Illuminated Manuscript of Aristotle, Averroes, and Ramon Llull Charging the Tower of Falsehood

Jonathan Greig (LMU Munich) posted the picture above to Twitter the other day, crediting Laura Castelli with finding it. It’s from a 14th Century illuminated manuscript by Thomas Le Myésier, Breviculum ex artibus Raimundi Lulli electum, and depicts Aristotle, Averroes, and Ramon Llull leading an army charging the Tower of Falsehood. I put a full resolution version here. It’s really amazing.

Here’s the (Google translated, too lazy to thoroughly revise, maybe I’ll get back to it if anyone needs me to) description from the university library at the University of Freiburg  (complete German original):

Miniature VI shows the army of Aristotle, which is advancing to destroy the tower of untruth, together with the commentator of Aristotle, Averroes (Exercitus Aristotillis ad destruendum turrim falsitatis cum suo commentatore). The tower of untruth is occupied by the messengers of untruth: left wickedness, inactivity, ignorance, weakness, confusion, fall, futility, nothingness; Right smallness, impossibility, hatred, untruth, punishment, contrast, emptiness, inflexibility, abundance, diminution (malitia, cessatio, ignorantia, debilitas, confusio, casus, frustra, nihil and parvitas, impossibilitas, odiositas, falsitas, poena, contrarietas, vacuum, difformitas, superfluum, diminutum). To the left of the tower, at the vanguard, the goal is stated: to take down the tower by destruction or distinction (Per interemptionem aut per distinctionem oportet dissolvere turrim). The shield affirms credible reasoning, the text above above the archer, excellent proof (Probabiliter arguo, Potissime demonstro).  The horse of Aristotle is rational reasoning (ratiocinatio), his lance represents “instruments abundunt in syllogisms”, the banner mentions the methods: consideration of the similar, exploring differences, use of proposition, distinction of diversity (consideratio similitudinum, inventio differentiarum , Sumptio propositionum, multiplicis distinctio). In the chariot, the five predicates or general statements of logic are found in the front: general genus, special kind, universal difference, peculiarity, accidental property (genus generalissimum, species specialissima, differentia generalis, proprietas, accidens). Behind this are the ten categories of ontology, ten simple principles of things (Decem rerum principia incomplexa). The text beside the lance of substance specifies “by itself, originally, first, by virtue of itself: per se, principaliter, primo, propter se subsistens.” The rest of the categories are characterized summarily on the banner by the fact that they are not in themselves, but are the substance in itself (Non sumus propter nos, sed ut sit substantia, see Ideo, quia ab ea dependemus, sibimet inhaeremus).

The following banner is carried by Averros riding on his horse, which represents the imagination (imaginatio). The principles of his philosophy of nature are listed on the banner: purpose, effect, form, material, deficiency (finis, efficiens, forma, materia, privatio). The three texts next to his lance read: To be perfect in speculation and to train in them is the highest happiness (Esse perfectum in speculativis and in ice exerceri summa est felicitas); The faith of the heretic Averroes is in every law (Fides Averrois haeretici in omni lege); The next to the lance of the first warrior it reads: body in its quantity, movement, time, external appearance, place, natural observation (Corpus quantum, motus, tempus, superficies , Locus, consideratio naturalis), next to that of the second, the Aristotelian saying (according to Metaphysics, 993b): just as the eye of the night owl is to sunlight, our intellect is related to what is evident in its nature. The Pope with a cross in his hands and the abbreviated “Te Deum” text, a bishop with a prayer (Deus misereatur nostri et benedicat nobis) and below a cardinal, restraining Averroes, with the texts: “Because the phenomena can not exceed the physical nature, your intellect is obscured for what is recognizable in the purely spiritual, Averroes! So that you do not lead us into temptation, we will curb your course, for it is a sacred duty that, when one must be elected among several friends, the truth is preferable.” The banner reads,”Socrates is a friend, but truth is more a friend” (Socrates amicus, sed magis amica veritas). In the lecture of the cardinal, which follows below, this is further elaborated by referring to the limits of Averroistic philosophy and the spiritual power of the Church with the pope, episcopate, clergy, religious and theologians. He concludes with the sentence: “Nevertheless, in physical and metaphysical truths, they allow you to advance on the tower of untruth, to destroy it together with others who wish to free truth from the dungeon of untruth with the help of God.”

