A New Semantic Characterization of. Second-Order Logical Validity

Transcription

1 A New Semantic Characterization of Second-Order Logical Validity Tomoya Sato Abstract A problem with second-order logic with standard semantics is that it validates arguments that can be described as set-theoretically valid rather than logically valid. A variety of properties of sets can be expressed in a second-order language, and a variety of arguments involving the expressible properties can be validated in the semantic system. The logical validity of some arguments depends on the existence or nonexistence of particular sets satisfying particular properties in the range of second-order variables. Such arguments can be verified based on some facts about the particular sets and the particular properties. In the present paper, I criticize this problematic feature of standard semantics in the light of the formality of logic and argue that it fails to identify second-order logical validity. Instead of standard semantics, I propose a version of Henkin semantics as a correct semantic system for characterizing logically valid arguments in a second-order language. 1 Problem with Standard Semantics An important feature of second-order logic with standard semantics (hereafter SOLS ) is that arguments can be validated by the existence or nonexistence of particular sets satisfying particular properties. Let us see two examples. Let ϕ 2 be the sentence u v[ X(Xu Xv)], where u and v are first-order variables and X is a second-order variable for unary 1

2 relations. ϕ 2 is true in any standard structure 1 whose domain contains at least two objects and false in any standard structure whose domain contains only one object. Let ϕ n (for n 3) be a sentence, which is written in a similar form, that is true in and only in standard structures whose domains contain at least n objects. Also, let ϕ be the sentence Z[ u v w(zuv (Zvw Zuw)) u( Zuu) u v(zuv)], where Z is a second-order variable for binary relations. ϕ is true in any standard structure whose domain is an infinite set and false in any standard structure whose domain is a finite set. Consider then the valid argument {ϕ 2, ϕ 3,...}, ϕ deriving ϕ from ϕ 2, ϕ 3,.... The validity of this argument, as ϕ itself states, depends on the existence of a binary relation in the range (D D) that is transitive, irreflexive, and serial. For any infinite domain, there exists such a binary relation on it. Therefore, the argument is valid. The second example is an argument whose conclusion indirectly expresses Cantor s theorem by the following sentence ϕ C : Z[ X u v(zuv Xv)]. ϕ C is true in all standard structures, and therefore the argument, ϕ C deriving ϕ C from no premise is valid. The validity of this argument is due to the nonexistence of a binary relation in the range (D D) satisfying the property described by the open formula X u v(zuv Xv). For any domain, there does not exist such a binary relation on it. Hence, the argument is valid. A variety of properties of sets can be expressed, explicitly or implicitly, in a secondorder language, and a variety of arguments involving the expressible properties hold in all standard structures. The validity of such arguments is evident. 2 However, whether or not they can count as logically valid is far from obvious. Consider the following argument. 1 A standard structure for a second-order language is a pair D, I of a domain D (a non-empty set) and an interpretation function I for extra-logical terms. The range of n-ary relation variables is the power set (D n ). 2 If an argument Γ, ϕ can be validated in SOLS, it is valid in the sense that it is impossible for ϕ to be false if all sentences in Γ are true, under the assumption that standard structures cover all formally possible cases and ways of assigning truth values to sentences. 2

