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Abstraction and Modelling

Peter B. Ladkin

Research Report RVS-Occ-97-04

Abstract: Engineers talk of abstractions and models. I define both, consistently with the way that engineers, computer scientists, and formal logicians use the terms. I propose that both abstractions and models have what Searle calls an agentive function, namely that both models and abstractions are models/abstractions of something, for a purpose It follows that both abstraction and modelling are most fruitfully thought of as ternary relations (or, when the purpose is forgotten, binary). A canonical example of abstraction arises in considering logical form. The criterion used to determine logical form are used mutatis mutandis to define abstraction (the purpose of logical form is given: to explicate a sentence's/proposition's role in inference; the purpose for a general abstraction remains to be selected by the abstracting agent). One may therefore consider a generalised, Wittgenstinian notion of logical form as what both abstractions and models have in common with their targets. Abstraction is closely related to the idea of describing, a relation between sentences and states of affairs; in fact it can be considered to be a precise version of description.


Synopsis


Abstraction and Models in Engineering

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It seems to be common in engineering to distinguish between real things (nuts, bolts, computer programs) and abstractions, which are themselves not always distinguished from models:

It is important to remember that a system [as conceived of in `systems theory'] is always a model -- an abstraction conceived by the analyst (Lev95, p137)

Software is merely the design of a machine; it is an abstraction - like an architect's plans or an electrical engineer's circuit diagrams - and has no physical reality (Lev95, p157)

`Physical reality' is here contrasted with abstraction, and models are abstractions. To be an abstraction is to be something that physical things aren't. This kind of model, being an abstraction, is not physical.

I shall argue that it is easier to make sense of the concept of abstraction as a binary relation, rather than as a feature that a thing has, and that it is most easily applied to things of similar type (in particular, things with similar syntactic form or structure) rather than things of heterogeneous type (for example, an assertion as an abstraction of `physical reality'). Second, I shall show how the sense of model in which an `abstraction' may be a model allows also for physical models and mathematical-logical models as in (logical) model theory. Third, the way in which `abstractions' may connect with `physical reality' is similar to the way that meaningful sentences obtain a truth value. Since assertions can be true or false (or vague or uncertain or imprecise ...), so can `abstractions'.

I shall conclude that abstractions are descriptions, and the agentive function (Sea96) of a model is, generally, also to be descriptive. (See (Lad97.1) for the application of the terminology and results of (Sea96) to engineering artifacts.)

Models - A First Quick Take

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To the second point first. There is a meaning of `model' in engineering in which models are physical things that are just as `physically real' as what they are modelling. A model airplane is just as physically real as a full-scale aircraft. A model devised for use in wind-tunnel experiments has many features in common with the aircraft it is modelling - although not its size. If `physical reality' is to be contrasted with abstraction, this sense of `model' is not accomodated by that from the quotation above.

What is a model? A tenth-scale aircraft is a physical thing, sits on the ground, partakes of physical reality just like the rest of us. What is it, then, that makes it a model, and something else, for example me, not a model? It's an artifact, built in order to fulfil a purpose, in order to be placed in a wind tunnel, because some of its aerodynamic properties are similar to those of a full-scale aircraft, unlike some of its other features such as that of having no cockpit. It has an agentive function (Sea96), stemming from the intention of those who built it, of predicting airflow over the shape of the full-size aircraft of which it is a model.

One notable characteristic of `concrete' models such as tenth-scale wind-tunnel airplane models is that many important features are very similar to or the same as the object they are modelling, for example the aerodynamic flow, whereas some features are very different, for example they usually do not have tenth-scale cockpits so that tenth-scale pilots may fly them from tenth-scale airports.

Does it make sense to say of something that it's a model, but it's not actually a model of anything? The wind-tunnel model is a model of a particular aircraft. It is not a model of my house or my car. Hence calling something a model cannot be simply elision. The form of these sentences is

A is a model of B

which denotes a binary relation, not a unary property, of the model-object, and the second argument is essential. (A possible exception might be that of a model aircraft or car which has been modified beyond all recognition by its youthful fabricator. We would probably still call the object a `model' because of its provenance, but it's no longer a model of anything in particular. I don't consider this a counterexample to the point made here.) We may define a unary predicate from the binary relation, say, by existential generalisation:

A is a model ==(df) (exists B)(A is a model of B)
but as we shall later see, this defines a trivial concept.

The relational nature of the modelling function comes from its agentive function. Since the model was constructed as a model by an intentional agent, it makes sense always to ask what purpose the model has, and that leads to asking what it is a model of, and how. Describing something simply as a model is thus a form of elision - one is simply omitting to say what it is a model of. To say of something that it is a model, then, is to ascribe to it some agentive function, and this agentive function comes from the similarity to some properties of an object which it is modelling. This agentive function must then affect the logical form of sentences ascribing modelhood (Sea96): to be a model, then, is to be a model of something and to share some but not all properties in common with that which is being modelled.

This concept of modelling does justice not only to the use of the term implicitly by Leveson, and also to physical models, but also to its use in logical model theory. But more of that later.

The Arity of Abstraction

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Turning to the first point, we can tell a similar story for abstraction. I can make sense of abstraction as a binary relation:

A is an abstraction of B

Let's consider for comparison one explanation of abstraction:

abstraction
Supposed process of forming an idea by abstracting out what is common to a variety of instances [...] The problem is that abstraction leads one to suppose that qualities such as substance, causation, change, and number may apply not only to the sensible bodies that give rise to our ideas of them, but also in [.... an]other domain quite outside the reach of experience. (Bla94)

Abstraction is defined here as a process, yielding a binary relation A is an abstraction of B as A is the result of abstracting from B (and other `instances'). Notions of abstraction as a binary relation may also be formalised in mathematics using, for example, category theory. Such notions are thus fairly well-formed.

However, Leveson takes `abstraction' to be a unary predicate of things in the quotation above, and contrasts being an abstraction with being physically real. This looks like a form of abstract/concrete distinction (see (Bla94), entry on abstract/concrete; (Aud95), entry on abstract entity; and Footnote 1). Models, in Leveson's sense above, would be thus abstract not concrete. A model aircraft, in contrast, is a physical thing. Leveson offers no further clue how an abstract/physical distinction could help in clarifying either of these notions of model (but that, of course, was not her purpose).

Here are two more quotations in which the notion of abstraction is used.