The text at the bottom left of the page is a lamentation of the truth: “In the dungeon of this tower, truth was incarcerated against its nature, and it languishes to be free from all the world and cry, crying and crying horribly:” Have mercy, have mercy With me, at least you, my friends! The hand of ignorance touched me, and in my place the unreliable opinion was crowned in public; I, on the other hand, which I fear from every angle, is entirely hidden from my will in the darkness and without light in the depths of the dungeon. Sad, deserted and almost desperate I die! There is no one to help me or give me comfort. On the contrary, many are more inclined to support the wrong opinion than to free me from the dungeon. All your philosophers, apart from God, I place all my trust in you, because you are true lovers of wisdom and truth, come to my aid, I beg you; Otherwise I must perish by inaction. Oh, Christian lords, how can you bear to be so oppressed by Jews and Saracens that I should fall from the top of the tower, which I should even pass, into the dungeon of this tower of untruth? ”

The miniature VII shows the approaching substitute army of Raimundus for the destruction of the tower of untruth and ignorance (Retrobellum et succursus exercitus domini Raimundi Lul de Maioricis ad corruendum turrim falsitatis et ignorantiae). The three trumpeters symbolize the three forces of the rational soul: recognizing, loving, honoring, referring to God, to the triads, to the Creator and the Savior, each in a permutating order (Deum cognoscamus, diligamus, recolamus, Unum Deum trinum diligamus, recolamus, Cognoscamus, Creatorum nostrum recolamus, cognoscamus, diligamus, nostrum redemptorem). Beside the trumpets are the three soul-forces: reason, memory of the will (intellectus, voluntas, memoria). The addition of another hand in the lower left shows that only one horse is represented here (Deberet esse unus equus tantum). An example of how Thomas Le Myésier inspected the finished images and corrected them in this case.

Lull’s horse bears the name of the right or good intent (recta intentio). The motto next to his lance reads: “He who wants to recognize the spiritual must pass over the senses and the imagination, and often himself” (Intelligent spiritualia oportet sensus et imaginationem transcendere et multotiens seipsum). On the banner, “we love God through the first intention and the greater goal” (Per primam intentionem et miioritatem finis Deum diligimus, Per secundam intentionem et minoritatem finis meritum spectamus). The eighteen principles of the Lullian Ars are recorded in the car: the nine absolute principles: goodness, greatness, duration, power, wisdom, will, virtue, truth, glory, bonitas, magnitudo, duratio, potestas, sapientia, voluntas, virtus, veritas, gloria).

To the absolute follow the nine relative principles: difference, agreement, opposition, beginning, middle, goal, greaterness, equality, minority (differentia, concordantia, contrarietas, principium, medium, finis, maioritas, aequalitas, minoritas); In the latter three the inscriptions and lances are missing. The fire column between the two groups, in reference to Exodus 13: 20-22, could symbolize the presence of God in the battle. The commentary text under the figure is: “Raimundus rides on the horse” good intention “(recta intentio) and follows the cross and the holy Catholic faith.) He sends three trumpeters ahead: the three forces of the reasoning elegans And the Son of God, Jesus Christ, who was crucified, who sought to recognize, love, glorify, and glorify God through His Ars For it is much appreciated by him, lovable, venerable, and worthy of gratitude, which must be our basic intention, a true intention, in contrast to that of Averroes, who did not know the truth, Not because he has disapproved of it as much as he can, but denies eternal life, asserts that the happiness is in the observation that it is perfect in the speculative sciences. He does not turn to the inner activity of God; As well as not of his creative outward activity, not taking care of the fact that every activity is directed towards the goal and the perfection. Neither did he care to recognize the nature of divine dignities or their activities, neither their unity, nor their personal distinction of activity, without which God would forever remain inactive in himself, and without any dignity. Consequently, in his whole nature he would be imperfect and ultimately unworthy to be God. But he himself has revealed himself as the most perfect, simple, uniquely, and purest act recognizing himself; But without one who knows, one can recognize one who is recognizable and the act of cognition, namely, the cognition itself, which can not be recognized by the one who recognizes eternally. Through this activity, we recognize necessarily the personal trinity as a unity in essence. Through their external activity, we recognize the creation of the world and the order of their parts, which God did not create infinite wisdom for no purpose and without goal, but arranged for the greater attainment of the goal. For God and nature do nothing in vain, as even the ancient philosophers and their first confess.”