3 A characteristic feature of logic is its formality. Logic is formal in the sense that the logical validity of an argument is independent of particular properties of particular individual objects, 3 and as a result independent of the existence and nonexistence of objects in domains satisfying them. If an argument is logically valid, it is supposed to hold, for example, in the domain D P of philosophers and also in the domain D F P of female philosophers. The logical validity of the argument does not depend on the property of maleness or the existence of males in domains. If an argument holds in D P but not in D F P, it cannot be regarded as formal, and therefore cannot be regarded as logical either. A similar argument can be made regarding the existence and nonexistence of particular sets satisfying particular properties. In a standard structure, second-order variables for n-ary relations range over the power set (D n ). As we have seen, the validity of some arguments depends on the existence or nonexistence of particular sets satisfying particular properties in (D n ). Now consider Henkin structures. 4 Suppose that an argument Γ 0, ϕ 0 holds in any Henkin structure whose range of second-order variables for binary relations contains binary relations satisfying the property P tis of being transitive, irreflexive, and serial, but does not hold in any Henkin structure whose range contains no P tis -relations. The inferential relationship between Γ 0 and ϕ 0 is affected by the existence of binary relations satisfying P tis in the variable range. We deny that the logical validity of an argument can depend on particular properties of particular individual objects such as maleness. It then seems to be consistent to deny, for the same reason, that the logical validity of an argument can depend on particular properties of particular sets such as P tis. Some might point out that P tis, unlike maleness, is a formal property in that whether or not a binary relation possesses P tis has nothing to do with what objects the binary relation is composed of. P tis -relations can be composed of male philosophers, of 3 This notion of formality has been widely held, particularly, among those who advocate the modeltheoretical characterization of logic such as Tarski, Sher, and others. A detailed discussion of this notion can be found in MacFarlane[4]. 4 A Henkin structure is a quadruple D, R, F, I, where R is a sequence of ranges for relation variables and F is a sequence of ranges for function variables. As with standard structures, D is a domain and I is an interpretation function for extra-logical terms. 3

4 apples, and of natural numbers. Thus, even if the validity of Γ 0, ϕ 0 is due to the existence of some P tis -relation, this does not mean that its validity contradicts the formality of logic. One thing to be noted in relation to this response is that formality is not an exclusive feature of logic: set-theory (in a broad sense), too, has the feature of being formal. Among formal properties are set-theoretically formal properties. 5 The logical validity of an argument is not allowed to depend on the existence or nonexistence of particular individual objects satisfying particular biological properties such as maleness. Analogously, it can be supposed that the logical validity of an argument is not allowed to depend on the existence or nonexistence of particular sets satisfying particular set-theoretically formal properties. If P tis is a set-theoretically formal property, the validity of Γ 0, ϕ 0 does not meet the formality of logic, although it does meet the formality of set-theory. Such an argument should be regarded as set-theoretically valid but not as logically valid. The idea underlying the criticism of SOLS along the argument above is a classification of valid arguments by kinds of properties of sets that their validity is based on: the validity of an argument is of a kind X if the properties of sets that it depends on are of the kind X. The validity that can be characterized by set-theoretically formal properties is set-theoretical validity. The problem with SOLS is that it validates not only logically valid arguments but also, wrongly, set-theoretically valid arguments. 6 Standard semantics is a semantic system to identify all formally valid arguments in a second-order language. It fails to characterize logical validity as a proper part of formal validity. How then can second-order logical validity be characterized? According to the form above, the logicality of a valid argument can be reduced to the logicality of the properties of 5 A precise definition of set-theoretically formal properties will be given in the next section. As will be seen, they can be characterized independently of any particular theory of sets. 6 This is just one among several problems with SOLS (for other problems, see, for example, Bueno[2] in which five major criticisms of SOLS are discussed). Quine[6] criticizes SOLS and describes it as set theory in sheep s clothing (See Rayo and Yablo[7] for what is and what is not meant by this phrase). SOLS are ontologically committed to sets in the sense that, for (many but not all) sentences in a second-order to be true, the ranges of second-order variables have to be supposed to contain certain sets satisfying certain properties. We do not claim that every case of the ontological commitment with respect to a sentence is problematic. But rather, we claim that the cases involving sets identified by set-theoretically formal properties are problematic in the light of the formality of logic. 4

5 sets that the validity depends on: if the validity of an argument is due to logically formal properties but not to set-theoretically formal properties, then the argument can count as logically valid. The characterization of second-order logical validity can be obtained by characterizing logically formal properties. In the present paper, we will approach second-order logical validity along this line. We will propose a characterization of logically formal properties in Section 2. For the new characterization of second-order logical validity, a new semantic system that validates all and only logically valid arguments in the sense above is necessary. We will provide the new system in Section 3. In the following sections, we will restrict our attention to a simple second-order language L 2 and characterize logical validity for it. L 2 is a formal language that contains, in addition to the standard logical constants, only unary relation symbols, binary relation symbols, and second-order variables for them. It does not contain n-ary relation symbols or n-ary relation variables for n 3. We also suppose that L 2 does not contain function symbols, variables for them, or the identity relation symbol =. Due to this restriction, our discussion will be able to avoid complexity that I think is not necessary for understanding the essential idea underlying the characterization of second-order logical validity. 2 Logical Property A logically formal property of sets is a special type of formal properties. For simplicity, we will call it a logical property hereafter. A formal property of sets is a property that sets satisfy independently of what their components are. A formal property P of sets has a characteristic function. The characteristic function f P of P is an operator assigning a truth value (T or F) to each set such that for any set X, X satisfies P if and only if f P (X) = T. We will characterize formal properties and logical properties by characterizing their characteristic functions. 5

