This is part of my series on success stories in philosophy. See this page for an explanation of the project and links to other items in the series.

Contents

1. Contents
2. Background
3. The problem of multiple generality
   1. How people were confused (a foray into the strange world of suppositio)
   2. How predicate logic dissolved the confusion
4. Definite descriptions
   1. How people were confused
   2. How predicate logic dissolved the confusion
5. The epsilon-delta definition of a limit
   1. How people were confused
   2. How predicate logic dissolved the confusion
6. On the connection between the analysis of definite descriptions and that of limit
7. “That depends what the meaning of ‘is’ is”

Background

Frege “dedicated himself to the idea of eliminating appeals to intuition in the proofs of the basic propositions of arithmetic”. For example:

A Kantian might very well simply draw a graph of a continuous function which takes values above and below the origin, and thereby ‘demonstrate’ that such a function must cross the origin. But both Bolzano and Frege saw such appeals to intuition as potentially introducing logical gaps into proofs.

In 1872, Weierstrass described a real-valued function that is continuous everywhere but differentiable nowhere. All the mathematics Weierstrass was building on had been established by using “obvious” intuitions. But now, the intuitive system so built up had led to a highly counter-intuitive result. This showed that intuitions can be an unreliable guide: by the lights of intuition, Weierstrass’s result introduced a contradiction in the system. So, Frege reasoned, we should ban intuitive proof-steps in favour of a purely formal system of proof. This formal system would (hopefully) allow us to derive the basics propositions of arithmetic. Armed with such a system, we could then simply check whether Weierstrass’s result, and others like it, hold or not.

So Frege developed predicate logic. In what follows I’ll assume familiarity with this system.

While originally developed for this mathematical purpose, predicate logic turned out to be applicable to a number of philosophical issues; this process is widely considered among the greatest success stories of modern philosophy.

The problem of multiple generality

How people were confused (a foray into the strange world of suppositio)

Dummett 1973:

Aristotle and the Stoics had investigated only those inferences involving essentially sentences containing not more than one expression of generality.

Aristotle’s system, which permits only four logical forms, seems comically limited¹ by today’s standards, yet Kant “famously claimed, in Logic (1800), that logic was the one completed science, and that Aristotelian logic more or less included everything about logic there was to know.” (Wikipedia).

Some medieval logicians attempted to go beyond Aristotle and grappled with the problem of multiple generality. As Dummet writes (my emphasis),

Scholastic logic had wrestled with the problems posed by inferences depending on sentences involving multiple generality – the occurrence of more than one expression of generality. In order to handle such inferences, they developed ever more complex theories of different types of ‘suppositio’ (different manners in which an expression could stand for or apply to an object): but these theories, while subtle and complex, never succeeded in giving a universally applicable account.
It is necessary, if Frege is to be understood, to grasp the magnitude of the discovery of the quantifier-variable notation, as thus resolving an age-old problem the failure to solve which had blocked the progress of logic for centuries. […] for this resolved a deep problem, on the resolution of which a vast area of further progress depended, and definitively, so that today we are no longer conscious of the problem of which it was the solution as a philosophical problem at all.

Medieval philosophers got themselves into terrible depths of confusion when trying to deal with these sentences having more than one quantifier. For example, from “for each magnitude, there is a smaller magnitude”, we want to validate “each magnitude is smaller than at least one magnitude” but not “there is a magnitude smaller than every magnitude”. The medievals analysed this in terms of context-dependence of the meanings of quantified terms:

The general phenomenon of a term’s having different references in different contexts was called suppositio (substitution) by medieval logicians. It describes how one has to substitute a term in a sentence based on its meaning—that is, based on the term’s referent. (Wikipedia)

The scholastics specified many different types of substitution, and which operations were legitimate for each; but never progressed beyond a set of ham-fisted, ad-hoc rules.

To show examples, I had to go to modern commentaries of the scholastics, since the actual texts are simply impenetrable.

