The analysis and decomposition of Begriffsschrift content

In §9 of the Begriffsschrift, Frege writes:

`Let us assume that the circumstance that hydrogen is carbon dioxide is expressed in our formal language; we can then replace the sign for hydrogen by the sign for oxygen or that for nitrogen. If we imagine that an expression can thus be altered, it decomposes into a stable component [...] and the sign, regarded as replacable by others. The former component I call a function, the latter its argument.’
To simplify later discussion I translate Frege’s idea into a contemporary framework.
Let L be a first order language of arithmetic with a standard syntax such that the sets of L-expressions, L-formulae and L-sentences are defined inductively. Also, a function can be defined which maps L-sentences to the expressions which occur in them.
Frege’s idea can then be generalized and formalized.
For any L-sentence \psi there are some L-expressions A_0, …, A_n which occur in \psi.

For any such set {A_0, …, A_n} there is an L-expression \Phi which results from the sentence \psi by replacing any A_i by a new free variable.
In this manner, for any L-sentence a set {\Phi,{A_0,…,A_n}} can be determined.
Taking up Frege’s terminology, A_0 to A_n are the arguments for the function \Phi.
Later, `analysis’ and `decomposition’ will be introduced as technical terms.
Therefore, I shall call the set {\Phi{A_0,…,A_n}} for a given sentence \psi the `division’ of \psi.
and \psi `divided’ into the function \Phi and the arguments A_0 to A_n.
Passage 3 now suggests that Frege held that

D. for any sentence there is more than one division.

According to Dummett, D is compatible with the content of sentences having a determinate structure
because only one division reveals the content. Dummett calls this the analysis of the sentence. Other divisions are mere decompositions, of which there are various.
This distinction needs explication.
The set of L-expressions is defined inductively.
There are simple and complex L-expressions. `1′ and `-’, for instance, are simple, `(-1)²’, again, is complex.
When Dummett says that the analysis of a sentence takes place in stages [citation!], he means that it results from an iterated division along the lines of its composition.
Complex sentences are composed of a logical constant and one or more sentences.
Since a given complex sentence can be divided in whichever expressions occur in it, there is one division into this main constant and the sub-sentences.
For example, the sentence `(-1)²=1 \rightarrow (-1)²+1=2′ can be divided into {\rightarrow,{(-1)²=1,(-1)²+1=2}}.
In such a division, the arguments are sentences themselves.
These again can be divided into their main logical constant and their sub-sentences.
This procedure is iterated until one arrives at atomic sentences.
Atomic L-sentences are construed out of a relation symbol and singular terms.
Consequently, for a given atomic sentence, among all of its divisions there is one into the relation symbol and the singular terms of which it is composed.
For most atomic sentences, however, some arguments are complex expressions themselves.
For example, the analysis of `(-1)²=1′ is the division {=,{(-1)²,1}}, where the first argument `(-1)²’ is complex.
In the definition of the set of L-expressions, complex terms are construed inductively from simple expressions.
Therefore, among the many divisions of a complex terms one corresponds to its composition.

For `(-1)²’, this division is {²,{(-1)}}.
Iterated division of a sentence along the line of its composition, into entirely simple expressions, results in what Dummett calls `analysis’.
For example, the analysis of `(-1)²=1 \rightarrow (-1)²+1=2′ is
{\rightarrow,{{=,{²{-,{1}},1},{=,{+{²,{-,{1}},1},2}}}}.

Since any sentence is composed of simple expressions in a unique way, analysis is unique.
This does not contradict D since next to its analysis, there still are various other divisions of a sentence.
These Dummett calls `decompositions’.
For example, the sentence `(-1)²=1′ is decomposed into {(- )²=1,{1}}.

MLitt Essay Seminar

Today we had a great seminar on the St Andrews MLitt, I enjoyed very much the work my fellow students presented.

I took the chance to present my case against Davidson’s argument for anomaly.

Dummett’s argument for why Analysis provides Thoughts/ Senses with a unique Structure

In a previous post I said that Dummett holds that analysis, but not so decomposition, provides senses with a unique structure.