6 In the contemporary model-theoretical approach to logic, the formality of the characteristic function of a property has been defined in terms of the notions of similarity relation and of invariance. To precisely describe formal properties of sets, we will define several concepts. An objectual structure of a domain D is a pair D, X, where X (D n ) (n = 1, 2), that is, X is a unary relation on D (a subset of D) or a binary relation on D. 7 A similarity relation is a collection of pairs of objectual structures. For domains D and D, if two structures D, X and D, Y are similar with respect to a similarity relation S, we say that they are S-similar. For a property P of sets, we will write its characteristic function acting on D as fd P P. The characteristic function f of a property P is said to be invariant under the similarity relation S if fd P (X) = f D P (Y ) for any S-similar objectual structures D, X and D, Y. For domains D and D of the same cardinality, a bijection η : D D determines a similarity relation S η such that D, X and D, Y are S η -similar if Y is the image of X under η. The collection of all bijections between domains also determines a similarity relation S bi : two objectual structures are S bi -similar if there exists some bijection η such that they are S η -similar. S bi is the union of all S η. We say that a property P is a formal property if f P is invariant under S bi. 8 According to this definition, for example, the property of cardinality 7 An objectual structure can be defined, more generally, as an (n+1)-tuple D, X 1,..., X n, where X i is an object of D of a finite relational type. However, for our purpose of characterizing of the second-order logical validity for the simple formal language L 2, we do not need to deal with such general objectual structures. 8 A possible concern regarding this definition is that the formality of a property is determined based solely on the invariance of its characteristic function. Even if two properties are expressed differently and therefore can be distinguished from one another, they can both be formal, provided that they share an S bi -invariant characteristic function. Consider, for example, the following two properties: (P 1 ) X contains eight objects; (P 2 ) X contains as many objects as the number of planets in the Solar System. Since there are eight planets in the Solar System, P 2 has the same characteristic function as the formal property P 1. Thus, P 2 is a formal property. However, some might wonder if P 2 could really count as formal; the condition P 2 has a content, namely, the number of planets being eight. What P 2 means is determined by a contingent fact. The problem here is about the way that a logical property is specified. Some might think that for a property to be formal, in addition to the invariance of its characteristic function, further conditions on how it should be described need to be satisfied. Other might claim that such conditions are not necessary. This problem has been discussed in the literature, in particular, in relation to the research on logical constants (See, for example, McCarthy[5], Sher[8], and MacFarlane[4]). However, we will not address the problem in the present paper, because even if the further conditions actually need to be imposed, and even if they can be correctly identified, they will not play any important role in our characterization of 6