Swiniarski 1970. Ockham’s Theory of Personal Supposition

Broadie 1993, which is Oxford University Press’ Introduction to medieval logic:

a term covered immediately by a sign of universality, for example, by ‘all’ or ‘every’, has distributive supposition, and one covered mediately by a sign of affirmative universality has merely confused supposition. A term is mediately covered by a given sign if the term comes at the predicate end of a proposition whose subject is immediately covered by the sign. Thirdly, a term covered, whether immediately or mediately, by a sign of negation has confused distributive supposition (hereinafter just ‘distributive supposition’). Thus in the universal negative proposition ‘No man is immortal’, both the subject and the predicate have distributive supposition, and in the particular negative proposition ‘Some man is not a logician’, the predicate has distributive supposition and the subject has determinate supposition. […]

Given the syntactic rules presented earlier for determining the kind of supposition possessed by a given term, it follows that changing the position of a term in a proposition can have an effect on the truth value of that proposition. In:

(10) Every teacher has a pupil

‘pupil’ has merely confused supposition, and consequently the proposition says that this teacher has some pupil or other and that teacher has some pupil or other, and so on for every teacher. But in:

(11) A pupil every teacher has

‘pupil’ has determinate supposition, and since ‘teacher’ has distributive supposition descent must be made first under ‘pupil’ and then under ‘teacher’. Assuming there to be just two teachers and two pupils, the first stage of descent takes us to:

(12) PupiI1 every teacher has or pupil2 every teacher has.

The next stage takes us to:

(13) Pupil1 teacher1 has and pupil1 teacher2 has, or pupil2 teacher1 has and pupil2 teacher2 has.

(13) implies that some one pupil is shared by all the teachers, and that is plainly not implied by (10), though it does imply (10).

In all this talk of supposition, we can discern a flailing attempt to deal with ambiguities of quantifier scope, but these solutions are, of course, hopelessly ad hoc. Not to mention that the rules of supposition were in flux, and their precise content is still a matter of debate among specialists of Scholasticism².

And now just for fun, a representative passage from Ockham:

And therefore the following rule can be given concerning negations of this kind : Negation which mobilizes what is immobile, immobilizes what is mobile, that is, when such a negation precedes a term which supposits determinately it causes the term to supposit in a distributively confused manner, and when it precedes a term suppositing in a distributively confused manner it causes the term to supposit determinately.

According to one commentator (Swiniarski 1970), in this passage “Ockham formulates a De Morgan-like rule concerning the influence of negations which negate an entire proposition and not just one of the terms.” I’ll let you be the judge. For at this point it is I who supposit in a distributively confused manner.

How predicate logic dissolved the confusion

The solution is now familiar to anyone who has studied logic. Wikipedia gives a simple example:

Using modern predicate calculus, we quickly discover that the statement is ambiguous. “Some cat is feared by every mouse” could mean

• For every mouse m, there exists a cat c, such that c is feared by m, i.e. $\forall m (M(m) \rightarrow \exists c (C(c) \land F(m,c)))$

But it could also mean

• there exists one cat c, such that for every mouse m, c is feared by m, i.e. $\exists c (C(c) \land \forall m (M(m) \rightarrow F(m,c))$.

Of course, this is only the simplest case. Predicate logic allows arbitrarily deep nesting of quantifiers, helping us understand sentences which the scholastics could not even have made intuitive sense of, let alone provide a formal semantics for.

Definite descriptions

How people were confused

The problem here is with sentences like “Unicorns have horns” which appear to refer to non-existent objects. People were quite confused about them:

Meinong, an Austrian philosopher active at the turn of the 20th century, believed that since non-existent things could apparently be referred to, they must have some sort of being, which he termed sosein (“being so”). A unicorn and a pegasus are both non-being; yet it’s true that unicorns have horns and pegasi have wings. Thus non-existent things like unicorns, square circles, and golden mountains can have different properties, and must have a ‘being such-and-such’ even though they lack ‘being’ proper. The strangeness of such entities led to this ontological realm being referred to as “Meinong’s jungle”. (Wikipedia)

The delightfully detailed Stanford page on Meinong provides further illustration:

