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Gregory Landini University of Iowa, Iowa City, Iowa. USA Fictions are all in the mind.

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1 Gregory Landini University of Iowa, Iowa City, Iowa. USA Fictions are all in the mind

2 Abstract. Poetic license is an essential feature of intentionality. The mind is free to think about any objects, even objects with logically incompatible properties. Some philosophers maintain that a theory that embraces an ontology of non-existent objects is indispensable to any account of the nature of intentionality. Any such theory, however, must face paradoxes whose solutions conflict with poetic license. In this paper, I propose a theory which rejects the argument that an ontology of non-existents is indispensable for any adequate account of intentionality. The theory maintains that the intentionality of thought is produced by the quantificational nature of the apparatus of thought. All de re ascriptions of propositional attitudes must quantify over concepts and respect simple-type stratification. There are no fictional objects; there are concepts which, in the impredicative reflections of quantificational thought, are presented as if objects of thought.

3 The problem of the objects of thought is very old, dating at least to Platos Theaetetus. It concerns the paradoxical matter of thinking about what is not. Perhaps the greatest challenge to any thinker is stating the problem in a way that will allow a solution. It was Russells most singular achievement in philosophy (in his 1905 On Denoting) to have shown that the question How do we think about what is not? is a complex question a question whose presumptions required for its intelligibility are false. There is no answer, since to answer would be within the parameters set by the presumptions. The presumptions must be rejected. We do not think about what is not! We think quantificationally, by means of all and some, together with negation and other logical operations. See p. 894 in Plato, The Collected Papers, ed. by E. Hamilton and H. Cairns, Pantheon Books, New York, 1961.

4 What happens in the brain when we form new concepts and think quantificationally remains the mystery of intentionality (if not also the mystery of consciousness itself). The mystery is inseparable from the nature of the origins of the innate transformational grammar studied by Chomsky and his followers, and the formation of the impredicative concepts omnipresent in mathematics. It is no great surprise that this mystery remains unsolved. Nonetheless, at the turn of the twentieth- century Russell showed how to avoid the conclusion that non-existents are indispensable to an account of intentionality (the directedness of thought to a specific object). Many philosophers today still have not understood the significance of his discovery.

5 Philosophers relish in discovering and contriving ontological conundrums. Many derive from the problems of intentionality and thus engage the many seemingly intractable questions of the nature of mind. Since there are, as yet, no viable answers to these questions, philosophers feel that their speculative ontologies have legitimacy as theories. What shall we make of this sort of argument from indispensability? To be sure, when every known theory faces anomalies (same in number and degree of seriousness), no one should be charged with irrationality for not abandoning their favorite. Aristotelians should adhere to the earth at rest at the center of the cosmos, mechanists should demand their rigid bodies and cling to causation as impact, Newtonians their forces, rectilinear inertia with absolute mass, time and space. But empirical anomalies do reach crisis proportions and empirical theories are rationally evaluated and some abandoned.

6 Logic, I should maintain, must no more admit a unicorn than zoology can; for logic is concerned with the real world just as truly as zoology, though with its more abstract and general features. …The sense of reality is vital in logic, and whoever juggles with it by pretending that Hamlet has another kind of reality is doing a disservice to thought. A robust sense of reality is very necessary in framing a correct analysis of propositions about unicorns, golden mountains, round squares and other such pseudo-objects. What of philosophical ontologies? Can they ever be rationally evaluated? It is not easy to see how. With a wry wit, Russell put the matter this way (Russell, Introduction to Mathematical Philosophy, 1919, p. 170 ):

7 What Not to say about What is not A robust sense of reality demands that embracing non-existent objects (whether possible or impossible) on grounds that they naively seem indispensable for intentionality is, as Russells softly puts it, doing a disservice to thought. Logic, Russell says in Our Knowledge of the External World, is the essence of philosophy. Logical muddles, complicated though they may be to unravel, are at the heart of all such indispensability arguments. Russell, OKEW, 1914, p. 42.