7 κ is a formal property of sets. Transitivity, irreflexivity, and seriality are formal properties of binary relations. How can the definition of formal properties be justified? Why can a property be regarded as formal if its characteristic function is S bi -invariant? The idea underlying the definition is that if two objectual structures D, X and D, Y are S bi -similar, then the sets X of a domain D and Y of another domain D can be thought of as formally similar. Generally, sets contain various objects and are distinguished by them. However, if they are of the same cardinality, there is nothing to distinguish one from the others from a formal point of view, because the difference between their components is not a formal difference. The existence of a bijection between D, X and D, Y, thus, means that D and D cannot be formally distinguished and also that X and Y cannot be formally distinguished. If the characteristic function of a property P is S bi -invariant, P is shared by all such formally similar sets: for any formally similar sets X and Y, it holds that X satisfies P if and only if Y satisfies P. In this sense, properties with S bi -invariant characteristic functions are formal. Logical properties can be defined in the same form as formal properties: a property is said to be logical if its characteristic function is invariant under a logical similarity relation among objectual structures. That is, a property can be taken as logical if and only if it is shared by logically similar sets. For the definition, what sets are logically similar or dissimilar needs to be determined. 9 Since a logical property is a special type of formal property, the logical similarity relation is supposed to be different from the S bi -similarity relation. In what follows, we will define two logical similarity relations: (i) The logical similarity relation among objectual structures D, X, where X (D); (ii) The logical second-order logical validity. There are normal formal properties such as P 1 and abnormal formal properties such as P 2. If you think that only normal formal properties should count as legitimate, you can suppose that we will deal only with them throughout this paper. 9 The problem of the logical similarity relation has been addressed in the context of the characterizations of logical operators and of logical constants. Several definitions have been proposed in the literature, but philosophers have not yet reached a consensus on which one is correct. In this paper, I will propose my own characterization and provide a justification. The definition that has been widely advocated is the one given using the concept of isomorphism (See Sher[8] and [9]), which actually is identical to S bi -similarity relation. Other definitions can be found, for example, in Bonnay[1] and Feferman[3]. 7

8 similarity relation among objectual structures D, R, where R (D D). Using these logical similarity relations, logical properties of unary relations and logical proprieties of binary relations will be identified. First, the logical similarity relation among objectual structures D, X, where X (D). For any domain D, we divide sets in (D) into three kinds: (i) the domain set D; (ii) the empty set ; (iii) others. Based on the division, the logical similarity is defined as follows: D, D is logically similar to and only to D, D, for any D ; D, is logically similar to and only to D,, for any D ; D, X is logically similar to D, Y, for any X (D) and Y (D ) other than D, D,. The reason that the domain set D and the empty set can be logically distinguished from other sets is that they can be characterized by logical properties of individual objects. A logical property of individual objects is a property that is shared by logically similar individual objects. To define the logical similarity among individual objects, consider pairs D, a, where a D, which can be seen as objectual structures of individual objects. A pair D, a is supposed to represent an individual object a of a domain D. Generally, individual objects a D and b D differ in various aspects (for example, one might be a concrete object and the other might be an abstract object). But, their difference is not a formal difference and therefore cannot be taken as a logical difference. Thus, objectual structures D, a and D, b cannot be logically distinguished by the difference between their second components a and b. The first components D and D can be formally distinguished from each other if they are of different cardinalities. However, the difference between their cardinalities has nothing to do with the difference between a and b. How a and b are similar or dissimilar is irrelevant to how many objects D or D contains. Therefore, D, a and D, b, as representing a of D and b of D, cannot be logically distinguished by 8

9 the difference between their first components either. Consequently, it can be supposed that objectual structures D, a and D, b are logically similar, for any D and D and for any a D and b D. A logical property of individual objects can be defined as a property whose characteristic function is invariant under this logical similarity relation. There are two kinds of logical properties of individual objects: (i) properties P such that f P D (a) = T for any D and for any a D; (ii) properties P such that f P D (a) = F for any D and for any a D. P is a property that is satisfied by any object in any domain, an example of which is the property of being identical to itself. P, on the other hand, is a property that is not satisfied by any object in any domain, an example of which is the property of being not identical to itself. The domain set D is the extension of P, and the empty set is the extension of P. They can be characterized by these logical properties. Other sets in (D) can be characterized by certain properties, but the properties are not logical. If P and P are supposed to be distinguished from other properties because of their logicality, then, analogously, their extensions should also be distinguished from other sets. A logical distinction should be made between these two special sets and others. Hence, D, D can only be logically similar to D, D, and D, can only be logically similar to D,. As for sets other than the domain sets and the empty set, we suppose that they are all logically similar to each other, even if they can be formally distinguished because of their different cardinalities. The reason for their logical similarity is that distinguishing sets by their cardinalities violates the following principle: For a similarity relation S, if D, D and D, D are S-similar, and if D, X and D, Y are also S-similar, then, for the complement D \ X of X in D and the complement D \ Y of Y in D, D, D \ X and D, D \ Y have to be S-similar. Note that this principle holds for the formal similarity relation (namely, S bi -similarity retaliation) and the identity relation (one kind of similarity relation). 10 The idea underlying the 10 If D, D and D, D are S bi -similar, and if D, X and D, Y are also S bi -similar, then there exists a 9

