Incompleteness, Inconsistency and Those Pesky Words!

With the exception of an image demonstrating an argument so utterly and awfully illogical, it deserves public shaming, this post largely works with abstract cases as examples in the hope of not upsetting quite as many people as it otherwise might.

We start with the background stuff …

Kurt Gödel, in 1931, dropped a bit of a bombshell on a mathematical and logical world (that was quietly believing the opposite) by showing that there are things that can’t be proven or disproven.  In other words, in all ‘vaguely normal’ systems, there are propositions that can be either true or false and it doesn’t really matter.  ‘Mathematics is incomplete‘.  In 1936, Alan Turing proved that there are problems that can’t be computed/solved and the rest of the computer science research community spent the next few decades realising that these were kind of the same thing.

Pretty disastrous, huh?

Well, no, not really.  The mathematical and computer logic world dusted itself off and got on with it.  And anyway, other branches of science – and beyond – had similar problems.  In physics, for example, there’s a limit to how closely you can measure something before you change what you’re trying to measure.  Turns out, in one form or another, ‘incompleteness’ is pretty normal in life.

So, no, incompleteness, isn’t that much of an issue.  (it just means we don’t know everything – in fact, can’t know everything: there’s some bits of science, philosophy, etc., we can’t do from our little three-and-a-half-dimensions backwater of the universe; or we’re not God, if you like.)

However, rather than ‘incompleteness’, something called ‘inconsistency’ would be a problem.  Why?  And what does that look like?

Well, it might be best – for now – to avoid rigid notions of ‘true’ and ‘false’; we’ll do that later.  Instead, and more simply, a new rule, or ‘fact’, introduced into a system, is ‘inconsistent’ if it contradicts anything already in the system.  Alternatively, two ‘facts’ are inconsistent if they contradict each other.  The propositions x > 2 and x < 2, for example are clearly inconsistent: they can’t both be true at the same time (for any x).  Or, in a more general sense, if you’ve got a whole system based on principles that something-or-other isn’t such-and-such; then you introduce new information to the effect that it is, then you’ve got problems; big problems as it turns out.

Now this ‘inconsistency’ thing might not seem like a big deal; probably something we really should take on the chin, along with incompleteness?  Another minor inconvenience, perhaps?  Just a further limitation of being human?  In fact, however, we can’t ignore this so easily; it’s way more important than incompleteness: inconsistency is an utter disaster for any system!  We just can’t be doing with it.  Why?

Well, because there’s a well-known result – presented in almost as many ways as there are logic textbooks – that says this:

If a system has (even a single) inconsistency, then anything can be proven.

Really?  That sounds like a big claim.   But it’s true.  However, instead of a formal proof, here’s an apocryphal dialogue:

• Bertrand Russell: “Any inconsistency in a system allows anything to be proved”
• Student: “OK, if 2+2=5, prove I’m the Pope!”
• Bertrand Russell: “Right, so 2+2=5?  Fair enough. But we also know (proven by extension of ‘Principia Mathematica’) that 2+2=4. Equating gives 5=4. Subtracting 3 from both sides gives 2=1. You and the Pope are two people; therefore you are one.”

Now, this might seem like wordplay.  (Indeed, it is to an extent and we’ll return to that later.)  But the principle is quite easily formalised – into set theory for example.  [Let S be the set containing, as elements, exactly the Pope (p) and the student (s) (and no more).  Then 5=4 ⇒ 2=1(=|S| say) ⇒ S={p,s}={ps}. The two elements must be the same element.]  There are even more formal proofs out there.

So, yes, if you’ve got (any) inconsistency in a system then any notion of ‘true’ and ‘false’ goes out the window.  In fact, any statement can be both true and false at the same time.  The system itself has become logically worthless.  (Another way of thinking about it for the logical philosophers is that our original incompleteness, previously isolated in remote pockets, has now taken over the whole system: really not good!)

No, inconsistency is entirely horrible: to be avoided at all costs.

Now, in real life unfortunately, we don’t always manage that.  The Law, for example is often hopelessly inconsistent, the result of new legislation being piled on old without due consideration for contradiction and consequences.  This ‘law X says this but law Y says that so which takes precedence?’ wrangle has kept lawyers in jobs in complex legal cases for years!  [It also gives a pointer as to why so many in the legal profession are politically centrist/liberal but that one’s left as an ‘exercise for the reader’; it’s not integral to this piece.]