Meinong tries to give a rational account of the seemingly paradoxical sentence “There are objects of which it is true that there are no such objects” by referring to two closely related principles: (1) the “principle of the independence of so-being from being” [“Prinzip der Unabhängigkeit des Soseins vom Sein”], and (2) the “principle of the indifference of the pure object to being” (“principle of the outside-being of the pure object” [“Satz vom Außersein des reinen Gegenstandes”]) (1904b, §3–4). […]

Meinong repeatedly ponders the question of whether outside-being is a further mode of being or just a lack of being (1904b, §4; 1910, §12; 1917, §2; 1978, 153–4, 261, 358–9, 377). He finally interprets outside-being as a borderline case of a kind of being. Every object is prior to its apprehension, i.e., objects are pre-given [vorgegeben] to the mind, and this pre-givenness is due do the (ontological) status of outside-being. If so, the most general determination of so-being is being an object, and the most general determination of being is outside-being. The concept of an object cannot be defined in terms of a qualified genus and differentia. It does not have a negative counterpart, and correlatively outside-being does not seem to have a negation either (1921, Section 2 B, 102–7).

In fact, as John P. Burgess writes:

as Scott Soames reveals, in his Philosophical Analysis in the Twentieth Century, volume I: The Dawn of Analysis, Russell himself had briefly held a similar view [to Meinong’s]. It was through the development of his theory of descriptions that Russell was able to free himself from anything like commitment to Meinongian “objects.”

How predicate logic dissolved the confusion

Russell’s On denoting, as the most famous case of a solved philosophical problem, needs no introduction. (Wikipedia gives a good treatment, and so does Sider’s Logic for Philosophy, section 5.3.3.)

Russell’s analysis of definite descriptions could have stood on its own as a success story. The tools of predicate logic were not, strictly speaking, necessary to discover the two possible interpretations of empty definite descriptions. In fact it may seem surprising that no-one made this discovery earlier. But as literate people of the 21st century, it can be hard for us to imagine the intellectual poverty of a world without predicate logic. So we must not be too haughty. The most likely conclusion, it seems to me, is that Russell’s insight was, in fact, very difficult to achieve without the precision afforded by Frege’s logic.

The epsilon-delta definition of a limit

How people were confused

As Wikipedia writes:

The need for the concept of a limit came into force in the 17th century when Pierre de Fermat attempted to find the slope of the tangent line at a point $x$ of a function such as $f(x)=x^{2}$. Using a non-zero, but almost zero quantity, $E$, Fermat performed the following calculation:

$$\begin{aligned} slope &= \frac{f(x+E)-f(x)}{E} \\ &= \frac{x^2 +2xE + E^2 -x^2}{E} \\ &= 2x + E \\ &= 2x \end{aligned}$$

The key to the above calculation is that since $E$ is non-zero one can divide $f(x+E)-f(x)$ by $E$, but since $E$ is close to $0$, $2x+E$ is essentially $2x$. Quantities such as $E$ are called infinitesimals. The problem with this calculation is that mathematicians of the era were unable to rigorously define a quantity with properties of $E$ although it was common practice to ‘neglect’ higher power infinitesimals and this seemed to yield correct results.

SEP states:

Infinitesimals, differentials, evanescent quantities and the like coursed through the veins of the calculus throughout the 18th century. Although nebulous—even logically suspect—these concepts provided, faute de mieux, the tools for deriving the great wealth of results the calculus had made possible. And while, with the notable exception of Euler, many 18th century mathematicians were ill-at-ease with the infinitesimal, they would not risk killing the goose laying such a wealth of golden mathematical eggs. Accordingly they refrained, in the main, from destructive criticism of the ideas underlying the calculus. Philosophers, however, were not fettered by such constraints. […]

Berkeley’s arguments are directed chiefly against the Newtonian fluxional calculus. Typical of his objections is that in attempting to avoid infinitesimals by the employment of such devices as evanescent quantities and prime and ultimate ratios Newton has in fact violated the law of noncontradiction by first subjecting a quantity to an increment and then setting the increment to 0, that is, denying that an increment had ever been present. As for fluxions and evanescent increments themselves, Berkeley has this to say:

And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

Kline 1972 also tells us:

Up to about 1650 no one believed that the length of a curve could equal exactly the length of a line. In fact, in the second book of La Geometrie, Descartes says the relation between curved lines and straight lines is not nor ever can be known.

