Singular Terms and Inferential Criteria

VERA TRIPODI Singular Terms and Inferential Criteria
1 1.1 “The syntactic priority thesis” and Inference At the heart of Frege’s philosophy of mathematics lies “the thesis of the priority of the syntactic over ontological categories” that Crispin Wright explains so: (SPT) «The question whether a specific expression is a candidate to refer to an object is entirely a matter of the sort of the syntactic role which it plays in whole sentences».1 This thesis is strictly connected to the Fregean Argument that, following Hale, runs as follows: (1) «If a range of expression functions as singular terms in true statements, then there are objects denoted by expressions belonging to that range; (2) Numerals, and many other numerical expressions besides, do so function in many true statements (of both pure and applied mathematics); Hence: (3) There exist objects denoted by those numerical expressions (i.e. there are numbers) ».2
I thank Prof. Cesare Cozzo, Valeria Giardino and Gianluca Paronitti for helpful comments on earlier draft of this paper. 1 Wright, 1983 51. 2 Hale 1987 11. VERA TRIPODI In general, the Fregean Argument establishes that an expression refers to an object if: (i) such an expression functions in certain sentence as a singular term; (ii) sentences in which it functions in this way are true. According to Frege, linguistic categories are prior to ontological categories: in order to explain what sort of thing an expression refers to, we need to explain what sort of expression it is. In order to understand the sort of thing to which the expression might refer, we need to understand how that expression functions in the sentence of which it is a part. The logical kind of an expression determines the ontological kind of its referent. More specifically, Fregean use of the ontological term “object” corresponds to his use of the linguistic term “proper name”: the primary use of a singular term in a sentence is to pick out a determinate object for which the term stands in the same way as a proper name stands for its bearer. As we can see, it is essential to the Fregean programme to characterize formally the class of singular terms. In Fregean ontology, syntactic criteria for singular term-hood are provided by inferential test for distinguishing the behaviour and the role of such terms in natural languages from terms that belong to other logical categories. Dummett’s point of view is a defence of the Fregean programme. At the centre of Dummett’s defence is a version of Frege’s famous dictum according to which «only in the context of a sentence does a word have any reference».3 According to Dummett, it is possible to characterize the logical kind of singular terms in «wholly linguistic terms»4 and to give clear and exact criteria for them by relating to their functioning within the language. Dummett suggests, in Frege. Philosophy of Language ,5 five criteria that are supposed to discriminate singular terms from expressions of other kind. These criteria are not based merely on conditions of grammatical correctness, because the place that a genuine singular 3 4 5 2 Frege 1988, § 62. Dummett 1981, 57. Ibid., chapter 4. SINGULAR TERMS AND INFERENTIAL CRITERIA term stands in a sentence can be occupied by other expression that expresses generality (such as “everything”, “nothing”, etc.). Thus, Dummett’s criteria are inferential criteria and are based on the speaker’s recognition of “the sentence as well-formed” and the «correctness or incorrectness of certain patterns of inference»6 at the intuitive level. So, given what Dummett says, any sentence that contains a singular term must admit a particular inference. 1.2 Dummett’s Criteria Dummett’s purpose is to describe the character of singular terms in general. His criteria reflect the idea that some simple patterns of inference are valid when singular terms occupy certain positions in their premises or conclusions. In fact, these criteria furnish a means by which we decide whether a given expression is to be considered as a singular term as it occurs in a particular sentence, and not whether some expression is or is not (considered in itself) a singular term. I first introduce the first three criteria all together and analyse them separately, then I explain the remaining two. Let t be an expression of a language, in this case the English language, which occurs in many different sentences. According to the first three conditions, t is a singular term only if: (i) For any sentence “A(t)”, the inference from “A(t)” to “there is something such that A(it)” is valid; (ii) For any sentences “A(t)” and “B(t)”, the inference from “A(t) and B(t)” to “there is something such that A(it) and B(it)” is valid; (iii) For any sentences “A(t)” and “B(t)”, the inference from “it is true of t that A(it) or B(it)” to “A(t) or B(t)” is valid. Criterion (I): The first criterion rules out those expressions that contains terms as “nothing”, “nobody”, “no-one” and expressions like “no philosophers” or “no Italian”. 