Of course, just to muddy this a bit further, we wouldn’t necessarily know if we were introducing inconsistency into a system: not at the time anyway; that might only become evident later when we started following different strands of the logic through and realised that something didn’t work.

Not only that, but being inconsistent is far from difficult.  The Greeks knew this.  You can even make a system inconsistent with pretty much your opening proposal (paraphrased from Epimenides):

• “Everything I say is a lie”

If that’s true, then it’s a lie, so it’s not true.  If it’s a lie, then it’s not true so you’ve got an inconsistent statement right there!  (Assuming you want your system to point towards truth?)  Hold that thought though because it’s an important one: it’s dead easy to say something that satisfies the rules of natural language but makes absolutely no logical sense at all.

It doesn’t have to be quite as daft as Epimenides, of course …

• “I think there should be an unequivocal ban on smacking children”
• “I agree. Except I think parents should be allowed to smack their children lightly, under certain circumstances”

Well, that’s not agreeing at all.  The system is inconsistent if we accept both statements into it.  The discussion is fatally flawed from here on in.  Smacking is simultaneously allowed and not allowed.

Still a bit daft?  Maybe, but the same problem can take subtler forms.  Suppose there was an international accord to the effect that “smacking of children should be banned unequivocally”.  It says no-one should smack children: ever; not even parents.  Let’s call it the ‘International Agreement on Smacking Children’ or ‘IASC’.  Now suppose you’re a bumbling ‘what’s-all-the-fuss-about?’ politician, trying to appease both sides of the debate.  It’s absolutely dead easy to say:

• “I can’t see what all the fuss is about’
• “I support the IASC without reservation”
• “I also think parents should be allowed to smack children occasionally”
• “There, wasn’t that easy?”

Easy, yes; but simultaneously complete nonsense.  It’s still inconsistent; the contradiction has just been filtered through a layer of obfuscation.  You can still get to any arbitrarily true/false position from here.

So, we get the idea: we really should try to avoid inconsistency where we can: it just isn’t helpful.

But we often don’t!  We really, really don’t.  Sometimes it’s accidental; sometimes deliberate; sometimes, well, who knows …

Before we dive into this though, we have to face another uncomfortable side issue: English is a bit crap!

No, of course it isn’t really.  No worse than any other than natural language, and bigger and better than most.  It’s beautiful.  It gives us poetry, declarations of love, stirs emotions with its descriptive power.  And that’s not all; properly handled, it can explain processes, describe complex concepts and we’d be generally knackered without it.  It’s just that, in terms of pure logic, particularly when that logic’s getting a bit tricky, particularly even when people are maybe trying to make that logic a bit tricky for their own ends, it’s, well … imperfect.

• • “Only an elephant or a blue whale gives birth to something that weighs more than 75kg”
  • “The President weighs 115kg”
  • “Therefore the President’s mother was either an elephant or a blue whale”

• • “You’re a fascist for saying that”
  • “You’re a fascist for calling me a fascist”

But we’re stuck with it.  It took Russell and Whitehead hundreds of pages of logical/mathematical notation to prove that 1+1=2.  That sort of pure logic has its place but we’re hardly going to manage (say) a Brexit debate using it.  No, English it’s going to have to be [note: other natural languages are available].  But we would be wise to be aware of its shortcomings as a tool for logical argument; particularly if we accept that not everyone’s motive might be entirely honest, and particularly, particularly in relation to this dreaded spectre of inconsistency.

Words are simultaneously powerful but imprecise.

Let’s take social media for example and the character-constrained format of a Twitter exchange …

When someone uses a big word (i.e. something vaguely erudite) on Twitter, aside from the fact that some people will have to look up what it means (although many won’t bother, which potentially biases involvement in the discussion), there are advantages and disadvantages:

• • A single word can often serve in place of many, which is efficient.  As a simple example, it’s much quicker to accuse someone of ‘schadenfreude’ rather than outline a case for how or why they may be enjoying another’s misfortune.  The power of the word then creates a base point for brevity in further discussion.
  • Often, however, a word’s meaning is either imprecise or its application in a particular case inexact: it’s easily detached from its (not terribly formal in the first place) definition.  Subsequent discussion may be hindered by this imperfect usage.  (And some people exploit this deliberately.)  ‘OK, so we’re saying that both X and Y are guilty of schadenfreude but have we applied the definition equally?’  As the discussion progresses, can they subsequently be assumed to have done effectively the same thing in all respects?  If the answer’s ‘no’ but the argument that follows (or perhaps just one side of it) assumes ‘yes’, then we’re in difficulty.