How predicate logic dissolved the confusion

Simply let $f$ be a real-valued function defined on $\mathbb{R}$. Let $c$ and $L$ be real numbers. We can rigorously define a limit as:

From this it’s easy to define the slope as the limit of a rate of increase, to define continuity, and so on.

Note that there are two nested quantifiers here, and an implication sign. When we remind ourselves how much confusion just one nested quantifier caused ante-Frege, it’s not surprising that this new definition was not discovered prior to the advent of predicate logic.

On the connection between the analysis of definite descriptions and that of limit

John P. Burgess, in The Princeton companion to mathematics, elaborates on the conceptual link between these two success stories:

[Definite descriptions] illustrate in miniature two lessons: first, that the logical form of a statement may differ significantly from its grammatical form, and that recognition of this difference may be the key to solving or dissolving a philosophical problem; second, that the correct logical analysis of a word or phrase may involve an explanation not of what that word or phrase taken by itself means, but rather of what whole sentences containing the word or phrase mean. Such an explanation is what is meant by a contextual definition: a definition that does not provide an analysis of the word or phrase standing alone, but rather provides an analysis of contexts in which it appears.

In the course of the nineteenth-century rigorization, the infinitesimals were banished: what was provided was not a direct explanation of the meaning of $df (x)$ or $dx$, taken separately, but rather an explanation of the meaning of contexts containing such expressions, taken as wholes. The apparent form of $df (x)/dx$ as a quotient of infinitesimals $df (x)$ and $dx$ was explained away, the true form being $(d/dx)f (x)$, indicating the application of an operation of differentiation $d/dx$ applied to a function $f (x)$.

“That depends what the meaning of ‘is’ is”

Bill Clinton’s quote has become infamous, but he’s got a point. There are at least four meanings of ‘is’. They can bec clearly distinguished using predicate logic.

Hintikka ‎2004:

Perhaps the most conspicuous feature that distinguishes our contemporary « modern » logic created by Frege, Peirce, Russell and Hilbert from its predecessors is the assumption that verbs for being are ambiguous between the is of predication (the copula), the is of existence, the is of identity, and the is of subsumption. This assumption will be called the Frege-Russell ambiguity thesis. This ambiguity thesis is built into the usual notation of first-order logic and more generally into the usual notation for quantifiers of any order, in that the allegedly different meanings of verbs like « is » are expressed differently in it. The is of existence is expressed by the existential quantifier (\exists x), the is of predication by juxtaposition (or, more accurately speaking, by a singular term’s filling the argument slot of a predicative expression), the is of identity by = , and the is of subsumption by a general conditional.

Philosophical problems are never solved for the same reason that treasonous conspiracies never succeed: as successful conspiracies are never called “treason,” so solved problems are no longer called “philosophy.”

— John P. Burgess

Contents

1. The consequences of defeatism
2. My approach
   1. Identifiable successes
   2. From confusion to consensus
   3. No mere disproofs
3. Successes: my list so far
4. Related posts

In this new series of essays, I aim to collect some concrete examples of success stories of philosophy (more below on quite what I mean by that). This is the introductory chapter in the series, where I describe why and how I embarked on this project.

Most academic disciplines love to dwell on their achievements. Economists will not hesitate to tell you that the welfare theorems, or the understanding of comparative advantage, were amazing achievements. (In Economics rules Dani Rodrik explicitly talks about the “crown jewels” of the discipline). Biology has the Nobel Prize to celebrate its prowess, and all textbooks duly genuflect to Watson and Crick and other heroes. Physics and Mathematics are so successful that they needn’t brag for their breakthroughs to be widely admired. Psychologists celebrate Kahneman, linguists Chomsky.