6 Ibid, 58. 3 VERA TRIPODI Examples. From “Nothing is wise”, it is not possible to infer “There is something such that it is wise”. From “Tom loves Mary”, it is possible to infer “There is someone that loves Mary”. Criterion (II): The second criterion rules out “something”, “somebody”, “some sheep” or many occurrences of indefinite substantival phrases such as “a policeman” when they occur in grammatical subject or object position. Examples. (i) «From “Something is red” and “Something is not red”, it is not possible to infer “There is something such that it is red and it not red”».7 (ii) «From “A policeman struck me” and “A policeman arrested me”, it is not possible to infer “There is something such that he struck me and he arrested me”».8 (iii) «From “I struck a policeman” and “I insulted a policeman”, it is not possible to infer “There is someone such that he struck me and he arrested me”». But this criterion fails to exclude an expression like “a policeman” or “a poet” when it occurs in complement position as predicates. Examples. (i) «From “Henry is a policeman” and “Peter is not a policeman”, it is possible to infer “There is something such that Henry is it and Peter is not it”».9 (ii) «From “Richard was born a poet” and “Henry has become a poet”, it is possible to infer “There is something such that Richard was born it and Henry has become it”».10 7 Wetzel 1990, 241. Hale 1979, 288. 9 Dummett 1981, 61. 8 4 SINGULAR TERMS AND INFERENTIAL CRITERIA Criterion (III): The third criterion excludes “everything”, “everybody” and “every sheep” and plural nouns, but it is not sufficient to exclude “undetected murders” as in the following example: Example. «From “it is true of undetected murders either that they are very rare or that they do not take place at all”, it is possible to infer “Either it is true that undetected murders are very rare or it is true that undetected murders never take place at all”».11 1.3 Some problems posed by Dummett’s criteria According to Dummett, these criteria establish only necessary conditions for an expression’s being a singular term. Nevertheless, as Dummett points out, these criteria require some emendations. In fact, even if they are sufficient to exclude various expressions whenever they stand in place of genuine singular terms, they are not sufficient to exclude indefinite noun phrases when they occur where singular terms cannot be. More explicitly, they do not exclude some expressions (e.g., “undetected murders” or “a policeman”) when they function as ordinary predicates (e.g. “Harry is a policeman”) and when they occur in sentences in which they are the subjects of second-level predicates (e.g. “Undetected murders are rare”). As Crispin Wright suggests, these first three criteria proposed by Dummett characterize at best a class of sentences as one in which a given expression occurs as an expression of level n, with predicates of level n+1. Nevertheless, they don’t distinguish a class of sentences in which an expression of, e.g., level 2, is predicated of an expression of level 1 from one in which an ordinary predicate is predicate of a singular term (an expression of level 0). Hence, we need some supplementary tests which allows us (i) to distinguish a difference in function or semantic role between those expressions that can refer to a particular object and those 10 Ibid., 60. Dummett writes about this example that it is «slightly unnatural but clearly intelligible». See Ibid., 1981, 60. 11 5 VERA TRIPODI which can not and (ii) to distinguish first-order from second (and higher) order generality. 1.4 Criterion IV: Aristotle’s dictum We need some further guide for distinguishing between firstlevel from second-level generality. According to Dummett, we can explain this difference only by means of the idea that «a proper name is standing for an object about which we are asserting that the predicate is true of it is».12 We have to explain the difference among different levels of expression and to clarify the difference between «generalization over objects from generalization over all the things that might be true of a first-level quality».13 The criterion proposed by Dummett is the following: Criterion (IV) «Suppose that we have a sentence “S(t,u)” involving two expressions “t” and “u” both of which pass the tests for [singular terms] which we have already laid down. We now inquire, which respect to “t”, whether we may assert “There is something such that S(it, anything) if and only if it is not the case that S(t, that thing)”: if we may, then “t” is not a [singular term]. Likewise, with respect to “u”, we enquire whether we may assert “There is something such that S(anything, it) if and only if it is not the case that S(that thing, u)”: if we may, then “u” is not a [singular term]».14 This criterion is based on Aristotle’s dictum that «a quality has a