It’s a small point but natural languages can be unreliable in this respect; worse, they can be open to deliberate abuse by those looking to mislead.  Anyway, that was a brief – but necessary – digression.  Back to the central plot: inconsistency … Right, where were we?

Hopefully, by now we’ve got the central message that inconsistency (which can be introduced accidentally or deliberately) should be avoided at all costs in a well-founded argument because everything falls apart at some stage after that.  The falling-apart doesn’t necessarily happen immediately in the argument or discussion but, at some point, it allows any old nonsense to pass for reason.  Ultimately true becomes false and vice-versa and we’d have been better off not bothering.  So, if we’re absolutely sure we’ve got that, then now might be a good time to turn that theory into practice …

OK, let’s take a real-world example.  Something not too controversial?  Er, … How about the International Holocaust Remembrance Alliance (IHRA) definition of antisemitism …?

Before we go any further, it really would be wise to read it first: it isn’t long, or complicated.  (People commenting on things they haven’t actually read is becoming a scourge of online debate: cf Turing’s 1950 paper on AI.)

On a first scan, the IHRA definition is well-written in simple language, with each of its paragraphs and bullet points clear and easy to follow.

But – taken all together – is it consistent?

In particular, how do the following extracts shape up against each other?

• … criticism of Israel similar to that leveled against any other country cannot be regarded as antisemitic. <1>
• … examples of antisemitism […] include, but are not limited to: <2>
  • Accusing […] Israel as a state, of inventing or exaggerating the Holocaust. <3>
  • […] claiming that the existence of a State of Israel is a racist endeavor. <4>
  • Drawing comparisons of contemporary Israeli policy to that of the Nazis. <5>

(The bits tucked away [in square brackets] aren’t to hide anything; they’re to focus on the potential problem text.  You can read them on the original link and see that what’s left is consistent with the original structure.  The <numbers> are just for reference in the discussion that follows.)

Now, it would be wrong to suggest that the answer to this question of consistency here is an easy one (as usual, the imprecise nature of the words themselves has to be taken into account) but there are certainly parts of this that don’t sit easily with each other.  Is criticism of Israel acceptable or not, for example?  Or what form might legitimate criticism take?

As a a more developed example, most people today would consider (say) South Africa, prior to the 1990s, a racist state – and that’s certainly a commonly stated view that generally passes off without objection.  On this basis <1> (above) would seem to suggest that a similar criticism levelled against Israel would be valid.  However, <4> is pretty clear that it isn’t.  Similarly, whilst no-one (well, very few people) would want to be compared to the Nazis, it’s an accusation that could be thrown at anyone (there have been more recent examples of ‘ethnic cleansing’, for example): on that basis, <1> and <5> are at odds here.  (We’ll swallow hard and consider the Holocaust later.)

The point is the IHRA definition might not be as clear-cut as we’d like it to be.  And the danger is, if we don’t deal with the inconsistencies immediately, they pop up to bite us harder later.  When the UK Labour Party was debating whether to adopt the full IHRA definition of antisemitism, including <1> and all of <2> (including <3,4,5>), although they eventually did, there were some concerned voices about possible inconsistency (and, yes, some objections also from those whose intent was less academically pure – that doesn’t help).  In response, though, this sort of counter-argument was all over the place on social media:

• “Why’s this so complicated?”
• “I support the IHRA definition without reservation”
• “I also think it’s appropriate to criticise Israel, where necessary, for its treatment of Palestinians”
• “There, wasn’t that easy?”

Compare with, of course:

• “I can’t see what all the fuss is about’
• “I support the IASC without reservation”
• “I also think parents should be allowed to smack children occasionally”
• “There, wasn’t that easy?”

Yes, easy to say, but, as we’ve seen, the Devil’s in the detail!  Saying something is simple; whether it makes any sense is an entirely different matter.

But it can get a whole lot worse …

• “The Israeli Lobby are saying …”
• “You can’t talk about the ‘Israeli lobby’. It’s antisemitic”     [not explicitly covered either way in IHRA]
• “Why not? I can talk about any other country having a ‘lobby'”
• “But no-one ever has/does. So it’s persecution”     [of whom?]

