisfy respectively t h e following conditions:
(7) (8)
We assume indeed t h a t an intended interpretation I+ is assigned to t h e
formalized language and we call t h e system
(9)
an interpreted language system (or a semantical system). As mentioned already above, we suppose t h a t there is a set of (possible) interpretations of such t h a t I+ Taking in consideration this last item, we shall construe an
in-terpreted language system as a system of t h e form (10)
where is a formalized language of t h e form (5), is a set of (possible) inter-pretations of and I+ is a designated element of called t h e intended
inter-pretation of
Definition 1.I is a C-preserving interpretation of in < > = Df
, where is the set of (primitive) constants of *
Definition 2. S is C-true in < > =D f [I is a
C-preserving interpretation of in < > (S is t r u e in I ) ] , where is the set of constants of and is the set of sentences of
Definition 3. S is C-false in < > = Df [I is a
C-preserving interpretation of in < > (S is false in I ) ] .
Definition 4. S is C-determinate = Df (S is C-true) V (S is C-false).
The usual interpretations of first-order or higher-order logic preserve t h e set of logical constants. Such interpretations are called standard interpreta-tions. More precisely the truth-functional interpretations are those which preserve t h e set of truth-functional connectives, t h e quantificational interpretations are those which preserves both t h e truth-functional connectives and t h e quan-tifiers, (and sometimes t h e i d e n t i t y sign), t h e set-theoretical interpretations are those which preserve t h e truth-functional connectives, t h e quantifiers, the identity sign, and t h e elementhood sign.
Correspondingly, t h e truth-functionally valid sentences are those which are true in all truth-functional interpretations, t h e quantificationally valid sentences are those which are true in all quantificational interpretations, and t h e set-theoretically valid sentences are those which are true in all set-theoretical interpretations.
Definition 5. C is a weakly quasilogical set of constants of < > = Df
[(S is C-determinate) (S is analytic in I+) ] .
We see t h a t t h e set L of ordinary logical constants is weakly quasilogical in t h e sense of Df. 5, since any logically t r u e sentence (or logically false) is b o t h L-true (or L-false) (in the sense of Df. 2 or Df. 3) and analytic.
We state now t h e basic theorems referring to any interpreted language system < > .
Theorem 1
[(I is C1-preserving) (I is C2-preserving)].
Proof: Let I be C1-preserving and C2 C1. Since I is C1-preserving, for every
c C1 we have I(c) = I+(c). But C2 being included in C1, c C2. Hence for
every c C2 we have I(c) = I+(c). We see t h u s t h a t I is C2-preserving. (Q.E.D.).
For example t h e set of truth-functional connectives is included in t h e set of quantificational constants, and t h e latter is included in t h e set of set-theore-tical constants. Thus every set-theoreset-theore-tical interpretation is a quantificational interpretation, and every quantificational interpretation is a truth-functional interpretation, b u t not conversely.
Definition 6. Cs = Df {c : (c occurs in S)}where is t h e set of
constants and t h e set of (closed) sentences of a (formalized) language Definition 7. I1 is -equivalent to I2 in < > , or for short, by
( S i s t r u e in I1) (S is
true in I2) V(S is false in I1) (S is false in I2)]
When = {S} we shall write ' ' instead of ' '.
Obviously -equivalence is an equivalence-relation, i.e. it is reflexive,
symmetric, and transitive. Further, if I1 I2 and , then I1 I2.
We make now t h e following (intuitively well-justified) assumptions:
Assumption I
Given a formalized language with a set of (closed) sentences ,
Since we study here only languages containing closed sentences, we have t h e right to assume t h a t . On t h e other h a n d , a closed sentence cannot consist exclusively of variables. We assume t h a t no sentence is e m p t y (i.e. contains an empty set of primitive symbols). It follows t h a t every closed sen-tence must contain at least one constant. Hence
Assumption II
This assumption is to the effect t h a t the truth-value assigned by an in-terpretation I to a sentence S depends only on t h e values I(c), c Cs, so t h a t
two distinct interpretations are S-equivalent whenever t h e y assign the same values to t h e elements of Cs.
Assumption III
This assumption is to t h e effect t h a t the truth-value of any sentence can be changed by re-interpretation. Theorem 2 -preserving)] (Q.E.D.) Theorem 3 S(S is -true).
Proof: Suppose t h a t there is a sentence S such t h a t S is -true. S is true then in every -preserving interpretation, and therefore, by Th. 2, in every I It follows t h a t S is also true in I+. Now by As. III there is an interpretation I
such t h a t . Hence S is false in t h e interpretation I. Therefore
S is not true in all interpretations. Hence S is not -true. (Q.E.D.)
Theorem 4
(I+ is C-preserving) ]
Proof: For any , I+ is C-preserving. Indeed
c [c 6 C I+(c) = I+(c) ]
is valid. (Q.E.D.)
Theorem 5
(I is Cs-preserving) ]
Proof: Given any sentence S, there is by As. III an interpretation I such t h a t . Now if I were Cs-preserving, then by As. II we would
have I I+. Hence I is not Cs-preserving. (Q.E.D.)