contrary but a substance does not».15 As Dummett points out, «for any predicate, there is another predicate which is true of just those objects of which the original predicate is false, and false of just those objects of which the original predicate is true».16 The underlying idea is that «we can not assume that, given any object, there is another object of which just those predicates are true which were false of the original object, and conversely».17 12 13 14 15 16 17 6 Ibid., 66. Ibid., 66. Ibid., 64 Ibid., 63. Ibid., 64. Ibid. SINGULAR TERMS AND INFERENTIAL CRITERIA Dummett’s thought is that, given any sentence from which a predicate is extracted, it is possible to extract from its negation, that is “it is not the case that Ft”, another predicate which is its contrary, that is “not-F”. Given any predicate as “wise”, it is possible to introduce an another predicate as “stupid” and to stipulate that – for every proper name “a” – “a is stupid” is to have the same truth-value as “It not the case that a is wise”. Nevertheless, given any proper name as “Socrates”, it not possible to introduce another proper name as “Not-Socrates” and stipulate that for every predicate “F(
)”, “F(Not-Socrates)” is to have the same truth-value as “It is not the case that F(Socrates)”. So, the idea is that whereas for any given predicate there is always a contrary predicate applying to a given object just in case the original predicate fails to apply, there is not, for a singular term, anything corresponding to this. The Aristotelian criterion, therefore, allows us to explain the difference between a proper name and a predicate as a difference between levels of expression. But, as Dummett observes, this criterion is not straightforward since its use involves a “double generality”. The Aristotelian criterion, hence, allows us to rule out from the class of singular terms every predicate that occurs in a sentence as predicate. For example, this criterion eliminates the predicate “man” in “Plato is a man”, but it doesn’t eliminate “undetected murders” in “Undetected murders are rare”.18 The reason is that «there is nothing of which just those things are true which are false of undetected murders, i. e., something which is frequent just in case undetected murders are rare, on the decrease just in case undetected murders are increasing or constant, etc.».19 But, since it fails to distinguish between different levels of generality, the Aristotelian criterion does not exclude expressions like “undetected murders” when they function as grammatical subject and some second-level predicate is predicated of them (e.g. “undetected murders” in “Undetected murders are rare”). We need, therefore, another criterion that allows us to distinguish from singular terms an expression like “undetected murders” when occurs as ordinary predicate 18 19 Ibid., 67. Ibid., 67. 7 VERA TRIPODI and, specifically, to distinguish when this expression serves to provide a first or a second level of generality. 1.5 Criterion V: the difference between first-order and second-order generality A characteristic of our natural languages is that they involve expressions that can indifferently be used to express first-level (over objects) and second-level generality (over properties or relations). For example, the expression “something” can be used to express generalization over things and over properties. The principal fault of the first three of Dummett’s criteria is that in their formulation the expression “something” expresses a first-level and second-level generality. We have, therefore, to propose a test for distinguishing the two different forms of generality. According to Dummett, valid inferences that - on the basis of the first three criteria - qualify adjectives and indefinite noun phrases as proper names are inferences in which the expression “something” in the conclusion expresses second-level generality. So, the inference from “Jones is a policeman” to “There is something such that Jones is it” is a valid inference that expresses a generalization over things that Jones would be or over a property Jones might be instantiate. To solve this problem, Dummett proposes an additional criterion, which Hale calls “the specification test”,20 for distinguishing first-order from second-order quantification and formulates it as follows: Criterion (V) «An expression expresses a second- or higher- level of generality “if a point may be reached where a demand for specification is still grammatically well constructed, but is nevertheless rejected as illegitimate».21 Dummett’s idea is quite simple: any existentially quantified sentence allows request for further specification. But the problem is that natural languages comprise ambiguous sentences that are 20 21 8 Hale 2001, 37. Dummett 1981, 68. SINGULAR TERMS AND INFERENTIAL CRITERIA subject to different readings: a first-order and a second-order reading. Dummett’s thought is that, when the generality