[...] Obviously there is something common to the argument `All men are mortal, Socrates is a man, so Socrates is mortal', and `All horses bite, Eclipse is a horse, so Eclipse bites'. This common form may be revealed by abstracting away from the different subject matter, and seeing each argument as of the form `All Fs and G; a is F, so a is G'. The symbols of symbolic logic simply represent such common forms and the methods of combining elements to make up sentences. [...]
(Bla94, definition of logical form)
when we see a white patch, we are acquainted, in the first instance, with the particular patch; but by seeing many white patches, we easily learn to abstract the whiteness which they all have in common, and in learning to do this we are learning to be acquainted with whiteness.
(Rus67)), reprinted in (MelOli97, p51))

Both of these quotations, as well as the definition of abstraction, describe a process of abstracting. The result of such a process could be called an abstraction. This suggests a definition of being an abstraction, the unary property, by existential generalisation as above:

A is an abstraction ==(df) (exists B)(A is an abstraction of B)

Given that I do not find an abstract/concrete distinction particularly helpful here (see again Footnote 1), I would be at a loss to explain what it might mean to ascribe abstraction as a unary property of something, other than through such a definition. Although I don't think it's the whole story, I think the binary notion of abstraction of is relatively unproblematic, and will help to give an account of modelling and abstraction which coheres in general with the uses of these words in engineering, as well as in the quotes above from philosophical logic.

The result of an abstraction may sometimes, in Wittgenstein's phrase, be shown but not stated, as argued for the case of logical form in (Lad97.1). However, depending on what properties one believes there `really are', this does not rule out that some abstractions could also be physically real. Here's the reasoning. Russell's example of whiteness, which is a property that, according to the quote above, is also an abstraction (being the result of abstracting), is exhibited in as `physically real' a manner as the objects which exhibit it. In fact, if one accepts an account of universals such as that of Armstrong (Arm78) (Lew83), and whiteness is a universal, it is physically present in each and every `particular' that is white. Russell considers it abstract because he considers that we may come to this notion by a process of abstraction. I don't see any contradiction between these two accounts. Russell's comment can be regarded as epistomological: it may indeed be that we can infer the existence of a universal of whiteness by observing objective similarity between white things, and engaging in a process of abstraction, which reliably brings us to the existence of a corresponding universal, provided we had been right in the first instance that the similarity we perceived were indeed objective. These questions, concerning particulars, universals and properties, are peripheral to my concerns here. They are also difficult. Distinguishing particulars from properties, however, is a complex task, with no uniformly accepted answer even for physical properties/particulars (MelOli97, Introduction).

To show the unsolved nature of this question, let's consider a couple of the answers. Frege held that predicates, which for him are special functions whose value is always a truth value, are, like all functions, essentially `incomplete' - they lack an argument (which, when filled in, yields a value); and distinguishes objects, which are `complete' - they have no fillable hiatuses. Frege would thus distinguish the property whiteness from all particulars. However, Frege's view identifies all properties that are coextensive (so, for example, the property of being a four-legged man is not distinguished from that of being a five-winged bird, both predicates having an empty extension) and also allows as real properties some that some philosophers consider odd (such as Goodman's grue, a property defined as `either blue at all times before 12.31 GMT precisely on 3 April 1997, or green at all times thereafter'). Ramsey, by contrast, argued that not all predicates correspond to a property, since then in the (pseudo-)sentence aRb, we could be able to distinguish the relation R which a and b have, from the property aR, which b has, from the property Rb, which b has, and Ramsey thought that was simply too many properties for one `fact'. And there are those who argue that Goodman's predicate grue does not correspond to a `physical' property, either, even though intuitively everything which fulfils the predicate grue is `physically real'. Frege also considered that the graph of any property, the set of pairings of objects with truth values, was in fact an object, even though the property itself was not (we know now, from Russell's paradox, that not all properties have such a graph).

Now for an intermediate view. Quine notes that one may explain away singular terms and names in favor of predicates, using Russell's translation in his Theory of (singular) Descriptions, and bound variables (suggested to be similar to pronouns). What there is, he suggests, or rather what we are committed to asserting the existence of, is whatever there needs to be as values of bound variables in order that our assertions should be true. Assertions should not be considered individually, but as part of an entire world view. So we may use predicates successfully without being able to infer that there are properties corresponding to them, speaking of red houses, red roses, and red subsets, without inferring there is something, `redness', which they have in common, although the predicate `is red' is true of all of those objects. If it follows from our worldview that there is something which red-houses, red-roses, red-sunsets, etc, all have in common, then we are committed to the entity redness, and if not, not. So some properties are presumably real, and some putative properties not (Qui48). These and further views on the reality of properties are displayed in (MelOli97, Introduction).

Lewis (Lew86, Section 1.5 on Properties) considers that other possible worlds that are not this actual world are just as real as this actual one, and defines various notions of properties and relations relative to this modal realism. He believes that the role that properties are to play has not been universally agreed upon; that there are conflicting roles, and different notions of properties will be required to fulfil those roles. He therefore defines abundant properties (which would include everything definable using classes and all possible distinctions, thereby including grue and all its correlates), and considers the roles of sparse properties (those properties that, for example, physics may need in order to explain physical features of our world, which he calls natural properties), as well as intrinsic (those properties that a particular has in virtue of the way it is in itself, such as size, mass, color) and extrinsic properties (those it has in relation to other particulars, such as distance from another object: basically, that it participates in some non-unary relation).

These distinctions are tricky. I conclude that the jury is out on whether Leveson's contrast between abstractions and physical things can be sustained. Given current thinking, the answer seems to be: for some, but not all, abstractions. This is one major argument for abstraction as a unary notion. Another is as an existential generalisation from a notion of greater arity. This method must thereby regard the multi-arity notion as more fundamental and the unary notion as derived. I believe a unary notion is less useful and the binary notion adequate for many purposes. However, I shall regard abstraction as logically a ternary notion, as we shall see. In Lewis's terms, to regard abstraction as an intrinsic property isn't helpful; regarded as an extrinsic property it is much more useful.

Modelling, the Meaning of Propositions, and Logical Form

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(Some of the material in this section is shared with (Lad97.1), on whose considerations on logical form the present analysis is based.)

Modelling has played its role not only in engineering but also in the philosophy of logic and language. Wittgenstein's influential picture theory of meaning (Wit22) ascribed the meaning of propositions to the `picture' they drew of a state of affairs, in much the same way as an arrangement of blocks on a table may represent an arrangement of cars on the road during a traffic accident (see the discussion in (Ric96)). The blocks represent the cars in the sense that the spatial relations amongst the blocks correspond directly and precisely to the spatial relations of the cars. Distances and measurements are transformed, but conformedly.