10 principle is that if the whole domains (D and D) are composed of similar parts (X and Y ) and of dissimilar parts (D \ X and D \ Y ) respectively, they have to be dissimilar. By applying the principle to the logical similarity relation, we obtain the requirement that sets X and Y can be regarded as logically similar only if their complements in their domains can also be regarded as logically similar. This requirement cannot be met if distinguishing sets by their cardinalities is allowed. Assume that a set A of cardinality κ A and another set B of another cardinality κ B could be logically distinguished because of the difference between their cardinalities. Let C be a set of an infinite cardinality κ C such that κ C is strictly larger than κ A and κ B. Consider then the domain D 0 = A B C. Since the cardinalities of A C and B C are both κ C, it seems that they can be regarded as logically similar. However, by our assumption, their complements (i.e., B and A) in D 0 are logically dissimilar. Thus, the requirement is violated. Distinguishing sets by their cardinalities is the only way to make formal distinctions among sets other than the domain sets and the empty set. But, the argument above shows that the only way is not available. Hence, D, X is logically similar to D, Y, for any X (D) and Y (D ). According to the logical similarity relation defined above, there are eight kinds of logical properties of unary relations: P 1 such that f P 1 D (X) = T if and only if X (D); P 2 such that f P 2 D (X) = T if and only if X (D) \ {D}; P 3 such that f P 3 D (X) = T if and only if X {D, }; P 4 such that f P 4 D (X) = T if and only if X (D) \ { }; P 5 such that f P 5 D (X) = T if and only if X {D}; P 6 such that f P 6 D (X) = T if and only if X (D) \ {D, }; bijection η : D D such that η(x) = Y. For such η, we have that η(d \X) = D \Y. Therefore, D, D \X and D, D \ Y are S bi -similar. It can be proven in the same way that the principle holds for the identity relation. 10

11 P 7 such that f P 7 D (X) = T if and only if X { }; P 8 such that f P 8 D (X) = T if and only if X. Formal properties other than these are all non-logical properties. We call them set-theoretical properties. Next, the logical similarity relation among objectual structures D, R, where R (D D) (i.e., R is a binary relation on D). Under R, objects in D are related to each other. Some object a D is related to all objects: that is, either a, b R for all b D, or b, a R for all b D. Some object is not related to any objects: that is, a, b / R and b, a / R for any b D. For a binary relation R, let R (1) (a) denote the set {b D : b, a R}. We call R (1) (a) the image of a under R. Similarly, let R (2) (a) denote the set {b D : a, b R} and call it by the same name the image of a under R regardless of the difference of the position of a. For R (i) (a), the restrictions of R to a, R (1) (a) {a} and {a} R (2) (a), are subsets of R. R is the union of the resections: R = a D[R (1) (a) {a}] = a D[{a} R (2) (a)]. Although there are various images of objects under R, they can be logically classified into three kinds, according to the logical similarity relation among sets in (D): (i) R (i) (a) = D; (ii) R (i) (a) = ; (iii) R (i) (a) D. Based on the classification, logical distinctions among restrictions can be made: R (1) (a) {a} and R (1) (b) {b} are logically distinguished from each other if R (1) (a) and R (1) (b) are logically dissimilar sets; similarly, {a} R (2) (a) and {b} R (2) (b) are logically distinguished from each other if R (2) (a) and R (2) (b) are logically dissimilar sets. As a result, restrictions R (1) (a) {a} and {a} R (2) (a) can be divided into three logically distinguishable kinds respectively. Let R (D D) and R (D D ) be different binary relations on different domains. We propose that R and R can be logically distinguished if they are composed of 11

