Professor Elizabeth S. Anderson Believes That All Scientific Theories are Political

i) Introduction

ii) Value Judgements and Background Assumptions

iii) The Epistemic Evaluation of Scientific Theories 

iv) The Political Applications of Scientific Theories 

v) Conclusion

Introduction

Elizabeth Secor Anderson is Arthur F. Thurnau Professor and John Dewey Distinguished University Professor of Philosophy and Women’s Studies at the University of Michigan. She has been published by — and featured in — The New Yorker, Jacobin, Chris Hedges’ Truthdig, 3:AM Magazine, Democracy, etc.

This is a commentary on Elizabeth S. Anderson’s paper, ‘Feminist Epistemology: An Interpretation and a defence’. I focus on a single — though important — part of that paper: her account of how (what she calls) “value judgments” and “background assumptions” impinge on (all) scientific theories. More specifically, I focus on Anderson’s account of how political and ideological value judgements and background assumptions do so. That supposition is at the heart of her paper.

Value Judgements and Background Assumptions

There is indeed a “logical gap” between observation/data/evidence and scientific theory. (In the philosophy of science, this is called the underdetermination of theory by data/evidence.) However, it doesn’t automatically follow that we need necessarily plug that gap with what Elizabeth Anderson calls “value judgements”. Of course this depends on what exactly Anderson means by those two words. Considering the rest of her paper, she doesn’t only mean those judgements as to what makes a scientific theory “simple”, “elegant”, “explanatorily powerful”, or “empirically adequate” (or a mixture of all these, as well as other criteria); which most scientists and philosophers of science accept as determinants of scientific theory. Instead, Anderson is talking about ideological and political value judgements. 

We all also need to know what Anderson means by “background assumptions”. These background assumptions would be — or would go alongside — the various value judgments; and those value judgments — or at least some of them — are seen to be be (at least in part) ideological and political in nature. Thus there are various background assumptions which will be “used to argue that a given observation constitutes evidence for a given hypothesis”. Again, need these (or any) background assumptions be political or ideological in nature?

Of course there’ll be a problem with that question, at least according to certain political theorists. And that problem arises from the view that all background assumptions (or value judgments) can’t help but be ideological or political in some — or even in many — ways. Here the political theorist’s argument will be about the necessity of the scientific theorist’s ideological or political background assumptions. However, wouldn’t the political theorist’s belief (about scientific theorists or theories) be a case of arguing in a circle if he/she already accepts the necessity of political or ideological background assumptions impinging on scientific theory-construction? If she/he already believes in that necessity, then she/he is bound to find such ideological or political background assumptions (or value-judgments). 

So what if some scientific theorists are anti-political, apolitical or just plain ignorant of politics (many are)? Here again the political theorist will back herself/himself up with the statement that “everyone is political” — even if that person (or scientific theorist) is non-political, apolitical or politically ignorant. The argument will be that such apolitical scientific theorists must accept the given ideologies and political realities of her/his milieu (i.e., simply because she/he doesn’t question them). Likewise, even the politically ignorant must be political or ideological in some kind of way — even if in a rudimentary kind of a way.

The other thing is that we’ll need to know what kind of scientific observations Anderson is talking about. Is she talking about observing the effects of particle interactions in a bubble chamber or the effects of having a low income on a family? You can argue that political or ideological value judgements (or background assumptions) may affect the latter; though what about the former? Indeed in principle — even if only in principle — ideological or political background assumptions needn’t necessarily impinge on one’s scientific observations of a poor family either. 

For example, what if we programmed a computer to do that “observing”? What if we used someone from a completely different culture or one who wasn’t poor, rich, powerful or particularly political? (Although, I said earlier, the political theorist will reject that final possibility and probably the former ones too.)

I may not be being entirely fair to Anderson by my pitting bubble-chamber observations with the observations of the effects of a low income on the lifestyle of a family. That’s because Anderson provides her own (more likely) example of how politics or ideology can impinge on scientific research and indeed on scientific observation. She cites the examples of evolutionary theory and work in genetics. Here she finds many examples of politics or ideology intruding on scientific theory. Yet this is evolutionary theory and genetics we’re talking about: both of which clearly and obviously impinge on both the human and the social. Despite that (again): what about observing the effects of particle collisions in cloud chamber or studying the behaviour of ants? (One can indeed “politicise” ant behaviour — in many directions.) 

