In classical logic formulae are contents with truth-conditions. ... The explanation of this a form of judgement is the following: the propositional ... an extended presentation of the Bolzanian notion of judgement, which results essential in the.
Presuppositions, Assumptions, Premises∗ Giuseppe Primiero
Abstract In Martin-L¨ of’s Constructive Type Theory (CTT) judgements are given in terms of assertion conditions. They, in turn, furnish an account of the key notions of presupposition, assumption, and premise. The analysis spells out required epistemic and alethic constraints on the notion of truth for categorical and hypothetical judgements. The distinctions drawn throw new light on the proper connection between truth, meaning, and assertion conditions, from a constructive point of view.
Constructive Type Theory, assertion conditions, constructive theory of judgement, information.
1
The Theory of Judgement in CTT
In classical logic formulae are contents with truth-conditions. In Martin-L¨of’s Constructive Type Theory1 the derivable objects are instead judgements2 . In this framework, a judgement made is the assertion of the truth of a certain proposition, as in this schema: z
judgement }| { A is true. |{z}
proposition
The explanation of this a form of judgement is the following: the propositional truth is defined as the existence of a proof of the proposition; truth for propositions is thus ultimately given in terms of proof-objects: A true = proof(A) exists. Thus, proof-conditions play the central role in the constructive interpretation of propositions3 . ∗ 1 The first published expositions of the theory are in Martin-L¨ of (1975) and (1982); a first complete version of the theory is in Martin-L¨ of (1984). See also N¨ ordstrom, Petersson, Smith (1990). These are introduction to the polymorphic version; for an introduction to the monomorphic version, far more complete that the one presented in the second and third sections of this article, see Primiero (2006). 2 For the here shortly introduced basic distinction between judgements and propositions in the constructive perspective, see Martin-L¨ of (1987) and (1996). 3 The form of judgement holding in CTT reflects therefore the passage from the Aristotelian form “S is P” to the Bolzanian one, “A is true”. For the fascinating history of judgement and inference, presented in a complete way, see Sundholm (2002), sec. VII-VIII and (2006). For an extended presentation of the Bolzanian notion of judgement, which results essential in the development of a constructive theory of judgement, see Sundholm (1998a) and Proust (1989) for a specific comparison with the Kantian notion.
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Our focus in this paper is directed instead towards the basic explanation of judgements: commonly, judgements in CTT assert, categorically or under assumptions - i.e. hypothetically, that certain objects are elements of certain types. The basic notion here is that of an assertion-condition: such conditions must be given respectively for categorical and hypothetical judgements, and they will be explained in terms of the key notions of presupposition, assumption and premise. The aim of the next sections is thus to provide a concise introduction to the formal structure of the theory, where possible by underlining its conceptual and philosophical aspects rather than the well-known technicalities: this furnishes an interpretation of the assertion-conditions for judgements in the framework just sketched, thereby stressing a still little-known difference between the alethic and epistemic constraints therein involved.
2
Types and objects: categorical judgements
The basic notion of categorical judgement holding in CTT is analyzed in this section by means of the introduction of two main elements: - assertion conditions: according to the constructive perspective, judgements are analyzed in terms of the conditions for them to be asserted; - the notion of type: it reflects the idea of concept, replacing the predicate in the classical form of judgement. The aim is here to present the formal categorical judgements of the theory by combining these two aspects and thus showing the entire set of conditions on which such judgements rely. The philosophical analysis will introduce in a next section (4) “presuppositions” as the specific form of basic assertion conditions involved by categorical judgements. The concept of type is at the core of the theoretical framework of CTT: it is heritage of a long and important philosophical tradition, starting with Aristotle and his notion of category, being developed all along the history of logic, mainly in terms of the modern notion of type due to the Russellian theory of hierarchy of predications, based essentially on the many previously tentative formulations of the notion of function4 . An essential point for the constructive interpretation is already present in the Aristotelian treatise Categories, where such a notion is interpreted as the common structure between “what is” (ontology) and “what is said” (semantics)5 : there is an essential relation between a being’s essence and the predications performed in relation to it; if essence corresponds to meaning, the latter is not just given by the category of substance (the first of the Aristotelian categories); rather, categories determine in general the meaningful predications which can be performed in relation to the predicate involved. Thus the (correct) forms of predication are those illustrating a thing’s essence: categories are in this sense not just a being’s structure, but also the way meaning is 4 For
this history see e.g. Laan (1997) and Primiero (2006). In the conceptual development of the notion of type major roles are played by Ramsey, Carnap, Lesnievski, Adjukiewicz and Church. 5 Aristotle (Cat), par.2; Martin-l¨ of has presented this connection between Aristotelian philosophy and type theory during a course given at Stockholm University in the academic year 2005.