12 logically dissimilar restrictions. For example, suppose that R (1) (a) = D for some a D and that R (1) (b) D for any b D. Then, R and R cannot be thought of as logically similar; the relation between a and other objects in D under R can be logically distinguished from the relation between b and other objects in D under R. They are, as binary relations, logically dissimilar. For R and R to be regarded as logically similar, they have to be composed of logically similar restrictions. We thus define the logical similarity relation among objectual structures D, R as follows. D, R and D, R are said to be logically similar if they satisfy the following conditions for i = 1, 2: (S-1) i For any a D, there exists b D such that D, R (i) (a) and D, R (i) (b) are logically similar; (S-2) i For any b D, there exists a D such that D, R (i) (a) and D, R (i) (b) are logically similar. Let us see two examples. Let D = {a, b, c} and D = {u, v}. And let R 1 = { a, a, b, b, c, c } and R 1 = { u, v, v, u }. R 1 and R 1 are logically similar. a b c R 1 R 1 a u b v c u v For i = 1, 2, and for any object x D, the image R (i) 1 (x) is a non-empty proper subset of D. Also for any object y D, R 1(i) (y) is a non-empty proper subset of D. Thus, D, R (i) 1 (x) and D, R 1(i) (y) are logically similar. Therefore, (S-1) 1, (S-2) 1, (S-1) 2, and (S-2) 2 are all satisfied. Let R 2 = { a, a, a, b, a, c } and R 2 = { u, u, v, v }. R 2 and R 2 are not logically similar. 12

13 a b c R 2 R 2 a u b v c u v We have that R 2 (2) (a) = {a, b, c} = D. But, R 2(2) (y) D for any y D. This means that there does not exist y such that the image R 2(2) (y) is logically similar to the image R 2 (2) (a). Therefore, (S-1) 2 is not satisfied. According to the definition, binary relations are divided into 13 kinds. 11 As a result, there are 2 13 kinds of logical properties of binary relations. We call a non-logical property a 11 For a binary relation R on D and for a D, the restrictions R (1) (a) {a} and {a} R (2) (a) are classified into three kinds based on the logical similarity relation among sets in (D). For each kind of logically similar restrictions, either R contains at least one restriction of the kind or contains no restriction of the kind. Thus, with respect to the restriction R (1) (a) {a} of R to a as the first component, there are seven (2 3 1) kinds of logically similar binary relations (the meaning of 1 is that it cannot happen that R does not contain any restriction of any kind of logically similar restrictions). Similarly, with respect to the restriction {a} R (2) (a) of R to a as the second component, there are seven kinds of logically similar binary relations. Although, as a result, 49 combinations of logically similar binary relations are conceivable, some are impossible. For example, if there is a D such that R (1) (a) = D, then for any b D, R (2) (b), because b, a R. We express the possible 13 combinations using the following notation: for a binary relation R (D D), R (i) (x), X means that there exists x D such that R (i) (x) = X. Let X 0 and Y 0 be sets in (D) such that X 0, Y 0 D. Then, any binary relation on any domain is logically similar to one of the following. 1. R such that R (1) (a), D and R (1) (b), X 0 and R (1) (c), and R (2) (b ), Y 0 ; 2. R such that R (1) (a), D and R (1) (b), X 0 and R (2) (a ), D and R (2) (b ), Y 0 ; 3. R such that R (1) (a), D and R (1) (b), X 0 and R (2) (b ), Y 0 ; 4. R such that R (1) (a), D and R (1) (c), and R (2) (b ), Y 0 ; 5. R such that R (1) (b), X 0 and R (1) (c), and R (2) (b ), Y 0 and R (2) (c ), ; 6. R such that R (1) (b), X 0 and R (1) (c), and R (2) (b ), Y 0 ; 7. R such that R (1) (a), D and R (2) (a ), D ; 8. R such that R (1) (b), X 0 and R (2) (a ), D and R (2) (b ), Y 0 and R (2) (c ), ; 9. R such that R (1) (b), X 0 and R (2) (a ), D and R (2) (b ), Y 0 ; 10. R such that R (1) (b), X 0 and R (2) (a ), D and R (2) (c ), ; 11. R such that R (1) (b), X 0 and R (2) (b ), Y 0 and R (2) (c ), ; 12. R such that R (1) (b), X 0 and R (2) (b ), Y 0 ; 13. R such that R (1) (c), and R (2) (c ),. R = D D in case 7, and R = in case

