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Monday 7 June 2021

Bad Form, What?

Architect Frank Lloyd Wright wasn't the only person ever who had a lot to say about form. It's something we've looked at a fair bit hereabouts but, for the most part, in very narrow context, and never quite as directly as I'd like. 

Here, I want to think very specifically about the notion of form, and how it applies in different areas of thought. It would be easy to think of these different contexts as talking about different things, but are they? Today's outing is more in the nature of cognitive housekeeping than anything, a means of tying together disparate threads of thought scattered throughout the blog. Thus, all of this has been covered before here, but never quite as directly or in as much detail, and not with the degree of structure I intend to impose here. I'm doing so because of where I want the whole tenor of the blog to go in the future. No drastic changes in content or anything, just the imposition of a little form. We have seen the beginning of this just prior to my hiatus, with In On The Secret (recently republished), so those who've read that will have some feel for what's coming.  

Strap in, dear reader, as we have some terrain to cover. Let me show you the map.

The idea of form has played a huge part in the history of thought, delivering ream upon ream of turgid navel-gazing, but also enough indicators across many disciplines to inform us that it's a good thing to think about.

A scan of any dictionary's definition of the word is a good indicator of how much it's informed (geddit?) our thought. From poetry to polynomials, the idea of form threads its way through the history of thinking in fundamental ways many and diverse, but all very closely related.

The foundation of our thinking about form, at least in the Western philosophical tradition, will be unsurprising to most, given that he's very much the bottleneck of Western thought; Plato.

As was often the case, the ideas were presented as part of his Socratic dialogues, so there's a degree of ambiguity in terms of what Plato himself thought. He certainly offered later critiques of some of the notions presented in the dialogues, which suggests that he at least changed his mind in some respects.

In a nutshell, the idea of Platonic forms is closely related to the notion of 'essence'. In Plato's scheme, there is an essential rabbit encompassing all the attributes of rabbitness, and the rabbit we experience isn't a real rabbit, it's an analogue of this real, essential rabbit, which is the thing that actually really-o, truly-o exists (hat-tip to one of my favourite commentators on philosophy).

I know, right? It's nuts. With a little care and a lot of cautious thought, it is actually possible to tease out a really wonderful idea from it (and this has been done) but, as presented, it's just daft.

It's certainly the case that our experiences of the rabbit are not the rabbit, and that what exists in our minds is a notional rabbit. It's also the case that all of our experiences are internally generated. In that sense, what's 'real' for an individual (as opposed to real in some ontological sense) is the form rather than the ding an sich (thing in itself), as Kant would have put it, but Plato's was an ontology, not a critique on perception and its relationship to the phenomenal world.

So let's talk about form for a bit, as it might prove interesting.

The obvious place for us to begin our journey into form is formal logical fallacies, since this is well-trod geography hereabouts and it's a highly approachable guide to just what we mean by form. It also conveniently allows me to put this all coherently into one place rather than scattered around the blog, which is nice.

A formal fallacy is a fallacy of form, which is to say it's precisely and unambiguously a problem in the structure of an argument. To see what that looks like, and to get a feel for the notation, let's lay out the root syllogistic argument forms for a clearer picture.

\(\dfrac{P\Rightarrow Q, P}{\therefore Q}\)

This form validates an argument. It doesn't prove the argument or otherwise show that it's correct, it merely shows that the conclusion follows validly from the premises and does not commit a non sequitur (lit. does not follow).

Let's set up a recyclable example so we can easily keep track. Here's one I prepared earlier:

Premise 1*: All men are mortal.
P2: Socrates was a man.
Conclusion: Socrates was mortal.

This form of argument is known as a 'syllogism'. It proceeds by distilling a hypothetical proposition to its atomic structure, the atoms of which are the 'antecedent' (P=man) and the 'consequent' (Q=mortal). These atoms must be binaries. Man or not (¬) man. Mortal or not mortal. 

Once we have our argument distilled to its atoms, it becomes really easy to see the logical structure. Because we've removed all the content of the argument, leaving only symbols representing the atoms of the proposition, it becomes easy to see that spotting a fallacy is incredibly simple, because it's all in the ordering and the operators.