Yes, this is where you can end up, starting from shaky foundations!  It’s not hard to see that, in this case, the argument is doubly daft*:

• Firstly, it’s factually wrong.  It’s impressive that we’ve almost not had to worry about anything quite this obvious in the discussion so far but this is just plain untrue.  Not only does web-searching for the term ‘lobby’ together with various countries or other federations (such as the EU) return copious results, there’s no evidence that any outrage was caused when the original comments were made.
• But, secondly, and way more interestingly, what would happen if someone did use the term in conjunction with another country, now (immediately) after this argument was tried on?  The initial reaction might be that it was contrived: a deliberately provocatively response to the argument.  And, whilst it might not be fair to consign this reaction wholesale to the ‘it’s all about me’ pile, the fact would remain that the term had now been used and we’d have to decide how to deal with it.  Logically, and consistently, there’d be three options: (1) It’s going to have to be OK to use the term ‘lobby’ in reference to any country, including Israel: the above argument is going to have to be retracted, (2) It’s not OK to use the term in reference to any country in future: we’ve banned certain combinations of English words in subsequent discussion, and not just in reference to Israel, or (3) Israel is going to have to be considered as a ‘special case’: uncomfortable, maybe, but we’re going to have to come back to it.

(* Actually, there may be another common, simpler dynamic at work here: the instinctive human desire not to be wrong: to find some response – any response, no matter how desperate – rather than the more honest, ‘yes, you might have point there: let’s reconsider that’.)

But the real cause of the problem here is the IHRA definition itself.  Yes, it’s starting to look problematic from a consistency point of view, but there’s also potentially an issue with the way it’s logically structured …

The IHRA definition, early on <1>, makes a very clear declaration of a principle by which something wouldn’t be antisemitic.  However, it then goes on <2> to present ‘examples’ of things that would be <3,4,5>.  Aside from the possibility that these rules themselves are contradictory/inconsistent, the fact that section <2> is presented as ‘examples’, suggests that there may be others not listed.  It’s not clear how these might be determined, or by whom, and it would be anyone’s guess as to whether they would or wouldn’t contradict <1>.  The above example (offence being taken at the use of the word ‘lobby’) does appear to contradict <1> and, if it wasn’t clear up to this point that we were being sucked into the quicksand of inconsistency, it certainly is once you see something this daft!

And then there’s the Holocaust.  Deep breath …

Right, we need to be really careful and clear here!

Our starting point, lest there be any doubt:

• The Holocaust happened
• It happened the way it’s generally presented as having happened, for the reasons generally given and with the results (including those obscene numbers) agreed by the vast majority.
• It was bloody awful and it should never happen again.
• There is massive documentary evidence of the Holocaust.
• There are people still alive with first-hand experience of the Holocaust.
• There is another generation now only one generation removed from those with first-hand experience, who will have been given direct evidence of the Holocaust.
• A ‘conspiracy’ on this scale is inconceivable.
• Holocaust denial is based on poor, flawed evidence and makes no sense.
• Those seeking to deny the Holocaust are the poorest sort of historians.
• They may have ulterior motives for their denial, including using the publicity for self-promotion, financial gain, etc.
• They may be antisemitic, and their denial may be a result of their antisemitism or it may be a tool of their antisemitism (or both).

But what is Holocaust denial in the abstract?  (OK, yes, that might actually be the problem here: that we can’t deal with antisemitism in the abstract: everything needs to be taken in context, and that might involve ‘special cases’.  We’re still coming to that; but one step at a time.)

Essentially, Holocaust denial is an alternative view or interpretation of history.  In the academic world, such a dispute would be permitted under the shield of ‘academic freedom‘.  In particular, ‘… the conviction that the freedom of inquiry by faculty members is essential to the mission of the academy as well as the principles of academia and that scholars should have freedom to teach or communicate ideas or facts (including those that are inconvenient to external political groups or to authorities) without being targeted for repression, job loss, or imprisonment.’

Bearing in mind there are charlatans both inside and outside of academia, and that there’s no precise definition of either an academic or whether someone is acting in an academic capacity (either within or outside of their recognised sphere of expertise), this ‘freedom to dispute’ can’t be ring-fenced: it’s not a uniquely academic privilege: it has to apply to ‘knowledge’ universally – and, in nearly every respect, across the world, in science and the arts – including history, it routinely does … except perhaps here?

Because, however stupid it might be, however much poor history it’s based on, whatever the self-promoting intent of the proponent, Holocaust denial remains a historical dispute involving a minority opinion against a majority one.  It may be unpleasant, upsetting and motivated by all the wrong drivers but that’s still what it is.