8 The problem of intentionality was a famous subject of Alexius Meinong ( ), an Austrian philosopher and psychologist working at the University of Graz. In 1894 he founded an institute of experimental psychology and supervised the promotion of Christian von Ehrenfels (the founder of Gestalt psychology). Meinong become infamous for maintaining that there are objects (of Intentionality) of which it is true to say they are not. Thus, to some philosophers, Russells robust sense of reality is simply a prejudice in favor of a given philosophical theory. Russellians find it odd indeed to be accused of prejudice in favor of what exists! In any case, Russells point is not to proclaim that only this or that exists. His position is not one of philosophical prejudice; it is simply a rejection of the argument from indispensability. There is no burden to demonstrate that Pegasus, Hamlet, Apollo and God do not exist, just as there is no burden to prove that there is no teapot orbiting the sun beyond the Kuiper belt. Those supporting the philosophical ontology of non-existent objects require an argument that they are indispensable to the directedness of thought (intentionality). Try as they might, no convincing argument has yet been produced.

9 Consider the sentence Ponce de Leon thinks about the fountain of youth. Meinong held that there must be some sense in which it is proper to infer from this that there is some fountain of youth that Ponce de Leon thinks about. These objects of intentionality must have sosein (they, in some sense, have the properties in virtue of which the intentional act is about them). But sosein is not sein (existence). These objects of thought are intentionally inexistent. Meinong did not mean to embrace a baroque ontology of non-existent objects. He meant that Phenomenology is altogether independent of ontology. We must be able, somehow, to speak of objects of intentionality as aussersein and not thereby commit ourselves to their ontological status. Meinong accepted the Principle of Intentionality of his mentor Franz Brentano. Brentano maintained that thoughts represent, are directed toward or about, objects (often other than themselves). Meinong held that the phenomenology of intentionality relies on a principle of the independence of sosein from sein. He thought that this is the only way to assure the directedness of intentionality. The sosein of the object of the intentional act of thinking, say, of the golden mountain assures a golden mountain in spite of the fact that no golden mountain exists.

10 In On Denoting, Russell remarked that Meinongs principle is apt to infringe the law of contradiction. But Meinong replied that existing as a determination of so-being (sosein) is not the same as existence (which is a determination of being). The bemused Russell had no more to say on this head. Russells explanation of the directedness of intentionality rejects Meinongs principle of the independence of sosein from sein. Its lesson is that thinking is fundamentally quantificational. How then do we think about what is not? We dont.

11 There is no master argument in favor of the indispensability of non- existent objects to intentionality. We can only look at some paradigm examples of the alleged indispensability. We shall see that, although some of these arguments are difficult to conclusively reject, none are ultimately telling. Consider the following (from Graham Priest): I thought of something I would like to buy you for Christmas, but I couldnt get it because it doesnt exist. At first blush, it may seem that we are pushed into holding that some object of thought, a particular Christmas present, is such that it doesnt exist. But an escape is readily available. We can put: Some property is such that I thought of buying you something that has that property for Christmas, but I didnt because everything fails to have that property. In this case, we have only to quantify over a property that is involved in directing thought toward an object.

12 The same technique is required in the following: St.Anselm worshipped the being a greater than which cannot be conceived. Theistic worshipping is an activity directed at an object in virtue of the quantificational nature of thought. Anselm directs his activities of worship by means of a concept of property that an entity exemplifies if and only if it is a being a greater than which cannot be conceived. Likely, Anselm also worshipped Pope Urban. The form is the same. But now Anselm redirects his activities of worship by means of a concept of a quite different property uniquely exemplified by Urban. The difference, Anselms ontological argument notwithstanding, is only that likely nothing exemplifies the former property.

13 Consider the more difficult example of the fact that Sherlock Holmes is more famous than any real detective. For a to be a more famous detective than b, there must be properties F and G such that F uniquely picks out a as a detective and G picks out b as a detective and there are more people who use the concept of the property F in directing their thoughts than there are people who use the concept of the property G. Thus, replacing Sherlock Holmes with the detective who, according to the Conan Doyle adventure stories, lived at 221 B. Baker Street, London, we have: The concept of a property F of a detective who, according to the Conan Doyle adventure stories, lived at 221 B. Baker Street, London, is such that there are more people who use it in directing their thoughts than there are people who use a concept G that is exemplified by a detective.