Where this `modelling' works, it seems I can always find some structure shared between representation and actual situation. For the blocks-representation, the shared structure would be the spatial relations and, maybe, uniformly-transformed distance measurements. For propositions, the `picture theory' holds that this structure obtains in common with the structure of `states of affairs' (a more inclusive philosophical term for `real world', which allows not only objects in the `real world', but relations between objects and so forth). This common shared structure was identified by Wittgenstein as the logical form. That is, what atomic propositions share with the states of affairs they picture is their logical form. Wittgenstein famously held that this could not be stated in a language, only `shown'.

Such a correspondence theory may obtain substance also from the notions of properties and relations. The basic states of affairs consist of particulars, and (sparse) properties and relations which obtain amongst those particulars. From these sparse properties, relations and particulars we can construct more complex properties and relations that also may be taken to be present in the state of affairs because of its structure. These complex properties and relations may not be `natural', but they certainly belong to the abundant properties. A proposition that correctly describes this state of affairs may be said to single out some complex relation present in the state of affairs. The logical form of the proposition represents this complex relation compositionally, by showing us how it is built up: and the structure of the state of affairs will faithfully mirror this construction. We have already noted that whether this complex relation is `natural' or not appears to be a separate question.

A lot has happened in philosophical logic and the theory of truth since the Tractatus. Why consider the Tractatus version? I have three reasons. First, in my experience, engineers seem to hold a version of the picture theory as being very near the truth. One talks about states of affairs, asserts they are one way or the other, and the `facts' `show' which sentences are true and which not. Second, some descriptions just picture `the way that things are', for example, the ubiquitous and sometimes iniquitous `boxes-and-arrows' diagrams in software engineering. One can then combine these simple descriptions with logical connectives, or their pictorial equivalent, if one wants, which is very much as a modified `picture theory' is supposed to work (see (Lad97.1)). One may suppose that a meaning-theory of pictures may follow easily from such a picture-theory of meaning. The third reason is that a picture-like theory may be justified by a correspondence theory of meaning or truth (meaning may itself be explained by a theory of truth-conditions), underlain by some notion of the reality of appropriate properties and relations. In my experience, engineers accept the reality of appropriate properties and relations, therefore it is appropriate to found engineering on such an theory. Philosophers may then care to explain away this apparent foundation, say by denying the reality of properties or of particulars, assimilating particulars to collections of properties, or vice versa, explaining all this by a theory of the use of signs, and so forth. But this should not interest engineers so much, as long as the common way of speaking is somehow well founded.

Making a picture and building a model are also very similar activities, both concerned with representing features of things in an artifactual way. However, most engineers do not think through their worldview to the depth that Wittgenstein and those that have subsequently considered these questions have done. It is therefore worth considering a Tractatus-like view to see how it may help to explain a worldview similar to that of many engineers (Footnote 2).

How To Use Composition

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One difficulty for the original `picture' view arises with negated propositions, and disjunctive propositions whose disjuncts are contraries (although there seems to be no such problem with conjunctive propositions). As suggested in (Lad97.1), one could restrict a `picture' correspondence to propositions which do not have the logical form of non-atomic sentences of predicate logic (call these logically simple propositions), and then build the correspondence of non-logically-simple (call them logically compound) propositions from the `picture' correspondence of their logically simple components, compounded according to their logical form. Although Wittgenstein held also that this logical form was a form of picturing, we do not need to worry about what it's called in order to explain the correspondence of logically compound propositions with states of affairs (Footnote 3).

Explaining the correspondence between logically compound propositions and states of affairs is to explain what states of affairs have to pertain for the proposition correctly to describe `the world' as it is. That is, for the proposition to be true. Explaining the correspondence, then, is to enumerate the truth conditions of the proposition. Supposing such a `picture' correspondence to the logically-simple propositions were feasible, this could be used to determine the truth conditions for logically compound propositions, in the following way. Consider a logically-compound proposition P, and construct a sentence S in a pseudo-language L by replacing the logically simple components of P by place-holder symbols of L (a pseudo-language has some natural language words and special symbols to act as `place-holders' for expressions of particular syntactic categories, as explained in the section Syntactic Transformation next. P then has the logical form of S, according to the definition in (Lad97.1)). We may suppose that we have rules which explain exactly how the truth conditions of S depend upon the truth conditions of its place holders (such is usual for a pseudo-language devised to show logical form, and is known as a compositional definition of truth conditions). Suppose we know the conditions under which each logically simple component of P obtains a truth value. Then we may determine the conditions under which P would be true

Part (b) is the same as in the compositional definition for S, since the relevant structure is identical, by construction of S from P. Therefore, if we know how to form truth conditions compositionally for a pseudo-language of sufficiently expressive form (part (b)), all we need to know is how to form truth conditions for logically simple expressions (part (a)) in order to determine the truth conditions for the complete sentence. These conditions for logically simple components could be given by a `picture' theory explaining the correspondence between states of affairs and the truth or falsity of the logically simple expressions. So one needs a `picture' explanation of truth conditions only for logically simple expressions. One could say that `picturing' is a general way of showing the correspondence between states of affairs and logically simple expressions.

As argued in (Lad97.1), it is problematic to consider logical form unique. That is, a given sentence or proposition corresponds to many other propositions, some in pseudo-languages, with which it shares logical form. This means that the notion of logically simple and logically compound component of a proposition or sentence P are as far as we can tell only well-defined relative to (pseudo-)languages in which there are sentences S with which P shares logical form. Suppose now that in a more expressive pseudo-language there is a sentence S' which has the logical form of both P and S. Then there will be logically compound subexpressions of S which correspond to place-holders in S and therefore to subexpressions of P whose truth conditions we have suggested could be found by `picturing'. However, in the language of S', part of that `picturing' correspondence is now explained by compositional rules, from the truth conditions of logically simple subexpressions of S'. So, on a `picturing'-compositional account, `picturing' is not an operation which is atomic in the sense of being indivisible, but itself may be partly explained by compositional rules when one moves to a more expressive pseudo-language. It follows that `picturing' must be consistent with, not necessarily disjoint from, compositional explanations of truth conditions. Since these compositional explanations follow logical form, it follows that `picturing' must be consistent with logical form.