It’s hard to think of any other situation in which an academic, or anyone else, could potentially get into so much trouble for questioning conventional wisdom.  A basic principle of knowledge, and its expansion, is that debate is to be encouraged – and it generally is, even if only to eliminate inappropriate models, interpretation or theory; and even if the same debate has to be revisited from time to time to continue to do so.  Bad science is just bad science.  Bad history is just bad history.  Debate puts it right.  Even if (deliberately silly example) someone was to suggest (on essentially no evidence) that the conquistadors weren’t responsible (either by force or introduction of disease) for the extermination of the indigenous ‘New World’ population – that instead they just chose not to reproduce, or sailed off somewhere, they might be roundly ridiculed, lose any credibility as a historian, be accused of having no higher object than making a quick buck through book sales, etc. … but they’re unlikely to be tarnished as racist (either anti-South American/Inca or pro-Spanish/Portuguese, etc.).

Unfortunately, once again, we’re just not being consistent here and by now we should know how dangerous that can be.  It may be ‘inconvenient’ in an academic freedom sense to point that out but we’re not.  And, once again, the IHRA definition, if it doesn’t actively hinder the unraveling of all this, certainly doesn’t help.  Because one of <2>’s explicit ‘examples’ <3> (full version), tells us that ‘accusing the Jews as a people, or Israel as a state, of inventing or exaggerating the Holocaust’ is antisemitic.  Well, yes, ‘accusing Jews as a people’ might be unfair (arbitrarily discriminatory) but ‘accusing Israel as a state’ seems to contradict <1> because the point is that Holocaust denial doesn’t have to involve either of these.  A Holocaust denier could simply be saying that ‘the whole world’s got this wrong’ with no attempt to associate fault or motive with any particular group whatsoever (or maybe it’s an Australian plot or a Chinese conspiracy or …).  If Holocaust denial makes no reference to either Jews or Israel then the preconditions of <3> aren’t satisfied so, by IHRA (or, at least, that particular rule), it isn’t antisemitic.  But it’s universally accepted these days as being antisemitic so what’s going on?  Or is this one of those implied but unwritten ‘other examples’ that imprecisely points to how exceptions to <1> might be arbitrarily conceived?  (But not how, why or by whom.)  It’s a mess.  It probably wouldn’t stand up as a piece of legislation or contract in a legal setting; it certainly fails abysmally as a piece of logic.

Or are we heading inevitably towards the solution that’s been pushing itself at us for several paragraphs now?

… that, because Jewish people have suffered so much, so horribly, for so long, the right has been earned for antisemitism to be treated differently to other forms of racism?

… that, in terms of what can and can’t be said, a connection be accepted between (in this case) Jewish people and the state of Israel that wouldn’t apply elsewhere?  (Translated into an equivalent scenario.)

… that we lose <1> from the IHRA definition?

In fact, are we implicitly doing that already?

Well, having finally, reluctantly introduced the idea, we’re just not going to go there!  Others can take that on: it’s a moral issue, not a question of consistency (which is what this post is supposed to be about) apart from noting that: it is a coherent argument in its own right but it does contradict IHRA as it presently stands.  it’s also related to the question of self-determination: in this case, whether the right to say ‘we’ll decide how discrimination is defined against us’ is being applied fairly and equally to everyone.  Whether treating antisemitism as a special case is an attractive proposition is probably best not discussed further here … except to point out that, based on a contradictory set of initial axioms, we’re seeing plenty of examples of getting close to its implicit adoption already out there.  Why?  Who’s to blame?  No, we’re really not going to try to answer that!

To conclude somehow, the examples used here are in no way, shape or form meant to detract from the entirely worthy and utterly necessary fight against antisemitism (together with all forms of racism and wider discrimination) everywhere in society – including, for example in the UK Labour Party.  But, for that fight to be effective, it needs to be built on strong foundations, not supposedly core principles, which themselves turn out to be at odds with each other.  Ultimately people with less honourable motives will exploit these weaknesses – these inconsistencies.  The best way to deal with them is always to shine a light on them, then remove them, or otherwise deal with them, rather than hide or ignore them.

OK, no-one’s perfect.  We each have our corners to defend and it can be tiresome when experts, ‘facts’ and logic don’t lead us (or more likely other people that we’re trying to convince) to just where we’d like; but a mature response is that that means we can’t always have exactly what we want: sometimes we’re just asking too much.  If we’re too stubborn in getting precisely our own way, then the system we build as a result becomes worthless: our own tool turns against us.  Yes, the ultimate danger here is that:

• We’ve already started treating ‘experts’ with scepticism
• We’re prepared to ignore ‘facts’ when they don’t suit us
• Are we going to boot good old logic out the door too?

Please, no.