14 Perhaps the most difficult case for indispensability was given by Geach. It is a case of the apparent witch about whom two distinct people Hob and Nob seem to share a reference. The example is this: Hob thinks a witch has blighted Bobs mare, and Nob wonders whether she (the same witch) killed Cobs sow. Does intentionality demand a witch, albeit a non-existent one, be accepted into ones ontology? Surely we can sketch some ontologically reasonable paraphrase avoiding the problem.

15 Does intentionality demand a witch, albeit a non-existent one, be accepted into ones ontology? Surely we can sketch some ontologically reasonable paraphrase avoiding the problem. The ascription reports de re on a concept of a property F employed by Hob in thinking some F has blighted Bobs mare. The property F is such that anything that exemplifies F is a witch. Nob employs a different concept of a property in directing this thoughts, namely, the concept G of an entity that is F and that blighted Bobs mare; and Nob wonders whether the entity which is G killed Cobs sow. Since paraphrase is required this is bound to leave out some of the connotations of the original. But it is not the full connotations that we are trying to capture. It is the truth conditions. Finding the right paraphrase is often difficult for want of a fully adequate philosophy of mind. But surely there is every reason to believe that a robust sense of reality ultimately will prevail. For a detailed account of this case within an intensional logic of nominalized predicates, see Cocchiarella, 1986b. For a discussion of this case within a theory of Guises, see Hector- Neri Castañeda, Reply to Burge in Tomberlin, 1983, pp

16 In evaluating an inference as deductively valid or invalid we are concerned with the logical form of the truth-conditions, not the grammatical form of the sentences involved. Validity is about structure, not about the content of what is said. The content of what is said, however, plays a very important role in exacting the truth-conditions and this is where empirical sciences and mathematics frequently come into play. Consider the following argument: Amphibians are disappearing in Panamas Omar Torrijos National Park. This frog is an amphibian in Panamas Omar Torrijos National Park. Therefore, this frog is disappearing. To decide whether the argument is valid, we must investigate the proper truth-conditions of the ordinary sentences that compose it.

17 It would be certainly naïve to regard the first premise as saying that everything is such that if it is an amphibian in Panamas Omar Torrijos National Park then it is disappearing! The truth-condition for Amphibians are disappearing (during temporal interval t in region R) is that the number of amphibians born during t in region R is less than the number of amphibians that died during t in region R. Though it appears well-formed grammatically, the argument is ill-formed (or equivocal). In the sense in which it is true that amphibians are disappearing we see that This frog is disappearing is ungrammatical.

18 Russell offers an analogous case. The statement Men are numerous has a form similar to the statement Amphibians are disappearing. Consider the argument Men are numerous Socrates is a man Therefore, Socrates is numerous. It would be perverse to regard the first premise to be saying that everything is such that if it is a man then it is numerous. The proper truth-condition is that the number of men is much greater than zero. The argument is ill-formed. The conclusion is ungrammatical. Russell, Logical Atomism, 1924.

19 Similarly, consider: Men exist Socrates is a man Therefore, Socrates exists. It would be perverse to think that the first premise says that everything is such that if it is a man then it exists. Moreover, it garbles the intent of the argument to present the first premise as saying that some man exists, for obviously a general first premise is supposed to be applied to the particular case of Socrates. The truth condition for Men exist is that the number of men is not zero.

20 Freges Foundations of Arithmetic had arrived at this position years earlier. But Freges concept-script requires that definite descriptions and proper names be genuine terms. It would not be of help to be told that the truth conditions for Pegasus exists are that the cardinal number of things equal to Pegasus is not zero. Pegasus still occurs in the phrase. Freges approach was to introduce a chosen object, say 0, for proper names that dont refer, and then just evaluate the truth conditions in virtue of it. But on that view, Pegasus exists and Pegasus is a number would be true, while Pegasus is a horse is false. Russell felt that this approach is plainly artificial. Russells theory of definite descriptions shows how to avoid this problem entirely. On Russells view, to give the truth-condition for Pegasus exists we replace the ordinary name Pegasus for an ordinary definite description. For example, let us replace it with the winged horse, who according to Greek mythology, was born from the beheaded Medusa. We then render the truth- conditions quantificationally as follows: Some unique winged horse was, according to Greek mythology, born from the beheaded Medusa.