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preserved and expressed. Similarly, in CTT the notion of type is a term of the predicative structure and it expresses the quidditas of a certain object: in this sense the concept of type involved by a judgement is prior to that of element of which the type is predicated, because there are no objects which are not categorized or typed; once an object is given, it is given in terms of the type to which it belongs. This yields an order of conceptual priority within the theory: 1. types are conceptually prior to their elements; 2. the explanation (definition) of the notion of type is given constructively in terms of knowing what an object of that type would be, i.e. what one has to know in order to formulate a judgement saying that an object belongs to that type; 3. moreover, one must be able to state when two objects of that type are equal. The last two conditions express the criteria defining types: - application criterion: establishing what it means for a certain object a to belong to type α, i.e. it is the categorical judgement a : α; - identity criterion: establishing what it means for two objects to be equal objects of a type, i.e. it is the categorical judgement of identity a = b : α. The correspondence between sets and propositions, which goes back to Curry and was later extended by Howard6 , holds in the theory; the extension due to Martin-L¨of provides the formal introduction of these objects in terms of the unique category of types: set : type prop : type. It is clear from this that the notion of type comes conceptually before that of set or proposition, so that these are defined in terms of the former7 ; and, in general, the category of types comes before the category of objects-of-types. By introducing the type of propositions (prop : type), the judgement A : prop is justified by knowing how to construct a canonical proof-object of A out of parts (i.e. consist in the ability of formulating the judgement a : A), this being the application criterion that justifies the assertion that A belongs to the type of propositions8 . The formulation of this latter judgement allows the following inference: A
a:A true.
6 It is the well-known “Curry-Howard isomorphism”; Curry and Feys (1958), Howard (1980). See also Nordstr¨ om, Petersson, Smith (1990), cap.2. 7 The introduction of the type of sets and of propositions represents the basic conceptual idea of the monomorphic version of CTT, which therefore treats these two notions as equivalent in terms of types. 8 See Martin-L¨ of (1987). An act of knowledge brings as its object the assertion of a judgement having as a content the stated truth of a proposition. Such act of knowledge is instantiated by a proof, which must fulfill the notion of validity or correctness in order to furnish a real object of knowledge, i.e. a proper, correct judgement.
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This inference is simply explained by saying that the judgement that A is true is justified by the element which instantiates the proposition A, namely the object a (its proof). The notion of type is therefore that of a category to which an object may belong, representing the predicate in a judgement of the traditional Aristotelian form (or the property defining the members of a set). Thus, according to the intuitionistic/constructive framework, truth is defined as existence of a proof. In the case of A : prop, the judgement a : A states therefore that A is true by showing the object a which is a proof of A, and in case of A : set it states the constructive non-emptiness of the property by exhibiting an inhabitant. The correspondence to the classical (Aristotelian) form of judgement (“S is P”) is given by the use of the copula, which is not a two-place relation, rather a way to instantiate the category “P” by the element “S”, or to predicate “P” of the subject “S” (to declare that “S” has “P-ness”.). The correctness of the judgement is based on the knowability of an instance, an element or similar. The application criterion defines therefore the basic condition for such a judgement, and let us state the truth of the proposition A (or the inhabitness of A when A : set). Proof-conditions establish therefore truth-conditions for propositions. The definition of type is then completed by a second condition, represented by the identity criterion: first, one shows that element (proof) such that a : A, and it must be also shown if for a certain b it holds that a = b : A. By considering the formulation of the assertion conditions for categorical judgements in the way intended by CTT, one has a clear and precise way to answer the question “what do I need to know, in order to know a type?”. Nevertheless, if one also takes into account the essential relation of priority of types over objects introduced at point 1. of the previous list, one seems to need a way to account for the introduction of types into the knowledge frame of an agent before its use within a categorical judgement: this means that one should consider the assertion conditions for something being a type independently from the judgement saying that something is in that type (i.e. providing a definition for it - point 2. and 3. of the same list), and therefore a distinct explanation of type introduction is needed. After extending the formal analysis of the theory to the introduction of hypothetical judgements in the next section, we will proceed in furnishing such a required analysis.