Theorem 6
{S is C-determinate I [ (I is C-preserving) ]} Proof: Any I is -preserving since c[c 1(c) = I+(c)] is valid.
Proof: First let S be C-determinate. Since by Th. 4 the interpretation I+ is
C-preserving, S must have in any C-preserving interpretation I t h e same
truth-value as it has in I+, i.e., I I+. On the other hand let I I+f o r
every C-preserving I. Then the sentence S has the same truth-value in any C-preserving I as it has in I+ which means t h a t S is either C-true or else C-false. Hence S is C-determinate. (Q.E.D.)
Theorem 7
S [ ( S (S is Cs-determinate) ]
Proof: Let S . By Th. 6 sentence S is Cs-determinate if I 1+ for all
Cs-preserving I. Let then I be Cs-preserving. It follows at once from As. II
t h a t I I+. Hence S is indeed Cs-determinate. (Q.E.D.)
Theorem 8
(S is C-determinate) ] Proof: Let C Cs= . By Th. 6, (S is C-determinate) is equivalent to
I [ (I is C-preserving) (I 1+) ]. We must therefore derive this
latter proposition. Now by As. III there is an interpretation I such t h a t
(I I+) . Of course I may not be C-preserving. But we can define a new
in-terpretation I1 such t h a t
We shall show t h a t I1 is C-preserving and (I1 I+) . Indeed, since I1(c)
= I+(c) for every c C, I1 is C-preserving. On t h e other hand, since I1(c) = I(c)
for and C Cs = , we have I1(c) = I(c) for every c Cs. Hence
by As. II we obtain I1 I. Since we have (I I+) , it follows t h a t
The proof of Th. 8 presupposes t h e following assumption:
Assumption IV
I3(c) - I2(c) ] }
Theorem 9
[ ( S i s C - t r u e S is t r u e in I+) (S is C-false S is false in I+)]
Proof: This is an immediate consequence of Th. 6. Theorem 10
[ (S is C1-true) (C1 C2) (S is C2-true) ]
Theorem 11
[ (S is C1-false) (C1 C2) (S is C2-false) ]
[ (S is C1-determinate) (C1 C2) (S is C2-determinate)]
Proof: Let S be C1-true and C1 C2. Since S is C1-true, S is true in all C1
-pre-serving interpretation. Now since C1 C2, every C2-preserving interpretation
is also a C1-preserving interpretation. Hence S is t r u e in every C2-preserving
interpretation. It follows t h a t S is C2-true. We prove similarly t h a t if S is
C1-false and C1 C2, then S is C2-false.
Theorem 12
S [ (S is Cs-true S is true in I+) (S is Cs-false S is false in I+) ]
Proof: This followsi from Th. 6 and Th. 9. Theorem 13
[ (S is C-determinate) (S is true in I+ S is C-true)
(S is false in I+ S is C-false) ]
Proof: Let S be C-determinate, i.e., either C-true or C-false. If S is C-true it is true in I+ (by Th. 9), and if S is C-false it is false in I+ (by Th. 9). On t h e other
hand, if S is both C-determinate and t r u e in I+, t h e n S cannot be C-false.
In-deed by Th. 9 every C-false sentence is false in I+. But S is true in I+. Hence
S must be C-true. Similarly we show t h a t if S is both C-determinate and false in I+, it must be C-false.
3. Determinacy-Classes, Skeletons, and Cores
Consider an arbitrary sentence S (of a formalized language ). By Th. 7, sentence S is Cs-determinate. Hence there is at least one set of constants C
such t h a t S is C-determinate. We know further by Th. 10 t h a t S is C-deter-minate for every C including Cs. We are now interested in finding out if there
are proper subsets of Cs in which the sentence S is also determinate. For this
sake we make the following definition:
Definition 8.
(S is C-determinate) }
The family of sets of constants (S) is called t h e determinacy class of sentence S.
Theorem 14
Proof: This is an immediate consequence of Th. 7.
A corollary of Th. 14 is t h a t every sentence possesses a non-empty deter-minacy class.
Theorem 15
is C-determinate]
Proof: This follows from Df. 8 by virtue of t h e law of concretion for class abstracts.
Theorem 16
Proof: This follows from Th. 10. Theorem 17
Proof: This follows by contraposition from Th. 8 and Th. 15. Theorem 18
Proof: This follows from Th. 3.
Definition 9. (S) = Df {C : 3 C1 [C1 (S) C = C1 Cs] }
We call (S) the strict determinacy class of sentence S. Obviously (S) (S), for any sentence S.
Definition 10. a) S is strictly C-true = Df (S is C-true) (C Cs)
b) S is strictly C-false = Df (S is C-false) (C Cs)
c) S is strictly C-determinate = Df (S is C-determinate) (C Cs)
Theorem 19
S [ CS ( S ) ]
Proof: This follows from Df. 9 and Th. 14.
It follows from Th. 19 t h a t every sentence has a non-empty strict deter-minacy class.
