11. Digging Deeper

11.1 Foundations for Knowledge

How are we to judge claims to scientific knowledge?

In metaphysical philosophy and natural philosophy there have been ideas on this topic which may be called foundational. In this chapter we present some ideas along these lines.

Principally this concerns foundations for logical and metaphysical knowledge, knowledge a priori, but I will touch upon foundational aspects of empirical knowledge.

Before entering into a positive account of foundations I want to say a few words about the role which such foundations are intended here to fulfill.

It may be useful to draw an analogy with the use of the term `foundation' in the construction of buildings. In the context a foundation provides a base solid enough for the construction of the desired building, so that the building will stand firm and will survive the stresses to which it may reasonably be expected to be subject.

For this a foundation does not need to be absolutely solid.

The construction of a foundation does not itself proceed in the same way as that for the building. One does not, in order to obtain a solid foundation, seek a yet more solid foundation on which to build the foundation (though sometimes bedrock serves this function). There does not arise in this way, an unsolvable problem of regress in the foundations of buildings.

There are two reasons

11.2 Logical Foundations

In keeping with the positivist tendency to which it belongs, our account of metaphysical positivism has been concerned primarily with underpinning and articulating methods and tools suitable not only for rigorous philosophical reasoning but for application in science and engineering.

In Aristotle's conception of first philosophy, utility is regarded with some scorn, and the insistence of positivists that philosophy should facilitate positive science leads to the idea that positivism is an anti-philosophical philosophy (which is consistent with seeing it as continuous with academic and pyrrhonean scepticism).

Metaphysics is the name by which those topics at the apex of philosophy as conceived by Aristotle is now known, and the name ``metaphysical positivism'' may therefore be read as hinting that the pragmatic orientation of our positivism does not involve a rejection of those more remote regions of philosophy whose connection with life seems most tenuous.

Nevertheless, in metaphysical positivism, locating a place for metaphysical investigation is not easy. Two kinds of defect which may be found in metaphysical (and other controversy) from a positivistic standpoint are meaningless claims and purely verbal disputes. It is characteristic of positivist to reject metaphysics as meaningless, and in metaphysical positivism I retain a concern for precision and clarity in language, which motivates some of the deeper concerns which we address here. However, it is the business of philosophy to address problems whose articulation is difficult, and that one philosopher does not find a conjecture or a definition meaningful does not suffice to establish that it is not.

In keeping with the graduated scepticism in metaphysical positivism meaningfulness is not taken to be an all or nothing affair. Languages (or idiolects) may be compared on two related kinds of scale. First they may be compared according to their expressiveness. A language A is as expressive as language B if everything which can be said in language B can also be said in language A. To compare precision of definiteness of a language we have to consider languages as having multiple possible meanings or interpretations and then compare the range of interpretations of two languages.

An easy and fundamental illustration of this kind of comparison may be found in axiomatic set theory. If we consider a specific theory. say ZFC, the axioms of the theory provide an implicit definition of the concept of set which is the subject matter of the theory. The truth conditions of sentences of ZFC can be made very definite by stipulating that a sentence is true in ZFC iff it is true in every model of the axioms. Truth will then correspond to provability, in consequence of the completeness of first order logic. It is also reasonable in this domain to take the meaning to be the truth conditions, so that ZFC becomes as definite in its meaning as first order logic is. Unfortunately when we look at the intended applications of set theory, of which the first is to the theory of arithmetic, we find that this conception of the meaning of set theory is unsatisfactory. The normal procedure in reasoning about arithmetic in set theory is to define the natural numbers as some convenient countably infinite set of representatives. The most popular has been the scheme for representation of ordinal numbers under which the natural number zero is represented by the empty set, and every other natural number is represented by the set of its predecessors, i.e. the set of all natural numbers which are less than that number.

Using this definition, together with definitions of the usual arithmetic operations over these representatives, we can derive the usual theorems of arithmetic in ZFC. More true theorems of arithmetic are provable in this way than is possible in the usual direct axiomatization of arithmetic in first order logic (known as PA, for Peano Arithmetic). However, it is known, as a result related to the incompleteness results proved by Kurt Gödel that not all the truths of arithmetic are provable in this way.

However, because of the completeness of first order logic, all the statements of arithmetic as expressed in the way indicated in ZFC which are true under that semantics. The arithmetic truths which are not provable in ZFC, are, under that semantics, with that manner of representation of numbers, not even true. We have failed to produce an adequate definition of the natural numbers.

This is not an avoidable defect in the Von Neumann representation of ordinals. It is easy to see that under the proposed semantics for ZFC the truths of set theory will be recursively enumerable, and therefore any decidable subset of those truths (the truths of set theory which happen to correspond to sentences of first order arithmetic) will also be effectively enumerable, whatever definition of natural number we start out with. But it is know that the truths of arithmetic are not recursively enumerable. It follows that the concept of natural number is not representable in ZFC under the given semantics.

Though the semantics is definite, it is not expressive.

We can make the semantics more expressive, at the cost of making it less definite, in the following way. The semantics we have been discussing involves acceptance of all models of ZFC, and its (semantic) incompleteness reflects the existence of models of ZFC in which the set defined to be the natural numbers is not what the definition is intended to give.