21 Some unique winged horse was, according to Greek mythology, born from the beheaded Medusa. That is surely false; there is no winged horse. Even if we could biologically engineer a winged horse, it certainly could not be the subject of any ancient Greek myth. When we think that Pegasus exists, we do not think about Pegasus. Similarly, when we think that Pegasus does not exist, we do not stand in some mysterious relation of intentionality to a non-existent entity Pegasus. Russell represents the quantificational truth-conditions as follows: Nothing is uniquely a winged horse who, according to Greek mythology, was born from the beheaded Medusa. In thinking that Pegasus does not exist, we are not thinking about Pegasus.

22 The application of Russells theory of definite descriptions is but one of several tools for finding the truth-conditions. Like the application of Newtons laws to the tides or Einsteins equations to explain the perihelion of Mercury, finding the truth-conditions can be very complicated. It may await the discovery of new empirical and philosophical theories. But the robust among philosophers will surely reject the thesis that non-existents are indispensable for an adequate theory of the directedness essential to intentionality.

23 The theory of definite descriptions is wonderful at making important scope distinctions. Consider the question as to whether Pegasus has a recent evolutionary ancestor in common with rhinoceroses. To be sure, all horses are currently thought to have such a biological ancestor, and Pegasus, according to the Greek myth, is a horse in a fully biological sense. But clearly this does not provide the truth-conditions. Importing the modern theory of equine evolutionary descent into the Greek myth seems quite inappropriate. There must be some sense in which Pegasus is a biological horse and yet has no ancestor in common with rhinoceroses. Let us replace Pegasus by the definite description the winged horse who, according to Greek mythology, was born from the beheaded Medusa. A primary scope yields this: Some unique winged horse who, according to Greek mythology, was born from the beheaded Medusa has an evolutionary ancestor in common with rhinoceroses. This is false since, of course, there is no such horse.

24 In the pragmatic context of a conversation, however, a more charitable secondary scope reading is natural. This yields: According to Greek mythology, some unique winged horse who was born from the beheaded Medusa has an ancestor in common with rhinoceroses. Now David Lewis points out that in determining what is true in consistent fiction we must import some knowledge about the world. All the same, he offers an important way of constraining, relative to the common beliefs of the community of people living when the story originated, what information is to be brought to bear on the question of truth in consistent fiction. Hence, even the charitable secondary scope is false. Though Greek mythology tells us that Pegasus is a horse in the biological sense, it would be odd indeed to import modern evolutionary theory to decide what is true according to the myth.

25 Surely if someone asks whether Pegasus is a horse, we shall be inclined toward a charitable secondary scope. We have: According to Greek mythology, some unique winged horse who was born from the beheaded Medusa is a horse. This is true. The primary scope interpretation is false, but it is out of place given the likely intent of the question.

26 In natural language, the intended scope of quantifiers can be clumsy to indicate and often relies heavily on context of usage. The notations of modern quantifiers are a useful tool in making scope distinctions clear. For example, consider the sentence All snakes are not poisonous The scope of the negation is not salient. We have: Not everything is such that if it is a snake then it is poisonous. ( x)(Sx Px) This is quite different from Everything is such that if it is a snake then it is not poisonous. ( x)(Sx Px)

27 Very often in natural language scope is not represented by syntactic markers at all. It is achieved by a speakers reliance on her listeners pragmatic knowledge of salient features of the context of her utterance. This is especially clear when definite descriptions are used. Consider the sentence: The present King of France is not bald We have (primary scope) Some unique present King of France is not bald ( x)(Ky y y = x.&. Bx) [ x Kx][ Bx]. (secondary scope) No unique present King of France is bald. ( x)(Ky y y = x.&. Bx ) [ x Kx][Bx ].

