On Scholastic Metaphysics: Me Against Aristotle

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Aristotle is my greatest philosophical hero. We’re talking about the guy that took Plato’s haphazard, mystic philosophy and turned it into a down-to-earth, rigorous, systematic investigation of all aspects of reality. Aristotle is the father of nearly every field of science, and every branch of philosophy. In the few cases where I think Aristotle was right (e.g. the Correspondence Theory of Truth), I wear my Aristotelianism with pride. So you can see why I’d be sympathetic to claims that Aristotle was fundamentally right, that Modern philosophy was wrong to reject virtually everything Aristotle said.

It is thus with great hope that I purchased Edward Feser’s magnum opus, Scholastic Metaphysics. This is the (small) book that’s supposed to show all those contemporary, analytic philosophers that they’re wrong and Aristotle was right. This blog-post series will be my reading diary of this book, my attempt to grapple with Feser’s arguments. As per my education in analytic philosophy, I’m opposed to his thesis – but I approach it not with fear he might be right, but with hope that he is! I would like nothing more than to see Aristotle vindicated.

Now, I disagree with Feser about, well, just about everything. But I do hope he is right, about the core of Aristotelian thought at least. With this in mind – let us read Scholastic Metaphysics!

  1. Feser vs. Scientism
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Bayesianism: Compound Plausibilities

[This post is part of a series on Bayesian epistemology; see index here]
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The last assumption of the core of Bayesianism is that the plausibility of (logically) compound propositions depends, in a particular way, on the plausibilities of the propositions that constitute them. I will write it in the following form* (using Boolean Algebra):
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Assumption 3: Compound Plausibilities: The plausibility of the logical conjunction (“A and B” or “AB”) and the logical dijunction (“A or B” or “A+B”) is a universal function of the plausibilities that constitute it, and their complements, under all relevant conditions of knowledge.
(A+B|X)=F[(A|X),(B|X),(A|X),(B|X),(A|B,X),(B|A,X),(A|B,X),(B|A,X),(A|B,X),(B|A,X),(A|B,X),(B|A,X)]
(AB|X)=G[(A|X),(B|X),(A|X),(B|X),(A|B,X),(B|A,X),(A|B,X),(B|A,X),(A|B,X),(B|A,X),(A|B,X),(B|A,X)]
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The functions are “universal” in the sense that they do not depend on the content of the propositions or the domain of discourse. The claim is that the plausibility of logical conjunction or disjunction – and thereofre of every complicated consideration of the basic propositions – depends on the plausibilities of the basic propositions, not on what they’re talking about.The assumption of universality is clearly correct in the cases of total certainty or denial. If A is true and B false, for example, we know that A+B is true – regardless of the content of the claims A and B, the topic they discuss, or so on. It is less clear why universality should be maintained for intermediate degrees of certainty. Some** suggest to consider it a hypothesis – let’s assume that there are general laws of thought, and let’s see what these are.

Another aspect of assumption 3 is that the universal functions depend only on their components. But assuming the functions are universal – what else can they depend on? They can only depend on some plausibilities. They cannot depend on the plausibility of an unrelated claim, for then it will not be possible to identify it in different domains of discourse. They must depend at least on the plausibilities of their components as they do under the extreme cases of utter certainty or rejection. It is perhaps possible to conjecture that in addition they may depend on some other compund proposition that is composed out of the basic propositions of the conjunction/disjunction, but this would surely be very strange. The decomposition into constituents therefore appears very simple and “logical” – I don’t know of any that object to it.

Let us proceed, then, under assumption 3.

It is a cumbersome assumption as each function depends on lots of variables. Fortunately, we can reduce their number. Consider the case where B is the negation of A, that is B=A. In this case
(A+B|X)=F[(A|X),(A|X),(A|X),(A|X),F,F,T,(A|X),T,T,F,F]
so that F depends on only two variables, (A|X) and (A|X). On the other hand, logic dictates that this plausibility must have a constant value,
(A+B|X)=(A+A|X)=(T|X)=vT
Assuming that the universal function F is not constant, the only way we can maintain a constant value when we change (A|X) is to change (A|X) simultaneously. We are forced to conclude that the plausibility of a proposition is tied to that of the proposition’s negation by a universal function,

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Theorem 3.1: The plausibility of A is tied to the plausibility of its negation by a universal function, (A|X)=S(A|X).
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We will determine S explicitly later. For now, it is enough that it exists.