In P1, we have a proposition that connects antecedent and consequent in some manner with an 'operator'. In this case, the arrow pointing right (\(\Rightarrow\)) means 'implies'. The atomic statement of P1 as represented in the notation, then, is 'man implies mortal', logically equivalent to the natural language statement. 

In P2, we have a specific instance of the antecedent, a man. If the proposition contained in P1 is correct, it must be the case that a specific case of P will fulfil Q, meaning Socrates must have been mortal.

This specific form is known as modus ponens, or 'the way that affirms by affirming'. What is it affirming? The antecedent.  

This gives us a 'rule of inference' we can add to our list of 'things to look for when logicking':

Affirming antecedent good. 

With a little imagination, it's not difficult to see how powerful this could get. Implication isn't the only thing we can do. With a few ifs, ands and buts, among other things, we can carry out complex logical operations on them easily. But I'm getting a couple of posts ahead of myself. Squirrels begone!

Here's another form:

\(\dfrac{P\Rightarrow Q, ¬Q}{\therefore ¬P}\)

This form validates the argument. Once again, it says nothing whatsoever about whether the conclusion is correct, only that the conclusion follows valid rules of inference in the route from the premises. Here's our recyclable example again.

P1: All men are mortal.
P2: Socrates is not mortal.
C. Socrates is not a man.

In content, this looks odd, but that's not an issue here, because what we're interested in is the form only. The content of the argument is irrelevant to that purpose.

In any event, when I tell you Socrates is the name of an asteroid discovered in the main asteroid belt in 1960 by a couple of Dutch astronomers, the content objection evaporates**.

Anyhoo, this form also has a name; modus tollens, or 'the way that denies by denying'. And what are we denying? The little 'not' symbol is a clue, because it's right next to the consequent in the second premise. We're denying the consequent. In the scientific arena, this is the logic underpinning 'falsification'. More on that in the links at the bottom.

Let's pop another rule into our book:

Denying consequent good.

So these two rules are looking good. We have affirming the antecedent and denying the consequent as valid rules of inference.

When an argument has true premises and follows these rules of inference, the conclusion is true. This is what it means, formally, to prove something. The same is true of mathematics, unsurprisingly, because this notational language and the logic on which it operates are exactly the same. Mathematics is, in fact, the language of all formal logic, though it may not always look like it. Indeed, in mathematics, what it means for something to be formal is for it to be cast in this language, and it's precisely because it is pure form; pure structure.

Those two options don't exhaust the possibilities of form, however. Here's another.

\(\dfrac{P\Rightarrow Q, ¬P}{\therefore ¬Q}\)

This form invalidates the argument. This is 'denying the antecedent', and it's a problem. A simple example should show why;

P1: All men are mortal.
P2: Socrates is not a man.
C. Socrates is not mortal.

With caveats relegated to footnote, while the conclusion may look OK, given that we're talking about an asteroid, I should acknowledge that I'm now talking about a different Socrates, my next-door neighbour's cat. The problem with this form is that, while P1 grants that all men are mortal, it does not grant that men are the only things that are mortal. This form is invalidating because it does something to premise 1 that isn't allowed. It smuggles in an assumption in the form of the implication, turning \(\Rightarrow\) into \(\Leftrightarrow\). That change in symbol is huge, because it reflects the implication back from the consequent to the antecedent. 

\(a\Rightarrow b = a\) implies \(b\).

\(a\Leftrightarrow b = a\) implies \(b\) and \(b\) implies \(a\)

Not all things other than men are not mortal, and this argument discounts all of them. 

Denying antecedent bad.

There's only one option remaining for form, and there's a reason I've kept this one for last, because this is the number one most committed fallacy in all of discourse. I'll come back to why that is shortly, but here it is:

\(\dfrac{P\Rightarrow Q, Q}{\therefore P}\)

We should trivially know what to call this given what's gone before. This is affirming the consequent. 

P1: All men are mortal.
P2: Socrates is mortal.
C: Socrates is a man.

This form invalidates the argument. By now, we can ignore the red flags because, once again, Socrates here refers to my next-door neighbour's cat††.

Affirming consequent bad.