is of second level, «the answer to a request for specification is treated as complete» 22 and the request for further specification is rejected. Example of second level generality: «From “Ten years ago undetected murders were very common” and from “Now undetected murders are rare”, it possible to infer “There is something which ten years ago was very common and now is rare”».23 According to the specification test, “something” in the conclusion of this inference supplies to express a generality of second level. As Dummett observes, if someone says “There is something which ten years ago was very common and now is rare”, someone can properly ask me “What is it that ten years ago was very common and now is rare?” and she may receive the answer “undetected murders”. But she can’t properly ask further “Which undetected murders?”, because the request for further specification clearly indicates a misunderstanding of my original assertion and can properly be rejected as illegitimate. Here is an example of first level generality. Example of first level generality: «If I say “George dropped something on his toe”, someone may ask “what was is that George dropped on his toe?” and I may reply “a hammer”. If she asks further “Which hammer?”, I can’t reject his request further specification as illegitimate because the question has an answer even if I don’t know what it is.»24 Hence, according to Dummett, the generality, involved in all his criteria relating to inference, shall be of first level. Unfortunately, as Bob Hale remarks, this test is flawed in opaque contexts. Here is a counterexample. Counter-example: «If I say, “Edward believes that someone is leaking official secret”, someone may ask me “whom does he believe to be leaking official 22 23 24 Ibid. Ibid., 69. Ibid., 67. 9 VERA TRIPODI secrets?” and I may reply “a member of his department”. The request for a further specification, “which member?”, may have an answer and in the same time may be rejected as illegitimate because it may betray a misunderstanding of the character of the belief attributed to Edward.»25 In the quantification involved in the formulation of Edward’s belief, “someone” expresses a first-level generality. It is possible to express this sentence, using the quantifier-variable notation, as follows: “Edward believes that (
x)(x is leaking official secret). This paraphrases is different from that of a sentence involving a higher-level quantification like “There is something that George is but Henry is not”, that we might write it as “(
F) (F(George) but ¬F(Henry)). To overcome this obstacle, as Hale illustrates, it is not necessary to introduce the general distinction between first and second level of generality occurrences of “something”. Indeed it is sufficient to apply the idea that «the quantifier contained in applications of those conditions should be first-order».26 Therefore, Hale proposes to emendate criterion V as follows: «a is a proper name only if the conclusion of an inference of the sort described in criterion (1) and (2) is not such that a point may be reached where a grammatically well-formed request for specification can be rejected as illegitimate ». 2 FURTHER DIFFICULTIES 2.1 Wetzel’s objections How can we recognize the validity of patterns of those inferences by means of which we could decide whether it is a singular term or not? As I pointed out earlier, Dummett maintains that his criteria are based on the capacity of recognizing the correctness or incorrectness of inferences “at the intuitive level”. If Dummett is right, what does it mean that we can recognize the correctness of certain patterns of 25 26 10 Hale 1979, 291. Wright 1983, 62. SINGULAR TERMS AND INFERENTIAL CRITERIA inference solely on our intuitions? Several philosophers don’t accept Dummett’s thesis in this respect. In her recent paper,27 Linda Wetzel maintains that this thesis is problematic in some points. More specifically, according to Wetzel, the thesis according to which “the recognition of simple patterns of inference is at the intuitive level” is absurd because it presupposes at the outset what should be explained (the notion of singular term) and because it seems to maintain the idea that a valid inference is one that preserves syntactic property. In my opinion, Wetzel’s point of view is not quite correct. Nevertheless, in order to determinate the validity of Dummett’s programme, it is crucial to analyze Wetzel’s objections and the reasons why, according to her, Dummett’s thesis is absurd. More specifically, I first examine Wetzel’s accusation about the supposed circularity of Dummett’s criteria and why the role that Dummett ascribes to the inference should not be in contrast with the “syntactic priority thesis” but also incompatible with Dummett’s theory of meaning. In the next paragraph, using an argument proposed by Bob Hale, I explore what is the correct way to understand Dummett’s thesis and argue that Wetzel’s accusation is unjustified. (i) Inference and Our Intuitions First, let us examine the thesis according to which the recognition of simple patterns of inference at the intuitive level is senseless and in contrast with the “syntactic priority thesis”. According to Wetzel, if this thesis were taken as involving merely the observation that we are intuitively able to pick out what we consider as the general notion of singular term, it would be absurd and in contrast with the fact that we have liable different intuitions about what is a singular term and whether a expression passes our tests. Examined in this prospective, letting recognition of the crucial patterns of inference be at intuitive level «leaves too much room for 27 See Wetzel 1990. 