Philosophy, on the other hand, like a persecuted child that begins to internalise its bullies’ taunts, has developed an unfortunate inferiority complex. As if to pre-empt those of the ilk of Stephen Hawking, who infamously pronocuned philosophy dead, philosophers are often the first to say that their discipline has made no progress in 3000 years. Russell himself said in The Problems of Philosophy:

Philosophy is to be studied not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves.

This view is very much alive today, as in Van Iwagen (2003):

Disagreement in philosophy is pervasive and irresoluble. There is almost no thesis in philosophy about which philosophers agree.

Among some writers, one even finds a sort of perverse pride that some topic is “one of philosophy’s oldest questions” and “has been discussed by great thinkers for 2000 years”, as if this were a point in its favour.

The consequences of defeatism

This state of affairs would be of no great concern if the stakes were those of a mere academic pissing contest. But this defeatism about progress has real consequences about how the discipline is taught.

The first is history-worship. A well-educated teenager born this century would not commit the fallacies that litter the writings of the greats. The first sentence of Nicomachean Ethics is a basic quantificational fallacy. Kant’s response to the case of the inquiring murderer is an outrageous howler. Yet philosophy has a bizarre obsession with its past. In order to teach pre-modern texts with a straight face, philosophers are forced to stretch the principle of charity beyond recognition, and to retrofit newer arguments onto the fallacies of old. As Dustin Locke writes here, “The principle of charity has created the impression that there is no progress in philosophy by preserving what appear to be the arguments and theories of the great thinkers in history. However, what are being preserved are often clearly not the actual positions of those thinkers. Rather, they are mutated, anachronistic, and frankensteinian reconstructions of those positions.” Much time is wasted subjecting students to this sordid game, and many, I’m sure, turn their backs on philosophy as a result.

The second, related consequence is the absence of textbooks. No one would dream of teaching classical mechanics out of Principia or geometry out of Euclid’s Elements. Yet this is what philosophy departments do. Even Oxford’s Knowledge and Reality, which is comparatively forward-looking, has students read from original academic papers, some as old as the 1950s, as you can see here. It’s just silly to learn about counterfactuals and causation from Lewis 1973 (forty-four years ago!). Thankfully, there is the Stanford Encyclopedia, but it’s incomplete and often pitched at too high a level for beginners. And even if Stanford can be counted as a sort of textbook, why just one? There should be hundreds of textbooks, all competing for attention by the clarity and precision of their explanations. That’s what happens for any scientific topic taught at the undergraduate level.

My approach

Identifiable successes

In this series, I want to focus on succcess stories that are as atomic, clear-cut, and precise as possible. In the words of Russell:

Modern analytical empiricism […] differs from that of Locke, Berkeley, and Hume by its incorporation of mathematics and its development of a powerful logical technique. It is thus able, in regard to certain problems, to achieve definite answers, which have the quality of science rather than of philosophy. It has the advantage, in comparison with the philosophies of the system-builders, of being able to tackle its problems one at a time, instead of having to invent at one stroke a block theory of the whole universe. Its methods, in this respect, resemble those of science.

Some of the greatest philosophical developments of the modern era, both intellectually speaking and social-impact wise, were not of this clear-cut kind. Two examples seem particularly momentous:

• The triumph of naturalism, the defeat of theism, and the rise of science a.k.a “natural philosophy”.
• The expanding circle of moral consideration: to women, children, those of other races, and, to some extent, to non-human animals. (See Pinker for an extended discussion).

These changes are difficult to pin down to a specific success story. They are cases of society’s worldview shifting wholesale, over the course of centuries. With works such as Novum Organum or On the Subjection of Women, philosophising per se undoubtedly deserves a share of the credit. Yet the causality may also run the other way, from societal circumstances to ideas; technological and political developments surely had their role to play, too.

Instead I want to focus on smaller, but hopefully still significant success stories, whose causal story should hopefully be easier to extricate.

From confusion to consensus

The successes need to be actual successes of the discipline, not just theories I think are successful. For example, consequentialism or eliminativism about caustion don’t count, since there is considerable debate about them still¹. Philosophers being a contrarian bunch, I won’t require complete unanimity either, but rather a wide consensus, perhaps something like over 80% agreement among academics at analytic departments.