So the problem noted earlier will arise here too: which kind of observations is Anderson talking about? Is she talking about all scientific observations and all scientific theories? Specifically, she says that 

“it is not unreasonable to use any of one’s firm beliefs, including beliefs about values, to reason from an observation to a theory”. 

Again, what kind of observations is Anderson talking about there? Is she talking about all scientific observations? Indeed which kind of “firm beliefs” and “values” is she talking about? Does she also include and allow beliefs and values which strongly clash with her own? (Parts of Anderson’s paper highlight the problems she has with certain beliefs and values.)

Because the phrase “value judgment” is so vague, it’s initially unobjectionable for Anderson to state that theories which “incorporate value judgements can be scientifically sound as long as they are empirically adequate”. But there’s a problem here too. Is it also a question of value judgements when it comes to deciding what actually makes a theory “empirically adequate” (or elegant, parsimonious, explanatorily powerful, highly predictive, etc.) in the first place? (Now does that work for or against Anderson’s stress on value judgments in science and epistemology?) If data, evidence or observations underdetermine theory (as the philosophical theory has it), then the fact that a theory is empirically adequate, etc. may not amount to much if hiding in the bushes behind that empirical adequacy, etc. are not only value judgments (or background assumptions), but also political or ideological value judgments (or background assumptions). If that’s the case, then surely empirical adequacy, etc. may not amount to much.

In any case, many philosophers of science have argued that empirical adequacy is easy (i.e., because data, evidence or observations always underdetermine theory). If that’s true, then perhaps background assumptions (or value judgments) really do take on an importance which we otherwise didn’t expect. Perhaps empirical adequacy, etc. weigh less on the scales than the prior value (political/ideological) judgments (or background assumptions) which are made and which then impinge on that empirical adequacy, etc. (or on what we believe is empirically adequate, etc.).

The Epistemic Evaluation of Scientific Theories

Anderson says that the logical gap between 

“the epistemic evaluation of theories cannot be sharply separated from the interests their applications serve”.

Yes they can… surely? So the argument is that, normatively, it is wrong to ignore such applications. (That too would depend on one’s normative stance on these issues.) Again, this would — or could — be more the case of a normative judgement (or epistemic evaluation) being applied — by the feminist epistemologist — to the otherwise neutral or apolitical epistemic evaluations of scientific theories. What’s being said by Anderson is normative in itself. It’s not a case of saying that scientific theorists indulge in epistemic evaluations which are sometimes (or always?) political or ideological. It’s more a case that Anderson believes that they should indulge in such political evaluations. That is, a scientific theorist can (and often does) separate his theory — and even his epistemic evaluations — form “the interests their applications serve”. Anderson is arguing that she/he shouldn’t do so.

This, then, ends up being less of a project in discovering the value judgements or background assumptions (specifically political and ideological ones) involved in scientific theory-construction, and more a case of a feminist epistemologist saying that the scientific theorist should have a political and ideological attitude towards the political applications of his theories. Not only that. The scientific theorist should have politically/ideologically acceptable (not the true, correct or empirically adequate) attitudes towards the applications of her/his scientific theories. Similarly, this also means that this is also all about scientific theorists having the wrong kind of political and ideological background assumptions and making the wrong kinds of value judgment. That is, it’s not just a case of scientific theorists simply having background assumptions and making value-judgements.

We’ve just crossed over another and wider logical gap: the gap between scientific theory and the outright political and ideological assessments of — or normative judgements upon — those scientific theories.

This account of Anderson’s views is correct because Anderson herself says that it is (i.e., at least indirectly). Anderson argues that feminist naturalised epistemology 

“rejects the positivist view that the epistemic merits of theories can be assessed independently of their ideological applications”. 

Here again it’s not a case of Anderson — or any other feminist epistemologist — discovering the scientific theorist’s political and ideological positions which impinge on her/his accounts of the “the epistemic merits” of his theories. It’s more a case of Anderson arguing that ideological and political considerations should impinge on her/his accounts of the epistemic merits of his theories. Not only that. It should be ideologically and politically correct considerations which do so.

