Frank Ramsey’s Redundancy Theory of Truth

F.P. Ramsey claimed that the statement

p is true

is logically equivalent to 

p

Both have the same truth-value. However, surely if that were true, then we wouldn't even require the concept of truth at all. The predicate "is true" is a useless or meaningless appendage to p. In fact, "we can say everything we want to say without it".

What is lost when we say p rather than "p is true" besides the linguistic predicate itself? Do the two words "is true" actually add anything to what's already there? However, if it were already there, then perhaps we haven't done away with truth after all. We can say that contained within p is the meaning that p is actually true. Alternatively, perhaps the predicate "is true" is simply a "reaffirmation" of the p which comes before it.

We can ask: What makes p true? Is it its truth-condition? Then we can pair the proposition with its truth-condition. The problem is that this will give "a different result for each proposition". The truth-condition of one p may be snow's being white and another may be ravens' being black. Why not ask, then, what it is that makes any proposition true? What do all true propositions share?

Perhaps they don’t share anything. 

Perhaps the question is "over-generalised". ("How much does anything weigh?") We can compare each proposition with its truth-condition. However, "there is no general truth about truth in the sense required by the traditional theories".

For example, are truths about the past the same as truths in mathematics or truths about what’s happening now? What about negative truths like the truth that Gordon Brown is not in my room now or the truth (if it is a truth) that God doesn't exist?

Instead, we should see truth as part of our language or as part of a language. As S puts it, truth

"does not have the magic property which nothing could have, of leading us out of language, into some direct of transcendental encounter with the world".

What would such a direct or transcendental encounter be like? It would be languages-less. Thus, we can't have such an encounter because everything we say about the world is said in some language or other.

And the predicate "is true" is unequivocally part of that language. If it is part of our language, then it is part of us. It belongs to our concepts, senses, categories, classifications and the like. It is polluted by minds and by language.

The problem has been that many metaphysicians have wanted something that is somehow language-less or even mind-less that can make truth something deep and more profound than just about everything else. They have wanted truth to be, then, a non-natural property or thing both in and out of the world. Something non-observable, non-concrete than can somehow belong to the world or be applied to the world. They wanted the best of both worlds. They wanted something transcendent to be applied to the immanent. Something abstract (or just non-spatiotemporal) to be part of - or be applied to - the concrete and spatiotemporal.

The problem is that we have become used to the predicate "is true". We use it everyday of our lives. Is it any wonder, then, that it would be hard to do without it in our everyday discourse? 

For example, take the locution

"The truth about Mozart’s death."

That certainly makes sense. We can say: "The facts about Mozart’s death." However, facts are almost as metaphysically controversial as truths. Not only that: there may be a strong relation between fact and truth.

What is a fact? 

It is something that is true? 

What makes something a fact? 

That something is true of the world and that makes it a fact? 

In addition, facts are said to correspond with true propositions or statements. 

Thus, there doesn’t seem to be a way out of this semantic circle. 

The same is true about "a story which is largely true". We can say: "A story that is largely factual." Here we have the same problems again. What makes this story factual? The fact that it is largely true!

According to Frank Ramsey’s theory, "how could you remove these words from these phrases"? However, the fact that we can’t remove the word ‘true’ from these phrases doesn't automatically mean that truth is a metaphysical property - or any property at all. We also use the word ‘Superman’. However, Superman doesn't exist. We can even use the phrase ‘"the round square"; though the round square doesn't exist (not even at a possible world or abstractly).

Similarly, we can say that ‘not’, ‘or’ and ‘and’ don't refer to anything. However, such words do have a use and they can be implicitly defined. Perhaps we can say that ‘true’ has a use and we can define its use in our discourse. However, it may have a use and also a definition which does not entail that truth is also a metaphysical property of some kind. It may function like ‘or’ or ‘not’ in our discourse. Alternatively, it may be closer to the word ‘yes’ or the word ‘stop!’.

Constructivism & Intuitionism

Constructivism

The primary position of constructivism is simple. The constructivist "believes that we have no conception of mathematical truth apart from the idea of proof".

