Sunday, December 30, 2007

On denoting and the laws of thought

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1. On First Cause
2. First steps: The General Principle of Equivalence
3. Building a sturdy ontology.
4. The cataclysm that attends the General Principle of Equivalence
5. On the structure available to possible worlds.

6. Boundary conditions of the world - the Cartesian view.
7. On denoting and the laws of thought.
8. Something for nothing, and why this is not absurd.

7. On denoting

A note for the faint at heart.
The following section is not strictly necessary for the general reader. However the more hard bitten among you will be aware of a problem in denoting things. For example, Bertrand Russell noted that one might say that the present king of France is bald. Under our wonderful logic, he either is or is not bald. But of course, there is no present king of France. The GPE enables us to consider a more neutral view, for we know that it is infallible, where the laws of thought (laws of logic) hold merely the status of being self-evident. And self-evidence is no solid foundation. Not so long ago it was self evident that time was as steady as a rock - then came Einstein.

My aim is to show that implicit within a rationalist Cartesian view, the oddities of formal logic may be overcome.

Russell on denoting
The challenge of denoting properly is considered in depth by Russell (1905). Other theories of reference and contributions have been advanced by Meinong, Frege, Mill, Devitt, Kripke, Strawson, Putnam, Quine and others (Reimer 2006), and fall into three categories – descriptive theories, causal theories, and hybrid theories. This has led to a lively discourse that is not essential to this development. It has been argued that Russell’s theory is at best incomplete (Donnellan in Reimer 2006), but this is not an aspect that is important here. I select Russell because it provides a useful background to consider the problems of denoting to enable me to develop a form of denoting that permits the Cartesian meditator to refer to omnets in a way that is not reliant on presupposition for certain cases.

Russell (1905) holds that the subject is of importance in logic, mathematics and the theory of knowledge, because it is denoting that permits access to certain kinds of knowledge. He considers the example of a reference to the centre of mass of the solar system at a definite instant as a definite point. He identifies that while we can affirm propositions about it, we have no immediate acquaintance with the point, which is known only by description. He holds that the distinction between acquaintance and knowledge about something is ‘the distinction between the things we have presentations of, and the things we only reach by means of denoting phrases’, (1905, n.p.). I agree, and note that in the Cartesian system our denotation of omnets and assets is not founded on acquaintance with any particular, rather it is founded on knowledge of the system.

For the Cartesian meditator, acquaintance is cut off. Presentations cannot be trusted. Knowledge of the world is important rather than acquaintance. Certainly, there is no way to determine, from the World-of-Seeming, whether an idea is a proper copy of its subject, which is an expression of internalism and the E\O divide. Yet we can know about the General Principle of Equivalence noumenon, and its necessary existence.

For Russell (1905), certain phrases denote unambiguously, although we have no acquaintance with what they denote (their referents). In the idea of omnets and assets, we find an echo of this, in that we recognise that whatever we have under consideration denotes an omnet at some level, within the domain of discourse. For the Cartesian meditator, the nature of acquaintance is not trustworthy. It was this that reduced the problem of self reference in Section Examples exist all around us, yet the real nature of these omnets does not carry into empirical understanding. For Russell (1905, n.p., his emphasis),
In perception we have acquaintance with objects of perception, and in thought we have acquaintance with objects of a more abstract logical character; but we do not necessarily have acquaintance with the objects denoted by phrases composed of words with whose meanings we are acquainted. ... All thinking has to start from acquaintance; but it succeeds in thinking about many things with which we have no acquaintance.
One aim of Russell’s efforts is to resolve the problem of non-referring expressions or negative existentials. His concern is expressed (1905, n.p.)
By the law of the excluded middle, either ‘A is B’ or ‘A is not B’ must be true. Hence either ‘the present King of France is bald’ or ‘the present King of France is not bald’ must be true. Yet if we enumerated the things that are bald, and then the things that are not bald, we should not find the present King of France in either list. Hegelians, who love a synthesis, will probably conclude that he wears a wig.
Russell sought to address the problem by introducing a complicated (by his admission) application of the Laws of Thought. Another solution presents if one recognises that the problem of denoting may not stem from denoting, but from making a method of denotation that conforms to the Laws of Thought, evident in Russell’s consideration of the present King of France. While Russell applies the Laws of Thought, to bring sense, he has introduced a kind of modal aspect, by providing three options: always, sometimes, and never. My contribution will be to shift the onus onto the ontology.