We’ve moved from discovering — or simply acknowledging — the role and importance of ideology and politics (in the construction of scientific theories) to the view that scientific theorists should be fully ideologically and politically aware all the way through the process of scientific theory-construction. To be more specific, the scientific theorist should always keep his eye firmly fixed on all possible future “political applications” (Anderson’s words) of her/his theories.

The Political Applications of Scientific Theories

We can characterise Anderson’s position by making these three points:

i) Anderson stressed the “logical gap” between observations/data/evidence and the theories which arise from them.
ii) She then noted the logical gap between the “epistemic evaluation of theories” and “the interests their applications serve”.

iii) Finally, Anderson also noted the logical gap between scientific theories and their political applications (which is obviously related to ii) above).

Just as Anderson discerns the possible ideological and political content of scientific theories which aren’t (to many others or the scientific theorists themselves) apparently ideological or political at all; so now she also tackles the political and ideological use of scientific theories.

Specifically, Anderson says that “a theory is [can be] used to support unpopular political programmes”. We’ll of course need to be clear what it means to use a theory; let alone what the words “unpopular political programmes” mean. Nonetheless, Anderson does say that such a use wouldn’t necessarily show us that “the theory is false”. That’s certainly true. Isn’t it the case that, for example, the theories and findings of quantum mechanics were used — directly or indirectly — to build atomic weapons? (In turn, they were then used for blatantly “political programmes” — from the bombing of Hiroshima to sustaining - some may argue - the Cold War.) However, just as my ballpoint pen can be used to stab someone’s eye out (but which can’t be blamed on either the inventor or manufacturers of ballpoint pens), so theories in quantum mechanics — or at least those who thought up the theories — can’t be blamed for Hiroshima or for the Cold War. (The earlier work in quantum mechanics began in the early 1920s: that was some twenty years before there was any research into nuclear weapons —i.e., roughly, in 1940.)

To tackle an example which Anderson herself (see later) cites.

In principle, even if a scientific theory or scientific research states that women are intellectually inferior to men (in whatever way you like — this is a what-if story), that too doesn’t automatically mean that it will be used to support political programmes of female oppression or even be used as a theoretical excuse to force women to stay at home. That possible theory of — or research into - the mental inferiority of women could be true and still not be used for forcing women to stay at home (or for any other discriminatory political or social practice). However, in this case at least, the reality is that any complete separation of theory and political application will be unlikely. That is, such scientific findings, research or theories will indeed be “politicised”. Yet much of that politicisation will be down to those who want to make sure that such findings, research or theories are never politically instantiated. Many political theorists and activists will even argue that such research be discontinued (i.e., because it’s politically or ideologically objectionable). 

Thus we firstly have the scientific findings, research or theories, and then we have the politicisation of such things. Nonetheless, Anderson’s argument is that politics and ideology are there from the very beginning — i.e., in the actual findings, research or theories. And if we accept that, then any politicisation which later occurs is only additional to the inherently political nature of science itself.

Anderson cites her own example of a bad political application of a (possibly true?) theory: the case of Professor Steven Goldberg. According to Anderson, he

“uses his theory of sex differences in aggression to justify a gendered division of labour that deliberately confines women to low-prestige occupations”.

As stated earlier, Anderson says that although there may be bad political applications of a scientific theory, that doesn’t necessarily make the theory false. And here too she talks about this further “logical gap”:

“The proponents of the programme [should respect] the logical gap between fact and value.”

However, even if the “facts” are applied (for example) to building nuclear weapons or sexist social policy, that is still — on the surface at least — only an application of the facts. It can still be argued that the value bit of the equation only comes in later on — when it’s directed at the appliers of the theory. (Such as the technologists or the political decision-makers.) It will of course still be argued (by Anderson and others) that scientific theorists should be fully conversant with the political applications of their theories… But should they? This doesn’t seem to be a question of the scientific theorist being burdened down with hidden or unacknowledged values (specifically political and/or ideological values). Rather, it’s really a question of values (specifically political and ideological values) being foisted or imposed upon them by feminist epistemologists, feminist philosophers of science, or even by people directly involved in politics.