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Dependent type and dependent object: hypothetical judgement
The theory of categorical judgements in CTT is completed by the introduction of dependent or hypothetical judgements: similarly to what has been done in the previous section, our aim here is to present assertion conditions for dependent types and dependent objects. In this way the clarification of a dependent judgement relays on the more basic explanation of which (kind of) conditions need to be formulated in order such a judgement to be known. The more important step is then accomplished in a later section (5), where one clarifies the nature of such conditions for dependent judgements, introduced in terms of “assumptions”. The basic case of a hypothetical judgement is that of a type declaration depending on one assumption (dependent type):
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[x : α] β : type saying that β is a type assuming that (provided that, under the condition that) the variable x is of the type α (i.e. that α is supposed to be inhabitated); this means, in turn, that provided an object a of type α, β[a/x] : type, i.e. β is a type whenever a is substituted for x in α. This expresses the application criterion for a dependent type. By the definition of a type given above in terms of the two criteria, one must also know that if a and b are equal objects of type α (a = b : α), then β[a/x] : type and β[b/x] : type are equal types (β[a/x] = β[b/x] : type). The identity criterion is thus stated as well. By this, it is clear that in order to clarify the nature of the dependent type of the form [x : α] . . . type one needs to state a novel form of assertion condition, one which is no more satisfied by the simple condition imposed over categorical judgements: in this second case, such a condition is built up out of the assertion conditions for α in connection to something being declared a type; this is obtained by formulating a hypothesis which assumes an element in the type α, but it does not state actual knowledge of any element a belonging to it. The assertion condition for a hypothetical judgement is therefore formulated in terms of a second judgement, previously understood but whose knowledge is of a special form, it being an assumption for the former (as it is shown by the variable in place of the proper proof-object). We want in the following to clarify the notion involved in such conditions: we will refer to it by using the term assumption (or, equivalently, hypothesis) and we explain it as an alethic notion: it expresses the value of a proposition maintained to be true in order the truth of another proposition to be formulated; moreover, it corresponds to the notion involved by sequents in natural deduction, to which one usually refers as antecedents. By means of this explanation in terms of assertion conditions, one understands moreover a dependent judgement as expressing the relation of consequence between an antecedent upon whose assertion the truth of a certain consequent proposition can in turn be asserted. The comparison with the usual formulation of hypothetical judgements is simple to show: A true ⇒ B
true,
is built up from the judgements “A true” and “B true”, together establishing the assertion conditions for the whole judgement (namely that if A is true, then B is true as well). This judgement shows the same schema as the previous formalized one: x:α⇒β
true.
In both cases, conditions for the dependent judgements to be true will satisfy the formal criteria expressed before: it must be established a certain b : B[x : A], such that whenever I know a : A, I am able to know b[x/a] : B[x/a] (for the sake of correspondence, A has been substituted for α and B for β). 5
Starting with hypothetical judgements with one assumption, it is possible now to generalize and to show judgements with an arbitrary number of assumptions (collected within contexts): their meaning is explained by induction on the number n of assumptions, supposing we understand the meaning of judgements with n − 1 assumptions. The hypothetical judgement [x1 : α1 , x2 : α2 , . . . , xn : αn ] α : type is understood by knowing that α is a type under substitution of x1 by a certain object a1 within the type α1 , and this under the assumptions x2 belonging to α2 up to xn belonging to αn , every of these assumptions holding under the substitution of xn−1 by a certain object an−1 . Once it is stated what does it mean for something to be a type under certain assumptions, we are able to state when two types are equal under the same assumption(s): [x1 : α1 ] α = β : type, means accordingly that for an arbitrary element a1 belonging to α1 , it holds that α[x1 /a1 ] = β[x1 /a1 ] : type. Once type declarations and equality judgements for types under assumptions are explained, the judgement that a certain object belongs to a type can also be made under an assumption (dependent object): [x : α] b:β means that for an arbitrary object a of the type α, b[a/x] is an object of the type β[a/x]. In the same way, we extend our dependent judgements by referring to the case where two objects are the same within the same dependent type, so that we can state the rule for substitution of objects in types: [x1 : α1 , . . . , xn : αn ] α[x1 /a1 ] = α[x1 /b1 ]. This rule for substitution states also respectively rules for substitution in equal types, in objects and equal objects9 . We are here interested in the meaning involved in the assumptions, in connection to the condition for the conclusion: whenever an hypothetical judgement is formulated, the hypothesis contains an assumed truth allowing to establish the truth of the consequent proposition, for which it therefore represents an assertion condition. Let us consider the following sentence: (1)
“If Plato is a man, Plato is mortal”.
Such a sentence is to be explained in terms of its assertion conditions: these can be expressed in the first instance by the hypothesis itself (assume Plato 9 See
also Nordstrom, Petersson, Smith (1990) for the complete formal aspects of the theory.
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to be a man, i.e. assume it is true that Plato is a man), and secondly by the understanding of the type involved in the conclusion (that we understand what does it mean to be mortal, which in turn amounts to a case of assertion condition of the form explained in the previous section). The truth of the dependent judgement can thus be clarified only by considering the connection between the antecedent and the consequent, so that it holds the relative consequence from the former to the latter (i.e. one is not considering separately the assertion condition for the hypothesis, that Plato is or is not a man). It is still to be clarified what kind of philosophical description is due for such kind of conditions, in order to have a fully developed and clear account of all the conditions mentioned.