Theorem 20
[C (S) C Cs ( S ) ]
Proof: This is equivalent to
(1) [S is C-determinate S is (C CS)-determinate]
Since C Cs C, it follows from Th. 10 t h a t
(2) [S is (C Cs)-determinate S is C-determinate]
Hence we must prove only t h a t :
(3) [S is C-determinate S is (C Cs)-:determinate]
In order to prove (3) let us assume t h a t S is C-determinate. Let then I be any (C Cs)-preserving interpretation. Define then (by help of As. IV) a new
Since I is a (C Cs)-preserving interpretation, I1 must be a C-preserving
in-terpretation. This follows from t h e fact t h a t C = (C — Cs) U (C Cs) and
t h a t C Cs — (C — Cs). We have then
It follows from As. II t h a t I1 I I+. Hence S is (C Cs)-determinate.
(Q.E.D.)
Theorem 21
[C (S) C Cs (S)]
Proof: This follows from Th. 20 and Df. 9. Theorem 22
[C1 (S) C1 C2 Cs C2 (S)]
Proof: This follows from Df. 9 and Th. 16. Theorem 23
S [ (S) ]
Proof: This follows from Df. 9 and Th. 17.
It follows from Th. 19 and Th. 22 t h a t the strict determinacy class of any sentence consists of a non-empty class of non-empty sets of constants.
Definition 1 1 . d (S) =D f < (S), >, (S) = Df < (S), >
(S) is t h e partial ordering generated by set inclusion in the determinacy class of S, and (S) is the partial ordering generated by set inclusion in the strict determinacy class of S.
Theorem 24
For every sentence S, (S) has exactly one maximal element, viz, the set of all constants of the language.
Proof: By Th. 14, Cs (S). Since Cs it follows from Th. 16 t h a t
D (S). Now a maximal element of (S) is an element of (S) which is not a proper subset of any element of (S). Since every element of (S) is
included in , it follows t h a t is a proper subset of no element of . Hence is a maximal element of is furthermore the unique maximal element
Theorem 25
For every sentence S, has exactly one maximal element, viz. The set Cs of all constants occurring in S.
Proof: By Th. 19, Cs (S).Since every other element of is a proper
subset of Cs it follows t h a t Cs is the unique maximal element of (S).
(Q.E.D.)
Theorem 26
For every sentence S, (S) is a subsystem of (S).
Proof: It suffices to prove t h a t (4)
The latter follows from Th. 20.
Theorem 27
For every sentence S, (S) and (S) have exactly t h e same minimal elements.
Proof: We can first show t h a t
(5) is a minimal element of C0 is a minimal
elem-ent of (S) ]
Indeed let C0 be a minimal element of (S).That means t h a t C0 (S)
and no proper subset of C0 is an element of (S). We must show t h a t C0 is
also a minimal element of (S). Indeed since C0 is an element of (S), it
follows from (4) t h a t C0 is also an element of (S). Let us assume now t h a t
C0 is not a minimal element of (S). Then there is a proper subset C of C0
such t h a t C (S) . Since C0 is an element of (S), we have C0 Cs
since we have C1 = C. Hence we see t h a t G (S). But this implies t h a t
C0 is not a minimal element of st (S) contrary to hypothesis. Hence C0 is a
minimal element of (S). (Q.E.D.)
of (S) since every other element of (S) is a proper subset of
Let us show now t h a t
(6) [C0 is a minimal element of (S) C0 is a minimal element
of ( S ) ]
For this purpose we assume t h a t C0 is a minimal element of (S). Then
C0 (S) and no proper subset of C0 is an element of (S). We have
(7) .
Now we must show t h a t C0 is also a minimal element of (S), i.e., t h a t C0
is an element of (S) and t h a t no proper subset of C0 is an element of (S).
By virtue of (7), C0 (S), if and only if C0 (S) and C0 Cs The first
holds by hypothesis. So we have to show only t h a t C0 Cs. Assume t h a t C0
is not included in Cs Now since C0 (S), sentence S is C0-determinate, i.e.,
for all Co-preserving interpretations I we have I I+. Define now a new
set of constants C1 = Df C0 Cs Then C1 is a proper subset of C0; we shall
show t h a t C1 (S). Indeed we have
(8)
whilst C0 (S) and C1 = C0 Cs Hence C1 (S). It follows t h a t C0 has a
proper subset which is an element of (S), so t h a t C0 is not a minimal element
of (S). But this contradicts our hypothesis. Hence C0 Cs We have
shown thus t h a t C0 (S).
It remains now to show t h a t C0 is a minimal element of (S), i.e., t h a t
no proper subset of C0 is an element of (S). We prove this last point as
follows: Let C be a proper subset of C0. Since C0 Cs, we have C Cs.
Sup-pose t h a t C (S). Then it follows from (4) t h a t C (S). But by hypo-thesis C0 is a minimal element of (S), i.e., C0 (S) and no proper subset
of C0 is an element of (S). Hence t h e proper subset C of C0 cannot be an
element of (S). Hence C is not an element of (S). It follows t h a t C0 is
indeed a minimal element of (S). (Q.E.D.)
Definition 12. C is a skeleton of S = Df C is a minimal element of (S)
Theorem 27 (a)
(C Cs) (C is a skeleton of S)]
Proof: Consider an arbitrary sentence S. Since every minimal element of (S) is a minimal element of (S), it follows t h a t if (S) has a minimal
element C
0, then C
0C
s. Furthermore it follows from Th. 18 that C
0We know that C
s(S). Hence if C
sis finite, (S) must have a non-empty
minimal element. This proves our theorem for the case of languages with
sen-tences of finite length.