11 VERA TRIPODI one’s ontological intuitions to colour the results».28 In philosophy of mathematics, for example, a realist and a nominalist have different intuitions about numerical terms and about which phrase has an “existential import”. Suppose, for example, that we have to establish whether the expression “two” in “Two is a prime number” is a singular term. Criterion (I) is a test to decide whether a given phrase has an “existential import”. According to a realist, since she assumes that by definition a singular term always denotes something, “two” has an “existential import” and it is a singular term. On the contrary, a nominalist would refuse this point of view. So, the notion of inference, to which Dummett appeals to, presupposes a certain general notion of singular term. For this reason, the thesis about “the recognition of simple patterns of inference at the intuitive level” is, according to Wetzel, not quite plausible because it fails to furnish a general notion of singular terms. The result of all this is, according to Wetzel, that Dummett’s criteria are circular and fail to delimitate formally the class of singular terms. Wetzel defends a new approach to ontology that is different from that of Frege because based on «more give and take between syntax and ontology».29 As Frege points out in his analysis of number’s definition, we can consider numbers as objects because we are forced to recognize numerical terms as proper names. At the same time, the approach defended by Wetzel is different from the approach - that Dummett ascribes to Geach - according to which ontological categories are prior to linguistic categories. According to Geach, we first apprehend that a linguistic term stands for a certain entity (i.e., to which ontological category it belongs), then we can assign that expression to an adequate logic category in accordance with the category of the entity it stands for. Therefore, we can consider numerical terms as singular terms because we are forced to count numbers as objects. Wetzel’s suggestion is to assume a strategy that is midway between Frege’s approach and Geach’s approach. Such an approach is able to explain the relationship between syntax and 28 29 12 Ibid., 252. Ibid., 254. SINGULAR TERMS AND INFERENTIAL CRITERIA ontology in terms of a development so that «sometimes our ontological intuitions inform our view of syntax and sometimes syntactic principles determine our view of ontology».30 But, if Wetzel is right, the account of the relation of inference proposed by Dummett in the formulation of his criteria doesn’t meet the requirements of the “Syntactic Priority Thesis” for singular termhood. Examined in this prospective, Dummett’s criteria fail to furnish a general and language-neutral notion of singular terms, which would be the original aim of the “Syntactic Priority Thesis”, and the notion of valid inference, to which Dummett appeals to, cannot be considered as a good guide to what there is because it fails to assure the adequate generality to our recognitions of a correct inference since it is based on linguistic notion that should be explained. I think that Wetzel’s objections are also correlated to the accusation of a certain form of linguistic and ontological relativism which can be tracked back to Crispin Wright.31 This accusation states that the linguistic relativism concerning singular terms infects also the objects that, in Frege’s framework, are thought of as their non-linguistic correlatives. Linguistic relativism puts in danger the Fregean notion of object. As I explained earlier, even if Dummett does not explicitly state it, his criteria are to be thought of as relative to a particular language, because any general and language-neutral characterization of singular terms is impossible. The point is that, if we don’t have a languageneutral notion of a singular term, it is impossible for a Fregean to supply any language-neutral notion of object and to defend his idea of “International Platonism”. We could not defend a general notion of object but at best, as Wright observes, we shall be able to explain the notion of an English-object, a German-object, an Italian-object, and so forth. The linguistic relativism in Dummett’s criteria, as Wright notes, puts in danger also the possibility to apply them to other languages, because they could be subject to the indeterminacy a là Quine and «minefield concerning translation».32 30 31 32 Ibid., 254. Wright 1983, 62-64. Ibid., 64. 