4
Presupposition: an epistemic notion
Application and identity criteria for categorical judgements have been explained in order to consider what does it mean to know a certain type. By combining this claim with the mentioned order of conceptual priority, stating that types come conceptually before their objects, it seems that any judgement requires the formulation of an additional assertion condition: we will introduce this additional condition in terms of something which needs to be formulated in advance, in order to proceed in stating something to be known. The nature of this additional condition is here formulated in terms of a condition of meaningfulness as essential to any proper predication. The kind of condition one is referring to in this case is radically different from the one considered for dependent judgements, whose assertion conditions will be re-considered in the next section. In fact, whereas in the case of dependent judgements a proposition is assumed to be true in order the truth of another one to be stated, in the present case we refer to a judgement which needs to be already known: when the judgement J2 is understood only under the condition of knowing J1 , the latter is a presupposition of the former. This notion of presupposition is a common one, but it has to be distinguished from the idea of a proposition being true provided another proposition is assumed as true. One usually understands presupposition theory as explaining the basically intuitive notion that certain things can only be said if other things are taken for granted: this has been formulated in a semantic account, according to which one sentence semantically presupposes another if the truth of the second one is a condition for the semantic value of the first one to be true or false10 ; or else in a pragmatic account, according to which a speaker presupposes something at a given moment in a conversation just in case he is disposed to act, in his linguistic behavior, as if he takes the truth of it for granted, and as if he assumes that his audience recognizes that he is doing so11 . In the following we will stress the epistemic nature of presuppositions holding in CTT, by enforcing the role of knowledge acts involved in formulating them, and especially stressing the distinction with the alethic value ascribed to assumptions12 . 10 Also known as Strawsonian Presupposition, see Beaver (1997), p. 948. This notion is very similar to assumptions as explained in the next section. 11 Stalnaker (1947). 12 One can say that the epistemic notion of presupposition holding in CTT is just the constructive interpretation of the semantic notion intended by Strawson.
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The notion of presupposition holds in CTT first of all on the basis of the order of conceptual priority explained above. Judgements in CTT have the forms of predicating an object of a certain type; let J2 be the following judgement: J2 = a : α. In order to understand such a judgement, the type α involved in it must be declared in advance as apt to be predicated. This means that in order to know the judgement J2 , I have to know already the following judgement J1 13 : J1 =< α : type > . The relation instantiated by the two judgements < J1 > J2 in no way has to be considered similar to that of a dependent judgement as presented in the previous section, i.e. a judgement made under assumptions. The judgement < J1 > states instead that α is a category apt to be predicated, i.e. it declares the meaningfulness of any judgement involving that category: the predication of an object into that type can then turn out to be rightly or wrongly made, can be predication of an object truly asserted to be of α, or instead not falling under it14 . Meaningfulness amounts to predication aptness, while one has the right to predicate an object of a type if one knows that it belongs to that type. Type-declarations of the form α : type are presuppositions for categorical judgements, thus introducing meaningful concepts within the theory: type declarations represent ways of stating meaningful predications, and their epistemic role is obviously different from the judgements defining the type therein involved (namely in this case J2 , and eventually another judgement stating the identity criterion). Type declarations like α : type are therefore the first kind of presupposition within CTT, knowledge of which is a condition for another judgement to be known. The second form of judgement allowed in CTT, that of identity of objects within types (a = b : α), requires certain other judgements to be known, therefore acting as their presuppositions: < α : type > a = b : α. This example shows that not only type declarations, but also categorical judgements can be used as presuppositions. Extending this analysis to the case of dependent judgements, the type declaration Γ : context is always a presupposition (< Γ : context >15 ) for stating judgements of the form (dependent type) 13 We
introduce here the symbols < . . . > to denote presuppositions. amounts to the condition of rightness of the proof, see Martin-L¨ of (1987), and introduces the notion of error: for a treatment of error revision in the constructive frame of CTT, see Primiero (2006b). 15 By the calculus of contexts, the same holds true also for type declarations of the form γ : environment. This calculus is developed by Martin-L¨ of in his (1991). 14 This
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[Γ] α : type and (dependent object) [Γ] a : α. We present here the structure of the immediate presuppositions for the judgements derivable in a context: Judgements
Immediate Presuppositions
∅
none
x:α
< α : type >
[Γ] α : type
< Γ : context >
[Γ] α = β : type
[Γ], [Γ] < α : type >, < β : type >
[Γ] a:α
[Γ] < α : type >
[Γ] a=b:α
[Γ], [Γ] < a : α >, < b : α >.