Aldo Antonelli’s last paper

Aldo Antonelli’s last paper, “Completeness and Decidability of General First-Order Logic (with a Detour Through the Guarded Fragment)” is now out in the most recent issue of the Journal of Philosophical Logic.

This paper investigates the “general” semantics for first-order logic introduced to Antonelli (Review of Symbolic Logic 6(4), 637–58, 2013): a sound and complete axiom system is given, and the satisfiability problem for the general semantics is reduced to the satisfiability of formulas in the Guarded Fragment of Andréka et al. (Journal of Philosophical Logic 27(3):217–274, 1998), thereby showing the former decidable. A truth-tree method is presented in the Appendix.

It is published together with a note on it by Hajnal Andréka, Johan van Benthem, and István Németi in the same issue.

Association for Symbolic Logic at the Pacific APA

The ASL Spring Meeting will take place on Wednesday and Thursday at the Pacific APA in Seattle! Note that the ASL Reception will take place on Thursday, April 13, 5:00–7:00 p.m. There will be snacks and wine!

Here’s the program:


Invited Speaker Session: MODALITY AND MODAL LOGIC
Chair: Audrey Yap

Peter Fritz (Universitetet i Oslo), A philosophical perspective on algebraic models for modal logics.

Fenrong Liu (Tsinghua University), Social epistemic logic.

Tamar Lando (Columbia University), Topology and measure in logics for point-free spaces.


Chair: Valeria de Paiva

Mark van Atten (Centre National de la Recherche Scientifique and Université Paris 4), Intuitionism and impredicativity.

Rosalie Iemhoff (Universiteit Utrecht), Quantifiers and functions in intuitionistic logic.

Joan Rand Moschovakis (Occidental College), Realizable extensions of Brouwer’s analysis.

THURSDAY MORNING, APRIL 13, 9:00 A.M.–12:00 P.M.

Chair: Richard Zach

Joachim Mueller-Theys, Defining and simplifying the second incompleteness theorem.

Valeria de Paiva and Harley Eades III, Dialectica categories for the Lambek calculus.

Ronald Fuller, First-order logic in 13th-century accounting systems.

Rachel Boddy (University of California, Davis), Fruitful definitions.

Michael McGrady, Garbage collection (GC), Gödel numbering, and periodicity in mathematical logic.

Fabio Lampert (University of California, Davis), On the expressive power of propositional two-dimensional modal logic.

Alexei Angelides (University of San Francisco), Weak arithmetics and the bar rule.


Special Session organized by the Committee on Logic Education: INCLUSIVENESS IN LOGIC EDUCATION
Chair: Alexei Angelides

Audrey Yap (University of Victoria), Symbolic logic, accessibility, and accommodation.

Fenrong Liu (Tsinghua University), Experiences and difficulties in teaching logic at Tsinghua University.

Nicole Wyatt (University of Calgary), The Open Logic Textbook.

Maureen Eckert (University of Massachussetts, Dartmouth), The Summer Program for Diversity in Logic: Some reflections.

Panel discusssion.

ASL Reception

New Textbook on Incompleteness

I made a textbook on incompleteness for my Logic III course. See it/read about it over at the Open Logic Project.