Simply, 

proof = truth

Or:

mathematical truth = proof

It follows that if truth = proof, then truth and proof are (despite Platonism) inventions of the human mind. Proof is all there is. More specifically, we can say the same about numbers. Numbers "do not exist until constructed, by operations which generate them in a finite number of steps". Mathematical operations don't just use numbers: they also construct them.

This leads us to a question: 

What did these mathematical constructions use before they constructed the numbers? 

What constituted the mathematical constructions before the numbers were actually created? Were numbers there from the beginning? In that case, who or what created them? Or, if they were there from the start, perhaps they weren't constructed (or created) at all and Plato was right after all.

The stark conclusion of constructivism is the ‘anti-realist’ idea that "all existing numbers are contained in the books and papers of the mathematicians" (384). Numbers aren't discovered or intuited by mathematicians. They're constructed or created. Thus if a number hasn't been constructed or proved, then it quite simply doesn't exist to be discovered or intuited. In addition, only numerals (not numbers) really exist. And to say "that numbers exist is to say that there are valid proofs involving numerals" (384). (This appears to be very like Hartry Field’s position.)

This position is very similar to that endorsed by Kant over a hundred and forty years earlier. Kant believed that mathematical propositions "are known a priori since we ourselves are the authors of them" (385). Is this mathematical idealism? Their a priori status is guaranteed simply because we don't need to look outside of our own minds to the empirical world (or even to a platonic realm) to discover numbers and their nature.

Intuitionism

Now we arrive at intuitionism, a variant on constructivism.

Here too proof is everything. However, there's a surprising conclusion to this emphasis on mathematical proof. We've already said that a 

"mathematical proposition is true only if there is a proof of it; similarly, it is false only if there is a proof of its negation" (385). 

But what if there is proof of neither? Does that mean that the proposition is neither true nor false? Perhaps it simply means that the proposition is "meaningless" or that it's not a genuine example of a mathematical proposition.

However, the intuitionists accepted one of these conclusions. The proposition may well be neither true nor false. It's still, however, a bona fide proposition. We must, therefore, deny the law of the excluded middle for such mathematical propositions. That is, we must deny the principle: either p or not-p. This means that such mathematical propositions must have a "third value". This third (truth?) value is often called "indeterminate".

There are more surprising conclusions one must accept if one is an intuitionist. For example,

"as Heyting demonstrated, we shall need an entirely new system of logic – which he called intuitionistic logic – in order to accommodate the constructivist vision of mathematical truth" (385).

The logicists tried to reduce mathematics to logic. Now we find that a discovery in mathematics will have a profound effect on logic itself. If mathematics requires a third truth-value (indeterminate), then so too will logic (which, of course, also deals with truth). Indeed a logical vision or system must ‘accommodate’ the new findings of constructivism or intuitionism. Does this in itself show us that logic is part of mathematics, rather than that mathematics is part of logic? Perhaps not in all cases.

Tarski's Convention T

Even though many philosophers believe that Alfred Tarski’s ‘theory’ of truth is not about correspondence, "he suggested [that it] captured the idea" of correspondence. This makes intuitive sense. In any case, he saw the notion of truth as foundational in logical discourse. He took this idea from Frege. Tarski’s "unspoken starting point was the account of reference proposed by Frege, in which truth features both as the aim of discourse, and as the semantic value of successful utterances" (109). This is also the position, it would seem, of Brandom’s inferential holism.


What are the three fundamentals of Tarski’s ‘semantic theory of truth’? –


i) That it should assign truth-conditions to each sentence of our language.


ii) That it should derive those truth-conditions from the semantic values of the parts of a sentence.


iii) That it should meet what he called a ‘condition of adequacy’, namely, that every instance of the following ‘convention’:


(T) s is true if and only if p.


should come out true.


What can we say about the schema above? We can replace the letter s above by a name. Or, more correctly, by ‘the name of a sentence’ (Frege said truth-valued sentences are names – names of truth-values?). Since it is a name of a sentence, and not a sentence itself, it will have inverted commas around it. In terms of the letter p, that will be replaced by the sentence itself – that is, without inverted commas. Now we can have:


(S) ‘Snow is white’ is true if and only if snow is white.


Because of his belief in object-languages and meta-languages, Tarski believed that


"truth could only be defined for each language taken on its own, and moreover that it must be defined not in that language but in another, which is called the “meta-language”’ (110).