Russell considers the proposition ‘A differs from B’ (1905, n.p.), explaining that if this is true, then there is a difference between A and B. In terms of the meditator’s system, this implies a difference in assets. He says that this may be expressed as ‘the difference of A and B subsists’. Otherwise, there is no difference, the statement is false, which may be expressed ‘the difference between A and B does not subsist’, (1905, n.p.). He asks, ‘ can a non-entity be the subject of a proposition?’ (1905, n.p.) because if so,
‘I think, therefore I am’ is no more evident than ‘I am the subject of a proposition, therefore I am’; provided ‘I am’ is taken to assert subsistence or being, not existence.
As such, he argues, it is always self-contradictory to deny the being of anything; but this in itself leads to contradictions. If A and B do not differ, a supposition that either there is, or there is not, such an object as ‘the difference between A and B’ seems equally impossible (1905, n.p.).
Russell asks that meditators not make up their minds against the argument until they have attempted to construct a theory of their own on the subject of denotation. In the light of the current effort, it is not my intention to critique Russell’s efforts. Instead, I want to describe the nature of denotation from a holistic perspective, in the light of my arguments regarding the nature of ideas and perceptions as these sit in relation to omnets, assets and the General Principle of Equivalence.

To do so, I will begin by describing denoting from a Cartesian perspective. Then I will discuss the Laws of Thought that to my mind are at the heart of the problem. I believe that sufficient development of these early concepts may provide a meaning for the action of denoting that informs knowledge. In so doing, this will provide a tool kit and a better understanding of logic to enable further consideration of the Cartesian world.

Denoting presupposes a bi-condition
One of Russell’s (1905) concerns is that while thinking begins with acquaintance (by his determination), we use words founded in acquaintance to create denoting phrases for meanings that are intended to refer to that with which we are not acquainted, for example the present King of France. If terms are grounded in the Cartesian method, a distinction can be made between a reference to supposed existing omnets, such as the present King of France, and omnets that have a well-founded ontology. I believe that this alters the role of acquaintance in denotation. To me the use of each denoting term presupposes two actualities. The meditator presupposes
1. That there is a well-founded omnet, to which the term refers; and
2. That the meaning of the term in the mind of the meditator corresponds to the omnet under consideration.
For this to work for Russell, the copy principle must hold properly, such that the idea referred to has some referent in the world other than the idea itself. However, this is not verifiable, by Russell’s own standards (1967), essentially due to the Epistemological to Ontological divide (the difference between how we think the world to be and the way it actually is).

The Cartesian meditator has already limited the world to assets having a positive ontology by accepting/letting in only that which is indubitable, into the actual ontology (as opposed to possible ontology). So the idea of the existence of a negative existential is not a valid mode of thought. This is not particularly problematic for denoting the present King of France in the light of the above bi-condition, because, while the meditator may have no knowledge either way as to the existence of the supposed king, he may have well defined parameters that he deems to be the assets of a king of France. Then the statement, ‘The present King of France is bald’ either has a referent and this matches with the intent of the meditator (other people’s minds being irrelevant due to internalism) or it has no referent and so cannot match with the meditator’s idea, and so is a false denotation. To make this clearer, a proposition, viewed with respect to the General Principle of Equivalence, is a denotation of an omnet. In the present example, ‘The King of France is bald’ is a reference to a person for which hair is not one of his assets. If there is no King of France, then there is no such omnet in the world beyond the idea itself, so the second condition fails.

The Law of Excluded Middle (hereinafter the ‘LEM’), for example that the King of France either is or is not bald, that was a central motivator for Russell’s (1905) efforts, does not enter into the equation, as will be implicit in the work on logic that I will now pursue.

On the Laws of Thought
If I denote a big ginger cat as being a big red cat, it is easy to see that the cat is red, yet not red. It may be big in one sense, but not in another. Also, what is the maximum height of a short man? We need a system that brings sense to such duality.