Dummett’s argument according to  IFP, pp. 272f:

1. L1 the sense of a complex sentence is composed out of the senses of the constituents given by its analysis
   1. Analysis reveals how the referent, i.e. the truth value of a complex sentence, depends on the truth values of the constituents. (definition)
   2. The sense of an expression of the language L is the way its referent is given to a competent speaker c of L. (premise)
   3. The way a referent of an expression e is given to c is the way in which c understands the referent of e to be determined. (premise)
   4. For complex expressions e, the referent of e depends upon the referents of which ultimate constituents are given by its analysis. (a)
   5. c understands how the referent of e is determined only if c understands how the referents of the ultimate constituents are determined. (d)
   6. The way c understands the referent of e to be determined involves depends on the way c understands the referents of the ultimate constituents. (e)
   7. The sense of e depends on the sense of the ultimate constituents. (f,b,c)
2. Any sentence can be analysed in only one way.
3. Consequently, the way the sense of a complex sentence depends on the senses of its ultimate constituents, is unique. (1,2)
4. In this respect, thoughts have a unique structure.

Beaney on Frege’s reasons to treat truth values as objects

The tremendous significance of the monograph Begriffsschrift [4, 6] for the development of modern logic rests in Frege’s invention of the quantifier-variable notation (§ 11). In motivation of it he replaces the traditional subject-predicate analysis of a sentence by the distinction between its function and its argument.

Any sentence, he claims, can be divided into a constant part, the function, and a variable one, the argument. For instance, in 1. ‘Hydrogen is lighter than carbon dioxide’

‘hyrogen’ could be replaced by ‘oxygen’ or ‘nitrogen’. Thus being considered variable it makes up the argument and ‘is lighter than carbon dioxide’ the function of (1). Equally, however, one may consider ‘carbon dioxide’ as the argument and ‘Hydrogen is lighter than’ as the function. In fact, there are various alternative ways to divide the sentence into functions and arguments. On the other hand, however, Frege holds that these alternative analyses have [6, p. 22]

[...] nothing to do with the conceptual content’.

If however, the conceptual content of a sentence remains constant for all its various reformulations, then it seems to lack a determinate interior structure.

In chapter Frege: Philosophy of Language[2] and more explicitly in chapter 15 of The Interpretation of Frege’s Philosophy [3], Michael Dummett advanced a solution to this apparent indeterminacy. He proposes to distinguish between the analysis of a sentence and its decomposition. Whereas a sentence indeed can be decomposed in more than one equally admissible ways, its analysis provides a unique structure. In other words, ambiguity is all with decomposition. Most recently, Peter Sullivan has connected with argued for Dummett’s view [8, § 5].

However, this view is controversial. Many scholars have denied its faithfulness to Frege. One recent example is [7].

In his [1], Michael Beaney explained Frege’s later treatment of the semantic values of sentences as logical objects [5] from the function-argument-analysis. The line of thought which he ascribes to Frege goes, roughly, as follows

1. A sentence can be divided into function and argument in different, equally appropriate ways.
2. Given the right arguments, different functions have the same value.
3. A sentence has conceptual content.
4. Different sentences have the same conceptual content if they are logically equivalent.
5. Therefore, the conceptual content of a sentence is the value of the function and argument into which it can be analysed

Furthermore, he suggests that Frege gave up on the notion of conceptual content because of the ambiguity of this analyses and ontological problems to which it gives rise.  Hence

1. The truth value of a sentence is the value of the functions and arguments into which it can be analysed.
2. The values of functions are objects
3. Therefore, truth values are objects.

Beaney thus traces Frege’s introduction of truth values as objects back to the analysis of sentences into functions and arguments. I wonder, however, whether Beaney overlooks the relevance of Dummett’s distinction between analysis and decomposition. In other words, I’d like to examine if the above reasoning still can be ascribed to Frege if one assumes that Frege distinguished between ways to divide a sentence into function and argument which leave the content untouched, and ways which reveal its internal structure.

References

[1]   Michael Beaney. Frege’s use of function-argument analysis and his introduction of truth-values as objects. Grazer Philosophische Studien, pages 95 – 123, 2007.

[2]   Michael Dummett. Frege: Philosophy of Language. Duckworth, 1981 edition, 1973.

[3]   Michael Dummett. The Interpretation of Frege’s Philosophy. Duckworth, 1981.

[4]   Gottlieb Frege. Begriffsschrift. Louis Nebert, 1879.

[5]   Gottlob Frege. Über sinn und bedeutung. Zeitschrift fï¿œr Philosophie und philosophische Kritik, NF 100:S. 25 – 50, 1892.

[6]   Gottlob Frege. Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In Jaen Van Heijenoort, editor, From Frege to Gödel. Harvard University Press, 1977.

[7]   James Levine. Analysis and decomposition in frege and russell. The Philosophical Quarterly, 52:pp. 195 – 216, 2002.