Of course we need to ask why Tarski thought that this should be the case. A sentence cannot predicate truth of itself. Therefore a language cannot predicate truth of itself?


We mentioned correspondence earlier. Now we can clarify why the convention explicates correspondence. Such schema "relate a sentence to the fact that it is used to express, by first naming the sentence, and then using it" (110). A sentence is used to express a fact. Why isn’t S using the term ‘truth-condition’ here? Does that mean that a fact is simply a truth-condition? Is there no difference according to Convention T? Anyway, in the jargon, when we write ‘snow is white’ we are naming that sentence (hence the quotes). When we write snow is white we are using that sentence, not naming it.


Because of the intuitive simplicity of convention T, or even its vacuity (according to some), we can know ‘a priori that the sentence “snow is white”… identified the very state of affairs… that makes the sentence “snow is white” true’ (110). We can know this a priori simply because the sentence used is simply the sentenced named with quotation marks. We can't go wrong! Thus this theory can entail every instance of (T) in a language (say, English). And this is "all that can be captured of the idea of correspondence: all that can be captured in language" (110). If someone asks what the correspondence theory of truth amounts to, we can say this:


(T) S is true if and only if p.


I said earlier that some philosophers have called this convention ‘vacuous’. S says that "Tarski simply returns us to the indisputable platitudes about truth" (110). What’s the point of platitudes when it comes to something as deep as truth? This is an alternative to ‘profound metaphysical theories’. Indeed ‘perhaps we should not ask more of a theory of truth’ (110). Perhaps this is all there is to say, even if it's basic. Anything more, one thinks, would be metaphysics, and perhaps that was Tarski’s point. He may have still had logical positivist sympathies, despite not being a member of that school.


Quine took this idea further by considering the predicate ‘true’. This doesn't ‘describe the metaphysical status of a sentence, but simply as what he calls a “predicate of disquotation”’ (111). Does that mean that the predicate ‘snow is white’ is simply disquoted to become snow is white? I mentioned naming a sentence and then using that sentence. In this case, ‘we pass from words quoted to words used: and that, indeed, is its function’ (111). That is the ‘function’ of what? The truth-predicate?


Again, the purpose of Convention T is in its "making the minimum metaphysical assumptions". That was the whole point. That is why it is so simple! Having said all that, Tarski came to believe


"that it was impossible, and that theories of truth could only be devised for artificial languages, and then always at the expense of constructing another language in which to discuss them".(111)


Why, then, did he think that a theory of truth is ‘by no means easy’. Indeed why did he fail in his task (in the case of natural, not artificial, languages)? Does that mean that there is something wrong with (T) above? In that case, what is wrong with it? Is it that, in the end, one can't leave out the metaphysics after all? Perhaps, then, rather than providing the requisite metaphysics, or failing without it, he should have given up on truth altogether and become a elimitivist or naturalist about truth.


Another problem with leaving out the metaphysics of (T) was that "minimalist theories could be embraced by defenders of correspondence and by defenders of coherence" (111). Alternatively, "maybe these are just rival descriptions of the same idea – the idea contained in convention T’"(111). We must ask, then, how the coherentist interprets Convention T. However, it seems pretty obvious how the correspondence theorist will interpret it. (Perhaps on a Tractarian model in which the picture theory tells us that parts of the world, the atomic fact, are pictured by the parts of the sentence.)

Steven Yablo's 'Identity, Essence, and Indiscernibility' (1987)

 

 

 

 

"If the requirements for being β are stricter than the requirements for being ά, then β ought to have a ‘bigger’ essence than ά…Thus, more is essential to the Shroud of Turin than to the piece of cloth [which was used as the Shroud], and the Shroud of Turin ought accordingly to have the bigger essence.” (Yablo, 1987)

We can admit that it's “necessary that the Shroud of Turin is the Shroud of Turin” (according to Steven Yablo’s paper), and that it wasn't necessary that the cloth of Turin (which was used as the Shroud) actually became the Shroud of Turin. (Therefore the Shroud has a property that the cloth didn’t have, according to Yablo, and so they aren't necessarily identical' but only “contingently identical”.) So isn’t it also necessary that the cloth of Turin was the cloth of Turin, in the same manner it's necessary that the Shroud of Turin is now the Shroud of Turin? If, on this count only, we can say that the Shroud hasn’t yet got a ‘bigger’ essence than the prior cloth.