This would suggest that all this is actually about the normative and political claim that scientific theorists should be politically and ideologically biased; rather than them actually being politically or ideologically biased (if often in the wrong direction). In other words, at this level (at least) all the ideology and politics is coming from one direction: from the feminist epistemologist (or from the feminist philosopher of science). Thus if scientific theorists don’t know — or care — about the political applications of their theories, then it’s hard to accuse them of using ideological or political value judgments or of having ideological or political background assumptions. All the politicising (or the making of political value judgments) seems to come later: from the feminist epistemologist and from the political appliers of scientific theories. Although (as stated earlier) Anderson (as well as other feminist epistemologists) may argue — and many political theorists do argue this way — that the scientific theorist not caring (or even not knowing) about the political applications of his theories is itself a deeply political and ideological stance.

Conclusion

To sum up.

Professor Elizabeth S. Anderson isn’t simply arguing that political and ideological value judgements and background assumption exist in all scientific theorising. Anderson is also making the political point that there are many politically-incorrect value judgements and background assumptions which underpin scientific theorising. In addition, scientific theories are often applied in ways that (to her at least) are politically objectionable.

This means that Anderson’s paper is just as much a work of politics as it is a work of epistemology (or of philosophy of science). Having said that, if Anderson deems the scientific theory/politics “binary opposition” to be false in the first place (i.e., it’s naive to separate science from politics), then my interpretation is hardly surprising. Indeed, by Anderson’s own lights, she must surely agree (at least in part) with my broad conclusion about her own position.

Finally, if Anderson is correct to argue (if often indirectly) that politics and ideology pervade all scientific theories, and that she additionally argues (again, often indirectly) that such politics and ideology should be politically acceptable (but to whom?), then scientific theory (alongside epistemology and philosophy of science) effectively becomes a political battleground. Despite all that, it’s probably the case that Anderson believes that this possible scientific theory/politics battle has always been the case anyway. 

So my broad conclusion is that Anderson advances the position (if somewhat indirectly) that all science is inherently political. (Historically, this was also the position of both the Nazi and Soviet states — see ‘Ideologically Correct Science’.) Thus it follows (to her at least) that epistemologists, philosophers of science and political activists/politicians (or at least those who share her own politics) must make sure that all science is both politically acceptable and politically correct. 

Kurt Gödel, Vacuous Paradoxes and Self-Reference

Kurt Gödel once wrote that

“[o]ur logical intuitions (i.e., intuitions concerning such notions as: truth, concept, being, class, etc.) are self-contradictory”.

And Gödel didn’t have any problems at all with self-reference either. He also wrote the following:

“Contrary to appearances, such a proposition involves no faulty circularity, for it only asserts that a certain well-defined formula … is unprovable. Only subsequently (and so to speak by chance) does it turn out that this formula is precisely the one by which the proposition itself was expressed.”

In the following it will be argued that it’s not simply a question of what Gödel called “faulty circularity”. Perhaps it’s also about whether anything at all can be affixed to (or said of ) such sentences as “This statement” or “This statement is not provable”. More specifically, if the words (or clause!) “This statement” are semantically and metaphysically empty, then perhaps we don’t even need to worry about Gödel’s faulty circularity.

For example, the problem is that the sentence

This statement is not provable.

is impeccable in terms of logic alone. (Or so the usual — often implicit — position seems to be.) Thus semantics and certainly metaphysics are irrelevant here. That is, the words “This statement” make a perfectly acceptable “word string” in terms of logic. That means that there’s no problem at all with affixing the suffix/predicate “is not provable” to it. So here this means that logic is completely independent of semantics and/or metaphysics — and that’s despite the use of language-language expression! But if the sentence “This statement is not provable” (or “All Cretans are liars”, for that matter) is used only to display purely “logical facts” (i.e, facts about paradox or self-contradiction), then why use natural-language expressions in the first place?

It may be this logical focus (i.e., the divorce from natural-language expressions) which help create the paradoxes and self-contradictions in the first place.

Despite that logical independence, Gödel also moved beyond mathematics, logic and even metamathematics (into philosophy) when he stated that

“our logical intuitions (i.e., intuitions concerning such notions as: truth, concept, being, class, etc.) are self-contradictory”.

All this is vaguely equivalent to Bertrand Russell’s attempt (in the early 20th century) to find what he called the “logical form” (a term which Russell first used in 1914) of faulty natural-language expressions (or, more importantly, statements). Thus logicians and philosophers (at that time at least) were actually finding the logical form of natural-language — and also philosophical - expressions such as “The King of France is bald”. But those “hidden” logical forms were simply the end product of a long process of a logical — and indeed philosophical — scraping away of everything that is contextual, semantic and metaphysical.