[8]   Peter Sullivan. Michael dummett’s frege. In The Cambridge Companion to Frege. forthcoming.

Davidson’s Argument against Psycho-Physical Laws (as in `Mental Events’)

After stating his thesis at the top of page 216, Davidson motivates his argument by
analysing the failure of defitional behaviourism. Mental terms cannot be defined in
behavioural vocabulary, he claims, because behaviour is explained only by the agent’s
intentions as a whole (p. 217). Davidson adds that for the same reason there are
no strict psycho-physical laws (ibid), but does not yet explicate this suggestion.
Instead, he considers how, in daily life, mental terminology is connected with physical
vocabulary. He does not deny that people generalise from particular coincidences and
formulate rules about how mental and physical events are linked. However, he points out,
such generalisations are stated only carefully, with explicit qualifications and clauses
to anticipate counterexamples. These rules of thumb, Davidson argues, therefore
have explanatory force only in virtue of strict laws working in the background (p.
219).

At this point, Davidson introduces a pair of terms to distinguish between two sorts of
generalisations (p. 219).
H. A general statement g in the vocabulary V is homonomic iff there are
V-formulae ϕ₀,…,ϕ_n such that ⌜g ∧ ϕ₀ ∧… ∧ ϕ_n⌝ is a strict law.
g in V is heteronomic otherwise.

Given this distinction, Davidson could derive that no psycho-physical laws are strict, if he
shows that no generalisations involving both mental and physical terminology is
homonomic.

To arrive there, Davidson now for the first time identifies a necessary condition for a general
statement to be a strict law (p. 219).

S. A law is strict²
only if it draws its concepts from a closed comprehensive theory.

Consequently,
1. a general statement in vocabulary V is homonomic only if it draws its
concepts from a closed comprehensive theory formulated in V ³.

Davidson does not explicate when a theory is closed and comprehensive, and how a concept is
drawn from such a theory. However, he gives an example (pp. 220f.). He assumes the
relation symbol ‘longer than’ to be part of the physical vocabulary. It is defined,
Davidson suggests, by the conjunction of two postulates L and M (p. 221). Whereas L
ensures that ‘longer than’ is transitive, M determines to which objects it applies. Since
⌜L ∧ M⌝ is formulated in the physical vocabulary as well, physics allows for homonomic
generalizations.

General statements in mental terms, however, cannot be homonomic, or so Davidson argues
in the remainder of the section. According to (Def.1), any statement in the mental
vocabulary ascribes some propositional attitude to an agent. Consequently, a general
statement in mental terms is homonomic only if there is a theory which determines the
ascription of beliefs, thoughts and desires similarly to how L and M determine the use
of ‘longer than’. At this point, however, Davidson importantly proposes that (p.
221)
R. any theory which ascribes propositional attitudes to an agent must let her
appear as rational as possible.

Furthermore, Davidson connects with Quine’s thesis that (p. 222)

I. translation is indeterminate.

From (R) he follows that translation plays a central role in the ascription of propositional
attitudes. Consequently, (R) and (I) entail that
2. there is no unique correct theory of the propositional attitudes of an agent.

(2) together with (1) precludes homonomic general statements in mental terms, Davidson
suggests (p. 222). Consequently, there cannot be strict laws in mental terms, especially no
strict psycho-physical laws.