13 VERA TRIPODI Nevertheless, if Linda Wetzel were right, the challenge itself to distinguish singular terms from other expressions would be absurd. Actually, it is easy to undermine Wetzel’s objection. Dummett’s criteria, in spite of the objection voiced by Wetzel, are not circular because they are based on a pre-theoretical notion of valid inference that is shared by speakers. As I discussed earlier, according to Dummett, the assigning of an expression to the adequate logic category depends on some simple feature of its use and the application of syntactic criteria for singular term-hood requires the capacity to distinguish a valid inference from an invalid one. Dummett’s definition of singular term tells us that an expression is a singular term if it functions in a “particular way” in inferential processes, which are accepted as valid by speakers, of drawing a conclusion from premises. So, the point is what we mean with the expression “accepted as valid”. What does it mean that an inference is “accepted as valid”? Does the expression “accepted as valid” mean “accepted as valid by the community of speakers” or “accepted as valid by the theorist”? What is it that establishes the validity of an inference? It’s obvious that we have to distinguish between the theorist’s point of view and the community’s point of view and between a theoretical notion and pre-theoretical notion of valid inference. So, even if Dummett does not expressly explain whether his criteria are intended to be semantic or syntactic, what he probably means is that the inference is not considered in semantic term, but in terms of what kinds of inference are accepted as valid by speakers. In order to be able to apply the criteria for singular terms, we need to be able to distinguish a valid inference from an invalid one. For this reason, we need to assume that the speaker understands the distinction between singular terms and other expressions if she understands the validity of certain patterns of inference. Therefore, it is in this sense –that we need the notion of inference to explain the notion of singular term without appeal to semantic notions. Nevertheless, what speakers accept as valid concerns the community's shared sense of a certain expression and not something that is theoretically maintained as correct. It’s evident that the application of Dummett’s criteria requires that a speaker understands the concepts used in their 14 SINGULAR TERMS AND INFERENTIAL CRITERIA formulation, in particular the quantifier “something”. It’s also evident that what Dummett means with the expression “inferences accepted as valid” is “inference that the community recognize as valid”. Consequently, Wetzel’s analysis clearly indicates a misunderstanding of Dummett’s use of the expression “inferences accepted as valid” and can properly be rejected as illegitimate. Examined in this prospective, Wetzel’s view according to which Dummett’s criteria are circular can be considered not justified. As to the question of the indeterminacy à la Quine, we can observe that that difficulty is (at least in theory) not so hard to overcome if we can arrive at a sort of isomorphism among translations of different languages. Now let us examine the second objection concerning the validity of inference. Validity is a semantic notion: an inference is valid if it impossible that its premises are true and its conclusion is false, i.e. if it preserves the truth. In order to evaluate whether an inference is valid, we have to know certain semantic facts, that is to say we have to know in what conditions its premises and its conclusion are true. Wetzel is not convinced that it is possible to know the conditions in which a sentence as “Middlemarch was written by George Eliot” is true «without already knowing if Middlemarch was a singular term or a predicate».33 In order to establish if “Middlemarch” passes Criterion II, we have to investigate if the following inference which goes from “Middlemarch was written by George Eliot” and “Middlemarch was written by Marian Evans” to “There is something that it was written by Marian Evans and it was written by George Eliot” is valid. Supposing that “Marian Evans” and “George Evans” are rigid designators, this inference is valid because it impossible that its premises are true e its conclusion is false. Nevertheless, according to Wetzel, to determine if this inference is valid, we would have to know someway that George Eliot is Marian Evans or that “George Eliot” is co-referential with “Marian Evans”. But, requiring this kind of knowledge would make the Frege-Dummett programme delimitate singular terms circularly, because it supposes that only linguistic analysis can establish what objects there are and not. According to Wetzel, this is a problematic point to which Dummett doesn’t give a solution. 