Let’s put all that another way.

Think now in terms of Gödel numbers.

We can assign a number to the words “This statement”. Then we can assign another number to the words (or predicate) “is not provable”. And once we have those numbers (or those arithmetical particles) assigned, then we can play all sorts of games with them.

However, the sentence “This statement is not provable” is a natural-language expression. That’s the case no matter what logical games we can play with it. And if we’re using a natural-language expression, then we must abide by the realities (or facts) of natural language — even if those realities (or facts) are far from being determinate, rule-bound or immune to philosophical dispute.

Thus we can now turn the sentence “This statement is not true” into the purely symbolic

¬p

Yet even here we’d need to clarify what the symbols p and ¬ mean in a natural language. So, in this case at least, can such symbols really be (as it’s often put about logical symbols) “drained of all meaning”?

Self-Reference

Gödel offered us his own self-referential formula. Or, more accurately and relevantly to this piece, we have this natural-language sentence:

A certain number, x, is not provable.

(The statement above doesn’t stop being a natural-language expression simply because it includes the variable x and the word “provable”.)

In some cases at least, Gödel showed us that the (Gödel) number x within a formula will happen to represent that very formula itself. Gödel himself said that “this formula is precisely the one by which the proposition itself was expressed”.

So if the symbol (or number) x represents the entire formula, then don’t we really have the following? -

A certain number, x (x = A certain number x, is not provable), is not provable.

In other words, what’s in the parenthesis above is actually a part of the whole sentence  — a necessary part of the whole sentence. To put that another way. The entire sentence has itself embedded within itself (as the symbol/number x) like a Russian doll within a Russian doll. Except, of course, one Russian doll must be smaller than the other; whereas x must be exactly the same as the sentence in which it is embedded. Yet surely this creates an infinite regress. That is, if

i) The x in the statement “A certain number, x, is not provable”

itself refers to the statement “A certain number, x, is not provable”, then don’t we have this? -

ii) A certain number, x… (x = A certain number, x… ((x = A certain number, x… (((x = A certain number, x… ((((x = A certain number, x… ((((( x = A certain number, x… ((((((… ad infinitum…)))))) is not provable.

Or it is simply this?

A certain number, x (x = a certain number), is not provable.

Now take this simpler statement:

This statement is not provable.

Again, is Gödel’s position about the whole formula (i.e., “This statement is not provable”) or is about the embedded sentence/clause (i.e., “This statement”)? It may seem that the whole sentence/formula is about itself. Therefore Gödel’s position applies to the whole of the sentence “This statement is not provable”, not simply to the embedded clause “This statement”.

The Metalanguage

On a similar theme, is the whole statement

i) This statement is false.

part of a metalanguage simply because the words/predicate “is false” is affixed to the words “This statement”? (One can ask here why “This statement” is a statement at all — see the section ‘Empty Paradoxes’ later.) Or is the longer sentence

ii) The statement “This statement is false” is false/true.

part of a metalanguage? Thus if everything is within within the same sentence (as in example i)), then how can it really be an example of a metalanguage (or a case of “language about language”)?

On the other hand, perhaps only the suffix/predicate “is false” is metalinguistic.

Take Alfred Tarski’s object-language/meta-language distinction. In this case, the meta-language is completely separated from the object-language. Yet in i) above we seemingly have both the object-language and the meta-language within the very same sentence (i.e., or sometimes within the same quotation marks). That certainly breaks Tarski’s own golden rule.

Of course that may simply be a question of grammatical layout. That is, perhaps

i) This statement is false.

is simply shorthand for the following:

ia) The statement “This statement is false” is true/false.

In many other self-referential or Gödelian statements the predicate/suffix “is provable” often occurs. That suffix/predicate hints at the possibility that all the proofs of mathematical systems — and the statements within them — must come from metalanguages. That is, they must exist outside the systems.

However, it’s not that proof exists outside the system. 

Statements within a system have be proved or are provable. So it’s the suffix/predicate “… is provable” that’s outside the system, not the proofs themselves. And that may simply be because the words “is provable” are from a natural language. That is, they’re not themselves mathematical or logical symbols. That said, all logical and mathematical symbols have a natural-language expression. So a further two questions can now be asked:

1) Do natural-language expressions of mathematical/logical symbols and symbolic statements/equations truly capture the whole logical/mathematical import of those symbols and statements/equations? 