We shall make the following simplificatory assumption:
Assumption V
S(C
Sis finite)
Theorem 28
Every skeleton is non-empty and every sentence has at least one skeleton.
Proof: Since a skeleton is an element of (S), it follows from Th. 18 that each
skeleton is non-empty. Now let S be any given sentence. By Th. 14, C
s(S).
Hence C
smust contain a minimal element of (S). Such a minimal element
is then a skeleton of S. (Q.E.D.)
We shall show now by means of an example that a sentence may have
more than one skeleton. Indeed consider e.g. the sentence S of the form
(9) 4 is odd 6 is even
We have then
C
s= {'4','6', , 'is odd', 'is even'}
The sentence S is obviously true since it is a conditional with false antecedent
and true consequent. Now we can easily see that
C
1= {'4', 'is odd', }
and
C
2= { , '6', 'is even'}
are both skeletons of S.
We shall use from now on 'Sk(C,S)' as a symbolic abbreviation of 'C is a
skeleton of S'
Theorem 29
Proof: First let Sk (C
1, S) and C
1C
2. If C
2were also a skeleton of S, then
contrary to hypothesis t h a t C1 is not a skeleton of S. Hence Sk(C2, S).
Se-condly let Sk(C1, S) and C2 C1. If C2 were a skeleton of S, then C2 (S).
But C2 is a proper subset of C1. Hence it follows t h a t contrary to hypothesis
C1 is not a skeleton of S. Hence Sk(C2, S). (Q.E.D.)
Theorem 30
[Sk (C1, S) Sk (C2 ,S) (C1 C2 Sk (C1 C2, S) Sk
(C1 C2, S ) ]
Proof: Let C1 and C2 be two different skeletons of S. Then none of the two
Theorem 31
{(I is C1-preserving) (I is C2-preserving)
[I is (C1 C2)-preserving]}
Proof: (I is C1-preserving) (I is C2-preserving)
I is (C1 C2)-preserving.
(Q.E.D.)
Theorem 32
[I is C-preserving I is C-preserving)].* Proof: Let be a family of subsets of Let I be C-preserving for every C Let U = C. We shall show t h a t I is then U-preserving, i.e., for every c U, we have I(c) = I+(c). Indeed let c U. Then there is a C such t h a t c C. Since I is C-preserving we see t h a t I(c) = I+(c). Conversely
* For typographical reasons and are used (instead of the ordinary and ) to denote respectively the union and the intersection of a family of sets. sets C1 and C2 is a subset of the other. Hence is a proper subset of both
C1 and C2, whereas C1 and C2 are b o t h proper subsets of . We see then
let us now take I to be U-preserving, and let C Then C Hence I is also C-preserving. (Q.E.D.)
Definition 13. S is irreducibly C-true (or irreducibly C-false, irreducibly C-deter-minate) = Df S is C-true (or C-false, or C-determinate) Sk(C,S)
Theorem 33
(S is irreducibly C-determinate S is strictly C-determinate) Proof: This follows from the fact t h a t the skeleton of any sentence S is included in Cs.
The converse of Th. 33 does not hold. For example sentence (9) is strictly Cs-determinate b u t not irreducibly Cs-determinate. On t h e other hand (9)
is b o t h strictly and irreducibly C1-determinate (as well as C2-determinate).
Definition 14. c occurs vacuously in S = Df
Definition 15. c occurs non-vacuously in S = Df (C occurs vacuously in S)
We have then
I [I is (Cs — {c})-preserving
Definition 16. Cr(S) = Df {c : c occurs non-vacuously in S}
Cr(S) is called the non-vacuous core, or for short, the core of sentence S.
Theorem 34
Proof: Let c Cr(S). Then c occurs non-vacuously in S. Assume c Cs.
Then Cs — {c} = Cs. Hence it follows from (10) t h a t
N o w i t follows from As. II t h a t if I is Cs-preserving then I I+. Hence
there is a Cs-preserving I such t h a t I I+. Therefore c Cs. (Q.E.D.)
Theorem 35
i.e., t h e intersection of t h e determinacy classes of a given sentence is equal to t h e intersection of the strict determinacy classes of this sentence as well as to t h e intersection of t h e skeletons of the same sentence.
Proof: By (4) we have . Hence (11)
B u t every element of Dst (S) is included in an element of D (S). Hence
(12)
It follows t h a t
(13)
On the other hand, a skeleton of S is a minimal element of . Hence we have first (14) Therefore (15) Hence (16) It follows at once t h a t (17) (Q.E.D.) Theorem 36
i.e., t h e core of any sentence is equal to t h e intersection of all the skeletons of this sentence.
Proof: We shall show first t h a t
(18)
For this purpose we shall prove t h a t
(19)
let C (S) and suppose t h a t Cr(S) is not included in C. Then there is a constant c such t h a t c Cr(S) and c C. Now since C (S) we have C is C-determinate and C Cs. On the other hand it follows from c Cr(S) t h a t
there is a (Cs — {c} )-preserving interpretation I such t h a t (I I+) .