The definition of the natural numbers is intended to give a set whose members are all the sets obtainable from the empty set by repeated application of the successor function (the function s(x) = x+1). The idea ``obtainable by repeated application'' cannot be directly expressed in first order logic, so the definition instead is given in terms of closure under the successor function. A set is closed under the successor function if for every member of the set, its successor is also a member. The natural numbers are then defined as the intersection of all sets which contain the empty set and are closed under the successor function.

Unfortunately, if we take an interpretation in which the intended set of natural numbers does not exist, in which every set which contains all the natural numbers also contains some other set, then when you take the intersection you get a set which contains that other set. This is a model with non-standard natural numbers, and such an interpretation will not get the truths of arithmetic right. Because our semantics allows these non-standard models, as well as models in which arithmetic is standard, statements of arithmetic which are true in the standard models but which are violated in some non-standard model will come out under the semantics as false.

To get the semantics on the nose for arithmetic statements we need to eliminate these non-standard models from the semantics. We cannot do this by adding another axiom, because the required constraint on the models is not expressible in first order logic. But we can add an informal stipulation to the semantics. We can specify the truth condition for sentences in ZFC as truth in all models of ZFC with standard natural numbers.

We now have a version of ZFC which is more expressive than the previous one. One in which the natural numbers really are definable (though not in the sense of this term which is used by mathematical logicians) and in which the sentences of arithmetic have the correct truth values. Though the semantics of this language are defined in part informally, the language can now be used to define the semantics of other languages in a formal way (relatively), whose semantics would not be definable in first order logic.

In this way we can define variants of the language of first order set theory which have progressively greater expressive power. This is done by using informal constraints on the class of intended interpretations of the theory. Such informal constraints can be placed in order of strength and the stronger the constraint is the more expressive the resulting language will be.

Beyond the constraint to models with standard natural numbers the following stronger constraints can be applied:

Constraining interpretations to be well-founded is strictly stronger than requiring standard natural numbers. It is as strong because in any model in which the set of natural numbers is non-standard it is not well-founded. It is strictly stronger because the same consideration applies to all limit ordinals (all transfinite numbers). In the absence of well-foundedness we can have models which give standard natural numbers but have a non-standard ordinal somewhere higher in the hierarchy. Well-founded models not only have standard arithmetic, but they also have standard ordinals all the way up. So under this semantics, but not under the previous one, the ordinals are definable.

Yet greater strength is obtained by requiring the full power set. All models of ZFC are closed under the formation of power sets. That means, that for every set in the domain of discourse the collection of its subsets (in the domain of discourse) is also a set in the domain of discourse. However, it is not the case that the power set is the same in every model, since not all subsets of the set are bound to be in the domain of discourse. The power set will include all the subsets which exist, but there may be subsets which don't exist (in this particular model). The constraint to full power sets eliminates any model in which there are missing subsets.

If we define truth in ZFC as truth in every model of ZFC in which the power set is full, then we get another language which is again strictly more expressive than the ones we have previously considered. Why is this? To understand this it is helpful to address the question why there exist non-well-founded models of ZFC.

There is an axiom in ZFC which is intended to assert that all sets are well-founded. This is called the axiom of regularity. Its intended effect is to deny that there are any infinite descending chains in the membership relationship, but this, like the obvious informal definition of the natural numbers, cannot be directly stated informally. The axiom of regularity states instead that every set has a minimal element. A minimal element of a set A is a member B of A which contains no member of A, such that A intersection B is empty.

This definition does ensure well foundedness if there are enough sets in the domain of discourse, which is the case if we have full power sets. Otherwise it does not, and there are non-well founded models of ZFC despite it having an axiom intended to deny their existence. Well foundedness is not expressible in first order logic, and so cannot be fully incorporated into the axioms, which is why the previous semantics relies on an informal constraint to well-foundedness.

If instead of requiring well-foundedness we stipulate the semantics in terms of models with full power sets, then since these are all well founded the resulting semantics is as expressive as the semantics based on arbitrary well-founded models.

That the semantics is strictly stronger can be seen from consideration of cardinal numbers. Cardinal numbers may be represented in ZFC as initial ordinals. An initial ordinal is an ordinal which has a greater cardinality than any previous ordinal. Two sets have the same cardinality if there exists a bijection between their elements. Such a bijection pairs up the elements in the two sets in a one-one manner so that they can be seen to have the same size. A difficulty with this definition of cardinality arises from the existence of models in which not all subsets of every set are present, in which we do not have full power sets. This is because the non-existence of a bijection might arise not because the two sets really are of different size, but because the bijection between their elements is just not in the domain of the model. The effect of this is that not all models of ZFC agree about which ordinals are initial, and consequently they do not agree about cardinal arithmetic.

Constraining the truth conditions to involve truth only in models with full power sets eliminates this source of disagreement between intended models of ZFC about cardinal arithmetic. Under this semantics but not under any of the previous semantics we can define the notion of cardinal number, so it gives a strictly more expressive language.

11.3 Empirical Foundations

Roger Bishop Jones 2012-09-23