Since I am sceptical as to whether there is a unique logical form for a proposition, I may remain agnostic on whether there are specific atomic `picturing' operations that cannot be further explained by compositional rules. Note, however, that that does not rule out that we may choose a set of picturing operations as `basic', and explain truth conditions by means of logically compositional rules from them. It means simply that it may be possible further to analyse `basic' operations, no matter which operations we take as `basic'.

A correspondence between states of affairs and a component of a proposition hitherto regarded as indivisible in the truth conditions for the complete sentence may be explained, in passing to an expressively richer pseudo-language, as compositional upon the (further-elucidated) logical form of the component. And there may be no atomic expressions to ground this process. This may help explain Wittgenstein's identification of `the' form of the picture correspondence with logical form.

Syntactic Transformation

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I now turn to considering abstraction in more precise form. We have seen that we may use a combination of `picturing' (correspondence) for logically simple expressions, along with a compositional explanation for a suitable pseudo language, in order to explain how a logically compound proposition corresponds to a state of affairs. But what kind of pseudo languages are suitable, and how may we related them to the language whose propositions we are analysing? I defined the notion of syntactic transformation between languages in (Lad97.1). This arose from the following considerations.

The logical form of a proposition expressed by a sentence is that structure which explains or shows how that proposition functions in inference. In order to show the similarity (or identity) of the logical form of the assertions `all men are mortal' and `all horses are four-legged', we may invent a pseudo-language which has some expressions in common with these, and some completely different (for example, individual words consisting of single capital letters, say F and G, which are not in the original language and which stand out as not being in the original language. We then use these symbols to take the place of words and phrases expressing natural kinds or unary predicates (`man', `mortal', `horse', `four-legged') which occur in the sentences, but which play no part (except that of being identical to or different from other occurrences of these expressions) in clarifying the role of the sentences in inference. Thus we may express the form of the examples by `all Fs are G'. The symbols F and G act as place-holders for genuine terms for unary predicates, and they are uninterpreted. They function to draw attention to themselves as place-holder symbols, and to take indicate where unary predicates occur which play no role in inference (except that of identity to and difference from other occurrences of unary predicates).

I call a language with these additional `place-holder' symbols a pseudo-language. It is also a semi-formal language (a formal language is one, all of whose symbols come from a precise vocabulary and which has precise rules of expression formation). Its purpose is to exhibit logical form more distinctly than a natural language (recall Wittgenstein's dictum that logical form can be shown, but not stated). This pseudo-language allows the definition of certain syntactical operations.

A natural language has syntactic categories: nouns, verbs, articles and so form. Similarly, the semi-formal language has syntactic categories, but not necessarily the same: variables, constant symbols, terms, formulas of arity n, and so forth. These categories are used to explain certain inferential functions. Items of each category may standardly be divided into atomic and compound symbols. Atomic symbols are those which have no components, such as `F', `man', and compound symbols may be further subdivided into meaningful components, `the king of France', `my car'. I say that a sentence A is syntactically transformable into sentence B if there exists an association of atomic symbols of A with compound symbols of the same category in B such that if these compound symbols are substituted for their atomic associates in A, sentence B results. Two sentences are syntactically equivalent if each is syntactically transformable into the other. Such a definition is applicable to any language with a categorial grammar (Footnote 4). Most natural languages have extensive sublanguages whose syntax may be captured with a categorial grammar, so these relations can be defined also on these sublanguages.

My semi-formal language has atomic symbols - words - which are partly natural-language words, and partly place-holder special symbols. Observe that `all Fs are G' may be syntactically transformed into `all men are mortal' and vice versa; and similarly with these two sentences and `all Hs are J'. Thus part of the notion of has the same logical form as, as developed in (Lad97.1) and as explained in (Bla94), coincides with that of mutual syntactic transformation.

Syntactic transformation is one ingredient of the notion of logical form in (Lad97.1). Another ingredient is an operation under which logical form should be invariant. This is inferential equivalence. Two sentences (language or pseudo-language, same (pseudo-)language or different (pseudo-)languages) are inferentially equivalent if they are logical consequences of exactly the same sentences and have exactly the same sentences as consequences. Since logical form is supposed to elucidate the role that a sentence (or proposition) plays in inference, two inferentially equivalent sentences play exactly the same role in inference. Thus a sentence also has the logical form of any of its inferential equivalents.

It is worth remarking that inferential equivalents do not have to share surface or semantic structural form. It was shown in (Lad97.1) that the Russell translation of sentences involving singular terms (that given by the Theory of Descriptions) and the surface syntax with the Strawson Truth-Value-Gap interpretation are inferentially equivalent, despite having two clearly different syntaxes (one is an existential statement containing a universal substatement; the other is in subject-predicate form).

A major thesis of (Lad97.1) is that it is relatively problem-free to say when two (pseudo-)sentences or propositions share logical form, but it is problematic to define a notion of the logical form of a sentence or proposition. There may be syntactically-incompatible pseudo-sentences which share logical form with the target, for example. Thus I concluded that the relation of having the logical form of was primary, in the sense that it is preferable to base explanations of logical form and the role of propositions in inference on this relatively unproblematic relation. We shall see that similar considerations hold for abstraction in general, since the operation of abstraction specialises to the operation of elucidating logical form.

The two ingredients for explaining the notion of logical form, then were syntactic transformation, and inferential equivalence; this latter being an equivalence relation under which sentences playing identical roles with respect to inference are grouped. A similar clarification will be given of the notion of abstraction: syntactic transformation along with an equivalence relation under which the process of abstracting is invariant.

Abstraction

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I considered in (Lad97.1) the notion of logical form in regard to sentences, propositions and states of affairs. The form in which propositions and sentences were presented was the usual natural-linguistic notion of a sequence of symbols or some sort. Whatever allows these to represent or denote, it seems to me highly unlikely that the considerations discussed in (Lad97.1) somehow presuppose such sequences as a `data structure'. Whatever makes the connection between (linguistic) assertional structures and states of affairs, it seems plausible that any broad analysis should apply also when other forms of data structure are used as the assertional objects.

Propositions expressed in sentences, as guided by our intuition, have a linear, discrete, structure (that linguistically may be thought of as a representation of a tree structure). Imagine now a visual syntax. Boxes-and-arrows diagrams, for example. Two-dimensional. I suppose that an even-higher-dimensional syntax is also possible. I might claim, based on an assumption that physics works within a specific number of dimensions, that it would be possible more easily and conformably to represent states of affairs than with a linear, `linguistic', syntax. Here's an example of how.