Both these formal fallacies can be critiqued in the same way in natural language by the simple expedient of pointing out that there's more than one Socrates, and these fallacious forms discount them in favour of a preferred outcome. 

So, we have four nice little rules, but what can we do with them? What do they give us? Well, with a little care, they give us the whole shooting match. These four little rules are the molecules of assessing claims and, with practice, they're really easy to spot.

Let's group them together just for the look of the thing.

1. Affirming antecedent good.
2. Denying consequent good.
3. Denying antecedent bad.
4. Affirming consequent bad.

It's really worth noting, very many informal fallacies reduce to either affirming the consequent or denying the antecedent. Exceptions are quite rare, and are usually technical fallacies like begging the question (reduces to \(P\Rightarrow P\) or bare assertion). Even many technical fallacies reduce to other informal fallacies which in turn reduce to one of our pair. The fallacy of equivocation, for example, wherein multiple definitions are used interchangeably despite having very different logical consequences, reduces to a red herring, a fallacy of relevance. Which of the formal fallacies it reduces to will depend entirely on the nature of the Clupidae in question, but it's most certainly the case that ALL fallacies of relevance, which comprise most of the landscape of informal fallacies, reduce to these two. It should be reasonably obvious why, but let's spend a moment unpacking.

Informal fallacies are those fallacies whose fallaciousness arises not from form, but from content. Remember form deals with validity, but validity is only one on the twofold path to logical nirvana. We also need soundness, which is about form and content. A sound argument is one which follows the correct rules of inference (i.e., is valid) and is founded on true premises.

My selection du jour for an informal fallacy to draw on to show this reduction is known as the fallacist's fallacy.

This fallacy is committed when we accept or reject the conclusion of an argument based only on the fact a fallacy was committed in the argument. 

Now, it's not immediately obvious to all this is even a problem, although regulars here will recognise it and the example syllogism I'm about to give.

P1. Jensen Button is German
P2. The moon is made of green cheese
C. Therefore, diamond is an elemental form of carbon.

It's clear P1 and P2 are both false statements, each entirely unconnected to the other logically, and the conclusion has nothing to do with either premise. No rules of inference have been followed and it's not so much non sequitur as nonsensical, yet the conclusion is a true statement. To reject this conclusion on the basis of the commission of a fallacy would be silly, hence the fallacist's fallacy. The simplest warning against this fallacy is to note that the argument is irrelevant to the truth of the conclusion in every way unless it proves it. The conclusion of a fallacious argument has three potential truth values -
 true, false, unknown - none of which can be ascertained from the argument. The most we can say is that the argument doesn't support its conclusion.

That it reduces to a fully formal fallacy is less obvious, but the sort of thing that you can't unsee once you see it, and possibly worthy of a Huxleyesque exclamation, to boot. Here's the fallacy cast in syllogistic form so it becomes clear:

Let's assign roles to the propositions underpinning the fallacy:

Let P be our antecedent 'sound argument' and Q be our consequent 'true conclusion'.

P1. A sound argument (\(P\)) implies (\(\Rightarrow\)) a true conclusion (\(Q\)).
P2. Your argument is not a sound argument (\(¬P\)).
C. Therefore, your conclusion is not true (\(\therefore ¬Q\)).

\(\dfrac{P\Rightarrow Q, ¬P}{\therefore ¬Q}\)

It should be apparent this is denying the antecedent which, as we've already seen,.

Another example is an old favourite, the argumentum ad ignorantiam or 'appeal to ignorance'. This fallacy is committed when a conclusion is asserted as true because you can't prove it isn't. An example is one we've met, the assertion that the resurrection must have occurred because people were willing to be persecuted in asserting they'd seen Jesus and why would they if it wasn't true? it should be pretty obvious what the underlying formal fallacy is here, but it's easy enough to formalise it, so let's do it.

Let P= resurrection and Q= willing to suffer defending it. That should, in fact, be all you need to see to spot the fallacy. That is, in fact, the beauty of reduction to atomic elements. All you need do is assign P and Q to the elements of a statement that affect the logical consequences, and how to do this will always be obvious in such a setting. Start just with how the implication relationship works. if the implication is unidirectional, it always flows from P to Q in this manner.