 

How do we decide these essences in the first place? (So as to thereby decide which object has the ‘bigger’ essence.)

 

For example, it might well have been necessary that the cloth could clean things (i.e., have a functional essence); otherwise it wouldn’t have been a cloth. (Let’s take the cloth of Turin to have been a cloth created to be used as a cleaning implement.) It's not necessary, on the other hand, that the Shroud can clean things because, after all, it's now a shroud and not a cleaning cloth. Therefore it must follow that the cloth had an essence or property that the Shroud doesn't have.

 

Similarly, it might well have been necessary (via the sortal cleaning cloth) that the cloth wasn't black; but white instead (i.e., so that it showed up the dirt). Again, surely it's not necessary that the Shroud is white rather than black.

 

Yablo extracts a ‘bigger’ essence from the Shroud by treating its function as part of its essence (i.e., the function sortalised by shroud for a dead body). He disregards the cloth’s own possible functions, one of which might have been cleaning. And even if Yablo’s Turin cloth was never a cleaning cloth (but only a piece of material used for garments), it would still have had an essence/property that the Shroud doesn't have which belongs to the sortal garment material.

 

For example, the cloth might have been used for garments (not shrouds) and therefore it shouldn't (or couldn't) have made its wearers itch. And it might have also kept them warm too. However, a shroud, or the Shroud of Turin, needn't have these properties because the dead don't suffer from itches or cold.

 

So not only is Yablo’s belief that the Shroud’s essence is ‘bigger’ than the cloth’s somewhat arbitrary, it may also be the case that all deemed ontological essences are always somewhat arbitrary and also stipulated via sortals rather than discovered ontologically.

 

The Yablo example somewhat parallels the oft-quoted Quine example of the rational, two-legged mathematician and cyclist.

 

Mathematicians are, in this example, deemed to be necessarily rational: sortalised by necessarily rational being. (Does this automatically make computers capable of difficult mathematical calculations and the discovery of new proofs rational?) Cyclists, on the other hand, are deemed to be necessarily two-legged – sortalised by two-legged beings. (Although a no-legged cyclist could free-ride down hills and push the cycle up hills.) However, what if we have a mathematician who's also a cyclist – a being who falls under the two sortals: rational being and two-legged being? Quine asks:

 

"Is this concrete individual necessarily rational and contingently two-legged or vice versa?" (1960)

Perhaps, according to Yablo, the mathematician cyclist has a ‘bigger essence’ than a mathematician who isn’t a cyclist. (Perhaps because he has no other interests either.) This mathematical cyclist would fall under the sortals: rational being and two-legged being. But Quine thinks all this is silly. He says:

 

"There is no semblance of sense in rating some of his attributes as necessary and others as contingent. Some of his attributes count as important and others as unimportant, yes, some as enduring and others as fleeting; but none as necessary or contingent." (Word and Object)

The essences of the cloth of Turin and the Shroud of Turin depend on sortal specification. The cloth turned out to have a smaller essence than the Shroud simply because Yablo didn't specify it in any way; except by saying that it was the cloth of Turin and that it became the Shroud of Turn. However, Yablo does specify the Shroud (via that very sortal shroud) by saying that it shrouded the dead Christ. Again, the cloth could be specified via its material makeup. A cloth must necessarily be made up of certain materials (e.g., wool, etc.), or that it must necessarily be woven or that it mustn't retain water. Quine, therefore, had this to say on essentialism:

 

"An object, of itself and by whatever name or none, must be seen as having some of its traits necessarily and others contingently, despite the fact that the letter traits follow just as analytically from some ways of specifying the object as the former do from other ways of specifying it…This means adapting an invidious attitude towards certain ways of specifying x…and favouring other ways…as somehow better revealing the ‘essence’ of the object." (From a Logical Point of View, 1953, pp. 155-6)

As Gibbard (1987) might have said: The Shroud is specified via two sortals: cloth and shroud. The cloth, on the other hand, is only specified via one sortal: cloth. So, in this scheme, essences come via the sortals of objects, not the objects themselves. Indeed the Turin Shroud could come to us (or we to it) via a sortal that Yablo didn’t use.