Something very important just happened – from the assumption that there are genreal rules for thought, we concluded that the plausibility of the negation of a proposition (A|X) is measured by the plausibility of the claim itself (A|X). It is therefore enough to just keep track of one plausibility, (A|X), to asses both. As we have said previously, this is an inherent part of the Bayesian analysis, and we see here that it is derived directly from the assumption of universality. The main alternative theory, the Dempster-Shafer theory, considers the measure of support that propositions have and requires a separate measure for the support of the proposition’s negation. The existence of S implies that Dempster-Shafer must reject the universality of their own theory! There cannot be a universal way to determine the support for compound propositions from the support we have for the basic propositions, and even within a particular domain if this can be done then the theory only reverts back to the Baysian one. Unsurprisingly, Shafer indeed doubts the existence of general rules of induction.

Let’s move on. The existence of S allows us to throw out the complements A and B from the parameters of the universal functions, as they are themselves a function of the propositions that they complement (A and B).

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Theorem 3.2: Simple Shapes: The universal functions F and G can be written without an explicit dependence on the plausibilities of the complements.
(A+B|X)=F[(A|X),(B|X),(A|B,X),(B|A,X),(A|B,X),(B|A,X)]
(AB|X)=G[(A|X),(B|X),(A|B,X),(B|A,X),(A|B,X),(B|A,X)]
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The functions are still rather complicated, but they can be made even simpler. Consider the case where A is a tautology. In this case there is no meaning for the expression (B|A,X) – no information in the world can determine that a tautology is wrong. This expression is just not defined. But that plausibility of (AB|X)=(B|X) must still be well defined! There must therefore be a way to write G in a way that does not depend on the undefined variable (B|A,X). We should notice here that the parallel variable (B|A,X) is actually well defined in this case, so G might still depend on it. A similar situation occurs for the pair of variables (A|B,X) and (A|B,X). We can therefore conclude that we can write the universal functions in a manner that does not depend on half of each of these pairs.
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Theorem 3.3: Simpler Forms: The universal functions F and G can be written without explicit dependence on the information that A or B are wrong.
(A+B|X)=F[(A|X),(B|X),(A|B,X),(B|A,X)]
(AB|X)=G[(A|X),(B|X),(A|B,X),(B|A,X)]
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These forms cannot be used when A or B are contradictions, but otherwise they should be applicable. With just four variables, they are simple enough to server as a basis from which we can prove Cox’s Theorem – the foundation of Bayesianism.
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* My variant on this assumption is somewhat more general than that usually given.

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** For example, van Horn.

Bayesianism: Consistent Plausibilities

[This post is part of a series on Bayesian epistemology; see index here]

In the previous post we set the foundational assumption of Bayesianism, which is that plausibility can be measured by a Real number. We now put forward the second assumption: consistency. We will assume that the rational plausibility of logically-equivalent propositions must be equal.

Assumption 2: Consistency: If A and B are (logically) identical propositions, then their plausibilities must be equal (A|X)=(B|X). They must also have the same affect as information, (C|A,X)=(C|B,X).
This is a trivial assumption – treating identical things differently is the height of irrationality. I don’t think anyone can object to it.

All tautologies are always true and therefore logically identical to each other and to the truth (T). Since nothing can be more certain than the truth, they must receive the maximal plausibility value (regardless of any information). The situation is similar in regards to contradictions, which are always false (F).

Theorem 2.1: If A is a tautology, then (A|X)=(T|X)=vT, where vT is the highest possible plausibility value, representing absolute certainty. If A is a contradiction, then (A|X)=(F|X)=vF, where vF is the lowest possible plausibility value, representing total rejection of the proposition.

In light of this, we can rewrite the definition we have provided for “further infomration” in a simpler manner:
Theorem 2.2: The plausibility (A|A,X) of proposition A under information A and the further information that A is correct, is that of the truth (A|A,X)=T. The plausibility (A|A,X) of proposition A under information X and the further information that A is false, is that of falsehood (A|A,X)=F.

These are pretty obvious results. The importancce of demanding consistency will be made more obvious soon, as we discuss complex propositions.

Bayesianism: Logical Propositions

[This post is part of a series on Bayesian epistemology; see Index]

Is it true that the Earth is round? Does Newtonian mechanics accurately describe the celestial mechanics? Before we answer such questions it seems that we first need to assume that they are meaningful. We need to assume that such propositions are true or false, in the sense that they accurately (or not) reflect and describe reality*. This is the first assumption of Baysianism.