J. Hornsby (1993) ‘Agency and Causal Explanation’

1. Hornsby argues for I Actions must not be viewed as links in causal chains, without considering the agent’s reasons.
2. In §2 H. argues that actions are events.
3. In §3 H. defends ‘[...] the claim that reason explanation is causal-explanation’ (p165).
4. I focus on §4 where H. develops Davidson’s argument for mentally described events not falling under laws.
5. H. adopts D.’s thesis that reason explanation is causal explanation ([2, p168]) and that the mental cannot be reduced (p169). However, she adjusts his argument for this since as it stands, it contradicts (I).
6. H. considers a common objection to D., namely that D. ‘[...] ought really to deny that rational explanation is causal-explanation’ (170). If described mentally, the event cannot be ascribed causal force to as well. Therefore, it is argued, AM is an epiphenomenalist position¹.
7. D. could respond: An event can be explained in more than one way and a mental event can play both a causal-explanatory and a nomological role.
8. However, H. elaborates on the objection (p171):
   1. Monism contradicts to rational explanation of action. Mental events being physical events does not cohere with rational explanation of action by giving the agent’s reasons. This critique is in line with (I).
   2. The argument for causal explanations of actions which H. finds in D. is ‘[...] our ability to frame [...] generalizations about pieces of behaviour [...]’ (p171).
   3. H. questions this argument. One explains an action by pointing at its role in the whole of a person’s attitudes and actions (171f), by rationalizing the agent. From this, H. derives a difference among causal connection: (a) purely causal vs (b) causal-explanatory connections.
   4. Possibly, action explanation would not be purely causal (172).
   5. Therefore, an event’s causal-explanatory and its nomological role would possibly fall apart, and D.’s response (7) would not hold. In other words (p173 above), H. criticizes D.’s step from causal explanations in mental terms to purely causal explanations².
9. H. concludes that D. provides no argument for ‘[...] subsuming actions in the impersonal world of causes’ (p174).
10. §5 H. supports her argument from §4 that the way actions are explained makes them inaccessible from the impersonal perspective.
11. In §6 H. turns her argument against Nagel’s position that actions can be viewed objectively ([4, p3]).
12. §7: Actions would not be part of nature since if there were no people, there would not be their actions (p183). H. points out that this conclusion on one hand corresponds well to D.’s argument for the anomaly of the mental.
13. On the other hand, however, it contradicts to D.’s monism. On this bases, H. charges D. with the implicit assumption that the principles of impersonal descriptions apply universally (183f).
14. H. considers the ontological consequence of her position that the naturalistic world is incomplete. Her response: Its not a portion of reality (space-time) but an aspect of it which is missing in the naturalistic picture (p184). The question how people and their actions emerge from nature would not be answered by monism, a deficiency awareness of which H. ascribes to D. himself ([1, 246]). Hence, giving up the universal application of physical/ impersonal descriptions is cheap (p185).

Questions/ Objections

1. H. introduces different meanings of ‘causal’. So the burden is on her to show that action explanation in mental terms is causal in a way different from explanations in physical terms. I don’t see a clear argument for this claim, although it must be somewhere on pp171-2.

References

[1]   Donald Davidson. Replies to Essays X-XII. In Essays on Davidson: Actions and Events, pages 242 – 52. Clarendon Press, Oxford.

[2]   Jennifer Hornsby. Agency and Causal Explanation. In Mental Causation, chapter 10, pages 161 – 188. Clarendon Press, 1993.

[3]   T. Honderich. The Argument for Anomalous Monism. Analysis, 1982.

[4]   Thomas Nagel. The View from Nowhere. Clarendon Press, Oxford, 1986.

Classical vs paracomplete routes to a solution of the liar paradox

Any theorist who seeks to understand the truth predicate must find an answer to the liar paradox. As a starting point of such an attempt it is advisable to look back at the locus classicus of modern work on the Liar. In his 1956, Tarski not only presents the paradox but also provides an analysis of possible resolutions. The text, however, does not lend itself to an easy interpretation because of its historical context when terminology and basic distinction were not fixed in the contemporary way. I will not attempt to engage in Tarski exegesis.

Therefore, I confine myself with the standard, or better perhaps the most common reading. According to it, Tarski identifies three factors which jointly lead to contradiction (). To avoid needless introduction of new terminology I  immediately turn to apply them to my above presentation. Thus the liar paradox occurs if
1.      Liar-sentences can be formulated, either by means of a labeling device as above or in some other way,
2.    the predicate involved obeys the principles of truth
and
3.     reasoning follows the usual patterns.

A glance back at the paradox shows that indeed all these conditions were involved in the derivation of `(21) is true and (21) is not true’.
First, the labeling of sentences by numbers made possible in the first place the formulation of the liar sentence (21).
Second, the principles of equivalence respectively intersubstitutivity allowed me to go from (21) to `(21) is true’ or vice versa.
Third, at many places in my argument for the paradox I deployed general rules of reasoning which I presumed to be valid. Among these, a prominent position was taken by the law of excluded middle.

Having distinguished between these three factors, three different resolutions to the paradox stand to reason. The first of these, however, is not considered an option within the present context. It might be argued that the labeling device I have been using is not genuinely part of the English language. Nonetheless, there are various alternative ways of constructing Liar sentences. Tarski chose a rather tedious route but Saul Kripke, for example, has made a strong case for self-reference as an unavoidable feature not only of natural but of mathematical languages as well [1975:693]. Chapter 4 below will show in detail how self-reference is a necessary consequence already of a fragment of arithmetic.

Therefore, two ways remain to address the paradox -
either one restricts the principles of truth, or the reasoning involved.
The first approach is usually dubbed `classical’ because it preserves the standard assumptions about validity.
The second one has been pursued under different names, but I adopt the nomenclature of the most recent work and call it the `paracomplete’ approach.