Such an interpretation I is also C-preserving, since C Cs — {c} (due to
c C). But S being C-determinate, it follows then t h a t I I+. We have
thus derived a contradiction. Hence Cr(S) is included in C. We have t h u s proved (19).
It follows from (19) t h a t Cr(S) is included also in the intersection of strict determinacy classes of S. But by Th. 35 this intersection is identical to t h e intersection of all skeletons of S. Thus we have proved also (18).
Conversely we shall prove t h e following:
(20)
For this purpose we shall show t h a t
(21) [c occurs vacuously in S Cs — {c} (S)]
Indeed let c occur vacuously in S. Then for every (Cs — {c})-preserving
inter-pretation I we have I I+. But t h a t means t h a t S is strictly ( Cs—
{c})-determinate (since Cs — {c} Cs). Hence we have Cs — {c} (S). This
proves (21).
We can now prove (20). Indeed by Th. 35, we can transform (20) into t h e equivalent
The latter means t h a t if any constant c belongs to every C (S), then c occurs non-vacuously in S. By contraposition this is equivalent to the fol-lowing statement: If any c occurs vacuously in S, then there is at least one C (S) such t h a t c C. Now let c be any constant of our language. Either c occurs vacuously in S, or else c does not occur vacuously in S. In the second case our statement is trivially t r u e . So let us take c to occur vacuously in S. Consider then t h e set Cs — {c}. It follows then from (21) t h a t Cs — {c}
(S). But obviously c Cs — {c}. (Q.E.D.)
Theorem 37
1 . A sentence S has a unique skeleton if and only if Cr(S) is a skeleton of S.
2 . A sentence S has a unique skeleton if and only if every skeleton element of S occurs non-vacuously in S.
3 . A sentence S has more t h a n one skeleton if and only if there is a skeleton element of S which occurs vacuously in S.
Proof: Let S have a unique skeleton C. Then by Th. 36 we have Cr(S) = C. Conversely let Cr(S) be a skeleton of S. Suppose t h a t S has a second skeleton C. Then it follows from Th. 36 t h a t Cr(S) C Cr(S). Hence Cr(S) C. But this implies Cr(S) = C. Hence S has exactly one skeleton. This proves 1.
Let S have again a unique skeleton C. Then by 1 we have C = Cr(S). Hence every skeleton element of S, i.e., every element of C occurs non-vacuously in S. Conversely let C be any skeleton of S and c any element of C. Let further c occur non-vacuously in S. Then Cr(S) = C. But
Cr(S) = C. Hence we have It follows t h a t S
has a unique skeleton. This proves 2.
We prove 3 merely by negating t h e two sides of t h e biconditional which expresses 2. (Q.E.D.)
From now on we shall denote by ' C " t h e complement of the set of constants C with respect to the set of all constants i.e.,
Theorem 38
[(c occurs vacuously in S)
I(I is{c}'-preserving I I+) ]
Proof: The theorem is equivalent to
(24) [S is (Cs — {c})-determinate) S is {c}-determinate]
Now we have Cs — {c} = Cs {c}'. Hence (24) holds by virtue of Th. 20.
(Q.E.D.)
As a consequence of Th. 38 we have
(25) [(c occurs vacuously in S) S is {c}'-determinate]
Taking into consideration (25) we shall generalize t h e concept of vacuous occurrence as follows:
Definition 17. The elements of t h e set of constants C occur co-vacuously in sentence S, or for short Cv(C,S) = Df
I [I is a (Cs — C)-preserving interpretation (I I+) ]
Theorem 39
[Cv(C,S) S is C'-determinate]
Proof: This follows from the relevant definitions and Th. 20. Theorem 40
Proof: This follows at once from Th. 39. Theorem 41
[c occurs vacuously in S Cv ({c}, S) ] Proof: This follows from the definitions.
It follows from Th. 40 t h a t the skeletons on any sentence S are the respec-tive complements of the maximal elements of t h e partial ordering
(26) < {C : Cv(C,S) }, >
4. C-Consequence
Definition 18. The class of sentences is true in interpretation I = Df S(S S is t r u e in I)
Definition 19. The class of sentences is C-true = Df
Definition 20. Interpretation I is a C-preserving model of the class of sentences = Df (I is C-preserving) ( is t r u e in I)
Definition 2 1 . Sentence S is a C-consequence of t h e class of sentences , or for short = Df
Remark: We shall write as short for
Theorem 42
Proof: Let K be C1-true and C1 C2. Then if I is a C2-preserving
interpreta-tion it is also C1-preserving. Therefore K is t r u e in I. Hence K is C2-true.
(Q.E.D.)
Theorem 43
[(I is a C1-preserving model of ) (C2 C1)
(I is a C2-preserving model of )]
Proof: Let J be a C1-preserving model of K and C2 C1 Then I is a C1
-preser-ving interpretation. Therefore I is also C2-preserving. Now is t r u e in I.
Hence I is a C2-preserving model of (Q.E.D.)
Theorem 44
Proof: Let and C1 C2. Then if I is a C2-preserving model of ,
it is also by Th. 43 a C1-preserving model of Hence S is true in Hence
(Q.E.D.)