CfP: Quantifiers and Determiners (part of ESSLI 2017)

QUAD: QUantifiers And Determiners

Toulouse, Monday  July 17 — Friday July 21:  17:00-18:30

ESSLLI 2017 workshop


deadline for submissions:  17 March 2017
submission website:
notification to authors:  15 April 2017
final version due: 19 May 2017
conference: 17-21 July 2017


The compositional interpretation of determiners relies on quantifiers  — in a general acceptation of this later term which includes generalised quantifiers, generics, definite descriptions i.e. any operation that applies to one or several formulas with a free variable, binds it  and yields a formula or possibly a generic term  (the operator is then called a subnector, following Curry). There is a long history of quantification in the Ancient and Medieval times at the border between logic and philosophy of language, before the proper formalisation of quantification by Frege.

A common solution for natural language semantics is the so-called theory of generalised quantifiers. Quantifiers like « some, exactly two, at most three, the majority of, most of, few, many, … » are all described in terms of functions of two predicates viewed as subsets.

Nevertheless, many mathematical and linguistic questions remain open.

On the mathematical side, little is known about generalised , generalised and vague quantifiers, in particular about their proof theory. On the other hand, even for standard quantifiers, indefinites and definite descriptions, there exist alternative formulations with choice functions and generics or subnectors (Russell’s iota, Hilbert-Bernays, eta, epsilon, tau). The computational aspects of these logical frameworks are also worth studying, both for computational linguistic software and for the modelling of the cognitive processes involved in understanding or producing sentences involving quantifiers.

On the linguistic side, the relation between the syntactic structure and its semantic interpretation, quantifier raising, underspecification, scope issues,…  are not fully satisfactory. Furthermore extension of linguistic studies to various languages have shown how complex quantification is in natural language and its relation to phenomena like generics, plurals,  and mass nouns.

Finally, and this can be seen as a link between formal models of quantification and natural language,  there by now exist psycholinguistic experiments that connect formal models and their computational properties to the actual way human do process sentences with quantifiers, and handle their inherent ambiguity, complexity, and difficulty in understanding.

All those aspects are connected in the didactics of mathematics and computer science: there are specific difficulties to teach (and to learn) how to  understand, manipulate,  produce and  prove quantified statements, and to determine  the proper level of formalisation between bare logical formulas and written or spoken natural language.

This workshop aims at gathering  mathematicians, logicians, linguists, computer scientists  to present their latest advances in the study of quantification.

Among the topics that wil be addressed are the following :

  • new ideas in quantification in mathematical logic, both model theory and proof theory:
    • choice functions,
    • subnectors (Russell’s iota, Hilbert’s epsilon and tau),
    • higher order quantification,
    • quantification in type theory
  • studies of the lexical, syntactic and semantic of quantification in various languages
  • semantics of noun phrases
  • generic noun phrases
  • semantics of plurals and mass nouns
  • experimental study of quantification and generics
  • computational applications of quantification and polarity especially for question-answering.
  • quantification in the didactics of mathematics and computer science.

Some recent relevant references:

  • Anna Szabolcsi Quantification Cambridge University Press 2010
  • Stanley Peters and Dag Westerstahl Quantifiers in Language and Logic Oxford Univ. Press 2010
  • Mark Steedman Taking Scope – The Natural Semantics of Quantifiers MIT Press 2012
  • Jakub Szymanik. Quantifiers and Cognition, Studies in Linguistics and Philosophy, Springer, 2015.
  • Vito Michele Abrusci, Fabio Pasquali, and Christian Retoré. Quantification in ordinary language and proof theory. Philosophia Scientae, 20(1):185–205, 2016.


The program committee is looking for  contributions introducing new viewpoints on quantification and determiners,  the novelty being either in the mathematical logic framework or in the linguistic description  or in the cognitive modelling. Submitting purely original work is not mandatory, but authors should clearly mention that the work is not original, and why they want to present it at this workshop (e.g. new viewpoint on already published results)

Submissions should be

In case the committee thinks it is more appropriate, some papers can be accepted as a poster with a lightning talk.