According to this list, presuppositions take the form of: - type-declarations α : type Γ : context acting as conditions for meaningfulness, or they are - categorical judgements, such as a:α b : α. The latter, will have however the respective type-declarations as their presuppositions. 9
A presupposition is therefore a notion characterized in epistemic terms: the judgement < J1 > is a presupposition for the judgement J2 if the assertion condition of J2 depends on J1 ’s being known, and this means that J2 is a judgement-candidate once J1 is known. According to this explanation it is therefore also possible to spell out the difference between two uses of the term judgement16 : 1. judgement-candidate: a judgment-candidate requests only its basic assertion condition(s) to be formulated; 2. judgement made - asserted proposition: it is the result of an act of judging, the realization of a judgement-candidate. The relation between 1. and 2. is realized in the first instance by the connection between judgements and (related) presuppositions: a judgement is a candidate if its presuppositions are known; alternatively, once meaningfulness for a certain element is established (α : type), that element becomes apt for predication in a judgement (a : α). By this analysis, it clearly appears that presuppositions are explained in epistemic terms.
5
Assumption: an alethic notion
The notions of dependent type and dependent object have been introduced in section 3 in order to analyze the concept of hypothetical judgement. It is our aim in this section to spell out some properties of the logical elements involved by the assertion conditions of hypothetical judgements. The explanation will in particular consider the role played by the notion of assumption within the knowledge frame. Let us recall, in first instance, that the form of a dependent judgement in the formalism of CTT is the following: [x1 : α1 , x2 : α2 , . . . , xn : αn ] β true. In the case of hypothetical or dependent judgements, assertion conditions are determined by the antecedent part of the judgement, its assumptions. The hypothesis expresses a proposition assumed to be true, which in turn establishes the assertion condition for the consequent. It is therefore the assumption that a certain judgement J1 is true which makes a conclusion J2 true: this clearly results in something different from the previous case, where the knowledge of J2 was established on the basis of the previous knowledge of J1 . Assuming the truth of an assumption as a condition for the truth of the conclusion establishes the holding of the consequence from the truth of the former to the truth of the latter: A true ⇒ B
true.
The relation involved represent a dependent object because its assertion condition is that for the consequent proposition in dependency of the conditions 16 This
terminology is borrowed from Sundholm.
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regarding the antecedent. In this sense their value is properly alethic, and therefore they are to be distinguished from presuppositions, epistemically determined as judgements known. Under the propositions-as-sets correspondence, an assumption can be seen either as the declaration of a free variable in a set, or as an ordinary logical assumption, i.e. the assumption that a variable is a construction for a certain proposition. In general within assumptions, types act as ranges of variables. Now it can be made clear what proper value assumptions have: an alethic assumption presupposes (i.e. has a presupposition in the previous sense) the predication aptness of the type contained in it (i.e. it requires the formulation of the related type-introduction), but it does not presupposes knowledge that a certain element belongs to that type (in fact, the assumption can turn out to be false, i.e. the type can be empty). Assumptions in use within CTT allow therefore the (conditional) truth of a certain proposition, without being actually based on a knowledge content in terms of a construction. This means in turn that in a dependent judgement the assumption provides the assertion condition for the conclusion, whereas the former does not rely on any proper construction satisfying its assertion conditions: the only condition satisfied will be that regarding its presuppositions (i.e. declarations for the type involved by the context); for this reason not (canonical) proof-objects but variables are sufficient in the declaration of an assumption. In this sense one is just here interested in the information received by such predication, representing the assertion condition for the truth of the consequent proposition17 . The alethic nature of assumptions in the form “assume A to be true” is clearly the usual understanding of derivations in natural deductions: when one aims to demonstrate a certain implication A ⊃ B, starting from the antecedent that A is true, this does not exclude in any way that the set of proofs for A may be actually empty; this is the case for example of an assumption using the type ⊥, for which obviously no proof-object in the proper sense can be provided18 . Assumptions recollect then the case of judgements assumed in order some other to be done, on the basis of a variable declaration. But a major source of error and misunderstanding in relation to the role and nature of assumptions, especially in connection to the previously presented notion of presupposition (and partially also with the notion of premise, next to be presented) is the special case in which an assumption expresses the more stronger case of assuming “something to be known”, which brings a quite different meaning to the involved logical notion. The first kind of assumption can be labelled as properly alethic assumption; the latter kind of assumption, namely an assumption that something is known, has clearly an epistemic value and therefore it is often conflated with the notion of something needed to be known for something else to be known (our notion of presupposition) or with the notion of something which is known and therefore brings knowledge of something else (our notion of premise): it can be labelled as a version of epistemic assumption 19 . To assume something to be 17 Starting from here it is possible to draw a constructive definition of the notion of information for knowledge systems. See Primiero (forthcoming). 18 Cf. Sundholm (2004), p.451. 19 Precisely this second kind of assumption is resembled by the commonly intended notion of presupposition as something required by the assertion of something else. Here therefore the distinction between these notions is of a remarkable importance. For the introduction of this distinction see Sundholm (2004).