Why we’re better of with`true’, despite redundancy theories

In the preceding section I have made a case for the particular usefulness of the word `true’. I have argued that it is not only used to designate true sentences but also to agree with unknown sentences and even to express infinite conjunctions. In sum, the short word `true’ would provide a competent speaker of English with a remarkable amount of expressive power.

However, I must concede that in the literature exists a strong, well developed theory that all this can be achieved without `true’ as well. It is the so called deflationary, minimal or redundancy view on truth and is traced back to Ramsey’s 1927.

Ramsey argues that a speaker designates the true sentences, or the sentences which are true according to him, already by merely asserting them. `True’ would be used only for stylistic reasons, as a means of emphasis, equivalent to other operators, such as `… is a fact 1927:44.

This idea in fact goes back to Frege who took declaratory sentences to have what he called assertoric force by themselves 1892, 1918.

Except for these allusions and some scattered remarks, however, no explicit elaboration or defense of the deflationary view is found in Frege. Ramsey’s deflationism as well remained fragmentary, due to the unfortunate early death of this still underestimated philosopher.

Nonetheless, I take the redundancy view seriously, indeed I am sure it casts light on an important aspect of the meaning of `true’. However, I also think that eventually it is perfectly compatible with my case. In fact, the aspect which the redundancy theorist rightly emphasizes I have given a central role to myself – it is the fact that the schema (T) above holds for whichever sentence one may wish to put replace for `\psi_0′. Certainly, the redundancy theorist may turn my reasoning upside down and argue that `true’ designates the true sentences because of its vacuity, because it could always be eliminated.

However, there are good reasons to remain skeptical about this claim.

First, the redundancy theory must account for the role of `true’ in expressing agreement as well. This step constitutes difficulties which Ramsey himself had to acknowledge 1927:45. Nonetheless, he suggests a way to paraphrase a sentence such as my (8):
13    For all p, if Jesus asserted p, then p
In this paraphrasis, the letter `p’ is supposed to stand for an arbitrary sentence.

However, (13) fails to be an adequate paraphrasis of (8), for simple reasons which become apparent if one considers an example instance of it. To connect with the above cases, presume that Jesus said `Snow is white’. The according instance of (13) would be

14    If Jesus asserted `Snow is white’ then `Snow is white’.

This, however, is anything but a grammatical English sentence.
The trained reader will be familiar with the distinction between the use and the mention of a word, respectively a sentence. The then-clause of (14) merely mentions `Snow is white’  and therefore lacks a verb in the first place.

Alternatively, the redundancy theorist might therefore try to insert the sentence without quotation marks, arriving at

15     If Jesus asserted snow is white then snow is white.

Clearly, however, in this case the if-clause fails to proper English, this time because it contains two verbs instead of one. In summary, the problem of schema (13) is that its if-clause requires the sentence replaced for the letter `p’ to be mention, the then-clause however requires its use.

The redundancy theorist again may try to put the difference between the if- and the then-clause in the schema itself, by putting it this way:

13′    For all p, if Jesus asserted `p’ then p.

However, this variant is even worse than the original. As famously argued by W.V. Quine 1970:13,
quoting the schematic sentence letter `p’ produces a name only of the sixteenth letter of the alphabet, and no generality over sentences

What is needed instead, is a device to both use and mention a sentence within one and the same sentence. However, the word `true’, used as in the above examples, being the truth predicate of the English language, lends itself to exactly this purpose. As I have explicated above, any speaker can, merely by his or her linguistic competence, argue from
16    `Snow is white’ is true
to
17    Snow is white
and vice versa. In other words, to apply the truth predicate to the mention of a sentence amounts to using this sentence. This is what Quine famously called the truth predicate’s `cancellatory force’ (1970:12). Because `true’ therefore is `[...] a device of disquotation’ (ibid) one can derive from (8) each sentence Jesus Christ indeed said.

This might sound as an argument for redundancy, however, it is important not to miss the difference between reasoning from (16) to (17) and dispensing with sentences like (16) in the first place. The redundancy theorist is right to the extent that she emphasizes the disquotational character of `true’. However, that any speaker always can get from (16) to (17) does not mean that `true’ would be superfluous.

