sign in sign up
<<
Home , Axiom of reducibility, Impredicativity, Propositional function

Principle of acquaintance and

Brice Halimi Université Paris Nanterre

15 mars 2018 Russell-the-epistemologist is the founding father of the concept of acquaintance. In this talk, I would like to show that Russell’s theory of knowledge is not simply the next step following his , but that his logic (especially the system of ) can also be understood as the formal underpinning of a theory of knowledge. So there is a concept of acquaintance for Russell-the-logician as well. Principia Mathematica’s logical types

In Principia, Russell gives the following examples of first-order propositional functions of individuals:

φx, ψ(x, y), (y) ψ(x, y),...

Then, introducing φ!zb as a variable first-order propositional of one individual, he gives examples of second-order propositional functions:

f (φ!zb), g(φ!zb, ψ!zb), F(φ!zb, x), (x) F(φ!zb, x),

(φ) g(φ!zb, ψ!zb), (φ) F(φ!zb, x),... Then f !(φb!zb) is introduced as a variable second-order propositional function of one first-order propositional function. And so on. A possible value of f !(φb!zb) is . . . φ!a. (Example given by Russell.) This has to do with the fact that Principia’s schematic letters are variables: It will be seen that “φ!x” is itself a function of two variables, namely φ!zb and x. [. . . ] (Principia, p. 51)

And variables are understood substitutionally (see Kevin Klement, “Russell on Ontological Fundamentality and Existence”, 2017). This explains that the language of Principia does not constitute an autonomous formal language. Principia’s propositional functions are either variables or dummy terms directly standing for “concrete” propositional functions, in contrast with what happens in modern logic à la Tarski. Principia’s propositional functions display all the possible particular functional forms that one can specify without mentioning a concrete propositional function. The type of a propositional function does not only display the number and respective orders of its arguments (real variables), but also the number and respective orders of its apparent variables; and, in case its arguments are functions themselves, it displays the number and respective orders of the arguments and apparent variables of those arguments, and so forth. To that extent, a type codes inductively a quite fine-grained logical structure. It can be formalized as:

a a r r ht1 ,..., tm; t1,..., tni

a where the tj ’s are the respective types of the apparent variables, and r the ti ’s are the respective types of the real variables, ordered according to their occurrences. Examples

The type of individuals is o := h−; −i.

The type of φ!x is (o) := h−; oi.

The type of (x) ψ(x, yb) is ho ; oi.

The type of (φ) f !(φ!zb, x) is h(o); oi.

The type of (φ) f !(φ!zb, ψ!zb) is h(o); (o)i.

The type of (φ) f !(φ!zb, ψ!zb, x) is h(o); (o), (o), oi. The order of a propositional function measures the depth of the presuppositions involved in its expression. It can be determined directly from the type of the function:

I the order of an individual is 0 a a r r I the order of a function ϕ of type ht1 ,..., tm; t1,..., tni is a r max(|ti |, |tj |)1≤i≤m + 1. 1≤j≤n

A propositional function is said to be predicative if its order is the lowest compatible with having its arguments. For instance, (φ) f !(φ!zb, x) is not predicative: it is of order 2 (because of the quantification over φ) whereas its arguments are of order 0. By extension, a variable is predicative when all its possible values are predicative. Russell’s “no loss of generality” claim

The Introduction of the (first edition of) Principia suggests that only predicative variables should be adopted in the formal language and that there is “no loss of generality” in doing so: It is important to observe that all possible functions in the above hierarchy [the hierarchy of functions of increasing orders] can be obtained by means of predicative functions and apparent variables. [. . . ] Thus we need not introduce as variables any functions except predicative functions. (Principia, pp. 53-54)

Apart from Church, most commentators acknowledge that all Principia’s variables are predicative. Yet this is puzzling. Firstly, Russell does not provide us with any clear justification of his claim.