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known, means in the first instance to make a stronger kind of alethic assumption, i.e. it refers to assuming something to be really true, i.e. an assumption of a knowable judgement or else the assumption about possessing a proof object for a certain content, and therefore it can be accounted as a special case of the proper alethic kind of assumption: in this case what is presented in a context for an hypothetical judgement is an expression of the form [a : A], from which a dependent judgement is built up. What let us to present these expressions in connection to hypotheses rather than presuppositions is the role they play: i.e. they provide assertion conditions for stating dependent judgements, thus representing special cases of assumptions. In natural deduction these expressions are equivalent to implications presenting closed derivations for the antecedent. Therefore, whereas in this last case of epistemic assumptions one is relying on the fact that it is really possible to provide a proof for the proposition used as antecedent, the case of properly alethic assumptions [x : A] does not necessarily involves the same property. As shown already in the previous section when the list of presuppositions for judgements was introduced, a hypothetical judgement is made in a context which collects the assumptions for that judgement. A list of assumptions can, on the other hand, have its own presuppositions. To make the distinction clear it is worth to present here an example within the formalization of CTT. The hypothetical judgement [x1 : α1 ] α2 : type states that α2 is a dependent type under the assumption that a certain object a1 substituted for x1 belongs to the type α1 . This assumption is itself based on a presupposition, namely the judgement < a1 : type > . The presuppositions on which every set of assumptions is based are those stating that any of the types involved in the context are apt to be predicated, where predication aptness indicates being at disposal for (right or wrong) predication. When one treats with alethic assumptions (and the related presuppositions), no assertion is made about the element involved, only their truth is assumed in terms of the meaningfulness of the expression: information is given which allows to get knowledge.
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Premise: clarifying the nature of inferential process
The basic notion of assumption introduced in the previous section plays a specific alethic role and it has a different epistemic counterpart, both needing to be clarified in terms of the logical relations they build up for hypothetical judgements and to be distinguished form others notions. This will also clarify the kind of assumptions one is using in CTT, and what else is involved by the notion of logical consequence. The structure explained for hypothetical judgements, in which assumptions are formulated, resembles that of an inference, preserving knowledge from premises 12
to conclusion. It is in the distinction between assumptions and premises that an essential ambiguity related to the term “inference” can be solved: the (different) epistemological values carried by these expressions will clarify two senses in which one refers to such a term20 . When we refer to assumed knowledge as a proviso for the knowledge of a certain conclusion, i.e. when we are using epistemic assumptions, we consider therefore the conditions for a certain inference to be valid, namely an inference mode (Schlussweise); an inference intended in this sense is an inference-schema: I − schema :
J1 . . . Jk J.
The truth of such a schema is reflected in the relation between the very same contents used as epistemic assumptions and the given conclusion. Quite differently, a proper inference is a discursive act of judging instantiating such a mode, which is valid in terms of preservation of knowability, and it is based on the formulation of premises, i.e. instances of those epistemic assumptions. Premises of an inference are to be thought of as judgements known, on the basis of such knowledge another judgement being known. The example (1) in section 3 presented an hypothetical judgement, in which the conditions for the consequent to be true was the truth of the assumption. The notion of inference shows instead the preservation of knowledge from premises to conclusion, so that it is clear that the well-known following inference says more than the mentioned example (1): (2)
“Every man is mortal. Plato is a man. Therefore Plato is mortal.”
Every premise within actual inferences are facts known on whose basis it is possible to assert the conclusion. The validity of an inference is established on the basis of preservation of knowability, and therefore we are here dealing with the epistemic value expressed by the known judgements contained in the premises. It is therefore clear that the act of inference .. .. . . J1 . . . Jk J says that the judgement J is known (knowable) given that (because) all of the premises J1 . . . Jk are already known21 . An inference is to be distinguished by the figure or mode (I−schema) according to which it is carried out: such a figure is based on epistemic assumptions, and it says that assumed certain knowledge, a resulting known judgement J is given; one can talk about the validity of the schema by stating that a certain judgement is known only under assumption of knowledge of other judgements (i.e. by satisfying epistemic assumptions); whenever an actual act of inference 20 In
Sundholm (2006), sec.1, p.5, the author explains this difficulty. is to be noticed that premises need to be realized by proper proof-objects, presented in the schema by vertical dots. 21 It