28 If a man says, I dreamt that the discoverer of Relativity did not discover Relativity, listeners would not take him to be an idiot, but rather to intend to say that some unique discoverer of Relativity was the subject of a dream in which he did not discover it. The intended scope is given by pragmatic assumptions of a shared understanding between speaker and listener. Speakers typically bring a great deal of descriptive information to bear when they use ordinary proper names in communication. But the apparatus of using and understanding proper names in communication involves something quite different from simply replacing the ordinary proper name with a definite description. Communication with ordinary proper names does not produce the sorts of ambiguities of scope that are involved in describing objects. This feature of communication became a favorite of Kripke who criticized Russells theory of definite descriptions. Kripke was working on systems of modal logic involving statements of possibility and necessity. It will not do to say, on behalf of Russell, that the communicative rule for using proper names is one of replacement with a definite description together with a particular convention on determining scope.

29 No irrationality seems to accompany the claim that archeologists now know that Greek warriors hiding in a wooden horse were not responsible for winning the Trojan War. But if all original information, e.g., from Homers Odyssey, requires that archeologists use the proper name the Trojan War as short for the definite description the war won when Greek warriors hid in a wooden horse they are irrational. We cannot state their historical discovery as saying the following: Archeologists now know that Greek warriors hiding in a wooden horse were not responsible for winning the war won when Greek warriors hid in a wooden horse. A primary scope seems powerless to correct the situation. We cannot state the point as this: Some unique war won when Greek soldiers hid in a wooden horse is such that archeologists now know that Greek warriors hiding in a wooden horse were not responsible for winning it. Upon hearing a historian say Greek warriors hiding in a wooden horse were not responsible for winning the Trojan War, the listener cannot proceed by using a simple replacement rule for proper names that tells him to replace the Trojan War with a definite description and adjust for a primary scope.

30 There is no reason to suppose that communicating with proper names involves the use of a simple replacement rule for them. Does this show that there is something wrong with Russells theory of definite descriptions as an endeavor to provide truth- conditions? Certainly not. The viability of Russellian treatment of ordinary proper names and definite descriptions as a theory of reference in communication is quite separable from its role as a tool rendering proper truth-conditions. For a spirited defense of a descriptivist theory of reference in communication which includes a theory of indexicals, see Francesco Orilia, Singular Referecne: A Descriptivist Perspective, 2010.

31 The Structure of Thought is Quantificational Ascriptions of belief to a person S are often de dicto that is, they are sensitive to the nature/structure of the thought believed by S. Consider Socrates believed that the morning star is a planet. We have the de dicto truth conditions: Socrates believe that some unique morning star is a planet. i.e., Socrates believed that [ xMx][Px] All the same, we do make ascriptions of belief to others without knowing the precise nature/structure of the thought believed. We have indirect speech. These belief ascriptions are said to be de re (about the thing) because the person ascribing the belief to another describes the object of the others belief in her own way, not in the way the other thinks of it. Now the lesson we exacted from the old conundrum of thinking about what is not was that the structure of thought is always quantificational. We think by means of all, some, the, and so on. But if Socratess thought must be quantificational, what would be a primary, de re, scope?

32 A straightforward rendition of the primary scope yields: Some unique morning star is such that Socrates believed that it to be a planet. [ xMx][Socrates believed that Px] But this violates our thesis that thought has a quantificational structure. To capture a primary scope truth-condition we have to represent the structure of Socratess thought as quantificational. Thus we have: Some unique morning star is such that some concept F picks it out uniquely and Socrates believed that some unique F is such that it is a planet. ( F)(E!( xMx).&. Fz z Mx :&: Socrates believed[ xFx][Px]) This is de re because the person ascribing the belief to Socrates does not say what concept F is employed by Socrates in thinking about the morning star. The ascription says only that Socrates employs a descriptive quantificational thought which employs a concept F which uniquely picks out the morning star.

33 The same point applies in generalizing de re from a statement of a propositional attitude. For example: Socrates believes some unique morning star is a planet. Thus, some thought is such that S believes it. Wrong: Socrates believed that [ xMx][Px] Thus, Some thought q is such that Socrates believes q. ( q)(Thought(q) & Socrates believes q) This de re representation obliterates the structure of Ss thought. To represent the truth-conditions properly we have: Correct: Socrates believed that [ xMx][Px] Thus, Some concepts F and G are such that Socrates believes that some unique F is a G ( F)( G)(Socrates believed[ xFx][Gx]) The English is not a very reliable guide in the matter. The structure of Ss thought must be represented in the de re ascription.