33 Wetzel, 1990, 253. 15 VERA TRIPODI (ii) Inference and the Analysis of Proposition On the basis of what Wetzel says, an opponent to Dummett’s programme could argue that if the thesis that “we can recognize the validity of an inference at intuitive level” were taken as involving that intuitively we pick out what we regard as the general notion of valid inference, it would be in conflict with the idea that the capacity of recognizing the validity of certain inferences depends on our being able to provide some sort of differentiation and to recognize the validity of inferential rules. If this is right, Dummett’s thesis “we can recognize if an inference is correct at intuitive level” should be also incompatible with one of his former claim: the recognition of valid (or invalid) inference requires at the same time also a capacity to construe a sentence as divisible in a particular way (the capacity, for example, to extract and to discern complex first or higher-order predicates within a sentence or functional expressions within a complex term). The argument against Dummett is the following. If we examine these criteria carefully, we see that they are not based on an intuitive recognition. According to Dummett, in order to settle the validity of an inference in which a sentence may be involved in premises or conclusion, it is essential that we are aware that «the constituents in which the sentence may be analysed may be complex incomplete expressions which we form from the sentence itself by omitting some other expression or expressions from it».34 Sentences are constructed in stages (step-by-step). A complex predicate is thought of as formed from a sentence by omitting some occurrences of a proper name. Therefore, it is essential to give «an account of inference in which it and some sentence involving the attachment of a sign of generality to the complex predicate in question both figure».35 An atomic sentence like “Socrates is wise” may be analysed in two different ways: as composed of the proper name “Socrates” and the incomplete first-level predicate “ is wise” or as composed of the first-level predicate inserted into the argument-place of another incomplete expression “
(Socrates)”. According to Dummett, it is also necessary to distinguish between two different notions of 34 35 16 Dummett, 1981, 65. Ibid., 29. SINGULAR TERMS AND INFERENTIAL CRITERIA predicate: «as it is required for Frege’s account of the expression of generality»36 and «as required in explaining the structure of atomic sentences».37 These two different notions of predicate reflect the idea that a predicate may be playing two different roles in a sentence. If we need to characterize the general form of an atomic sentence, the notion of simplex predicate is necessary and sufficient. For example: “Socrates is wise” is composed of the proper name “Socrates” and the incomplete first-level predicate “
is wise”. Otherwise, it is not possible to considerate a sentence like “Someone is wise” as composed of the expression “Someone” and the predicate “
is wise”. The sentences “Someone is wise” and “Socrates is wise” have two different logical forms. We need this notion to give «an account of inference in which it and some sentence involving the attachment of a sign of generality to the complex predicate in question both figure».38 The notion of complex predicate is necessary not to explain the way in which its constituents determine the sense of an atomic sentence, but to explain certain inferential uses of sentences. This kind of analysis allows us «to explain the general scheme of inference»39 to which we appeal and «state the general principle to which we are appealing when we recognize an inference»40 (e.g., from “Brutus killed Caesar” and “Anyone who killed Caesar is an honourable man” to the conclusion, “Brutus is an honourable man”). Complex predicates are necessary to explain the generality expressed by sentences. Examined in this prospective, the validity of an inference, given what Dummett says, seems to be based on a particular analysis of the proposition and not on the recognition of correctness or incorrectness of some patterns of inferences “at the intuitive level”. The recognition of the validity of inference seems to require that a speaker possess an understanding of the concepts used in their formulation, specifically an understanding of the quantifier “something”. If that were right, the validity of inference would 36 37 38 39 40 Ibid., 27. Ibid. Ibid., 29. Ibid. Ibid. 17 VERA TRIPODI depend on the possibility to preserve some syntactic and not semantic property. This is the subject of the next paragraph. 