Relatedly, there needs to have been actual debate and/or confusion about the topic, previous to the success story. This is often the hardest desideratum to intuitively accept, since philosophical problems, once solved, tend to seem puzzlingly unproblematic. We think “How could people possibly have been confused by that?”, and we are hesitant to attribute basic misunderstandings to great thinkers of the past. I will therefore take pains to demonstrate, with detailed quotes, how each problem used to cause real confusion.

No mere disproofs

In order to make the cases I present as strong as possible, I will adopt a narrow definition of success. Merely showing the fallacies of past thinkers does not count. Philosophy has often been able to conclusively restrict the space of possible answers by identifying certain positions as clearly wrong. For example, no-one accepts Mill’s “proof” of utilitarianism as stated, or Anselm’s ontological argument. And that is surely a kind of progress², but I don’t want to rely on that here. When physics solved classical mechanics, it did not just point out that Aristotle had been wrong, rather it identified an extremely small area of possibility-space as the correct one. That is the level of success we want to be gunning for here. For the same reason, I also won’t count coming up with new problems, such as Goodman’s New Riddle of Induction, as progress for my purposes.

Successes: my list so far

Here are the individual success stories, in no particular order:

1. Predicate logic: arguably launched analytic philosophy, clarified ambiguities that had held back logic for centuries
2. Computability: a rare example of an undisputed, non-trivial conceptual analysis
3. Modal logic and its possible world semantics: fully clarified the distinction between sense and reference, dissolved long-standing debates arising from modal fallacies.
4. The formalisation of probability: how should we reason about unsure things? Before the 1650s, everyone from Plato onwards got this wrong.
5. Bayesianism: the analysis of epistemic rationality and the solution to (most of) philosophy of science.
6. Compatibilism about free will (forthcoming)

It’s very important to see these five stories as illustrations of what success looks like in philosophy. The list is not meant to be exhaustive. Nor are all five stories supposed to follow the same pattern of discovery; on the contrary, they are examples of different kinds of progress.

Related posts

These posts don’t describe success stories, but are related:

• What success in philosophy sometimes looks like
• The failures and successes of conceptual analysis

In online discussions, the number of upvotes or likes a contribution receives is often highly correlated with the social status of the author within that community. This makes the community less epistemically diverse, and can contribute to feelings of groupthink or hero worship.

Yet both the author of a contribution and its degree of support contain bayesian evidence about its value. If the author is a widely respected expert, the amount of evidence is arguably so large that it should overwhelm your own inside view.

We want each individual to invest the socially optimal amount of resources into critically evaluating other people’s writing (which is higher than the amount that would be optimal for individual epistemic rationality). Yet we also all and each want to give sufficient weight to authority in forming our all-things-considered views.

As Greg Lewis writes:

The distinction between ‘credence by my lights’ versus ‘credence all things considered’ allows the best of both worlds. One can say ‘by my lights, P’s credence is X’ yet at the same time ‘all things considered though, I take P’s credence to be Y’. One can form one’s own model of P, think the experts are wrong about P, and marshall evidence and arguments for why you are right and they are wrong; yet soberly realise that the chances are you are more likely mistaken; yet also think this effort is nonetheless valuable because even if one is most likely heading down a dead-end, the corporate efforts of people like you promises a good chance of someone finding a better path.

Full blinding to usernames and upvote counts is great for critical thinking. If all you see is the object level, you can’t be biased by anything else. The downside is you lose a lot of relevant information. A second downside is that anonymity reduces the selfish incentives to produce good content (we socially reward high-quality, civil discussion, and punish rudeness.)

I have a suggestion for capturing (some of) the best of both worlds:

• first, do all your reading, thinking, upvoting and commenting with full blinding
• once you have finished, un-blind yourself and use the new information to
  • form your all-things-considered view of the topic at hand
  • update your opinion of the people involved in the discussion (for example, if someone was a jerk, you lower your opinion of them).

To enable this, there are now two user scripts which hide usernames and upvote counts on (1) the EA forum and (2) LessWrong 2.0. You’ll need to install the Stylish browser extension to use them.