On Aristotelian Logic

In Περὶ Ἑρμηνείας Aristoteles defines the basic terms of his logic to be single predicates, each of single subjects see for example the Stanford Encyclopedia of Philosophy

This excludes all predicates of higher arity, e.g. two-place relations. What effects does this restriction imply for the statements that can be logically analysed and discussed?


As every predicate has only one argument, then if it holds at all, it must hold eternally, because there is no possible second parameter to serve as a timestamp or interval of validity.

So statements like Socrates is alive, coffee is ready, the emperor of france is bald are either true or false; and if true at the time when their validity is checked, then they must be true eternally.

The discourses using this logic are restricted to timeless expressions. It is impossible to examine processes.

Unrelatedness and separation

Since only unary relations are allowed, statements such as Socrates is married to Xanthippe are inexpressible. Aristotle is therefore unable to infer the valid Xanthippe is married to Socrates, even if he would somehow know the general rule If A is married to B, then B is married to A, (or “Being married to” is a symmetric relation, which is expressible in his logic).

Subjects can only be treated in isolation. In the resulting ontology they are cut off from each other and from all context by the restrictions of the underlying logic.

Inexpressability of basic mathematical concepts
Euclid's Geometry

Geometry concerns intersections of two (or more) lines, angles formed by two lines, figures enclosed by three or more lines, etc. For example Euclid's fifth postulate begins with if two straight lines lying in the same plane intersect a third [...]. The relation “line A intersects line B” cannot be formed from one-place predicates. Although Aristotle uses these in his examples in Prior Analytics, Posterior Analytics and Meteorologica, he'd be unable to formulate them in his Logic.


The fundamental concept of a function cannot be expressed in Aristotelian logic, because it requires at least two-place relations (“f at x has value f(x)”). So even eternal laws of Physics — like Newtons law of gravity — can (ironically) not be stated with the logic of the author of Φυσικὴ ἀκρόασις.

Equivalence Relations

Since the late 19th century, many mathematical structures are formally defined by equivalence relations on sets or classes see for example The Search for Mathematical Roots, 1870-1940 by Ivor Owen Grattan-Guinness, Princeton Univ. Press ,chapter 5, Paragraph 5.3.6.

These are two-place relations which are

  1. reflexive : for every a: R(a,a)
  2. symmetric : R(a,b) ⇒ R(b,a)
  3. transitive : R(a,b) and R(b,c) ⇒ R(a,c)

For example the elements of the cyclic group of numbers modulo thirteen are defined as the sets of all numbers having the same residue after division by 13. So the thirteen elements are the quotient sets Z/R

  • n : R(n,0) : (..., -13, 0, 13, 26, 39, ...)
  • n : R(n,1) : (..., -12, 1, 14, 27, 40, ...)
  • n : R(n,2) : (..., -11, 2, 15, 28, 41, ...)
  • ..
  • n : R(n,11) : (..., -2, 11, 24, 37, 50, ...)
  • n : R(n,12) : (..., -1, 12, 25, 38, 51, ...)

as defined by this relation.

Since two-place relations cannot be used, most of modern mathematics is outside the possible subjects of Aristotelian logic, although practically all mathematical statements are timeless.

In the light of these findings, the broad adoption of Aristoteles' logic in Western European theology and philosophy could be judged an impediment to reasoning about practically any non-trivial subject.

This may sound harsh, but it was remarked before by much more distinguished writers:

The doctrine of the individual independence of real facts is derived from the notion that the subject-predicate form of statement conveys a truth which is metaphysically ultimate. According to this view, an individual substance with its predicates constitutes the ultimate type of actuality. If there be one individual, the philosophy is monistic; if there be many individuals, the philosophy is pluralistic. With this metaphysical presupposition, the relations between individual substances constitute metaphysical nuisances: there is no place for them. Accordingly — in defiance of the most obvious deliverance of our intuitive 'prejudices' — every respectable philosophy of the subject-predicate type is monistic. The exclusive dominance of the substance-quality metaphysics was enormously promoted by the logical bias of the mediaeval period. It was retarded by the study of Plato and of Aristotle.
A.N.Whitehead, Process and Reality(1929), p. 137


Such investigations show very soon that traditional Aristotelian scholastic logic is quite inadequate for this purpose [of finding a constitutive theory].
Neurath, Carnap and Hahn in Wissenschaftliche Weltauffassung. Der Wiener Kreis (2012), F. Stadler.