 

For example, the Shroud could have been specified via the sortal objects that bear an imprint. This is a genuine sortal because there are other members of the sortal, objects that bear an imprint, other than the Shroud (e.g., white walls with their hand prints). Again, the Shroud could be specified via a sortal that the cloth certainly didn’t have: historical artefact. However, as has been said, the cloth of Turin could have come to us, or we to it, via sortals not specified by Yablo, say, cleaning cloth or garment material.

 

Frege: Fun With Classes

The Equivalence Class

There's a logical idea used in the Fregean definition of number: the equivalence class.

Say that we want to define the geometrical term "the same direction". Take also the notion of a line in Euclidean geometry. Does

         ab have the same direction as cd?

We can answer the question

         if and only if ab and cd are parallel.

ab and cd, therefore, must be an example of the well-known geometrical parallel lines. What has this to do with the equivalence class or, indeed, with Fregean number theory? It does so because the direction of ab can be seen in terms of a class or of classes. That is:

            ab is the class of all lines which have the same direction as ab.

In other words, the class of ab (or the direction of the line of ab) has as its members all that have the same direction as ab. So, in that case, perhaps cd is a member of the class ab. It is so, as said, because it is parallel to ab. If any other relation, say yz, is parallel to cd, then by definition it must be parallel to ab as well. So both cd and yz (amongst many other lines) belongs to the class ab – they are its members; though they aren't classes themselves.

It would be better to give our ‘ab’ a better symbolic expression so as to distinguish it from ‘cd’, ‘xy’ and every other member of it.

The equivalence class, then, will "fully identify the extension of the concept: direction of ab". That is, its extension includes all examples of direction which are equivalent to ab or parallel, in this case, to ab. In terms of the concept or predicate expression "direction of ab", the class ab (or the class of ab’s) has as members the extension of that concept or predicate expression. Or both cd and yz fall under the concept [direction of ab] just as horses fall under the concept [horse], etc.

All the above is an example of all the directions instantiated by parallel lines. What of other directions or of "direction in general"? These too can be defined in terms of classes. Instead of the class direction of ab (or the class of all parallel lines), we can now have the class of classes which are equi-directional. In other words, this higher-order class has as its members the members that instantiate direction in general (or are all examples of direction). However, didn’t we say earlier that classes within classes were disallowed on pain of paradox and infinite regress? Now we have a class of classes, and these member-classes themselves must also have their own members and so on. We would have yet another case of the infinite regress or indefinite inflation of classes in Fregean number theory (or his class theory). In addition, the equi-directional class must be an infinite class for another reason. That reason is that it's surely the case that there's an infinite (or innumerable) amount of actual or possible directions in general; especially bearing in mind that one line (or direction) may start off being perfectly straight but then, for example, take a diagonal turn and so on. The permutations of a given line or direction must surely be infinite or innumerable. So this strange meta-class – the class of equidirectionality – must give rise to many infinite regresses and paradoxes, not just the one brought about by having other classes as examples of some of its members.

Numbers

Let’s get back to defining the concept NUMBER.

As we've said, we can define number as classes of equinumerous classes. In the case of the number 6, this number is defined in terms of all six-membered classes (whether the class of six horses or the class of six black persons and so on). Though, again, this equinumerous relation between classes mustn't rely on numbers or counting. Instead, that is, in terms of the one-to-one correspondences between all the members of a six-membered class and a different six-membered class. And this relation of correspondence is brought about by using the logical ideas equinumerosity and the equivalence class.

What of the problem of classes within classes?

To sum up. All the above includes examples of the logicist or Fregean attempt to

"complete the definition [of number] using only logical concepts: concepts whose meaning and extension are determined by the elementary laws of thought" (388).

In other words, it's an example of the many attempts to reduce mathematics (or simply arithmetic) to logic. As we know, this project failed and it was later generally accepted that logic is in fact a branch of mathematics, not vice versa.

0

Let’s go into detail and define zero (or 0) in purely logical terms.