34 In this case we know the quantificational structure of Ss belief. In some cases, however, we may not know it. Thus we need a means of representing such a structure without knowing exactly what sort of quantification it is. Happily, Frege showed the way. Thoughts, understood as quantificational structures, embody a simple type hierarchy (that Frege called levels of concepts). On this view, concepts have only a predicational capacity. They must always occur in predicate positions. They are not objects, which always occur in subject positions. For example, the first-level predicate … is a man represented as (…), fits together with the second-level concept Everything is such that ….it… represented as ( x)(…x…), to form Everything is such that it is a man which is represented as ( x)(Fx). Frege adopts a special variable M and would write For all M, M x Fx to quantify over all structures of this type. The notation solves the difficulty above.

35 Consider the sentence Ponce de Leon thinks about the fountain of youth. Here we do not know the structure Ponce employs. So we put Some concept F is such that an entity has F if and only if it is uniquely a fountain of youth and for some quantificational structure M, Ponce de Leon thinks M x Fx. There is no intentionally inexistent object of Ponce de Leons thought. Nonetheless, Ponce de Leon thinks about the fountain of youth. Ponce is just using a fountain of youth concept to direct his search.

36 Poetic License with a Vengeance and the impossibility of Intentional objects The thesis that there are intentionally inexistent objects of thought has led to different Meinongian attempts to form consistent theories of such objects But in countenancing intentional objects, they all stumble over the problem that the motivating principle for such objects is the poetic license which lies at the foundation of intentionality. We seem to think about all manner of objects of which it is true to say they are not. Even fantastical and inconsistent thoughts and stories about the Russell class (i.e., the class of all classes not members of themselves), round-squares, and existent golden mountains have to be accommodated. The mind allows the free play of full poetic licensepoetic license with a vengeance. To capture intentionality, one must capture this complete lack of restriction in the realm of the intentional objects. Yet a consistent theory of intentional objects requires restrictions on what are admissible objects of thought.

37 The objects of Intentionality are a rich source of paradox. There are paradoxes of intentionality which purport to show that in strange contingent circumstances a contradiction can come about by self- referentially directing ones thoughts. Suppose a person S at a given time were to believe exactly the thought that all of his thoughts are false. The expression of the existence of S involves a characterization, de re, of a thought. It says that there is a person S such that the following two conditions hold: (1) S believes that for all thoughts p, if S believes p, then p is false. S believes {( p)(S believes p p)} (2) All thoughts q are such that if S believes q then q equals the thought For all thoughts p, if S believes p, then p is false. ( q)(S believes q q = {( p)(S believes p p)} ) The paradox seems captivating because we imagine that there could contingently be such an S. Any one of us might try thinking just such a thought. How can it be that such a person as S is logically impossible?

38 S believes {( p)(S believes p p)} To generate the contradiction, the de re quantifier all thoughts p must involve a circular loop including in its range the thought Every thought p is such that if S believes p then p is false. The paradox is not solved by a general ban on quantificational loops! When paired with the poetic license of thought, our thesis that thought is fundamentally quantificational demands that the range of quantification be wholly unrestricted. That is, we must not adopt the position that quantification involved in thought is ramified in a hierarchy of orders.

39 That is, we must not adopt the position that quantification involved in thought is ramified in a hierarchy of orders. Ramification yields the following: Intentionality is not structured in that way! The quantifier all, which is essential to intentionality, is looped. It is Impredicative. If we rule out impredicativity by adopting ramification, we couldnt develop ordinary mathematical concepts such as the least of all upper bounds, or mathematical induction, the ancestral relation. I couldnt form the concept that the inventor of the Landini Cadence is among my ancestors. We couldnt think a person to be in self-refutation when he asserts that there are no truths. We couldnt enjoy the joke of a man who says, in speaking to a guest with an inordinately long nose, when it comes to inordinately long noses, it is a duty not to mention them.