Tue, 17 May 2022
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I've got a little list …

At 36C3, Erwin Ernst Steinhammer gave a Talk (in German) on lists of suspects suspected of being gay that the German police collected well before the Nazis' rise to power.
The police claimed that they had those lists just to keep their eye on the milieu.

When the nazis came to power they almost immediately used the lists to find and deport people to concentration camps where most of them were killed.

Ernst's point in the talk is that a benevolent government must take care not to aid a possible future extremist government by compiling lists of their prospective victims.

This reminded me of the Holocaust documentation center in Oslo. It has a fantastic piece of art just beside the entrance: a gigantic Hollerith punchcard.
It is an art installation by Arnold Dreyblatt based on work by William Selzer on genocide and statistics.

Deutsche Hollerith-Maschinen Gesellschaft Ad from the forties

What the Germans did after the occupation and installment of a puppet-regime under Quisling was to conduct a census of the Norwegian population. And they used DeHoMAG, that is, IBM equipment.

(The Nazis did not need to search the census data to find, deport and kill Norwegian jews — jewish religious organisations provided the Nazis with membership lists.)

The punchcards from the census became interesting when the Germans were looking for young Norwegian men to conscript into labour service or the army.
And this is where history gets (even more) exiting. The Norwegian resistance knew about the German plans and the punchcard technology. They tried to destroy the database i.e. the collection of punchcards, but failed. They then proceeded to sabotage all of the IBM 405 tabulating machines in Norway.

The story of this fight against the abuse of census data is told by Oslo university's Jon Bing in Protecting personal data in wartime: The destruction of the alphabetic tabulators in Oslo.

Dreyblatt's artwork has its own book Innocent Questions with texts by Willam Selzer on the data gathering and subsequent analysis that typically preceeds genocides.

Thu, 12 Nov 2020
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Sparql, Smarql, Omsk and Tomsk

Task: Use Wikidata to find all Russian or Ukrainian or Belorussian towns/cities/... with names ending in "sk" (old Tom Lehrer reference:
I have a friend in Minsk
who has a friend in Pinsk
whose friend in Omsk
has friend in Tomsk
with friend in Akmolinsk...

Turns out not to be so easy:
not every city (Q7930989) is a human settlement (Q486972), so take the UNION.

"Located in country" (P17) seemed the only reasonable relation of cities and countries, so iterate that over Russia (Q159), Belarussia (Q184) and Ukraina (Q212) but it turns out that at least Omsk (Q898) is not in the country of Russia (Q159) but instead is an "instance of" (P31) an "administrative territorial entity of Russia" (Q192287), so UNION over "administrative territorial entities" of Russia and Ukrainia.

Turns out that there is an "administrative territorial entity of Crimea", so who knows what other non-orthogonal classifications people used to enter ex-Soviet cities into WikiData. Not exactly promising.

  { ?city wdt:P31 wd:Q486972. }
  { ?city wdt:P31 wd:Q7930989. }
  { ?city wdt:P17 wd:Q159. }
  { ?city wdt:P17 wd:Q212. }
  { ?city wdt:P17 wd:Q184. }
  { ?city wdt:P31 wd:Q192287. }
  { ?city wdt:P31 wd:Q203323. }
  ?city rdfs:label ?cityLabel.  FILTER( LANG(?cityLabel)="en" )

  FILTER(STRENDS(?cityLabel, "sk"))

(Click to run the query)

UPDATE: Omsk (Q898) is infact "in the country" (P17) of Russia (Q159), my oversight.
UPDATE: FILTER( LANG(?cityLabel)="en" ) is much wiser than to use the translation SERVICE.

Mon, 21 Jan 2019
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Why I like .onions

TOR's hidden services are an extremely cool feature.

Not because people can hide their illicit websites (the Warez community managed to do that decades before), but for other reasons:

Firstly, .onion addresses name services, not host interfaces. Tying interface addresses of hosts to names and re-using them in URLs to point at services is a misdesign which leads to such kludges as the Server header in HTTP/1.1 where the application transmits which name it was using when initally connecting to the service. So URLs map services to hostnames which map to IP addresses which have interfaces which have bound services which get the unresolved names again on the application layer to find out which service was actually addressed. This makes it very complicated to move a service without fiddeling with DNS. An .onion name does not have to ultimately resolve to a globally visible interface address. Instead it identifies the tunnel-entry for a service which can be moved from machine to machine as long as the hidden_service configuration is carried along.