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is performed instantiating that schema, one will speak about known premises leading to a known conclusion. What is realized by an act of inference is thus the preservation of knowability from premises to conclusion, like in the case of an act of inference instantiating a modus ponens figure: ` (A true ⇒ B true) ` A true ` B true where the first A is an assumption within the hypothetical judgement A ⇒ B (if A true, then B is true): the whole judgement (A true ⇒ B true) works as a premise for the inference; the second A is a proper premise, given (we know that) A (is) true, then the conclusion that B is true may validly be drawn22 . To sum up the various forms of consequence relation, the holding of a consequence (B is a consequence of A, or A entails B)23 , (A ⇒ B) holds, demands for its verification a functional object f of the following type: f : (P roof (A))P roof (B) : type. The case of a conditional statement (if A is true, then B is true, involving an assumption)24 is different, demanding for its verification a dependent proof b of B provided that x is a proof of A, i.e. it requires for its validation an object of the following dependent type: P roof (B)(x : P roof (A)) Finally, also a different object is what requested for the validation of an implication (A therefore B), which is structured as a propositional connection of the following form (A ⊃ B) true, whose validation is provided by a canonical proof-object of the form ⊃ I(A, B, (x)b) : A ⊃ B. It is thus clear that hypothetical judgement, implication and inference are different objects satisfied by the same conditions (but not in the same way). In fact they are equi-assertible notions, i.e. if one of them can be stated, the others (containing corresponding propositions or equivalent judgements) can be as well: an implication can be transformed into the related hypothetical judgement or consequence but, as shown, the assertion conditions for them are not the same25 . A solution of the mentioned ambiguity between different uses of the 22 The explanation of sequents in terms of the validity of inference and the possible problem of infinite regress have been spelled out in Sundholm (2002). 23 What in Sundholm (2006) is called “closed consequence”. 24 This is also what in Sundholm (2006) is called an “open consequence”. 25 Sundholm (2006) considers moreover that they are refutated by the same counter-instance, namely a pair composed by a proof-object of the antecedent together with a dependent proofobject which makes true the open consequence from the conclusion to the absurdity.
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term “inference” is thus explainable also on the basis of a correct understanding of the notions of assumption and premise. The two terms refer to different epistemological states, the former carrying an alethic value and having a quite particular epistemic counterpart; the second must be considered in the light of a more stronger condition, namely in terms of knowledge already acquired. In the sequel some related issues are presented, connected to the analysis of truth and knowledge within the constructive viewpoint.
7
Final Remarks
The main aim of this paper was to analyze the meaning of terms often ambiguously used in the philosophical study of logic. Such considerations can be regarded independently from the framework of CTT in which they have been exposed. Nonetheless, it seems clear that such frame has furnished a proper formalization which let us draw the right distinctions in terms of the epistemic properties of those notions. Such properties can be summarized as follows: - a presupposition expresses a basic condition for further knowledge; this condition has to be considered epistemic in that it is knowledge required by the assertion of another judgement, and it coincides (in its basic property) with stating meaningfulness; - an assumption makes explicit the condition on whose basis one is allowed to declare some hypothetical truth: here the constraints are related to truth transmitted and preserved, and the value attached to assumptions is purely alethic; a particular kind of assumption can be recognised as epistemic in the particular case of assuming something to be known; alethic and epistemic assumptions are distinguished also in the kind of inferences one can build from them; - a premise produces the actual instantiation of an epistemic assumption; premises are judgements (actually) known, transmitting knowability to new judgement(s). This structure clarifies a deep problem related to the conflating of logical objects such as inference, consequence and implication, showing that misunderstandings are due to an incorrect determination of knowledge and truth, and of knowledge and information content. The constructive frame represents in this way a solid basis for explaining and clarifying how these notions must be understood26 . CTT in particular offers a rigorous formalization and a deep philosophical justification, by means of which the definition of truth in terms of knowledge of proof(-objects) let to consider also assertion conditions dependently determined. In developing a demonstration, knowledge is processed and augmented, and this has its alethic counterpart in the validity of the process in terms of truth preserved. CTT clarifies the suggested ambiguity of the term inference by distinguishing the elements involved: the knowledge states resulting from the use either of proof-objects in terms of premises or rather of assumptions for dependent objects are quite different. This analysis suggests a final consideration: assertion conditions for judgements do not coincide with existence of proofs, 26 See
Sundholm (1998).
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which instead determine uniquely truth-conditions for propositions. Assertion conditions as analyzed within the constructive frame extend the epistemic basis on which truth is defined in terms of proofs, by means of a formal analysis of meaningfulness and hypothetical truth as the informational support on which knowledge is acquired.
References [1] Aristotle: Cat Categories. [2] Beaver, D.I.: 1997 Presupposition chapter 17 in Handbook of Logic and Language eds. Van Benthem, J., Ter Meulen, A., pp.939-1008 North Holland, Elsevier, Amsterdam, The MIT Press, Cambridge, Massachussets. [3] Curry, H.B., Feys, R.: 1958 Combinatory Logic, vol.I Amsterdam.