2) Do these natural-language expressions add something (problematic) to those logical/mathematical symbols and statements/equations?

The Cretan Liar and the Barber Paradox

The Cretan liar paradox also provides us with a perfect example of self-reference.

In 1869 Thomas Fowler expressed the Cretan liar paradox as follows:

“Epimenides the Cretan says that ‘All the Cretans are liars’, but Epimenides is himself a Cretan; therefore he is himself a liar.”

The upshot is this.

Epimenides stated: “All Cretans are liars”. He was a Cretan. Therefore he was a liar. That means that his statement “All Cretans are liars” must be a lie (or false).

Put differently. All the above means that if what Epimenides says is true (i.e., that all Cretans lie), then it must be false because he’s a Cretan and all Cretans are liars. That is, if a Cretan (as one of the set Cretan Liars) says that “All Cretans are liars”, then that statement must be false.

Furthermore, if the statement “All Cretans are liars” is itself a lie (i.e., false), then that must mean that at least some Cretans must tell the truth. So is this particular Cretan (i.e., Epimenides) one of those Cretans who tells the truth? After all, the statement “All Cretans are liars” doesn’t need to translate into “No Cretans are liars” simply because Epimenides himself may be an exception. It may simply have been that some Cretans are liars. So was the Cretan who made this statement himself a liar or a truth-teller? If he’s a truth-teller, then it may be the case that all Cretans are liars… But he is a Cretan himself!

The Cretan liar paradox also highlights a problem which runs through this piece. That problem is one of the application of logic to natural-language statements or expressions. Or, inversely, the problem occurs when the logical form of natural-language statements/expression is (as it it used to be put) “discovered”. (This was mentioned in the introduction.) If these things are done, then they often throw up paradoxes or self-contradictions.

Basically, one possibility is that the statement

“All Cretans are liars.”

should really be

“Except for myself, all Cretans are liars.”

That is, the above is a more natural and less problematic natural-language expression of the words “All Cretans are liars”.

However, the universal quantifier “all” ( or ∀ in logic) is somewhat negated by the proceeding clause “Except for myself”. This also has the consequence that it is no longer paradoxical or self-contradictory.

This all hinges on the quantifier “all” and the problems (or difficulties) self-reference throw up. In logic, it’s often agreed that quantifiers nearly always have a restricted range (or domain) which is determined by specific contexts. So does the word “all” in “All Cretans are liars” have a restricted range? Is the speaker of the words “All Cretans are liars” that very restriction (or exception) himself? If we take the word “all” literally, then he can’t be. However, if we take the word “all” contextually or as a quantifier with a restricted range, then he may well be that very exception. After all, in natural-language terms (therefore in terms of context), many people would be happy to accept that when a person says that “All people are evil” (or says that “All people are nice”), then he may well be exempting himself from that statement. Indeed if someone were to say (out loud) that “All people always remain silent”, then (by definition) he must be an exception to his own universal statement.

So what about the Barber paradox?

This paradox isn’t about a single self-referential statement. That is, it can only be established through a chain of arguments. It is self-referential, however, in that it deals with the question of whether the barber does or doesn’t shave himself. Nonetheless, it’s not about a self-referential statement (or sentence) — as in the Liar paradox.

Of course the Barber paradox can be (partly) summed up in a single sentence. For example:

“I shave everyone who doesn’t shave themselves.”

Despite that, the paradox is still not a self-referential sentence like “I am lying at this very moment”. It’s about a self-referential situation (as it were); though, unlike the Lair paradox, it isn’t about a single sentence referring to itself (or a person referring to what he himself is currently saying). Thus the Barber paradox is about a possible (or impossible) state of affairs; not about a self-referential statement.

Empty Paradoxes and Semantic Content

Many logicians have argued that (semantic) content isn’t required when it comes to self-referential statements like “This statement is false”. Yet specifically in reference to the Liar paradox, one logician wrote:

“[The Liar paradox] led to the collapse of logicism and indirectly to Gödel’s incompleteness results (i.e., that in a formal system like Zermelo-Frankel set theory you can derive (G(F) = ‘This sentence cannot be proved in F’.)”