2.2 Hale’s defence of the Frege-Dummett programme Hale tries to defend Dummett against the accusation of linguistic and ontological relativism with the following argument. Hale states that it is necessary, in order to defend the syntactic priority thesis and to give an adequate characterization of singular terms, to distinguish between two different questions: (i) What is it for an inference to be valid? (ii) Which inferences are valid? How can they be recognized as such? Appealing to the “principle of necessary preservation of truth”, Hale argues that it is possible to give a general answer to the question (ii): an inference is valid if it is impossible for its premise (or premises) to be true while the conclusion is false. From a logical point of view, we need an account of language only as it relates to truth, because the notion of the validity of an inference relates precisely to truth. To satisfy this purpose, it is sufficient to provide an account of the structure of the sentence by a specification of its truth-conditions as connected with that structure. Nevertheless, this answer is neutral from a linguistic point of view and it does not supply an adequate guide for discriminating a valid inference from another. To do that, what we need is some guide which is strictly connected to a specific language and that makes use of information about the truth-conditions of a sentence of that particular language. According to Hale, the validity of inference thought of as language-relative is neither a threat nor an obstacle, because it is on the basis of this general account of (ii) that we can pick out and construe criteria for recognizing valid inference expressed in one particular language. Likewise, Hale states, the relation between (i) and (ii) is similar to the relation between the following questions: (iii) What is for an expression to function as a singular term? 18 SINGULAR TERMS AND INFERENTIAL CRITERIA (iv) How these expressions, which play such a function, have to be recognized? It is possible to give a general answer to question (iii): the primary function of a singular term is to refer and to identify an object. Several philosophers accept this general thesis. But this thesis is still not sufficient to pick out singular terms from other linguistic expressions. It is necessary to formulate specific tests for different languages. Even in this case, our choice in formulating inferential tests for singular termhood is suggested by our «general conception of a singular term as one whose function is to identify an object».41 Hale’s idea is that we are guided by general answer to (iii) to pick out singular terms in a unitary sense. It is the general answer to question (iii) that regulates the selection of specific sets of criteria for different languages. The point is that, Hale notes, the recognition of the validity of inference requires that a speaker possess an understanding of the concepts used in their formulation (specifically an understanding of the quantifier “something”), but a speaker can possess such understanding and at the same time lack «any general criterion by means of which he could decide, for any arbitrary expression of the language, whether if is a quantifier or not».42 So, according to Hale, Dummett seems to establish the following analogy. Understanding a sentence requires knowing its truthconditions and the capacity to evaluate the inference in which that sentence figures, but, generally, possession of this knowledge doesn’t demand a capacity to explain how the truth-conditions are determined. Likewise, the recognition of an inference as valid doesn’t require the capacity to furnish a theoretical account of the particular form it has. Hence, according to Hale, judging whether an inference is valid doesn’t require an ability to explain why it is so. To sum up: I’m not pessimist about the possibility of explaining the inference relation as Dummett suggests and, in spite of the objections voiced by Wetzel, I’m convinced that Hale succeeds in defending Dummett against the accusation that his criteria are circular and in conflict with the “syntactic priority thesis”. 41 42 Hale, 1987, 44. Hale, 1979, 289. 19 VERA TRIPODI References Dummett 1981: M. DUMMETT, Frege Philosophy of Language, Cambridge (Mass.) 1981. Frege 1988: G. FREGE, Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung über den Begriff der Zahl, Hamburg 1988. Hale 1987: B. HALE, Abstract Objects, Oxford 1987. Hale 1979: B. HALE, Strawson, Geach and Dummett on Singular Terms and Predicates, «Synthese», 42 (1979), 275295. Hale 2001: B. HALE, Singular Terms (1), in B. HALE - C. WRIGTH [ed. by], The Reason’s Proper Study. Essays towards a Neo-Fregean Philosophy of Mathematics, Oxford 2001, 31-47. Hale 2001: B. HALE, Singular Terms (2), in B. HALE - C. WRIGTH [ed. by], The Reason’s Proper Study. Essays towards a Neo-Fregean Philosophy of Mathematics, Oxford 2001, 48-71. Wetzel 1990: L. W ETZEL, Dummett’s Criteria for Singular Terms, «Mind», 99 (1990), 239-254. Wright 1983: C. WRIGHT, Frege’s Conception of Numbers as Objects, Aberdeen 1983. 20