Priest & Tanaka (2004) provide an analogy that suggests a useful mental stance, for the meditator in his considerations of logic, through Bohr's theory of the atom. By this account, an electron orbits the nucleus of the atom without radiating energy. Yet Maxwell's equations – an integral part of the theory – requires that an electron which is accelerating in orbit must radiate energy, and so should spiral in toward the nucleus. This seems to imply that Bohr's account is inconsistent. Priest and Tanaka explain that the inconsistency can be resolved if the two apparently inconsistent views are seen as paraconsistent. Logic is ‘paraconsistent’ if and only if its relation of logical consequence is not explosive; where ‘explosive’ means that a proposition and its negation (A,~A) implies that the proposition ‘B’ is true for any B (Priest 2004). The notation used here is the common form, as it presents in Priest (2004). Paraconsistent logic exists to accommodate inconsistent but non-trivial theories (trivial meaning that A, ~A is always true). Essentially, paraconsistent logic denies, qualifies, or makes conditional the Law of Non-Contradiction (hereinafter the ‘LNC’). As such, LNC is not a law at all. Then another sieve would be useful to make sense of propositions.

Stepping back from Priest and Tanaka’s description, consider that if, say, both Bohr’s description and Maxwell’s equations have an actual foundation in a well-founded omnet, both describe the same condition, then these prescribe boundary conditions upon the system, and the dichotomy is more properly viewed as a duality. The scientist (as opposed to the meditator for the sake of simplicity) is placed in a position where she or he ought to recognize that there is some underlying reality which anchors the model. Normally, one is not in a position to assign necessity to apparently dichotomous propositions, because one cannot be certain of the truth of either, for the reasons discussed in Chapter 2. But this is not the case for the world of the Cartesian meditator. In that world, two well-founded propositions can only be consistent, even if seemingly inconsistent.

Then, if there are two boundary conditions, and these are apparently contradictory, the problem lies in the way the meditator is conceiving the propositions, because the statements are paraconsistent. It may be that the Laws of Thought have reached the end of their proper domain.

It then lies with the meditator to pursue an interpretation of the system that brings meaning to terms used. For surely there can only be a singular Totality, else it would not be the Totality, which follows from the founding discussion and from the General Principle of Equivalence. It is this sense in which I consider the Totality, and wonder what interpretation of the concept enables evolution of the system while not contradicting the idea of there being a Totality omnet. The epistemology needs to account for such apparent contradictions as a feature of the Cartesian system.

I am not arguing that paraconsistency be employed, as it is for empiricism, until the more general view which accommodates the apparent contradiction is discovered, as was, strictly speaking the case for the atom. I am saying that, for the Cartesian meditator, the nature of omnets and assets begins as a windowless view, so terms essentially stand in for whatever is the case. As such the meaning of these terms is illuminated by an understanding of the system as a whole. Then, if all possibilities imply a singular result, then there is no contradiction and paraconsistency becomes consistency by our deeper understanding of the conditions that the terms were provided to denote.

For empiricism the normal method is to condition or limit previously held theories as soon as a more consistent view presents. Such theories are limited to truthlikeness (Oddie 2001). Consider the case of Bohr’s atom. The electron simply does not crash into the nucleus. The empirically based solution presents in the form of quantum mechanics. One might argue that this replaces Bohr’s view. Another interpretation, the one that I am advocating, is that quantum mechanics better informs the idea of an orbit (in the context of an electron) in terms of energy levels. Then one is justified to argue that the term ‘electron orbit’ (in context) and the ‘electron energy level’ denote the same state. Another example is the wave particle duality of quanta. We no longer consider this to be a dichotomy, for which either one or the other is incorrect. Rather, we accept that the idea is paraconsistent. The reason I discuss Bohr’s atom is because of my concerns regarding the meaning of the Totality, meaning ‘what is’.

Now turning to the Totality, I see a similarity of concept between the apparent duality of the idea of Totality, and Heraclitus’ claim: ‘We step and do not step into the same rivers; we are and we are not,’ (Robinson in Priest 2004). The contradiction is absorbed if one recognizes that the ontology includes a distinction between the river as omnet at a particular time, and the river as omnet over an extended period. Such a resolution might interpret the Totality as including all states of the world over all of time, or more simply, all states of the world, which equates to all assets (as world states) that go to make up the omnet that I refer to as Totality.