Secondly, if all schematic letters are variables and all variables are predicative, non-predicative terms are neither variables nor possible values for variables (there are no variables appropriate to them), so what is their purpose?

An easy way out consists in claiming that all terms other than individuals are predicative. (This is Gregory Landini’s reading of the formal syntax of Principia.) Two questions

The problematic existence of non-predicative terms that could not instantiate any variable prompts two questions: 1. Are all terms predicative? 2. Are all predicative terms (predicative) variables?

Landini answers “yes” to both questions. (“Circumflexion was not a term forming operator in Principia.”)

According to Landini, Russellian matrices φ!(x), f !(φ!zb, x), . . . , are free variables in schematic letters’ clothing.

Non-predicative wffs (well-formed formulas) can be formed by quantifying some variable but, since only predicative variables are allowed, such wffs are not terms and correspond to terms only by virtue of the axiom of reducibility. The axiom of reducibility (AR)

` ( E ψ) φx ≡x ψ!x It has much to commend it:

I Without (AR), quantification and types would be much more complex: there would be as many second-order variables as there are types of first-order propositional functions.

I Defining identity would be impossible, because all first-order propositional functions do not make up a legitimate totality (Principia, p. 49 and p. 57). Thanks to (AR), quantification over all predicative first-order propositional functions amounts to quantification over all propositional functions whatsoever.

I The theory of classes is basically equivalent with the adoption of (AR): φx ≡ x  z˘(φz) However, the axiom of reducibility is not a justification of Russell’s “no loss of generality” claim, or only retrospectively. (It is an axiom, not a rule of syntax.) A puzzle

It will be seen that “φ!x” is itself a function of two variables, namely φ!zb and x. [. . . ] (It should be observed that when φ is assigned [?], we may obtain a function whose values do involve individuals as apparent variables, for example if φ!x is (y) ψ(x, y). But so long as φ is variable [?], φ!x contains no apparent variables.) (Principia, pp. 51-52)

If quantification is substitutional, how to conceive of the “assignment” of φ to a propositional function involving apparent variables? Moreover, if all variables are predicative, how could their possible values be non-predicative? A solution may come from the epistemic contextuality of logical analysis. Nicholas Griffin, “Russell on the Nature of Logic (1903-1913)” (1980): [. . . ] Russell’s is to some extent context sensitive: for example, an item which, in one context, may be taken to be simple may, in another, turn out to be complex; and thus terms like ‘individual’ or ‘first-truth’ are not stable across contexts [. . . ]. An example occurs in connection with the word ‘Socrates’ which when used by Socrates himself denotes a simple individual of Socrates’ acquaintance; whereas, when used by someone who has never met Socrates, it is a complex hidden description to be analyzed by Russell’s theory of descriptions. Thus Socrates is a possible value of /xb is an individual/ only for Socrates himself, for others, with no acquaintance with Socrates, Socrates is not a possible value for that function [. . . ]. In general, since different people are acquainted with different items, the range of total variation for functions like /xb is an individual/ will be different for different people. Thus it is intolerable [to Quine and Sommerville] to treat such functions as propositional functions of logic. Russell indeed distinguishes between “Socrates is human” as uttered by Socrates himself, and as uttered by us who only know Socrates by description and thus for whom the sentence contains apparent variables.

In the same way, Russell suggests that “Napoleon is a great general” can be analysed either as G(N) or as (φ) f (φ!zb) ⊃ φ!(N) where “f (φ!zb) stands for “φ!zb is a predicate required in a great general.” Epistemic realizations of logical types

Let’s consider for instance the expressing that a certain individual a is green. If I am acquainted with both the individual a and the universal Green, I shall say: (1) Ga (elementary truth, first-order proposition).

Having only access to a as to “the F:” (2) ( E x) (y) Fy ≡ y = x & Gx (first-order proposition).