In addition to the problems remarked upon above, there are a number of further problems with the redundancy approach, a comprehensive list of which is found in GCB.
The observations so far, however, already provide me with sufficient reasons to conclude that even if the word `true’ might not be indispensable, it is still extraordinarily useful because it comprises expressive means which otherwise require lengthy, technical solutions. Furthermore, I can rest my case on the shoulders of a giant, as Quine concludes that [ibid]
The truth predicate in its general use [...] is eliminable by no facile paradigm.

How `true’ allows us to express infinite conjunctions

In this post, I’d like to show how the short word `true’ allows speakers of English can express their agreement with the whole of a theory at once.

Generally, agreement with a large or indeterminate number of sentences can be expressed by means of the word `true’ in the following way:

A  Whatever F is true.

where `F’ denotes a property characteristic for the sentences with which one attempts to agree.

To connect with my example from above, a speaker can use the schema (A) to agree with whichever prediction belongs to Newtonian mechanics.
11    Whatever is predicted by Newtonian mechanics is true.
For example, let classical mechanics predict that my pencil will impinge on the floor at exactly 2:32pm (presume that I am about to knock it from my desk). Whoever utters (11) therefore especially says the sentence “Jönne’s pencil is at the floor at 2:32pm’ is true’. Given the meaning of `true’ as the truth predicate of English, this allows him or her to infer `Jönne’s pencil is at the floor at 2:32pm’. Thus, whoever states (11) expresses his or her agreement with any such a particular prediction as well.

So far, this case coincides with the role which the preceding paragraphs have  ascribed to `true’. However, at a closer analysis, the application of a schema such as (A) to a theory like the Newtonian has a remarkable consequence which in fact makes up an additional distinct function of the truth predicate.
To extrapolate from my pencil, for a given object o Newtonian mechanics predicts for any point in time t, a place in space x where o will be located. In other words, for any o and t, there is a sentence \ulcorner At t, o is at x \urcorner which is part of Newton’s theory.
Especially, given such a t, it will predict o’s place for t+n for any natural number n. Thus, Newtonian mechanics includes for any natural number n a sentence \ulcorner At t+n, v is at x \urcorner .
As there are infinitely many natural numbers, Newtonian mechanics consequently includes infinitely many sentences.

Similarly, however, it has been demonstrated above that from a statement such as (11) all sentences can be inferred which are part of the theory.
Therefore, a phrasing of the form (11) allows any competent speaker of English to infer infinitely many sentences. Equivalently, (11) allows a speaker to agree with an infinite number of sentences. The short, quotidian word `true’ thus extends English by the means of expressing infinite conjunctions, at least in a mediate way.

Kripke’s picture of learning `true’

I presume that ordinary reasoning with the natural language truth predicate is not affected by the paradoxes.  I don’t think ordinary reasoning has become trivial only there is that Liar sentence. The reason is, I think, that the liar sentence is not taken seriously. Rather, it’s kind of sorted out before the principles of truth are applied. This idea is fundamental for most of the research during the last decades and also for the paracomplete approach I’m currently working on. Its seminal formulation it has found in Kripke’s `Outline of a Theory of Truth’. Kripke’s idea can be spelled out as follows.

Imagine a competent speaker of the fragment of English based on the English vocabulary minus the predicate `true’ (English). This person then is confronted with sentences involving this new, unknown word. However, she is told that `true’ would be a predicate applying to sentence, and is given the intersubstitutivity principle from above. Based on this, the following learning process will turn her into a competent speaker of full English, as well.

In the beginning, this word is thoroughly unintelligible for her. She does not understand any sentence containing `true’ in this early stage and accordingly does not know when she is entitled to assert these sentences or not. However, from her competence in English- she knows which of the sentences not involving `true’ are assertible. Be  one of these sentences. By the given principle, she concludes that she can assert  is true as well. Be this sentence called `A_0′ More generally, she has learned that `true’ applies to all those English- sentences which are assertible themselves.

Thus, however, she has already reached the next stage of her learning process, because she now can apply the principle again to the newly understood sentences. This step she can repeat again and again. For any n, if the speaker knows that A_n is the case, she can infer that she is entitled to assert A_n+1=`A_n is true’. Since there are infinitely many English sentences, this proceeding will continue for all natural numbers n. Thus, the speaker of English- continuously extends her understanding of `true’ through all finite stages.

However, in this idealized picture the speaker’s inference at transfinite stages can be determined as well. At the first transfinite stage omega, having derived An for all finite n, the speaker will understand when she can assert  `For any n, An is true’.

From there, she can proceed as before. For successor ordinals she continues as at finite stages, at limit ordinals  she undertakes a similar generalization as at stage omega.