40 Human thought essentially involves comprehension of impredicative concepts. To take an example from Whitehead and Russells Principia Mathematica, its comprehension axiom introduces an concept P that an entity x has if and only if x has all the attributes that are held in common by great generals. In symbols this is: Px ( )( ( y)(Gy y) x). Now suppose that x has P. Then it follows that ( )( ( y)(Gy y) x). The quantifier ( ) is said to be impredicative because it ranges over all attributes--- including P itself. This theory is consistent (relative to ZF set theory).

41 The impredicative concepts are looped, but they involve a good loop– a loop essential to mathematics and to even the most ordinary concepts. This good loop is not jeopardized by the requirement that the structure of thought be quantificational (and so simple-type stratified). Indeed, the poetic license required for any viable account of intentionality is left entirely intact when such quantificational (i.e., simple-type) stratification is imposed. In contrast, Ramification would utterly destroy it.

42 But what precisely makes the circular loop of Ss belief logically impossible? Our answer: Thought is necessarily impredicative and (simple-type) quantificationally structured. De re quantification into a propositional attitude (such as thought) requires respect for the quantificational (simple-type) representation of the structure of the thought in question. This is a simple type structure which leaves quantifiers unrestricted. It is not a ramification. Once the quantificational structures are set forth, the paradox cannot be formed.




46 Impredicativity as the Foundation of Mathematics and Intentionality

47 The following involve self-reference in very worrisome way. Claim: the only acceptable self- reference is that of impredciativity. All other features of cognition (thought) are derived from impredicativity. 1) Indexical self-reference: This sentence is false. 2) Descriptive self-reference: The sentence in the seventh line of the slide after the picture with two faces in Landinis talk is false. 3) Ontological self-reference: Every proposition is self identical. S believes exactly the thought that every thought S believes is false.

48 (NeoKantian) Thesis: Both mathematical logic and Intentionality have their foundation in impredicative quantificational concept formation. Impredicative quantificational concepts enable concepts such as (limit, ancestral, number). Without these concepts there is no way to generate genuine learning -- something needed to solve the disjunction problems: 1) Fodors disjunction problem, 2) Dretskes problem of misrepresentation 3) Searles problem of original intentionality. (Chinese Room /Indeterminacy of function of a of physical process).

49 Without impredicative concept formation, animals have only very disjunctive tracking tied to an environmental niche and so their content is not determinate. They cannot be said to form representations of specific objects, and hence they cannot misrepresent specific objects. That doesn't mean to imply that everything shows up in the disjunction for a given species. But it does mean that there is a precise finite disjunction fixed for the species in question. The animals in that species can never get free of their response patterns; and it is such fixed (though dynamic) patterns that are all the theory of evolution can generate. But with impredicative concept formation, an animal is free to form ever new concepts and is not locked into any environmental niche. Only such animals can be said to find genuine patterns in the world. This requires impredicative concepts (such as ancestor, limit, series) and this alone which enables non-disjunctive tracking--- genuine representations and misrepresentations of objects of the world.

50 This a new wedding of the philosophy of logic and mathematics with the philosophy of mind. Unlike former marriages of this sort– e.g., marriages arranged by Plato and Kant, this one has its foundation in impredicativity. Unlike Platos Rationalism, we do not embrace mathematical objects in a way that makes epistemic access occult. Unlike Kant, we do not make mathematics transcendentally ideal– necessary only for the phenomena. Unlike neo-Empiricism, we do not hold mathematics hostage to mechanisms for tracking features of the environment that can evolve in a process of selective adaptation.

51 This takes impredicative concept formation ideal (the scaffolding of Intentionaltiy). We know a priori that Intentionality involves a strange loop (Hofstadter). Reflective self-awareness is a given in Descartes Cogito. Sartre held that every intentional act is non- positionally self-reflexive. My pain and my awareness of my pain are one. Intentionality is a strange loop. But what sort of loop is it? My thesis is that it is the loop of impredicativity. The self or I concept is generated by impredicatively characterized fixed points of world maps.