Suppose I choose to `assert' a particular state of affairs S, which, say, includes spatial features, by using that very state of affairs, with everywhere the red surfaces replaced by blue ones, to denote S. Call this `denoting' state of affairs S' I take it that (a) the two states of affairs are not identical (crudely, S has red where S' has blue, and an assertion that something is red and that something is blue are distinct assertions, as shown in Footnote 5); (b) the conformance between the state of affairs S and the higher-dimensional proposition S' representing it will intuitively be understood, easily, by the reader, no matter what her favored philosophical underpinnings (providing both that she allows the presupposition of realism, to enable there to be states of affairs such as I have supposed, and that she allows a general method of introduction of signs sufficient for me to use S' as a denotation of S). I won't attempt to state what that conformance is, remembering the slogan, `shown, not statable'. I hope it's also plausible that they share very similar logical form: everything the same, in fact, up to color specification (Footnote 6).

So the points that I made with respect to propositions and states of affairs carry over to such a case. The notion of syntactic transformation remains the same. The definition of syntax is somewhat broadened, and I see no problems with this. Very little of the technique or commentary here or in (Lad97.1) is dependent on the presumption that linguistic objects form linear discrete sequences. Syntax for two-dimensional linguistic assertive objects is common in the computer world. In fact, so are translations of this syntax into the linear sequential syntax form (via `visual representation languages'). So it could even be that linear sequential form is a sufficient data structure for all linguistic assertional structures, including the higher-dimensional example we are now considering, in which case the notion of logical form would carry over without modification to all these cases. But I have no need here to demonstrate such a thesis.

As in the case of semi-formal languages considered above, I can imagine constructing a semi-formal language to show the `logical form' of the higher-dimensional language. I could leave the color ascriptions unspecified, by inventing a suitable sort of place-holder. And this would share the structural form of both situations above. I admit that we are less used to thinking of place-holders in such situations - imagine seeing in color, except for all the red things, which are now mysteriously...grayish, like a black-and-white TV - but I see no problems in principle. The notion of syntactic transformation remains as before.

The Definition of Abstraction

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I can now clarify the notion of abstraction using the notion of syntactic transformation for this general case. I start with a criterion of abstraction.

A is an abstraction of B *if*B is a syntactic transformation of A, in the extended sense of syntax indicated above

The quotations above concerning abstraction speak of abstraction as a process, as a human process. It may be a process of invention or discovery, but in any case such activity has a purpose. This purpose is missing from the criterion above. I can abstract a sentence token with respect to its logical form, or with respect to its typography. Both are abstractive processes, but lead to very different results. It is not incorrect to speak of A being an abstraction of B - it is just incomplete. Given the purpose, we can consider the equivalence relation of things that are equivalent with respect to this purpose. Let us call this the abstraction-purpose equivalence, or purpose equivalence for short. For example, the notion of logical form can be described, as it is in (Bla94, definition of logical form), as that structure which shows the role that a sentence/proposition plays in inference. The notion of inference equivalence is the relevant purpose equivalence. Purpose equivalence becomes also a criterion for abstraction:

A is an abstraction of B with regard to purpose P *if*there is a purpose-equivalent C to B, and A is an abstraction of C
The two criterion form jointly a criterion for abstraction. I would like to say that they suffice, but provide no argument for that here. It is a consequence of this criterion that logical form is an instance of abstraction, as asserted implicitly in (Bla94, definition of logical form). The complete criterion is thus:
A is an abstraction of B with regard to purpose P *if*
  • B is a syntactic transformation of A; or
  • there is a P-equivalent C to B, and A is an abstraction of C
This is as it stands a recursive criterion (sometimes called `inductive' - it has a `base' clause and an `inductive clause). However, since the notion of syntactic transformation is transitive (Lad97.1), and so is an equivalence relation by definition, this criterion is easily seen to be equivalent to
A is an abstraction of B with regard to purpose P *if*
  • B is a syntactic transformation of A; or
  • there is a P-equivalent C to B, and C is a syntactic transformation of A

According to this criterion, abstraction is thus a ternary relation between abstrahiens, abstrahiendum and purpose. I have very little to say about `purpose' in this essay. My point is simply to clarify the form of the notion of abstraction. If either the purpose is understood, or one is only interested in the criterion of syntactic transformation, it becomes a binary relation. If abstrahiendum is understood, we could claim that the abstrahiens is `an abstraction' with respect to the purpose. If both are understood, we could say that something is `an abstraction'. However, I submit that such a situation presupposes a tightly circumscribed context, just as the interpretation of a sentence such as `he did it' requires a tightly circumscribed context in order to know who `he' is and what `it' refers to.

The example given above of higher-dimensional syntax started from states of affairs (`reality', somehow) and imagined an equally `real' but different, state of affairs as a proposition corresponding to it. I then endeavored to explain the `abstraction' by suggesting a physical unreality (`replacing' the color by an other-wordly `gray', similar to the replacement of natural-linguistic `real' words by completely-dissimilar schematic sentence letters in the semi-formal language I used to explain the notion of logical form). The `abstraction' with my other-wordly gray looks to be in this story very much a `thing', even if not somehow `real'. So it looks as if there can be `things' which are `abstractions', that are not `real'. This would reify the notion of abstraction, as in the use of the term by Leveson. I recall, however, Quine's observation (Qui48) that appropriate use of a property or relational term does not automatically presuppose that there is a `thing' corresponding to the property or relational term, just as appropriate use of a definite description such as `the king of France' does not presuppose there has to be a king of France. However, we could call something A an abstraction if were an abstrahiens: that is, there were to be a purpose P and an abstrahiendum B such that A is an abstraction of B with respect to P. This is the definition of a unary property by existential generalisation. I shall now show how useless such an attempt to define a unary property would be.

Every Pseudo-Language Sentence is an `EG-Abstraction'

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I define the unary notion of EG-abstraction:

A is an EG-abstraction with purpose P ==(df) (exists B)(A is an abstraction of B with purpose P)
Every pseudo-language sentence is then an EG-abstraction, for any purpose P. The argument is as follows.

All sentences in a semi-formal language include place-holder symbols. For every sentence which includes place-holder symbols, there is another sentence which is a syntactic transformation of it, therefore of which it is an abstraction (with respect to any purpose). Consider any sentence A. It has atomic components, which are either atomic subsentences (if it is a sentence in an appropriate pseudo-language) or predicate terms. I consider these two cases separately. Suppose it contains atomic sentential symbol P. Let B be the sentence obtained by replacing P throughout by P \/ Q, where Q is a sentential symbol not occurring otherwise in A. B is then a syntactic transformation of A and A is thus an abstraction of B (with respect to any purpose). Suppose now that A contains no sentential symbols. Select any predicate. Form B by replacing the predicate P by the predicate P and True, where True is the predicate symbol of the same arity as P whose value for any choice of arguments is always (the logical constant) True. P and True is a compound predicate, using the standard definition of compound predicate: for example, when the arity is 1 then (P \/ Q)(x,y) == P(x,y) \/ Q(x,y). QED.