14 set-theoretical property. For example, transitivity is a set-theoretical property. Although R 1 and R 1 above are logically similar binary relations, R 1 above is transitive while R 1 is not. In addition to transitivity, many other major properties of binary relations such as reflexivity and symmetricity are set-theoretical properties. Remember the principle that if two sets are logically similar, then their complements also have to be logically similar. Our definition of the logical similarity relation among binary relations holds the principle. Let R (D D) and R (D D ) be logically similar binary relations. Also, let R and R be their complements respectively. Then, R is composed of the restrictions (R (1) (a)) {a} (or {a} (R (2) (a)) ) for a D, and R is composed of the restrictions (R (1) (b)) {b} (or {b} (R (2) (b)) ) for b D. We have that if R (i) (a) and R (i) (b) are logically similar, then (R (i) (a)) and (R (i) (b)) are logically similar. Therefore, D, R and D, R satisfy all the conditions (S-1) 1, (S-2) 1, (S-1) 2, and (S-2) 2. 3 New Definition of Logical Validity If the validity of an argument is due to the existence of sets satisfying logical properties, then the argument is logically valid. If the validity is based on the existence or nonexistence of sets satisfying some set-theoretical properties (i.e., non-logical properties), then the argument cannot be regarded as logically valid: it is set-theoretically valid. To characterize secondorder logical validity in these forms, we will consider the following two conditions: (i) the condition that logical validity is allowed to depend on the existence of sets satisfying logical properties (the dependence condition); (ii) the condition that logical validity is not allowed to depend on the existence or nonexistence of sets satisfying set-theoretical properties (the independence condition). Recall that our characterization of second-order logical validity is restricted to the logical validity for the simple language L 2. Throughout this section, we will express a Henkin structure for L 2 as a quadruple D, D(1), D(2), I, where D(n) is the range of second-order 14

15 variables for n-ary relations. Let us first consider the independence condition. Consider, for example, the settheoretical property P κ of cardinality κ (0 < κ) that can be applied to sets in (D). For a valid argument of SOLS, in order for its validity to be regarded as independent of the existence of sets satisfying P κ, it has to hold not only in all standard structures D, I but also in all Henkin structures D, D(1), (D D), I such that D(1) does not contain any set of cardinality κ. On the other hand, for the validity of the argument to be regarded as independent of the nonexistence of sets satisfying P κ, it has to hold in all Henkin structures D, D (1), (D D ), I such that D (1) contains at least one set of cardinality κ. A similar argument can be made for any set-theoretical property. Let P s be a settheoretical property of n-ary relations (n = 1 or 2). Also, let Ext D (P s ) be the extension of P s on D. Then, that the logical validity of an argument is not allowed to depend on the existence or nonexistence of sets satisfying P s means that the argument has to hold in any Henkin structure D, D(1), D(2), I such that D(n) includes (D n ) \ Ext D (P s ). The independence condition then can be formulated as follows. Let P 1, P 2,... be set-theoretical properties of unary relations (i.e., subsets of domains). And let Q 1, Q 2,... be set-theoretical properties of binary relations. For the validity of an argument to be independent of these set-theoretical properties, it is required to hold in any henkin structure D, D(1), D(2), I such that D(1) includes (D) \ Ext D (P i ) and D(2) includes (D D) \ i Ext D (Q j ). j Let us turn to the dependence condition. Consider, for example, the logical property P domain of containing all objects in the domain. The only set in (D) satisfying P domain is the domain set D. Some Henkin structures have D in the range D(1) and others do not. There are arguments that are invalidated in some Henkin structure of the latter kind because of their validity being based on the existence of the domain set in the range. However, such arguments cannot be regarded as logically invalid. For, according to the dependence condition, the logical validity of an argument can depend on P domain. If the domain set is 15