Having only access to green as to “the colour of grass:” (3) ( E φ) (ψ) CoG(ψ) ≡ ψ = φ & φa (second-order proposition). Combining both descriptions:

(4)

E ( E x) ( φ) (ψ) (CoG(ψ) ≡ ψ = φ & (y) φy ≡ y = x & φ!x (second-order proposition). In the third case, ‘green’ is nothing but an incomplete symbol. Nevertheless, its occurrence in any context will generate a second-order quantification. In view of this fact, green may be identified with a non-predicative second-order propositional function with one individual argument. The sentences (1)-(4) above express involving propositional functions of different types, even if these propositions correspond, in some way, to the “same” state of affairs.

Principia’s ramified type theory allows us to distinguish among between the sentences (1) and (4), even between (3) and (4) —hence the utility of fine-grained types.

The logical structure of the situation will vary from subject to subject, and so will, more generally, the realization of the whole theory of logical types. Epistemic diagrams

Depending on the epistemic perspective, the stock of individuals and predicative terms will vary. The epistemic diagram of one epistemic subject will be:

I Individuals: a, Being a great general

I Predicative propositional functions of individuals: x is a great general

I Second-order propositional functions of individuals: x is “green,” i.e., x has the color of grass, whereas that of another subject will be:

I Individuals: a, Green

I Predicative propositional functions of individuals: x is green

I Second-order propositional functions of individuals: x is “a great general,” i.e., x has all the predicates that make a great general. (The quotation marks are meant to express the fact that the subject does not speak for herself, but only uses a predicate that she borrows from another subject’s language.) The puzzle solved

It will be seen that “φ!x” is itself a function of two variables, namely φ!zb and x. [. . . ] (It should be observed that when φ is assigned, we may obtain a function whose values do involve individuals as apparent variables, for example if φ!x is (y) ψ(x, y). But so long as φ is variable, φ!x contains no apparent variables.) (Principia, pp. 51-52)

Formally considered, φ!x here is a variable, even though its possible counterpart in some epistemic realization could be (y) ψ(x, y) —where ‘ψ(x, y)’ here is not another variable but stands for some actual . Distinguishing two contexts —the and its application to some epistemic realization— allows one to shed some light on a passage otherwise very obscure. (The bracketed sentence is actually a typical example of mixing.) It also makes it possible to better understand the status of non-predicative propositional functions: Non-predicative terms as the equivalent, within one system, of what can be seen, within another system, as a predicative function.

For example, what an epistemic subject S1 mentions as a predicate, being a great general, can only be, to another subject S2, a non-predicative second order functional term, namely the property of having all the predicates required in a great general. The syntactic possibility of non-predicative propositional functions accounts, not for non-predicative entities, but for the shadows that other epistemic perspectives leave upon mine. Non-predicative terms as translation patterns

Imagine that some epistemic subject, other than me, mentions (what are to her) predicative attributes with which, however, I am not acquainted. I am nonetheless able to interpret these in my epistemic realization, but, in doing so, I cannot but understand them as involving some quantification that makes them non-predicative. I will then use a non-predicative functional term F (say, of type t) only as a proxy for what is, in some other epistemic perspective than mine, a predicative attribute. Accordingly, any formula (ϕt ) A(ϕ) containing an apparent non-predicative variable of type t is interpreted as asserting that every formula A[F/ϕ] is true for any non-predicative such term F. The idea is thus to defend a substitutional semantics for non-predicative variables, based on a “translational” semantics for non-predicative complex terms, based itself on an epistemic understanding of type theory. In that view, the wffs of Principia keep track of properties that some subject cannot identify with any predicate that would be available to her.