Final versions of accepted papers may be slightly longer. They will be published on line. We also plan to publish postproceedings

Programme committee:

  • Christian Retoré (Université de Montpellier & LIRMM-CNRS)
  • Mark Steedman (University of Edinburgh)
  • Vito Michele Abrusci (Università di Roma tre)
  • Mathias Baaz (University of Technology, Vienna)
  • Daisukke Bekki (Ochanomizu University, Tokyo)
  • Oliver Bott (Universität Tübingen)
  • Francis Corblin (Université Paris Sorbonne)
  • Martin Hakl (Massachusetts  Institute of Technology, Cambridge MA)
  • Makoto Kanazawa (National Institute of Informatics, Tokyo)
  • Dan Lassiter (Stanford University)
  • Zhaohui Luo (Royal Holloway College, London)
  • Alda Mari (CNRS Institut Jean Nicod, Paris)
  • Wilfried Meyer-Viol (King’s college, London)
  • Michel Parigot (CNRS IRIF, Paris)
  • Anna Szabolcsi (New-York University)
  • Jakub Szymanik (Universiteit van Amsterdam)
  • Dag Westerstahl (Stockholm University)
  • Bruno Woltzenlogel Paleo  (University of Technology, Vienna)
  • Richard Zach (University of Calgary)
  • Roberto Zamparelli (Università di Trento)

Jack Howard Silver, 1942-2016

From the Group in Logic and the Methodology of Science:

It is with great sadness that we announce that Professor Jack Howard Silver died on Thursday, December 22, 2016. Professor Silver was born April 23, 1942 in Missoula, Montana. After earning his A.B. at Montana State University (now the University of Montana) in 1961, he entered graduate school in mathematics at UC Berkeley. His thesis, Some Applications of Model Theory in Set Theory, completed in 1966, was supervised by Robert Vaught. In 1967 he joined the mathematics department at UC Berkeley where he also became a member of the Group in Logic and the Methodology of Science. He quickly rose through the ranks, obtaining promotion to associate professor in 1970 and to full professor in 1975. From 1970 to 1972 he was an Alfred P. Sloan Research Fellow. Silver retired in 2010. At UC Berkeley he advised sixteen students, three of whom were in the Group in Logic (Burgess, Ignjatovich, Zach).


His mathematical interests included set theory, model theory, and proof theory. His production was not extensive but his results were deep. Professor Silver was skeptical of the consistency of ZFC and even of third-order number theory. As Prof. Robert Solovay recently put it: “For at least the last 20 years, Jack was convinced that measurable cardinals (and indeed ZFC) was inconsistent. He strove mightily to prove this. If he had succeeded it would have been the theorem of the century (at least) in set theory.”


He will be greatly missed.

Jack’s contributions to set theory, according to Wikipedia (used under CC-BY-SA):

Silver has made several deep contributions to set theory. In his 1975 paper “On the Singular Cardinals Problem,” he proved that if κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Prior to Silver’s proof, many mathematicians believed that a forcing argument would yield that the negation of the theorem is consistent with ZFC. He introduced the notion of master condition, which became an important tool in forcing proofs involving large cardinals. Silver proved the consistency of Chang’s conjecture using the Silver collapse (which is a variation of the Levy collapse). He proved that, assuming the consistency of a supercompact cardinal, it is possible to construct a model where 2κ++ holds for some measurable cardinal κ. With the introduction of the so-called Silver machines he was able to give a fine structure free proof of Jensen’s covering lemma. He is also credited with discovering Silver indiscernibles and generalizing the notion of a Kurepa tree (called Silver’s Principle). He discovered 0# in his 1966 Ph.D. thesis. Silver’s original work involving large cardinals was perhaps motivated by the goal of showing the inconsistency of an uncountable measurable cardinal; instead he was led to discover indiscernibles in L assuming a measurable cardinal exists.

[Photo ©Steven Givant. Picture taken on the occasion of the Tarski Symposium at UC Berkeley in 1971.]

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Legendary mathematicians: They’re just like us!