16 not available for second-order variables in a Henkin structure, an argument does not have to hold in it in order to be logically valid. As we have argued, with respect to the logical similarity relation, unary relations are divided into three kinds, and binary relations are classified into thirteen kinds. Accordingly, there are 2 3 kinds of logical properties of unary relations and 2 13 kinds of logical properties of binary relations. One of the 2 3 kinds is the logical property that any unary relation does not satisfy and one of the 2 13 kinds is the logical property that any binary relation does not possess. For all logical properties other than these empty logical properties, the same argument as the one for P domain above can be made. Let P l be a logical property of n-ary relations (n = 1, 2) such that Ext D (P l ) for some D (that is, P l is not an empty logical property). Also, let H = D, D(1), D(2), I be a Henkin structure such that Ext D (P l ) and Ext D (P l ) D(n) =. This feature of H says that there are n-ary relations on D satisfying P l but they are not in the range D(n): n-ary relations satisfying P l are not available in H, although they exist. What the dependence condition means is that whether or not an argument is logically valid has nothing to do with whether or not it holds in such a Henkin structure H. The independence condition specifies Henkin structures in which logically valid arguments have to hold, while the dependence condition determines Henkin structures in which logically valid arguments do not have to hold. These conditions jointly identify a special kind of Henkin structures. We say that a Henkin structure D, D(1), D(2), I is a prime Henkin structure if it satisfies the following condition: for any logical property P l of n-ary relations, either Ext D (P l ) = or Ext D (P l ) D(n). A prime Henkin structure is a Henkin structure whose range contains at least one n-ary relation satisfying P l if possible. A prime Henkin model of a set Γ of sentences is a prime Henkin structure in which all sentences in Γ are true. We define logically valid arguments as follows: An argument Γ, ϕ is said to be logically valid if every prime Henkin model of Γ is also a prime Henkin model of {ϕ}. 16

17 The two arguments that we have seen at the beginning of the present paper are not logically valid. 12 In addition to them, various arguments that are valid in SOLS can be invalidated by some prime Henkin structures. What valid arguments of SOLS do still remain on the list of logically valid arguments? If there is a complete proof-theory for the new second-order logic, a full specification of second-order logical validity will be possible. However, whether or not the new logic has the completeness property, and also whether or not it has other important mathematical properties such as the Löwenheim-Skolem property, are not know yet. 12 An example of prime Henkin structures that invalidates the argument {ϕ 2, ϕ 3,...}, ϕ is H 1 = N, (N), D(2), I, where D(2) only contains the following binary relations: R 1 such that n, 0 R 1 for any n N, such that m, m R 1 for any odd number m, and such that n, m / R 1 for any n N and for any positive even number m ; R 2 such that n, 0 R 2 and 0, n R 2 for any n N and such that n, n R 2 for any positive natural number n; R 3 such that n, 0 R 3 and n, n R 3 for any n N; R 4 such that n, 0 R 4 for any n N; R 5 such that n, n R 5 for any positive natural number n N; R 6 such that 0, 0 R 6 and such that n, m R 6 for any positive natural number n and for any odd number m; R 7 = D D R 8 such that 0, n R 8 for any n N, such that m, m R 8 for any odd number m, and such that m, n / R 8 for any n N and for any positive even number m ; R 9 such that 0, n R 9 and n, n R 9 for any n N; R 10 such that 0, n R 10 for any n N; R 11 such that 0, 0 R 11 and such that m, n R 11 for any positive natural number n and for any odd number m; R 12 such that n, n R 12 for any n N; R 13 =. Each R i above is logically similar to the binary relation of the same number given in Footnote 11. However, none of them is transitive, irreflexive, and serial (in particular, R 1,..., R 12 are not irreflexive, and R 13 is not serial). Therefore, H 1 is not a model of {ϕ }, although it is a model of {ϕ 2, ϕ 3,...}. An counterexample of the argument, ϕ C is a prime Henkin structure H 2 = D, D (1), D (2), I, where D = {0, 1, 2}, D (1) = {, {0}, D }, and D (2) = (D D ). The binary relation { 1, 0, 2, 0, 2, 1, 2, 2 } in D (2) satisfies the property expressed by the open formula X u v(zuv Xv). Hence, H 2 is not a prime Henkin model of {ϕ C }. 17

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