We can do so, to begin with, by considering the predicate "not identical with itself". Clearly, then, there's not a single thing in our universe (or even at any possible world) that isn't identical to itself. It follows, therefore, that nothing falls under the concept or predicate "not identical with itself". In addition, Frege also believed that every predicate has as its extension a particular class and its members. What if nothing falls under "not identical with itself"? We can now say that its extension has precisely no members. Thus we can have an extensional class for our own number zero or 0. We can even say, a la Frege, that the number 0 is the class of all no-membered classes. Thus 0 too has its own extension!

That extension-class is often called the null class. More technically, the number 0 can be defined with variables instead of the numeral ‘0’. We can say that the

"number zero can be defined as the number of things x, such that x is not identical with itself (alternatively, as the class of all classes of things which are not identical with themselves)". [388]

This is another way of saying that there's no x which isn't self-identical; which is itself a logical truth. In terms of classes rather than the variable x, there are no classes of classes that aren't self-identical. However, according to Frege there is a class of classes of things (or members) that aren't self-identical. The point is, all these classes are empty, for the reasons just given. Or surely we must add that the class of all classes of non-self-identical things does indeed have members; though these members must only be classes - if only a single such class. It doesn't, then, have particulars or individuals as members, only an empty class – the null class.
 
This is an even more stark and blatant case of a class that has other classes as members. Not only that: its own member-classes are all empty! So, as with the infinite-regress argument, an empty class that's a member of the class of all empty classes will itself have at least one member-class as its own member. And so on. However, can we really say ‘and so on’ when this ostensible infinite regress is a result of empty member-classes and an empty meta-class – though that too, as said, may or must itself be a member of a higher or ‘larger’ class in a case of infinite inflation rather than infinite regress.

Frege required these empty classes (these null classes) in order to find an extension for the number 0, and therefore complete his reduction of arithmetic to logic, etc. Can we even make sense of an empty class?

S provides us with a logical schema to define the number 0, thus:

                           0 = df. ¬(x = x)

Put in a stark logical language, the proposition of non-self-identity seems even more ridiculous and absurd, if not contradictory or paradoxical. In natural language, the schema above can be translated thus:

For a definition of the number 0, we can say that it's not the case that all things equal themselves. Indeed there's at least one thing that doesn't equal itself.

It follows that according to Frege we can define the remaining numbers in the way we defined the number 0 – we can do so recursively. It appears strange that we can define 1 in terms of 0; or, more precisely, in term of the empty or null class. As S puts it:

"1 is the class of all classes equal in number to the null class (for it is a logical truth that there is at least one and at most one null class)." (388)

1

It seems strange, prima facie, that the number 1 is compared with 0 (or is ‘corresponded’ to the null class). We think this because empty classes seem strange or even impossible. And there we may think that there isn’t a single null class. As we saw earlier, Frege himself believed that there is a null class, otherwise we wouldn't have a logical extension for 0.

So 1 is the class of all classes with no members. And there's precisely one null class, according to Frege. Thus,the member-class of the null class is made to correspond to the class of all one-membered classes. It can do so because although the null class has no members (except, perhaps, another class), the null class is still a class of sorts even without members. So it's still a one or a unity. Despite the fact that there's one or a single null class, the members of all one-membered classes can still be taken to correspond, literally one-to-one, with the null class. It follows, therefore, that the number 1 also has its own extension – the null class or the class with no members. However, again, it's still a one (or a unity) and it can therefore still be used to correspond with all the one-membered classes which themselves belong to the class 1 (or the number 1).

In that case, we must also say that the meta-class (the class of all one-membered classes) must itself contain the null class (alongside all the one-membered classes as its other members). It is, again, a member of the class 1 not because it has one member; but because the empty class itself is a one or a unity. The class 1 contains two different classes: one-membered classes as well as the singular null class. This situation is replicated recursively with the other higher numbers.

2

Now take the number 2.

Instead of saying that 2 is the class of all two-membered classes, we should instead say that 2 is the class of all classes equal in number to the class whose only members are the null class and the class whose only member is the null class. Perhaps we must put it this way because if we talk of ‘one-membered’ or a ‘two-membered’ class we're using numbers in our definitions of numbers. And that's not allowed; primarily because numbers aren't logical but mathematical objects.