Abstractions and Models Defined

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We have seen that speaking of `abstractions' and `models' as properties does not do justice to the concepts. Models are models of something (for a purpose, generally) and abstractions are abstractions of something (also for a purpose, generally): let's call this `something' the target of the model or abstraction. This relation need not be functional: abstractions may have many targets, and so may models; in particular if the purpose is included. As we have noted, models share some but not all features with their targets: the model aircraft shares shape and relative size but not absolute size with the real aircraft. However, abstractions share all their features with their target. It's plausible to think that one may form an abstraction from the common features of a model and its target, for example, `shape-of-an-F18' would be an abstraction whose targets are both the aircraft and its tenth-scale model; whereas the actual aircraft is the target of the model. Thus the relations `abstraction-target-of' and `model-target-of' are different.

The difference between abstractions and models may thus be explained: abstractions share all their features with their targets (but not conversely); whereas models only share some, their salient features. This difference appears also in the use of these terms in mathematics. A vector space is an abstraction of vectors over Euclidean space; but vectors over Euclidean n-space, n-tuples of real numbers, have features (namely, that they are n-tuples of real numbers) that the abstraction may not be shown to have.

We may summarise the discussion thus. Abstractions describe what they are abstractions of: every feature of the abstraction is faithfully reflected in the target; every true sentence concerning the abstraction is similarly true of the target, under the appropriate interpretation of the predicates and names. Models describe their targets only in certain ways: true sentences about the model may not be true of the target, but the salient ones are. It should also be true that model and target have a common abstraction.

These considerations lead to the following definition.

A is a model of B with regard to purpose P *if*
    (exists C)
  • (C is an abstraction of A with purpose P; and
  • C is an abstraction of B with purpose P).

Three caveats. First, none of this explains how abstractions or models are created. Second, because the languages of mathematical logic are pseudo-languages, the relation of theory and model in mathematical logic is that of abstraction and target. There is no obvious equivalent in pure mathematics to model and target; but in applied mathematics, continuous dynamical systems are modelled by numerical discretisations. Thus both modelling and abstraction occur in mathematical behavior. Third, since abstraction and target have all common features in common, my explanation allows them to be model and target - in fact, either way around: abstraction-target is target-model, or is model-target. This may not be intuitive, but I am uninclined further to specify the concepts merely to conform to intuition, since the distinguishing criterion fulfils my purpose.

It also follows from the definition that the relation of modelling is symmetric. That means that the full-size aircraft is just as much a model of the tenth-scale aircraft as vice versa. Logically, I see no problem with this. However, model-making is purposeful behavior, and the (meta-)purpose may dictate the direction of the modelling relation (Footnote 7). For example, I have one object of a modelling pair for a purpose P and may wish to create another one, which has certain particular properties that fulfil my meta-purpose.

It seems that in engineering the difference between models and abstractions is not always remarked. For example, I would be inclined to interpret the quotations from Leveson as stipulating that software is an abstraction of a physical computer system on which the software is implemented. A design is an abstraction, just as Leveson says. Therefore (cf. the third caveat) it can also be a model, also as Leveson says. However, one should not infer from this that there is necessarily something inexact about it. This inexactitude would arise only from true assertions about the model that are untrue of its target; and when the model is an abstraction, there aren't any.

The definition of model, along with the observations that

can now be put together to show that everything has a model and is a model. That does not of course mean that everything is a useful model.

How Do Abstractions Come About?

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A fundamental question is: is the notion of an abstraction of `reality' coherent? I take it that the notion of a mathematical abstraction of a mathematical structure is at least as coherent as that of a mathematical model of a mathematical theory: I have already pointed out that a mathematical-logical theory is an abstraction of all its mathematical-logic models. But mathematics may be one thing, and the `real world' another. For example, it has been doubted whether objects putatively named in an abstraction have any independent reality (Ben65).

There are many different explanations of how variables in mathematical theories may get their values. But while one may doubt whether such values are `real' in the same sense in which the name `Peter B Ladkin' denotes me, there is almost general consensus that when someone says now, `Peter B. Ladkin is sitting typing at his computer' that is both a successful use of the name to denote, and a successful description of what I'm doing now. Of course, I'm doing more: I'm writing about denotation, I'm finishing a paper, I'm considering the explanation of abstraction, and I'm drinking espresso off and on. All these are successful predications about me now. They are black-and-white veridical, not almost-true. And they are successful predications, the sentences they form are true, because of the meanings and denotations of the subexpressions they contain. Were `Peter B. Ladkin' to denote someone else, the predications may not be true. Were `writing' to denote another activity, such as sneezing, and `drinking' to mean `washing in', the predications would not be true.

In line with the many different explanations of how variables may get their values, there are many different explanations of how names can denote and how sentences or propositions can get truth-values. But every such account must be compared against the normal usage of these concepts, and no such account could be taken seriously unless it allowed the unmitigated truth of the above predications, the completely successful predications about a person who is me. We may say that the business of explaining how names may denote and how sentences may obtain their truth-value is partly a normative activity. A successful account must allow that we are mostly doing what we think we are doing, must allow for successful and unsuccessful predication and naming, and explain what the distinctive role played by true sentences and how they get that role, in conformance with a more-or-less standard account of truth. A more-or-less standard account of truth explains conjunction, disjunction and negation in terms of their truth functions, as well as explaining any illocutionary acts that may be performed by asserting these sentences. Such a normative account must allow that (and explain how) a sentence which corresponds (by the mechanism indicated above) to the formal sentence ~(p & ~p) is true, no matter what sentence p is.

Let us return to the successful predications about me. I claim that these sentences successfully describe me, because they attribute certain activities to me at a certain time and I am in fact engaged in those activities at that time. Somehow, the sentences describe reality. They are not, however, a complete description of me. They say nothing of what clothes I'm wearing, where I'm sitting, how tall I am, or what I'm digesting in the way of breakfast. Not only that, but to be successful, they do not have to say anything about these aspects. They successfully describe me, without necessarily distinguishing me from Joe Blow next door, who may satisfy all the same predications.