Secondly, .onion addresses deliver what https URLs failed to, namely mapping public keys to services uniquely. There are no multi-rooted hierachies of CAs behind the name-to-key bindings, no obscure ASN.1 based certificate schemes. An .onion address uniquely and automatically identifies the service with the public/secret key pair involved in the key exchange. There has been at least one attempt to build something similiar into IPv6 addresses (RFC 3972), but implementations are either missing or hidden in the darknet.
And because connections inside the tor network are always encrypted, one could even safely run a telnet daemon inside a hidden service.

As a result of Secondly, Thirdly, .onion addresses are a barrier-free global namespace, without absurd fees charged for bits in config-files, trademark disputes and the like.

I run at least one hidden service on each relevant machine to provide a MITM-safe entry point to services.

Tue, 20 Dec 2016
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Postscript Fibonacci Squares and Logarithmic Spiral

Fibonacci Squares and Logarithmic Spiral

This relatively small piece of PostScript code draws Fibonacci Squares and a Logarithmic Spiral.

Sat, 03 Sep 2016
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Blue eyed naïvité in leading economist

The Bank for International Settlements (BIS) hosts an annual conference that brings together central bank governors, leading academics and former public officials to exchange views.

At the 13th Annual Conference a research paper was presented by Bengt Holmstrom of the MIT titled Understanding the role of debt in the financial system on the mechanisms of the financial market in connection with the 2007-now crash.

The paper includes the following fascinating statements:

[..] But it is hard to believe that investment bankers would be colluding to defraud investors [by issuing opaque securities].

Probably as hard to believe as that investment bankers would be colluding to defraud investors by manipulating the London interbank offered rate (they did). Or by manipulating foreign exchange rates (they did). Or by manipulating the ISDAfix Interest Rate Derivative Index (they did).

Or that a publicly held, international corporation would massively invest in the expansion of the Auschwitz concentration camp (they did) .
An economic reality is that white-collar crime has a higher Return on Investment than most legal activities, so the imperative of increasing profits enforces criminal behaviour, specially when the the rate of detection+prosecution+conviction is near zero.

And shortly after:

[..] But it equally hard to believe that hard-nosed profit-hungry investment bankers and traders would be ignorant out of ignorance.

The crash did not harm the profits of aforementioned bankers and traders at all. So there is no incentive to smarten up (Holmstrom is supposed to be an expert on incentives).

Later we read:

Invoking the empirical sucess of the EMH [Efficient Market Hypothesis] (in a variant they call relative EMH), Gilson and Kraakman (2014) among others have advocated [...]

What the crash of 2007 very empirically proved was the failure of the Efficient Market Hypothesis. If prices reflect all available information, and still fluctuate by more than 50 percent in a single day, then that reflecting property is worthless.

That leading academics show such naïvité at the motivations of criminals and cling to unrealistic assumptions is just depressing.

The paper goes on to show that collateral-backed debt is an extremely stable investment, and information-insensitive (because 1. it is backed and 2. the debtor might recover before the debt contract ends). Having more transparent collateralisation, Holstrom argues, would affect the traders' belief system as to the value of the lending bank, thereby endangering the stability of banks, which is posited as a common good. In other words:

  1. market participants are not rational, they have “belief systems”

  2. market efficiency is bad for market participants

  3. market efficiency is not a necessity, it can easily be avoided by publishing less information

This reasoning could be called anti-circular, and I'd suspect that there is no other field of academics where conclusions negate the premises used to draw the conclusions.

Ernst-Ludwig von Thadden's attached commentary at the end of the paper shows some hope, as he points out the aspect of time (mostly ignored by economists, because differential equations are just too hard) in the handling of debt, i.e., debt based vehicles rely on a rollover of short-term debt over time. So they're not so risk-free over a longer term.

Wed, 17 Feb 2016
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Advanced German for Pirates!

Study this to improve your piratical German.

Fri, 20 Dec 2013
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Deutsch für extrem weit Fortgetretene

Bewurstlosigkeit :
Like unconsciousness, but without the sausages.

Tue, 04 Oct 2011
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