\(\dfrac{P\Rightarrow Q, Q}{\therefore P}\)

P1. Resurrection implies willing to suffer.
P2. Willing to suffer.
C. Therefore, resurrection.

Clear affirming the consequent which, as we've seen,

In the final analysis, affirming the consequent is easily the most common fallacy committed in discourse, by orders of magnitude. Some of that is due to a quirk in how we tend to order things in our minds, meaning that the way specific instances of informal fallacies are phrased means that the tendency is toward affirmation rather than denial, and this tendency is reflected in affirming the consequent being far more common than denying the antecedent. Ultimately, the phrasing of the informal fallacy and its context and content will determine how it reduces and, as I say, fallacies of relevance always reduce in this manner.

Another important thing to keep in mind is the vast majority of fallacies, whether formal or informal, are very specifically deductive fallacies, and thus only apply to deductive reasoning. In this instance, we have a fallacy that is both deductive and inductive in nature. This is not obvious. It's deductive because it renders an irrefutably true conclusion if the premises are correct and the reasoning valid, but it's also inductive, because it's a fallacy arising from ignoring the problem of induction. This latter is the definition of the fallacy of induction.

And one final critical note here; even sound, valid arguments can be fallacious. This is a very subtle point of logic, but it's an important one. The obvious example of this is circular reasoning. 

It's worth a paragraph or two on this, because circular reasoning is deeply misunderstood by many. First, there's some dispute about whether circular reasoning and begging the question are different or the same thing. In reality, neither of these is true. In fact, circular reasoning is a subset of begging the question, and it's easy to see why with a little thought. Let's look at the distinction to see how it works. 

Circular reasoning is reasoning in which the conclusion is contained in the premises. It's called circular reasoning because, as in all arguments, the conclusion relies on the premises. If the conclusion relies on the premises and the premises contain the conclusion, then all we've done is assert the truth of the conclusion on the basis of 
the truth of the conclusion. And we know it's true because the conclusion is true. You see where this leads. It's nonsense. It reduces to \(P\Leftrightarrow P\).


We know the bible is the word of god because it says so in the bible, and the bible is the word of god.

This does no more than assert the conclusion, and it's easy to see why it's called 'circular'.

Begging the question is a fallacy committed when the truth of your conclusion relies on the truth of some proposition. it's often not well grasped, and this is obvious from the number of sources telling us that begging the question is a type of circular reasoning, when in fact it's exactly the other way around. All circular reasoning is question begging, but not all question begging is circular reasoning. 


P1. The word of God is true.
P2. The bible is the word of God.
C. The bible is true.

This is clearly not circular, yet there is just as clearly a question being begged. The truth-value in this syllogism, which is logically valid (affirming the antecedent ), is entirely dependent on the first premise being true. The conclusion isn't directly contained in the premises, so it isn't circular, but there is a question being begged, namely whether the first premise is true (among others; whether god exists, whether there's any such thing as the word of god, whether P2 is true, whether the bible really is, etc.). 

Back to that subtle point of logic, namely that a sound, valid argument can still be fallacious. These two fallacies both employ valid reasoning. I mean, in the case of circular reasoning, it could hardly be otherwise. If your premises are true and your conclusion is contained in your premises, then the conclusion must be true, and the route from premises has to be valid because the statement reduces to a tautology.

The same is true of begging the question. If the premises are true, and the reasoning valid, the conclusion must be true. The question being begged, then, in both cases, is 'are the premises true'. 

For more on different types of reasoning, I'll link an earlier post at the bottom. If you haven't read, it's worth doing so, because it highlights some things about even formal fallacies, specifically in the context of the domain of applicability. All else aside, even 'affirming the consequent' can be a valid argument, just not in a deductive setting. This becomes extremely important when we start talking, for example, about scientific hypotheses.

In the above cases, it should be reasonably clear there's an underlying assumption, and this assumption is in turn rooted in one of the above formal fallacies in selecting a conclusion at the expense of all possible alternatives.