To get back to Russell’s example,

G(N) (“Napoleon was a great general”), as uttered by some subject S1, is understood by some other subject S2 as:

(φ) f !(φ!zb) ⊃ φ!(N). where the propositional function f !(ψ!zb), which stands for “being a predicate required in a great general,” is a predicative second-order propositional function available in S2’s epistemic perspective. The bracketed non-predicative term [...]S2 spawn by the rendition (logical analysis) of “being a great general” works as a translation rule: [ ! ] :≡ [(φ) !(φ! ) ⊃ φ! )] G x S1 x f zb x S2

This rule introducing “G!x” adds no non-predicative terms to S2’s perspective: G is only emulated from S2’s point of view.

Contrary to f !(ψ!zb) (which is part of S2’s perspective), the complex term translating G is introduced in S2’s language as a mere symbol in order to account for some of S1’s sentences.

Still, S2 needs to be able to quantify over such symbols, which means using variables having in fact non-predicative terms as possible values, but only in a substitutional way and with respect to some translation rule. Otherwise put, non-predicative functional terms (obtained by circumflexion) are only the nominal equivalent, within one epistemic perspective, of what is accessible, within another epistemic perspective, as a predicative function. Accordingly, non-predicative variables are introduced, but only to render predicative quantification as used in some other epistemic realization. In the realm of any epistemic realization, predicative terms (of whatever order) and proper names correspond to the epistemic basis (individuals and universals or attributes) of that realization —hence, to objects of acquaintance. Non-predicative terms only provide for the phenomenon that arises as soon as the match-up of two different epistemic perspectives is expressed. Still, they are perfectly legitimate terms in the formal system of ramified type theory. The two questions answered

1. Are all terms predicative? 2. Are all predicative terms (predicative) variables?

Answering these questions requires distinguishing between the formal system of Principia and its epistemic realizations:

epistemic realizations formal system Question 1 Yes No Question 2 No Yes

(AR) is generally understood as a comprehension axiom, of the form: ( E ϕ(t)) (x t ) ϕ(t)(x t ) ≡ A(x t ) where ϕ(t) is a predicative variable not free in A. One could allow for comprehension axioms with non-predicative

variables ϕ, such as: ( E ϕh2;(o)i) (x o) ϕh2,(o)i(x o) ≡ A(x o)

In contrast, (AR) is a comprehension axiom where non-predicative terms A occur as dummy wffs matching the order of a predicative term. The axiom of reducibility (AR) reconsidered

Remember the translation rule about “being a great general:”

[ ! ] :≡ [(φ) !(φ! ) ⊃ φ! )] G x S1 x f zb x S2

I simply claim that (AR) ` ( E ψ) ψ!x ≡ φx should be understood along such lines. In that light, the (AR) states that every non-predicative term corresponds to some object of acquaintance in some (other) epistemic perspective. In other words, (AR) is a principle of surrogate acquaintance. A complication

Let us consider again the rule

[ ! ] :≡ [(φ) !(φ! ) ⊃ φ! )] (?) G x S1 x f zb x S2 (The equivalence holds even though, among all the predicates φ quantified over in (?) from S2’s point of view, there is some for which S1 does not have any predicative counterpart.)

Suppose that f !(φb!zb), available to S2, also belongs to S1’s epistemic perspective. Then, for any particular predicate F also belonging to that perspective, G!x ⊃x f !(F!zb) ⊃ Fx has to be valid in S1’s realization.

If, on the contrary, f !(φb!zb) is not part of S1’s own stock, then: [ !(φ! )] :≡ [Θ(φ! )] (??) f zb S2 φ zb S1 where Θ is the higher-order complex propositional function which constitutes S1’s equivalent of f . This second translation rule (concerning f ) adds to the first one (concerning F), in spite of going in the reverse direction. [ ! ] :≡ [(φ) !(φ! ) ⊃ φ! )] (?) G x S1 x f zb x S2 [ !(φ! )] :≡ [Θ(φ! )] (??) f zb S2 φ zb S1