Gödel’s “This sentence cannot be proved in F” isn’t like the Liar paradox — at least it’s not precisely the same.

Put it this way. The following

Statement S in system A is true though it can’t be proven to be true in A.

isn’t like a single sentence which refers to itself. The above states that a statement (or mathematical truth) within a system is true even though it can’t be proven within that system. This means that Gödel’s “S in x” is true — just not proven to be true within the system to which it belongs. The problem with the Liar paradox, on the other hand, is that it can’t be established if it’s true or false (or if it’s both) at all.

Now take the following:

(A) The sentence A is false.

Isn’t it the case that statement A has no semantic content? Perhaps it’s not a genuine statement (or proposition) at all. Nonetheless, since many logicians and philosophers don’t take this view, let’s take it as they take it — as being a genuine (if paradoxical) statement.

The first thing to say is that it’s self-referential. Again, just like the Liar paradox, it’s about itself. (Here’s a list of other self-referential paradoxes.)

We have the sentence “(A) The sentence A is false”, which includes the symbol A. And A stands for the sentence it is in or the words which surround it. That means that a symbol (i.e. A) within a sentence refers to the sentence which it is in. Thus we have this again:

The sentence A… (A = The sentence A… ((A = The Sentence A… (((A = The sentence A… ((((A = The sentence A… ((((( A = The sentence A… ((((((… ad infinitum…)))))) is false.

Now what, precisely, is true or false? Sentence A is true or false. What does sentence A say about itself? It says that it’s “false” — and that’s it. It doesn’t say its subject-term (or phrase) is false: it says that the whole sentence is false.

So if we take out the A from the original statement “The sentence A is false”, then what do we have left? This:

The sentence… is false.

Since A only refers to the sentence itself, then why can’t we take A out? And if we do that, then what are we left with? It’s already been said that “The sentence A is false” is without content: so it’s even more the case that “The sentence… is false” is without content.

This can be boiled down even more.

We’ve already removed the A: now we can also remove the words “is false”. After all, the predicate “is false” must be applicable to something else. So what is the “is false” (in “The sentence A”) applicable to? That’s right, the “is false” suffix/predicate/clause is applicable to “The sentence”! So the two words “The sentence” are meant to be either false. Yet how can the two words “The sentence” be either true or false when they say precisely nothing?

To recap. It’s being argued here that the statement

(A) The sentence A is false.

is a pseudo-statement with no semantic content. Another way in which the same thing can more or less be said is to say that it’s malignly self-referential. Or, more correctly, that truth can’t be applied self-referentially (as Tarski argued) — especially when the statement has no content in the first place. Indeed it’s self-referential precisely because it has no content.

So what about a sentence which has a sentence embedded within itself which does have content? Take this example from Tarski:

(S) The sentence “Snow is white” is true iff p.

Now that’s not really a single sentence (or statement) at all. It is in fact two sentences. We have the embedded sentence 'Snow is white' as well whole sentence “This sentence ‘Snow is white’ is true iff p”. Thus it isn’t self-referential. The meta-language “The sentence ‘Snow is white’ is true iff p” is being applied to the object-language’s 'Snow is white'. The statement 'Snow is white' clearly has content and the whole sentence “The sentence ‘Snow is white’ is true iff p” isn’t self-referential either because there’s both a meta-sentence and an object-sentence.

Despite all that, logicians defend the sentence “The sentence A is false” for two main reasons:

i) The words “is false” are an acceptable English predicate.

ii) The whole sentence is grammatically “unassailable”.

Is it grammatically unassailable? I don’t think it’s either logically or philosophically unassailable (or acceptable). And now the grammar can be rejected too.

Again, the argument is that we can grammatically assert the sentence “The statement A” and grammatically apply the words “is false” to it. But that depends on what’s meant by “we can grammatically assert the sentence”. Can we? Grammar, unlike logic, is largely about what is acceptable to say in order to make sense (or communicate) in certain contexts. Now “This sentence” (or “This sentence A is false”) isn’t grammatically acceptable for precisely the reasons given. (It’s roughly equivalent to saying “I walk down” or “This is”  — and no teacher of English grammar would accept this locution without the speaker or writer also supplying some sentential or semantic context.)

So we can conclude by saying that this is why the philosophy of logic is over and above pure (or formal) logic.