North Holland,
[4] Heyting, A.: 1930 Die formalen Regeln der intuitionistischen Logik Sitzungsberichte der preussischen Akademie von Wissenschaften, Phys.-Math. Klasse, pp.42-56. [5] Heyting, A.: 1956 Intuitionism: an introduction North Holland, Amsterdam. [6] Howard, W.A.: 1980 The formulae-as-type notion of construction in Seldin J.P., Hindley J.R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and formalism pp.479-490 Academic Press, London. [7] Kolmogorov, A.N.: 1932 Zur Deutung der intuitionistische Logik Mathematische Zeitschrift, n.35, pp. 58-65. [8] Laan, T.: 1997 The Evolution of Type Theory in Logic and Mathematics PhD Dissertation Thesis, Tecnische Universiteit Eindhoven Printed and bound by Print Partners Ipskamp, Enschede. [9] Martin-L¨of, P.: 1975 An intuitionistic theory of types: predicative part in Rose, H.E. and Shepherdson J.C. (eds.) Logic Colloquium ’73, pp.73-118 North Holland, Amsterdam. [10] Martin-L¨of, P.: 1982 Constructive Mathematics and Computer Programming in Cohen L.J., Lo´s J., Pfeiffer H. and Pdewski K.P. (eds.) Logic, Methodology and Philosophy of Science IV, pp.153-175 North Holland Amsterdam NY -Oxford, PWN Polish Scientific Publishers, Warszawa. [11] Martin-L¨of, P: 1984 Intuitionistic Type Theory Bibliopolis, Napoli. [12] Martin-L¨of, P.: 1987 Truth of a proposition, evidence of judgement, validity of proof Synthese 73, pp.191-212. [13] Martin-L¨of, P.: 1991 Contexts and Environments unpublished lectures held in Stockholm, notes by P.M¨aenpa¨a and A.Ranta.
16
[14] Martin-L¨of, P.: 1994 Analythic and Synthetic Judgements in Type Theory in Parrini, P. (ed.), Kant and Contemporary Epistemology, pp. 87-99 Kluwer, Dordrecht. [15] Martin-L¨of, P.: 1996 On the Meanings of the logical constants and the justifications of the logical laws lectures given in Siena, 6-9 April 1983 at (Scuola di Specializzazione in Logica Matematica - Universit´a degli Studi di Siena) in Nordic Journal of Philosophical Logic, vol. 1, n. 1, pp.11-60 Scandinavian University Press 1996. [16] Martin-L¨of, P.: 1998 Truth and knowability: on the principles C and K of Michael Dummett H.G.Dales and G.Oliveri (eds.), Truth in Mathematics Clarendon Press, Oxford. [17] Nordstr¨om, B., Petersson, K., Smith, J.M.: 1990 Programming in MartinL¨ of ’s Type Theory: an Introduction Oxford University Press, Oxford. [18] Primiero, G.: 2006 Information and Knowledge - A constructive Approach to the Philosophy of Information and a Solution to the Problem of Analyticity PhD Dissertation, Dep. FIERI, University of Palermo. [19] Primiero, G.: 2006b Belief Revision in Constructive Type Theory Logica 2005 Yearbook (M. Bilkova & O. Tomala eds.), pp.177-188 Philosophia Publ., Czech Academy of Science, Prague. [20] Primiero, G.: f orthcoming An epistemic constructive Definition of Information under revision for publication. [21] Proust, J.: 1989 Questions of Form - Logic and the Analytic Proposition form Kant to Carnap University of Minnesota Press, Minneapolis. [22] Ranta, A.: 1994 Type-theoretical Grammar Clarendon Press, Oxford. [23] Smith, J.M.: 1983 The identification of proposition and types in MartinL¨ of ’s Type Theory Foundations of Computation Theory, Proceedings of the Conference, pp. 445-456. [24] Stalnaker, R.: 1974 Pragmatic presuppositions in M.Munitz and P. Unger (eds.) Semantics and Philosophy, pp.197-213 New York University Press, New York. [25] Sundholm, B.G.: 1998 Inference, consequence, implication: a constructivist’s approach Philosophia Mathematica, 6, pp.17-194. [26] Sundholm, B.G.: 1998a Inference vs. Consequence in Logica Yearbook 1997 (Timothy Childers, ed.), pp.26-35 Philosophia Publ., Czech Academy of Science, Prague. [27] Sundholm, B.G.: 2002 Varieties of Consequence in The Blackwell Companion to Philosophical Logic (D. Jacquette ed.), pp.241-255 Blackwell Publishers, Oxford. [28] Sundholm, B.G.: 2004 Antirealism and the Roles of Truth in Handbook of Epistemology, Niiniluoto, Sintonen and Wolenski eds., pp.437-466 Kluwer Academic Publishers, pp.437-466. 17
[29] Sundholm, B.G.: 2006 A Century of Judgement and Inference: 1836-1937 in L. Haparanta (ed.) The History of Modern Logic Oxford University Press, Oxford.
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