The additional rule (??) can be said to be a package of the original rule (?). It is part of the “unraveling” of (?) in S1’s perspective. This unraveling has to be consistent and to come to an end: the process through which S1 and S2 come to agree about a given translation cannot be infinite. The axiom of reducibility tells us that any non-predicative propositional function, presumably associated to a consistent unraveling, corresponds to an actual predicate in some other epistemic realization. Epistemic realizations are the semantic counterpart of the theory of logical types, and that distinguishing the latter from the former is the only way to understand the status of non-predicative terms —and in particular to understand their syntactic possibility despite the fact that one never finds variables of non-predicative propositional functions. It should be added that epistemic subjects mean nothing else here but complete epistemic diagrams, NOT actual cognitive subjects. There is an issue about whether it is possible or impossible, in Russell’s view, that objects do exist without belonging to any epistemic realization. I leave that question aside. A few remarks

I The existence of a plurality of epistemic realizations is perfectly compatible with the purity of logic. It enables Russell to leave aside the problem of a single integral analysis of reality. Such an analysis which otherwise would seem to be presupposed by Principia without being likely to be given by them. Russell’s account, on the contrary, is bound to a neutrality which explains that the only propositional functions to which the Principia are committed are the variables of propositional functions.

I Russellian universalism does not balk at meta-theoretical considerations. It even turns out to be compatible with a semantic perspective.

I This semantics is epistemic in nature, which in turn explains why Russell did not simply append his theory of knowledge to his logic: theory of knowledge is part of logic, since logical terms articulate a propositional function as analysed from a certain epistemic perspective. The “no loss of generality” claim explained

Let us take stock. The formal system of Principia’s ramified type theory lends itself to different epistemic realizations, in which each type is assigned to the of all concrete propositional functions of that type in the epistemic realization that is considered.

For example, the concrete propositional function xb is green I will correspond to a universal if if it turns out to be an object of acquaintance in the epistemic perspective under consideration.

I Otherwise, it will correspond to a definite description, for instance “the colour of grass” or “the colour of my neighbor’s pants”, whose logical analysis will involve higher-order quantifications. Epistemic realizations are a natural thing to bring up as soon as Principia’s type theory is understood as a formal scheme to be applied in a natural language environment. Besides, describing epistemic realizations like so many interpretations of type theory is a way to give Russell’s view of logic and knowledge a Leibnizian flavour that is not irrelevant after all: cf. the notion of subjective “perspective” brought up in “The Relation of Sense-data to Physics” (1914). Two different realizations of ramified type theory are like two sections of the same universe, and types correspond, in each section, to reference marks for the logical representation of that section: predicative terms correspond to the frontal sides, and non-predicative ones to the dotted lines that occur in a drawing in perspective. Non-predicative terms (made by circumflexion) work as patterns for connections between epistemic realizations, as translation rules from an epistemic realization to another. Each epistemic realization contains what the subject is acquainted with (individuals and predicates as universals) A non-predicative term is only the counterpart, within one epistemic realization, of an attribute F that is not available to the subject S2 of that realization, but that another epistemic subject S1 is acquainted with.

In such a case, S2 can still render F through a definition that may involve higher order quantification and thereby turn F into a non-predicative term [F]S2/S1. On that basis, non-predicative variables can be introduced (in some epistemic realization) to match predicative quantification in some other epistemic realization, but their interpretation is only substitutional.

Moreover, even though S2 is able to assert true sentences about F, the term [F]S2/S1 is devoid of any ontological import: its reference does not exist in any realization, but only in some realization from the point of view of another one. Surrogate acquaintance and quasi-values

If a subject S1 knows an individual only as “the G,” she will assert

something like F(the G), i.e., (5) ( E x) (y) Gy ≡ y = x & Fx without being able to indicate any possible instantiation for x.