Suppose I were to formalise all the description of me above, and ask of what objects in what circumstances it would be successful, it would yield true sentences. One way of doing this would be to introduce a name g without saying what the name is to denote; and a series of unary predicate symbols P, Q, R S and T. My criterion that the predication be successful is just that P(g), Q(g), R(g), S(g) and T(g) should all be true; equivalently, that the sentence (P(g) & Q(g) & R(g) & S(g) & T(g)) should be true. This is an abstraction: it shares common form with the predication concerning me, and there is no form which it exhibits which is not shared with the predication concerning me. I can also say this using the notion of syntactic transformation: if a name for me, say `Peter B. Ladkin', were to be substituted for g, and the predicates, ` - is sitting at a computer, `- is drinking espresso' were to be substituted for P, Q, etc., then the expression in the formal language would be transformed into the original natural language expressions. Furthermore, any general account of how the truth of the formal language components may be determined, say by evaluating the truth of its component phrases using the notion of what it is to be a successful predication of a object denoted by a name, will yield an account of how the truth of predications of me could be determined in a similar manner.

Using this approach, we may determine that ~( has-five-feet(Peter B. Ladkin) & ~ has-five-feet(Peter B. Ladkin)) is true. The predicate ~( has-five-feet(-) & ~ has-five-feet(-)) is successfully predicated of me. The argument proceeds as above: one moves to an abstraction, and demonstrates that under usual accounts (classical or constructivist, it doesn't matter) of how the truth of a sentence is built up from the truth of component expressions, the predication of me, indeed of anything else, must inevitably be successful.

Thus is the role of abstraction different from that of modelling. The use of abstraction may enable one to definitively decide, or at least get insight into, the truth of statements and the success of predications. Whereas, when modelling, one must in addition attempt to decide somehow whether the predication being attempted falls within or outside of the aspects of the target being modelled.

In summary, whatever account one gives of denotation, predication, or truth, or in what we believe `reality' consists, the relation of abstraction allows results from considering the abstraction to be tranferred salva veritate to the target. And the kind of abstractions I have considered consist in assertions and predications of objects. I shall assume that we may successfully predicate of and refer to objects in the `real world', and I then may use formal languages, with their simple relations of denotation and predication (obtained by fiat) to infer about the success of predications in the `real world'. And for this, I do not need to explain in what the `real world' consists, merely to assume that there is some account of how to predicate and refer in this realm, whatever it may be.

I have not yet said how one may obtain useful abstractions, whether they exist `out there' or are entirely the product of human intention, or both. Neither do I wish to address this general Platonist vs. constructivist debate. I wish only to make a point about language use. Rather than talk about an abstractions, and oppose them to `real things', as Leveson does, I would prefer to remain agnostic about the ontological status of mathematics and how mathematical statements obtain their truth. I assert only that any such account will also allow the notion of syntactic transformation; that the notions of abstraction and modelling are best thought of as binary; and that abstraction is syntactic transformation, albeit in more types of languages than we are normally used to using; and that modelling is having a common (useful) abstraction. However, I will profess a disposition, like Maddy (Mad90), to see (finite) sets actually in concrete physical situations. And rather than talk about abstractions, I prefer to call them descriptions of reality, to emphasise the tight binding to truth.

Conclusion

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Twentieth-century philosophical concerns since Frege have focused on explaining the role of language (natural and formal), on assertions, both true and false, and their form and meaning, in order to clarify a variety of puzzling problems, some of which have been around since Aristotle. Accordingly, one may no longer talk about things-in-themselves, primary and secondary qualities, perception, the construction of physical reality, without paying close attention to the form which assertions involving those concepts have, and the role of that form in explaining the truth and falsity of those assertions, and their logical links to other such assertions. Frege's initiation of the philosophy of language was followed by the development of the science of formal languages (started by Frege's solution of the problems of Aristotle and the schoolmen in logic) and concentrated on true and false assertions. Wittgenstein broadened interest to a wide variety of human endeavor he called `language games', and construed many philosophical questions as `grammatical', in a broadened sense of `grammar'.

Austin called attention to the use of sentences in natural language for purposes other than assertion; Searle has developed this theory to explain also intention and human convention. This is by no means the only thread of direct relevance: others have developed formal and philosophical logic, the theory of meaning and truth, and the logical analysis of action, all paying attention to logical form, and the form of logic. Accordingly, it is virtually obligatory nowadays to inquire into the logical form of assertions or other speech acts, the use and meaning of which one wishes to explain.

I have shown elsewhere that logical form is best thought of as a relation rather than a function, and defined it using the notions of syntactic transformation and inferential equivalence. I have argued here that the notion of abstraction is very closely related: one may use a similar definition, but substituting a different equivalence relation for that of inferential equivalence. Models (mutual models) are then just those things with a common abstraction under a particular purpose. However, the activity of modelling is empowered sociologically by meta-purpose; that is, one deliberately constructs models of artifacts for sociological or engineering purposes, and such purposes are different from the technical notion of purpose involved in the definition of model of, whose point is to denote an equivalence relation. About the choice of equivalence relations for the (technical) purpose, and about meta-purposes, I have said little or nothing. The topic of this paper is simply to discuss the logical form of the notions of abstraction and modelling.


References

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(Arm78): D. M. Armstrong, Universals and Scientific Realism, Cambridge University Press, 1978.

(Aud95): Robert Audi, ed., The Cambridge Dictionary of Philosophy, Cambridge University Press, 1995. Back

(Ben65):Paul Benacerraf, What Numbers Could Not Be, Philosophical Review 74(1):47-73, Jan. 1965. Also in (BenPut83). Back

(BenPut83):Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics, 2nd edition, Cambridge University Press, 1983. Back

(Bla94): Simon Blackburn, The Oxford Dictonary of Philosophy, Oxford University Press, 1994. Back

(Lad97.1): Peter B. Ladkin, The Success and Failure of Artifacts, at http://www.rvs.uni-bielefeld.de, March 1997. Back

(Lad97.3): P. B. Ladkin, Logical Form as a Binary Relation, Technical Report RVS-RR-97-03, available at http://www.rvs.uni-bielefeld.de Back

(Lev95): Nancy G. Leveson, Safeware: System Safety and Computers, Addison-Wesley, 1995. Back

(Lew83): D. Lewis, New Work for a Theory of Universals, Australasian Journal of Philosophy 61(4):343-377, December 1983. Also in (MelOli97). Back