Theorem 45
[( is C-true) (S is C-true)]
Proof; Let be C-true and Then if I is a C-preserving interpreta-t a interpreta-t i o n is interpreta-t r u e in I. Hence J is a C-preserving model of Therefore S is true in I. It follows t h a t S is C-true. (Q.E.D.)
Theorem 46
is (C1 C2) -true]
Proof: Let be C1-true and Then by Th. 42 we have is (C1
S is (C1 C2)-true. (Q.E.D.)
Teorem 47
Proof: Let be t r u e in I+ and Then S is t r u e in every C-preserving
model of But is t r u e in I+ and I+ is C-preserving. Hence I+ is a
C-preser-ving model of Therefore S is true in I+. (Q.E.D.)
Theorem 48
Proof: means t h a t S is t r u e in every C-preserving model of Now I is a C-preserving model of if and only if I is C-preserving and is true in I. But is true in every interpretation, since is true in I means t h a t
which holds trivially. Hence means t h a t S is true in every C-preser-ving interpretation, i.e., S is C-true. (Q.E.D.)
Theorem 49
(I is a -preserving model of is t r u e in I)
Proof: J is a -preserving interpretation of means t h a t J is a -preserving interpretation and is true in I. But every interpretation I is -preserving. Hence "I is a -preserving model of " is equivalent to " is true in I " . (Q.E.D.)
Assumption VI
( is true in I) ( is false in I)]
Theorem 50
Proof: Let S . By Th. 49 a -preserving model of is any I such t h a t is true in I. Hence let I be any -preserving model of Then S is true in I (since S ). It follows t h a t Conversely let Then S is true in every I in which is true. So let I be an interpretation in which is true and suppose t h a t S . Then by As. VI there is an interpretation I such t h a t is true in I whilst S is false in I. Hence S (Q.E.D.)
Remark : -consequence is (by Th. 44) the strongest consequence relation. entails , for any
Theorem 51
Proof: This follows from Th. 49. Theorem 52
Proof: This follows from Th. 50 and Th. 44. Theorem 53
Proof: is -true means t h a t
Now by Th. 18 no sentence is -true. Hence if is -true, t h e n Conversely if we have (Q.E.D.)
Remark: The e m p t y set (of sentences) is b o t h C-true and C-false, for any set of constants C.
Theorem 54
Proof: By Th. 48, is equivalent to S is -true. But by Th. 18 no sentence is -true. (Q.E.D.)
Definition 22
Theorem 55
(I is C-preserving model of I is a C-preserving model of )]
Proof: Let . Then every element of is true in every C-preser-ving model of . Hence every C-preserC-preser-ving model of the first set is C-pre-serving model of the second set. Conversely if every C-preC-pre-serving model of the first set is C-preserving model of t h e second set, then every element of t h e second set is true in every C-preserving model of the first set,hence
(Q.E.D.)
Theorem 56
Proof: Let S , and . Then S , so t h a t S is true in every C-pre-serving model of . Hence . (Q.E.D.)
Theorem 57
(reflexivity of )
Proof: This results immediately from Th. 56. Theorem 58
Proof: Let and . Then if I is a C-preserving model of t h e first set, by Th. 58, I is also a C-preserving model of the third set. But then 1 is also a C-preserving model of t h e third set. Hence the third set is a C-con-sequence of the first set. (Q.E.D.)
Definition 23 Cnc ( ) = Df {S : S}
Theorem 59
Proof: This follows from Df. 22 and Df. 23. Theorem 60
Proof: This follows from Th. 59 and Th. 57. Indeed by Th. 59 our assertion becomes which holds by virtue of Th. 57. (Q.E.D.)
Theorem 61
Proof: By Th. 59 our assertion is transformed into Cnc [ C nc( ) ]
Now let K1= Cnc ( ) a n d = Cnc ( ). We have then K1 C nc( ) and
Cnc (K1). The latters are transformed by Th. 59 respectively into
Then we obtain, by Th. 58,
But = Cnc [Cnc ( )] . Hence we get
(Q.E.D.)
It follows from Th. 60 and Th. 61 t h a t for every C the operation C nc
Theorem 62
Proof: means t h a t
I [(I is -preserving) ( is true in I) (S is true in I) ]
B u t I+ is the unique C-preserving interpretation. Hence our statement is equi-valent to
( is t r u e in I+) (S is true in I+)
(Q.E.D.)
Definition 24.
Now I1 is C-preserving. Furthermore is true in I1 (by virtue of As. I I ) . Hence,
since , it follows t h a t S is true in L. B u t t h e n (by As. II) S is true also in I. (Q.E.D.)
Definition 2 5 . dp(I) = D f{ c : I(c) = I+(c)}
dp(I) is t h e domain of preservation of interpretation I. We have obviously:
and
Theorem 63
Proof: Let C1 = {S}. Since C C1 C,
is holding. So we must show only
Let . Let furthermore I be a (C C1 )-preserving interpretation such
[(I is C-preserving) C dp(I)] Definition 26. I is maximally C-preserving = Df dp(I) = C
We see t h a t I+ is the unique -preserving interpretation, and t h a t if I is
both maximally C-preserving and C2-preserving, then C2 C1.