The class of all two-membered classes (or the class with these particular concepts that aren't counted, etc.) must include not only the null class but also the class whose only member is the null class. That last class, as we've already seen, is the class we used to define the number 1; just as we used the null set to define it, and, in turn, the null set alone was used to define the number 0. We now have:

              i) the class of two- membered classes

              ii) the class of one-membered classes the null class

As we've already seen, the classes enclosed within the meta-class must also contain classes ad infinitum.

What we correspond isn't the type of member-classes, but only classes qua classes qua members of classes. It doesn’t matter that the class 2 is a class that itself contains disparate classes - including the class whose only member is the null class and the class whose only member (not members) is the null class. What matters are the classes, not the types of classes (as the later Russell might have put it). In terms of counting, we only count classes as members qua classes, not types of classes. In terms of corresponding classes or members one-to-one, we also only take member-classes qua classes, not qua types. And so on. This ‘so on’ is a recursive ‘so on’, as it were.

As S puts it: We ‘“build” the numbers from the null class, while making no ontological assumptions whatsoever’ (388). This is another way of saying that Frege’s theory is concerned with classes qua classes, not classes qua types of classes. As with an axiomatic deductive logical system, we derive or ‘infer’ the rest of the numbers from the ‘axiomatic’, as it were, 0, rather than from logical axioms or premises. This is no surprise considering that it was part of Frege’s attempt to reduce mathematics to logic. This deductive system of number is itself logical in nature, not just the definitions of the individual numbers themselves.

The Successor Relation

We can also mention the "successor relation". We define this by means of the existential quantifier -∃. More correctly, for the

"number of the Fs is one more than (i.e. successor to) the number of the Gs if there exists an F such that the rest of the Fs are the same number as all the Gs". [388]

In other words, the members of class F are one more in number than the members of class G. In addition, if class F contains a member or individual F such that when that is excluded, all the other members of class F in fact are the same in number as the previous class G. So, in terms of one-to-one matching of every member of class G with every member of class F bar one. That single anomaly makes class F a different class.

However, in the above we appear to be using number-terms such as ‘number’ itself, ‘one more than’, the ‘same number’, and so on. I thought that number-terms couldn't be used in these Fregean definitions of numbers. However, S writes: "Remember 'same number as' is defined without reference to number.” So S must have used number-terms to translate and simplify the purely logical definitions that, instead, rely only on logical ideas or terms like ‘corresponds with’, ‘equivalence’, and ‘equinumerous’. Without the above translation using numbers, it may be difficult to understand the purely logical definitions of number offered by Frege and others.

In terms of logical recursion in number-definition, we can use a variable and say that

"x is a natural number if it falls under every concept which zero falls under and which is such that any successor of whatever falls under it also falls under it". [388]

In other words, x is a natural number if it falls under a concept which nothing falls under, or has as its extension the null class. This, clearly now, is a reference to the definition of the number 0. In terms of recursion, we can now talk about other concepts or classes under which at least one thing falls or has as its extension a single class. That is, the number 1. It's a recursive definition because the null class is a member of every ‘successor’ concept or successor class. That is, the class 1 contains both the class with a single member, the null class, and the null class with no members. In that case, successor numbers have been defined in terms of prior numbers or prior classes that are now contained in new classes and thus recursively generate newly defined higher numbers. As S puts it:

"… every concept which zero falls under and which is such that any successor of whatever falls under it also falls under it". [388]

To conclude, we logically define the number 2/3 by including a class which itself contains classes as members, such as class 2, class 1, and the null class. Or classes with two members, classes with one member, and a single class with no members – the null class.

Peano's Postulates

This Fregean definition was actually used by Peano to derive his own postulates. In turn, from Peano’s he also derived the rest of arithmetic. Both done recursively. Further, Dedekind and Cantor showed how to derive the whole of number theory from arithmetic. Therefore we have derived mathematics from logic. 

We can represent this recursive edifice in terms of a foundational schema thus:

         i) the whole of number theory

        ii) arithmetic

       iii) Peano’s ‘postulates’

       iv) Frege’s number theory (logical terms)

Or instead:

       i) the whole of mathematics

       ii) the whole of number theory (Cantor, Dedekind)
    
       iii) arithmetic (also Peano)

       iv) Peano’s ‘postulates’

       v) Frege’s number theory

       vi) logical terms and the ‘constants’

We can therefore condense these schemas into the simpler:

      i) mathematics

      ii) logic