Assumption 0: Domain of Discourse: We are concerned with the truth value of a set of basic propositions (A,B,C….) about the world, each of which can be either True or False, and with the truth of all the complex propositions that can be constructed from them.

Most explanations of Bayesianism don’t even bother listing this as an assumption, but it actually isn’t quite as trivial as it appears to be**. The most important objection to it is that thinking of descriptions of the world – propositions about it – as either “true” or “false” is generally simplistic and naive. The Earth isn’t really round – it’s shape is somewhat oblong, it vibrates, and if you insist on perfect accuracy you will find no physical object has a well-defined shape due to quantum effects. But clearly, saying that the Earth is round is more correct than saying that it is flat. The “truth” of a statement is therefore not a simple matter of “true” or “false”. There are grades of truth, grades of accuracy. The same description can even be less or more true in different ways (e.g. more accurate, but less complete). And there is good cause to think that we would never have a complete and fully accurate description of reality, Truth with a capital “T”, as that would require an infinitely detailed description and our minds (and records) are finite.

While all of this is true, it is nevertheless clearly useful to think in such dichotomies. Indeed, even when we aren’t we’re really only thinking about the truth of more elaborate propositions, like “It is more accurate to say that the Earth is round than that it is flat”. Even in computer applications and mathematical derivations attempting to make use of more “fuzzy” thinking*** the programs and theorems are written down as they “truly” are, not just fuzzily.

I therefore believe that this objection ultimately fails in practice. Bayesian thinking indeed refers to rough dichotomies, but in many cases and in many ways such thinking is perfectly valid and acceptable. The limited nature of this discourse, however, should be borne in mind, and ideally we should find ways to adapt and change our discourse as greater accuracy and precision are needed.

Another objection along these lines is that meaning is holistic. This line of thought objects to the entire attempt to determine the truth of some set of individual propositions. The proposition “the Earth is round”, for example, is meaningless on its own, without an understanding of what “Earth”, “round”, and so on mean. Meaning is only conferred by a network of intertwined concepts, and judging the truth of individual propositions in isolation is therefore impossible. Instead of trying to ascertain the truth of propositions, we should be considering the mental structures and representations underlying them.

I find this objection to be even weaker than the first. While meaning is indeed holistic, there is nothing wrong with approaching the problem of knowledge by considering specific propositions. They still represent truth claims about the world, and as such can be judged to be true or false. Understanding how to consider logically- and conceptually-related propositions should certainly be a concern, but this does not undermine the Bayesian approach.

The division into basic and complex propositions is not problematic, as far as I can see. The moment you admit to a domain of discourse containing true and false claims, you automatically can generate complex claims from it and it is only reasonable to want to consider their truth as well. As we shall see, it is precisely the construction of this complex structure that allows Bayesianism to proceed.

* This is the “Correspondence Theory of Truth”, and implicitly presumes “Realism”. Formally at least, Baysianism can be advanced without this metaphysical burden – merely having two truth values is enough, regardless of your theory of truth. In practice, however, I (and every Baysian I’ve read about) uses Truth in the correspondence sense and believes in Realism, so I’ll leave it at that.

** The only one I know that discusses it, however, is Mark Colyvan, in “The philosophical significance of Cox’s theorem”.

*** Allowing truth values to be continuous between 0 and 1, rather than 0 or 1, leads to “fuzzy logic”.

Bayesianism: Introduction

How to tell what’s true? What should we believe in? One growingly influential approach can be called Bayesian epistemology. It claims, in broad strokes, that rational beliefs and changing belief should conform to a mathematically precise probabilistic framework.

I admit I was rather skeptical of it, but after reading the first chapters of E.T. Jaynes’ book “Probability Theory: The Logic of Science” I’ve given it more respect and thought and now think that at least on some levels these Bayesians may be right. So I’ve decided to write this series of posts to examine it more closely. I will be focusing on examining its underlying assumptions and, therefore, its validity – but also on deriving its conclusions at a level of rigor and in a manner that appeals to me.

This post serves as an index to the series. I will update it as more posts are added. The intended series can be divided into several parts:

Cox Theorem: Proving the ‘rational’ beliefs must conform to probability theory.

  1. Logical Propositions
  2. Real Plausibility
  3. Consistent Plausibilities
  4. Compound Plausibilities

[and more…]