These examples, should give some insight into how formal fallacies are smuggled into informal fallacies. More importantly, while there can be a degree of ambiguity in which informal fallacy has been committed (in fact, it's fairly rare that informal fallacies crop up in isolation; generally, multiple informal fallacies are committed at the same time), there is never any ambiguity in which formal fallacy underpins it. As a general rule of thumb, if instinct tells you a fallacy has been committed but you can't pin down which one, look for the underlying formal fallacy, because it will almost always be there. There are theoretically an infinite number of quite distinct informal fallacies, tiny variations on themes that make them functionally impossible to classify comprehensively. On the other hand, there are exactly two formal fallacies, which renders classification slightly more tractable a problem.

I should note that it's become de rigeur in debate to simply drop the name of a fallacy and consider the argument won. That practice is this particular fallacy elevated to stupidity, and really doesn't reflect a solid command of logic or good debate skills. If you're going to accuse somebody of committing a fallacy, you should at least expose the fallacy so that somebody has a chance to learn from it, or you're just flinging peanuts.

What should be fairly apparent from all of the foregoing is formalising arguments removes the pathologies of natural language. 

It had been my intention to discuss a lot more about form, and especially in different areas in which plays a major part; mathematics (whose notation we've been using in our formal syllogisms, but whose notion of form is very different in one important respect, but for reasons deeply intertwined with what we've discussed here), form and space in architecture, form and composition in art, music, photography, etc., but I'm aware that this post is getting on the lengthy side, and moving on to another area will mean another 1500 words or a massive reduction in quality. 

I'm going to hold those over for another post.

Alongside this post, I've been working on another post about what limitations, if any, there might be to knowledge. Keep an eye out for Completely Incomplete in the coming days.

Thank you for reading and for your continued support (especially that one guy, who knows who he is but would not want me to say more). Space for comments, critique and questions down below.

I'm told by purported experts in such matters it's critical to keep the length of my posts down, make my titles between x words and y words long (far too may words even at x for my liking), but I couldn't identify one of these 'experts' that wasn't desperately trying to sell me something, which always renders advice suspect. I may not have many readers, but they're all class, and I can't thank you all enough for staying with me. Besides, you and I know the topics we cover here aren't clickbait, and are certainly not conducive to brevity, even were I not an inveterate gobshite who could ramble on for days. 

Besides, I like to keep things informal.

Deduction, Induction, Abduction and Fallacy Different types of reasoning and a more broad treatment of fallacies.
In On The Secret - The first in a series attempting to demystify archaic notation. More to come on notation in the very near future, including a more comprehensive treatment of the notation in this post.

*Properly, this should be spelled 'premiss', which is a term specific to logic whose meaning largely reflects the word 'premise' as we'd use it in the vernacular, but with very subtle implicational differences, mostly narrowing the definition to avoid ambiguity and, more importantly, to couch it in terms of something assumed arguendo rather than erected with the full force of a truth statement. It's an important subtlety in the study of logic, but rather less than important for our purpose here, which is just to distil the atoms of the notation. As long as we operate with the caveat that we're working with propositions rather than truth statements about reality, we can largely ignore the content of the arguments and focus just on the things that the notation allows us to do, namely study the structure in isolation so that we can remain pragmatic about it.

** In fact, there's no good evidence Socrates, teacher of Plato, ever existed. The only source we have for him is Plato, and there's an increasing body of thought suggesting Socrates was invented by Plato as a didactic device. I have plans to expand on this a little in a future post, in which I counter one of the common apologetic tropes, namely the notion that we have different standards of evidence for Jesus and Socrates. In the event, of course, Socrates non-existence would cause major informal problems with the first argument as presented here, but we're only using him as a conventional example.

† I want to plant a red flag here about that conclusion. The way I'm presenting these ideas follows a fairly well-worn convention in the exposition of syllogistic forms, but it does have the inherent danger that monikers carry baggage. We've covered 'what's in a name' in at least three different guises hereabouts, long-winded and extremely wordy expositions of the pitfalls of natural language and terms of art. Remember that the content is entirely irrelevant here. I can't express strongly enough how important it is to focus only on the form. The examples given are purely a guide to the form of the logic.

†† Regulars will be aware that I made this up. My next-door neighbour's cat is actually named Heidegger.

‡ Satire VI is the source of quite a few famous phrases, but the most famous two are this and "Sed quis custodiet ipsos custodes?"  - "Who watches the watchers?"

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