52 Impredicative concepts may well not be recursively definable. But this is nothing like the occult nature of a non-well-founded object, be it a set or a self-referential res-cognitans.

53 Concept-Correlation To see how self arises in such a system. There is one more feature of cognition which we must address: concept-correlation. I hope to show in further work that his feature is derivable from imprediative quantification itself. But for the present I will take it to be a separate cognitive faculty. Concept correlation is the foundation of representation and it is to be analyzed in terms of impredicative mapping.

54 With Fregean structured variables we have a language which can express both simple type regimentation of quantificational concepts and type-freedom with respect to the objects of thought. Frege naively thought that concepts/attributes of any level can be correlated with objects, so that for each attribute there is a distinct object– its concept-correlate. It is easy to interpret this as a sort of naïve type-free theory of classes. It yields the Russell contradiction of the class of all classes not members of themselves. Concept-correlation is properly understood a cognitive act of representation– not as a metaphysics of new objects (classes, etc.) We embrace object of the process of concept correlation only as pretended objects that live in propositional attitudes.

55 That is, a propositional attitude provides a containment field within which simple-type freedom (and even contradictions) can live. This perhaps why Meinongians feel that thinking about something entails that there is something (an object) which is the object of thought. There are objects-of-thought about which it is true to say that they are not.

56 We can use concept-correlation to formally model cognitive acts of reflective self- awarenss. From the inside (of a propositional attitude) we seem to have objects of thought which are type-free (and even contradictory). But from the outside there are no such objects. Thoughts are essentially quantificational and thereby constrained by simple type theory (with predicates in predicate positions only).

57 Level 1 ( M)(Ansel [Mx x] Ansel [ú u ú Mx (x u) ]) Level 2 ( )(Ansel M [ (Mx x )] Ansel M [ú Mx (x u) ź ( ( ú u z))] ]) Note that Ansel [Mx x)] is of the form (Mx x) hence by the impredicative quantification of level 2 we have: Ansel M [Ansel [Mx x)] ] Ansel M [ú Mx (x u) ź (Ansel ( ú u z))] ] This is one (of many possible) representations of self (as ersatz object) to oneself.

58 Let us look at Level 1 more closely: Ansel [Mx x] Ansel [ú u ú Mx (x u) ] ú u and ú Mx (x u) are ersatz objects that live in the computational process that constitutes Ansels mind. Notice that ú u ú Mx (x u) has the form z z. And it is in virtue of this that we have the above result by impredicative quantification together with concept-correlation.

59 The cognitive process of concept-correlation is itself an impredicative (looped) quantificational process and it catches itself. That is, Ansel x [ x.. x ú u] Ansel M [ Mx x.. ú u ú Mx (x u)] And so on. Recalling that Ansel x is a system of quantifiers, universal instantiation yields Ansels discovery of Russells paradox: Ansel x [x x.. x ú(u u) ] Ansel [ú(u u) ú(u u).. ú(u u) ú(u u) ] The cognitive process of concept correlation (together with impredicativity) leads Ansel to entertain contradictions.

60 An important illustration of the power of impredicative concepts is in the proof of the fixed point theorem and the Schröder-Bernstein theorem. The theorems are useful for a theory of original intentionality because they involve impredicatively defined fixed points generated by injective mappings. This notion is relevant to any theory of mind that employs the notion of a map and a conception of self situated at a location on the map. Edelman and Tononi, for example, surmise that sheets of neurons in the brain produce reentry maps. This is a promising approach, but only if it is a neural foundation for impredicative conceptual processes, else (as Searle is apt to point out) Edelman and Tononi are not entitled to help themselves to the notion of a map. To avoid Searles problem, these maps must be intrinsically maps. Gerald Edelman and Giulio Tononi, A Universe of Consciousness (New York: Basic Books, 2000).


62 K =df {z: ( C)( z C & (u)( A-gB u.&. (x u x gfx u) : : C u )) } hx = fx if x K A h B g -1 x if x K A A-gB gf(A-gB) gfgf(A-gB) K = Kn, K0 = A-gB n 0 K1 = gf(A-gB) Kn = gfKn n 0


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