(Lew86): D. Lewis, On the Plurality of Worlds, Oxford, Basil Blackwell Publishers, 1986. Back

(Mad90): Penelope Maddy, Realism in Mathematics, Oxford, Clarendon Press, 1990. Back

(MelOli97): D. M. Mellor and Alex Oliver, eds., Properties, Oxford University Press, 1997. Back

(Qui48): W. V. O. Quine, On What There Is, Review of Metaphysics 2:21-38, 1948. Also in (Qui53) and (MelOli97). Back

(Qui53): W. V. O. Quine, From a Logical Point of View, Harvard University Press, 1953. Back

(Rus67): Bertrand Russell, The Problems of Philosophy, Oxford University Press, 1967. Back

(Sea96): John R. Searle, The Construction of Social Reality, Simon and Schuster, 1995; Penguin, 1996. Back

(Wit22): Ludwig Wittgenstein, Tractatus Logico-Philosophicus, German+English, trans. C. K. Ogden, Routledge, 1922, trans. D. F. Pears and B. McGuinness, Routledge & Kegan Paul, 1961. German only, Suhrkamp Taschenbuch 501, 1984. Back

(Wol89): Sybil Wolfram, Philosophical Logic: An Introduction, Routledge, 1989. Back

(Ric96): Thomas Ricketts, Pictures, logic and the limits of sense in Wittgenstein's Tractatus, in (SluSte96) Back

(SluSte96): Hans Sluga and David G. Stern, eds., The Cambridge Companion to Wittgenstein, Cambridge University Press, 1996. Back


Footnotes

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Footnote 1:
I'm not sure that such a distinction is very helpful here. One major problem with an abstract/concrete distinction is to determine what things are supposed to be abstract and what things are concrete. First, according to Quine's dictum (QUiXX) that `to be is to be the value of a bound variable', all things which there are are on an equal ontological footing. An integer is just as `real' as you or I, providing our ways of dealing with the world requires arithmetic. So saying something is abstract is not to deny that it is real.

Physical objects are supposed to be concrete. But are all concrete things physical objects? If calling something abstract is merely to say that it is not a physical object, then sentences (both sentence tokens and sentence types) are `abstract', whether written or spoken. And it would follow from Leveson's comments that such things `have no physical reality'. But it would then follow that a spoken sentence has no `physical reality', and that would seem a strange use of the words `physical reality'. The spoken sentence, after all, derives directly from a pattern of physical movements creating sounds and describable by physics. And we are able to discriminate amongst sentences created in this physical way, by physical means. So an interpretation of `concrete' as meaning `physical object' leads via Leveson's comment to its contrary `abstract' as `having no physical reality', and therefore to the conclusion that things which are not physical objects have no physical reality. This conclusion would seem unsustainable, therefore we must give up the supposition, which is that there are things which are not physical objects are abstract.

So there are `concrete' things which are physical objects, things which are not physical objects but have `physical reality', and things which are abstract and have no `physical reality'. What is then supposed to be the argument for `systems' belonging to the third category, rather than the second?

An interpretation of `concrete' to mean `has physical reality' (that is, that includes both physical objects and those things which are not but which nevertheless have physical reality) leads to an equivalent question: what is the argument that `systems' are not `concrete' in this sense?
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Footnote 2:
It is also interesting to note that the view held by some engineers towards claims that program verification is in principle impossible is justified by arguments that may appear to be close to those favored by the later `language-game' Wittgenstein.
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Footnote 3:
One way of justifying this distinction would be to assimilate logically-simple propositions to assertions that a `natural' property or relation holds of certain particulars identified by proper names or direct demonstratives.
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Footnote 4:
A categorial grammar classifies sentence elements into distinct, disjoint sets, called categories. Grammatical rules specify in terms of these categories how items from these categories may be combined to form grammatical phrases. For example, we may have as categories nouns, verbs and articles, and in each category numerous different words. We may specify that a verb followed by an article followed by a noun forms a verb-phrase, and a noun followed by a verb-phrase forms a sentence. Generative grammar is categorial grammar, and there are also categorial grammars that are not generative.
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Footnote 5:
Consider the two propositions `I see red all over' and `I see yellow all over', which seems to be a different proposition. My field of vision can't be red-all-over and yellow-all-over at the same time, which rules out one truth-value combination. It can be red-all-over and not yellow-all-over, yellow-all-over and not red-all-over, or neither red-all-over nor yellow-all-over. That is, three out of four truth value combinations are physically possible. The two propositions cannot therefore be identical, else only two out of four combinations (both, or neither) would be possible; and the impossible combination (both) would be possible.
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Footnote 6:
One might puzzle over how these two physical structures could be `exactly the same, except that one has red where the other has blue'. Physics tells us that there must be some difference in textural structure, at the level of the atoms at least, in order for a surface to be a different color. The colors are caused by the physical structure in now well-known ways. So there would have to be more differences. A sceptic may now begin to wonder whether a coherent interpretation could be given to my curt phrase `exactly the same, except....'.

While acknowledging the point, I see the difficulties here as analogous to the difficulty in determining that, say, I am the same person as I was 10 years ago. I share very few molecules with myself, my hair color and skin texture is not the same, and I'm somewhat heavier, all of which makes for large physical differences, if we're speaking on the level of identity and difference of atoms and molecules and their relative arrangements. Bits that use to be part of me are now spread across the globe as I have travelled.

It would be a radical criterion of identity that determined that I could not be identified across this time span. This criterion may in the future be argued for and generally accepted. The current status is that very few would argue for such a radical criterion. We acknowledge these identity problems across time and try to explain how nevertheless such personal identity across time may be accomplished. Whatever story is told, the same approach may be used to explain my situation `exactly the same, except for the blue where there is red'. The coherence of my example is of a par with the temporal identity problems of physically-evolving entities, and addressing that problem is not part of my theme in this essay. I am therefore content for the reader to assume whatever solution she would prefer.
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Footnote 7:
One should not confuse meta-purpose with the purpose P included in the definition of the modelling relation. I have not explained what this purpose P in the definition may be, or what its role is, and shall not do so in this paper. What is meant by the difference between purpose and meta-purpose may be illustrated thus: my tenth-scale aircraft and full-size aircraft are models with purpose to have similarity of aerodynamic shape. However, the meta-purpose of the tenth-scale model is to figure out the airflow over the full-size aircraft by wind-tunnel experiments. To this meta-purpose, a tenth-scale model was built to fulfil the purpose of same-shape. While purpose is a logical relation, meta-purpose is more sociological.
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