Now, if another subject S2 knows “the G” by acquaintance as being a, then a becomes a quasi-value for x in (5). Every subject indeed resorts to an entity variable whose values do not boil down to her own actual objects of acquaintance, but also include all the virtual objects of acquaintance that she is provided with by any other trustworthy subject, and in particular the lacking quasi-value of x in (5). OF COURSE, Principia only provides the formal patterns of surrogate acquaintance, not the grounds on which such surrogacy can be justified (the grounds on which a subject can be said “trustworthy,” to begin with). Everything

In the suggested view, everything is the result of some kind of cooperation among all subjects.

The admission of a, as a quasi-value for an entity variable, into S1’s epistemic domain, hinges on S1’s believing that S2’s identification of “the G” with some actual entity a is reliable. Granted that the G actually exists, “the G” becomes what Russell calls a logical constructions: This means that this definite description then “has (speaking formally) all the logical properties of symbols which directly represent objects” (Principia *14.18): It can, by extension, instantiate an individual variable. So, in the end, logical variables x, while ranging various domains of values (depending on the epistemic perspective), can also be said to range an all-inclusive domain of values and quasi-values. This point echoes a difficulty about the substitutional interpretation of quantification —a difficulty raised by Russell despite his endorsing such an interpretation: Suppose we consider some proposition “f (x) is true for every x”, e.g., “for all possible values of x, if x is human, x is mortal”. We say that if a is a name, “f (x) is true for every x” implies “f (a)”. We cannot actually make the inference to “f (a)” unless “a” is a name in our actual vocabulary. But we do not intend this limitation. We want to say that everything has the property “f ”, not only the things that we have named. There is thus a hypothetical element in any general proposition; “f (x) is true of every x” does not merely assert the conjunction f (a) f (b) f (c) ... where a, b, c ... are the names (necessarily finite in number) that constitute our actual vocabulary. We mean to include whatever will be named, and even whatever could be named. (Inquiry into Meaning and Truth, p. 203) The axiom of reducibility as a principle of surrogate acquaintance for predicates

In the case of individuals, any object introduced by a definite description and assumed to exist (as opposed to “the actual king of France”), must correspond, up to a shift in epistemic context, to an actual individual. This is a kind of quasi-acquaintance principle. Once the epistemological background of Principia is given some attention, the axiom of reducibility can be construed as the same kind of principle, but for predicates instead of individuals: for any propositional function, there is always an epistemic realization in which that function, as a symbol, corresponds to a genuine predicate. A similar suggestion was already made: The axioms of acquaintance and reducibility postulate (respectively) the possibility of knowing individuals and classes in terms of functions that possess a certain epistemic transparency, a transparency embodied by acquaintance in the one case and the absence of complex forms of quantification in the other. Classes occur in the hierarchy only under the guise of predicative functions, which are the means by which they are known. Reducibility thus postulates a concordance between mathematical reality and our knowledge of it that the ramified theory is otherwise unable to demonstrate. (Peter Clark & William Demopoulos, “The of Frege, Dedekind and Russell”, 2005)

HOWEVER, contrary to what Clark and Demopoulos suggest, Russell does not claim any logical “pre-established harmony” to operate: the reducibility axiom does NOT postulate any “concordance” of our knowledge with mathematical reality, but expresses the concordance of two different epistemic perspectives with one another. Conclusion: Transfering acquaintances

The idea of surrogate acquaintance points to a fundamental issue, namely the transfer of acquaintance, more generally the establishment of connections between bodies of information, and the logical representation of such connections. Bodies of information need to be connected. Even if they are not based on Russellian acquaintance, this is what François Recanati’s mental files illustrate as well, with the notions of “conversion:” Conversion is the process through which information stored in a file is transferred into a successor file when the ER relation [the “Epistemically Rewarding” relation which enables the subject to gain information] which sustains the initial file comes to an end. (François Recanati, Mental Files, p. 81) See also the notions of “linking” (integration of information distributed in distinct files) and of “clustering” (merging informations derived from different ER relations).

I think that connections between bodies of information based on specific cognitive relations, lend themselves to being formally represented. But this is another story. At any rate, formal logic need not end when theory of knowledge begins.