5. Quasilogical Sets and Logical Constants Definition 27. is analytic in I+ = Df
S(S S is analytic in I+)
It follows from Df. 27 t h a t t h e empty set is analytic in I+.
Definition 28. C is a quasilogical set of constants = Df
is analytic in I+) ( is t r u e in I+)
(S is analytic in I+) ]
Theorem 64
C (C is quasilogical C is weakly quasilogical)
Proof: Let C be quasilogical in the sense of Df. 29. We shall show t h a t C is also weakly quasilogical in the sense of Df. 5. Indeed let S be C-true. Then by Th. 48 we have S. But is b o t h analytic in I+ and true in I+. Hence
(since C is quasilogical) S is also analytic in I+. It follows t h a t any S which is
C-true is analytic in I+, i.e., t h a t C is weakly quasilogical. (Q.E.D.)
Definition 29. = Df {C : C is a quasilogical set}
is the class of quasilogical sets (of a given interpreted language system).
Definition 30.
is the partial ordering generated by set inclusion in the class of quasilogical sets (of a given interpreted language system).
Theorem 65
Proof: By Th. 50 our assertion is equivalent to
[( is analytic in I+) (S ) (S is analytic in I+)]
It follows from Th. 65 t h a t the partial ordering has exactly one minimal element, viz. the empty set
Theorem 66
Proof: Let C1 and C2 C1 Let further be analytic in I+ and true in
I+, and . Then by Th. 46 we have also . But C1 . Hence
S is analytic in I+. Therefore C2 . (Q.E.D.)
Theorem 67
(S is true in I+ S is analytic in I+)
i.e., the set of all constants of a language is quasilogical if and only if all true sentences are analytic.
Proof: means t h a t
VS [( is analytic in I+) ( is true in I+) ( ) (S is analytic
in I+)]
By Th. 62 the latter statement is equivalent to
[( is analytic in I+) ( is true in I+) ( is true in I+ (S is
true in I+) (S is analytic in I+)]
But this is equivalent to
[( is analytic in I+) ( is true in I+) (S is true in I+) (S is
analytic in I+)]
It follows t h a t if every true sentence is analytic, then . Conversely we shall show t h a t if then every true sentence is analytic. Indeed let , but suppose t h a t there is a sentence S1 which is true in I+ but not
analytic in I+. Applying universal instantiation with and S = S1 to the
definiens of we get ( is analytic in I+) ( is true in I+)
(S1 is analytic in I+)
This is equivalent to
(S1 is analytic in I+)
(S1 is -true) (S1 is analytic in I+) and by Th. 20 t o (S1 is Cs1-true) (S1 is analytic in I+) and by Th. 20 to (S1 is Cs1 -true) (S1 is analytic in I+) This in t u r n reduces, by Th. 11, to (S1 is true in I+) (S1 is analytic in I+)
Now by hypothesis S1 is true in I+. Hence, contrary to hypothesis, S1 is analytic
in I+. It follows t h a t every sentence which is true in I+ must be analytic in I+.
(Q.E.D.)
Definition 3 1 . The set of constants C is compact = Df
Definition 32. C is strongly quasilogical = Df
(C is quasilogical) (C is compact)
Definition 33. = Df {C : C is strongly quasilogical}
Definition 34. = Df < >
Theorem 68
Proof: By Th. 65 the empty set is quasilogical. We shall show t h a t it is also compact. Indeed by Th. 50, is equivalent to S . Let us take
t h e n = {S}. We have obviously
Hence is finite, and .Therefore is compact. Hence is strongly quasilogical. (Q.E.D.)
Indeed let C1 S and C2 C1. Then C1 is quasilogical and therefore, by Th.
66, C2 i s quasilogical. L e t . Then b y Th. 4 4 w e have . But C1
is compact. Hence there is a finite such t h a t . But does not follow.
Compactness is hereditary neither for subsets nor for supersets, i.e. both (C1 is compact) (C2 C1) (C2 is compact) (C1 is compact) (C1 C2) (C2 is compact) seem to be invalid. Theorem 69 is compact Proof: Let . By Th. 62 we h a v e : (i) ( is true in I+) (S is true in I+)
We shall show t h a t there is a finite such t h a t (ii) ( is true in I+) (S is true in I+)
Indeed is either finite or infinite. If is finite we can choose simply = . So consider the case t h a t is infinite. Now S is either true in I+ or false
in I+. If S is true in I+ we can choose simply = {S1} for any arbitrary
S1 . For then b o t h (i) and (ii) will hold in this case. Consider then the case
t h a t S is false in I+. The set of sentences is itself either true in I+ or not.
then both (i) and (ii) do not hold. If is not true in I+ then we can choose
= {S1} where S1 is an element of which is false in I+. Indeed in this last
case both (i) and (ii) are holding. (Q.E.D.)