However this suggests a creation of new assets without limit, for if there is a continuity, then one might expect a whole new set of assets for each change of the universe, one more pure asset for every connection between the previous state and the next (whether this be a continuous change or not). Of course such a solution is just a guess, and is not, strictly speaking, accessible to the Cartesian meditator, for whom the particulars of the world are not properly informed by empiricist acquaintance.

For the Cartesian meditator, at this point in the discussion, there is a Totality in which he must be embedded (but this has only vague meaning at present), there is one or more pure assets, whether empty or not, and there is the General Principle of Equivalence. How this can accommodate change, in fact why one should expect change, will become evident in due course.

On the nature of truth and logic, a Cartesian perspective.
The Principle of Identity can be replaced under the General Principle of Equivalence with the condition of having all and only one’s assets in compresence as distinct from non-compresence. Identity then is a term that focuses on compresent assets, which was central to Leibniz’s (1995a) consideration of the Principle of Identity of Indiscernibles, in that he assumes compresence implicitly in properties going to make up an object as a unity – that is, being ‘in esse’.

Under the General Principle of Equivalence there is no foundation for an actual contradiction in the ontology, because the system is founded on there only ever being one omnet that can have all and only it assets. This is consistent with the view that there can be no two true statements (meaning statements that describe a reality), that are contradictory (see Glanzberg 2006 for an explanation of the central stances relating to idea of truth). An apparent contradiction springs from the epistemology when one describes an omnet using two propositions and finds that there is at least one asset (in the predicate) supposed of the omnet (the subject) referred to in one proposition, say A, and at least one asset supposed of the omnet by another proposition that is ‘not A’ or ‘other than A’.

The issue is informed by the ontology of the world as implied by the General Principle of Equivalence. Firstly I want to make a distinction between ‘not’ and ‘other than’, which has the same assets if the Law of the Excluded Middle (LEM) is necessarily true. Of course the LEM was banished to doubt hood by the Cartesian method.

‘Not A’ may be taken as representing the negation of A. What does this mean in the Cartesian world for which all assets have a positive ontological status? The meaning of words returns us to issues of denotation. Two interpretations present. Firstly, the meditator might be presupposing an asset A that an omnet (say B) with positive ontological status does not have or cannot have. This interpretation does not assume a proper ontological foundation for that asset A at all, given the positive ontology. It will be useful to denote interpretation symbolically as /A, meaning ‘there is no such asset A (in B)’. Alternatively, a reference by the meditator to ‘Not A’ may refer to some omnet of the Totality other than A. This interpretation might be denoted symbolically ~A (meaning ‘an asset of the Totality other than A’). So the first supposes the absence of an asset ‘in esse’, the second supposes the existence of an asset ‘ex esse’.

A further distinction may be made between the condition ~A referring to a particular B of the Totality and the general form of ~A referring to more than one B of the Totality, up to the limit case of all ~A of the Totality. For this thesis the simplest solution for the present is to ensure the intended target of the denotation is made clear in the description of system under consideration. While this may seem burdensome, in reality it expresses a very simple ontology, that there is a Totality, it has its assets and no two assets have exactly the same pure assets, which is just what the General Principle of Equivalence expresses.

My sense is that this is not a case against the LNC or the LEM, for they are an extension of ideas of equivalence and inequivalence. Rather, my concern is that the LNC and LEM are somewhat opaque to the meditator until these are considered and moderated by the meditator using the General Principle of Equivalence in the Cartesian system. As such, I feel that the LNC and the LEM may be treacherous at this stage due to the problems of denotation that I have just described. Then it is better to apply Ockham’s razor, which is essentially the effect of Cartesian doubt. This is not to be mourned. The LNC and LEM have long been a source of contention in philosophy, which is evident in concerns over denoting phrases. Other serious concerns are identified by Priest (2004). For these reasons, accounting for the notion of negative existentials through denotation in terms of assets and omnets assists the meditator in understanding the world.

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