Remark ; We have proved t h a t b o t h and are compact. Hence if compact-ness had been hereditary either relative to the subset relation or relative to t h e superset relation, it would follow t h a t every set of constants C is compact. Such a consequence seems rather counterintuitive. So we have a new reason to conjecture t h a t compactness is not hereditary relative to either or
Definition 35. = Df {C : C is a weakly quasilogical set}
Definition 36. = Df < >
Remark : We can generalize the concept of skeleton for the case of a C-implica-tion between and S. (If is holding, we say t h a t C-implies S.)
Definition 37.
is t h e determinacy class of the implications between and S. Definition 38. C is a skeleton of the implications between and S, or for short Ski (C, , S) = Df C is a minimal element of the partial ordering
Theorem 70
(M is a maximal element of )] i.e., every weakly quasilogical set is included in a maximal weakly quasilogical set.
Proof: Using Zorn's Lemma we shall show t h a t every chain {Ci}i in is
Ci and Ci Ci + 1. Now U if and only if.
S(S is U-determinate S is analytic in I+).
By Th. 20 this is equivalent to
S[S is (U Cs)-determinate S is analytic in I+].
The latter becomes, by virtue of U Cs = (Ui Ci) Cs = Ui (Ci Cs),
(27) S{S is [Ui (Ci Cs) ]-determinate S is analytic in I+}
Now since Cs is finite and Ci Cs Cs there can be only a finite number
of distinct sets of t h e form Ci Cs, say Ci1 Cs, ... , Cik Cs with
We have then , and therefore, Ui (Ci Cs) = Cik Cs.
The index ik depends obviously on t h e sentence S. Hence ik = f(S), or
Cik = Cf(S). Furthermore we have
since each Cik is a member of chain in . In this way (27) is transformed into
(28) S {S is [Cf(S) Cs]-determinate S is analytic in I+}
By Th. 20, we can transform (28) into
(29) S [S is Cf(S)-determinate S is analytic in I+]
But since Cf(s) it follows at once t h a t (29) is valid. But (29) is equivalent
to U . (Q.E.D.)
Remark : In case t h e set of all constants of the given language is finite, Th. 70 is obvious. Recourse to Zorn's lemma is needed only in case is infinite. On the other hand Th. 70 establishes by no way t h e existence of any non-empty element of . We do not discard (by means of a new assumption) the case of trivial languages for which we have
i.e. languages having as quasilogical sets only the empty set. But we shall define a language possessing logical constants as one having at least one non-empty quasilogical set, and furthermore such t h a t
(30)
Then t h e partial ordering is a complete lattice, viz. t h e lattice defined on t h e power set of the union of all quasilogical sets. We have then
(31)
i.e.
(32)
Now the necessary and sufficient condition for (32) is simply (30). Hence (30), (31), and (32) are pairwise equivalent. When any of these conditions is satisfied then and only then has a unique maximal element, viz. t h e union
Definition 39 The set of logical constants L = Df
It follows from Df. 39 t h a t a language possesses logical constants if and only if on the one hand t h e language has a non-empty quasilogical set and secondly the union of all t h e quasilogical sets of t h e language is itself quasilo-gical.
Definiion 40. S is logically true = Df S is L-true
S is logically false =Df S is L-false
S is logically determinate = Df S is L-determinate
Since no sentence is -true, or -false, or else -determinate, it follows t h a t in a language in which t h e set of logical constants L = , there are no logically true, logically false, or logically determinate sentences at all.
Theorem 71
S (S is L-determinate S is analytic in I+) Proof: This is equivalent to
(33)
Now (33) is true in all languages, and not only in those possessing logical constants. Indeed in a language without logical constants we have L = But . Hence L . On t h e other hand in a language with logical constants we have
(34)
and at the same time (33) is satisfied. (Q.E.D.)
The converse of Th. 71 does not hold, i.e., there are languages (in particular t h e ordinary language) containing analytic sentences which are not logically determinate. We shall show this by means of the following example from or-dinary language. Consider the predicates
' R ' = Df 'is red all over', ' G ' = Df 'is green all over'
' F1' = Df 'emits or reflects only light of frequency f1'
'F2' = Df 'emits or reflects only light of frequency f2'
where f1 and f2 are the characteristic frequency ranges of red and green light
respectively. Let then
C3=C1 C2 = {'R', 'G', ' F1' , ' F2' , }
and
where 'a' designates a particular uniformely colored surface.
Now sentences S1, S2, S3, and S4 are all true in the intended interpretation
I+ of ordinary English. Furthermore S1, S2, and S3 are analytic in I+, whereas
S4, is not analytic in I+.
Consider sentence S3. In this sentence only the constant 'a' occurs
va-cuously, so t h a t
Cr(S3) = C3
Hence C3 is t h e unique skeleton of S3, and S3 is irreducibly C3-true. We
shall show now t h a t the unique skeleton C3 of the analytic sentence S3 is not a
quasilogical set. Indeed sentence S4 is true b u t not analytic in I+. In S4 also
only ' a ' occurs vacuously. Hence
Cr(S4) = C3
and C3 is t h e unique skeleton of S4. Therefore S4 is (irreducibly) C3-true. But
S4 is not analytic. Hence C3 is not quasilogical. (Q.E.D.)
Our previous example illustrates how we can show t h a t a given set of constants is not quasilogical, and therefore, a fortiori, not logical too.
As a conclusion we give the following definition:
