# Modal Logic! Propositional Logic! Tableaux!

Lots of new stuff in the Open Logic repository! I’m teaching modal logic this term, and my ambitious goal is to have, by the end of term or soon thereafter, another nicely organized and typeset open textbook on modal logic. The working title is Boxes and Diamonds, and you can check out what’s there so far on the builds site. This project of course required new material on modal logic.  So far this consists in revised and expanded notes by our dear late colleague Aldo Antonelli. These now live in `content/normal-modal-logic` and cover relational models for normal modal logics, frame correspondence, derivations, canonical models, and filtrations. So that’s one big exciting addition. Since the OLP didn’t cover propositional logic separately, I just now added that part as well so I can include it as review chapters. There’s a short chapter on truth-value semantics in `propositional-logic/syntax-and-semantics`. However, all the proof systems and completeness for them are covered as well. I didn’t write anything new for those, but rather made the respective sections for first-order logic flexible. OLP now has an `FOL` “tag”: if `FOL` is set to true, and you compile the chapter on the sequent calculus, say, you get the full first-order version with soundness proved relative to first-order structures. If `FOL` is set to false, the rules for the quantifiers and identity are omitted, and soundness is proved relative to propositional valuations. The same goes for the completeness theorem: with `FOL` set to false, it leaves out the Henkin construction and constructs a valuation from a complete consistent set rather than a term model from a saturated complete consistent set. This works fine if you need only one or the other; if you want both, you’ll currently get a lot of repetition. I hope to add code so that you can first compile without `FOL` then with, and the second pass will refer to the text produced by the first pass rather than do everything from scratch. You can compare the two versions in the complete PDF. Proofs systems for modal logics are tricky; and many systems don’t have nice, say, natural deduction systems. The tableau method, however, works very nicely and uniformly. The OLP didn’t have a chapter on tableaux, so this motivated me to add that as well. Tableaux are also often covered in intro logic courses (often called “truth trees”), so having them as a proof system included has the added advantage of tying in better with introductory logic material. I opted for prefixed tableaux (true and false are explicitly labelled, rather than implicit in negated and unnegated formulas), since that lends itself more easily to a comparison with the sequent calculus, but also because it extends directly to many-valued logics. The material on tableaux lives in `first-order-logic/tableaux`. Thanks to Clea Rees for the the `prooftrees` package, which made it much easier to typeset the tableaux, and to Alex Kocurek for his tips on doing modal diagrams in Tikz.

# ISSOTL 2017 Presentation of Student-Oriented Logic Course

Aaron had a poster presentation at last week’s ISSOTL conference in Calgary, presenting the results of our evaluation of his intro logic course using some novel delivery techniques and the Calgary remix of forall x. issotl-poster Download the poster here.

# Logic Courseware?

Kit Fine asked me for suggestions of online logic materials that have some interactive component, i.e., ways for students to build truth-tables, evaluate arguments, translate sentences, build models, and do derivations; ideally it would not just provide feedback to the student but also grade problems and tests. There is of course Barwise & Etchemendy’s Language, Proof, and Logic, which comes with software to do these things very well and also has a grading service. But are there things that are free, preferably online, preferably open source?

• First we have David Kaplan’s Logic 2010. It’s written in Java, runs on Windows and Mac, is free but not open source, and has a free online grading component. It goes with Terry Parson’s An Exposition of Symbolic Logic, which is also free. To use the software and grading service, you’d have to make arrangements with David. The text does propositional and first-order logic including models and Kalish-Montague derivations. I haven’t tried the software, but it’s used in a number of places.
[Free software Free book Online ✗ Open source ✗]
• UPDATE: Carnap is an open source framework for writing webapps for teaching logic written by Graham Leach-Krouse and Jake Ehrlich. It comes with a (free, but not openly licensed) online book, and currently can check truth tables, translations, and Kalish-Montague derivations (and they are working on first-order models). Students can have accounts and submit exercises. The software is written in Haskell and is open-source (see Github). It’s used at Kansas Sate and the University of Birmingham.
[Free software Free book Online Open source ]
• Kevin Klement is teaching logic from the (free) book by Hardegree, Symbolic Logic: A First Course. (There’s a newer version that doesn’t seem to be freely available.) He has an online component (exercises and practice exams) with multiple-choice questions, truth tables, translations, and Fitch-style derivations. I’m not sure if the backend code for all of this is available and could be adapted to your own needs. He has provided a version of the proof checker that works with the Cambridge and Calgary versions of forall x, and that code is open source, however. I’m not sure if it’s possible to add the functionality he has on the UMass site for saving student work. Neither the book nor the online exercises cover models for first-order logic.
[Free software ✓ Free book ✓ Online ✓ Open source ?]
• The Logic Daemon by Colin Allen and Chris Menzel accompanies Allen and Michael Hand’s Logic Primer. It can check truth-tables, models, and Suppes-Lemmon derivations, and generate quizzes. The interface is basic but the functionality is extensive. There doesn’t seem to be a grading option, however. Software seems to be written in Perl, I didn’t see the source code available.
[Free software ✓ Free book ✗ Online ✓ Open source ✗]
• Then there is Ray Jennings and Nicole Friedrich’s Project Ara, which includes Simon, a logic tutor, and Simon Says, a grading program. The textbook is Proof and Consequence, published by Broadview (ie, not free). It does truth-tables, translations, and Suppes-style derivations, and also no models. It requires installing software on your own computer, but it’s free and runs on Windows, Mac, and Linux. The software is free but not open source. I haven’t tried it out. (That website though!)
[Free software ✓ Free book ✗ Online ✗ Open source ✗]
• Wilfried Sieg’s group has developed AProS, which includes proof and counterexample construction tools. I don’t think these are openly available, however. It’s used in Logic & Proofs, offered through CMU’s Open Learning Initiative. According to the description, it’s available both as a self-paced course and for other academic institutions to use for a self-paced format or for a traditional course with computer support. Not sure what the conditions are, whether it’s free or not, and have inspected neither the texts nor tried out the software.
[Free software ? Free book ? Online ✗ Open source ✗]

Do you know of anything else that could be used to teach a course with an online or electronic component? Any experience with the options above?

# Graphing Survey Responses

As I reported last year, we’ve been running surveys in our classes that use open logic textbooks. We now have another year of data, and I’ve figured out R well enough to plot the results. Perhaps someone else is in a similar situation, so I’ve written down all the steps. Results aren’t perfect yet. All the data and code is on Github, and any new discoveries I make will be updated there.

What follows is the content of the HOWTO:

As part of two Taylor Institute Teaching & Learning Grants, we developed course materials for use in Calgary’s Logic I and Logic II courses. In the case of Logic I, we also experimented with partially flipping the course. One of the requirements of the grants was to evaluate the effectiveness of the materials and interventions. To evaluate the textbooks, we ran a survey in the courses using the textbooks, and in a number of other courses that used commercial textbooks. These surveys were administered through SurveyMonkey. To evaluate the teaching interventions, we designed a special course evaluation instrument that included a number of general questions with Likert responses. The evaluation was done on paper, and the responses to the Likert questions were entered into a spreadsheet.

In order to generate nice plots of the results, we use R. This documents the steps taken to do this.

## Installing R, RStudio, and `likert`

We’re running RStudio, a free GUI frontend to R. In order to install R on Ubuntu Linux, we followed the instructions here, updated for zesty:

• Start “Software & Updates”, select add a source, enter the line
``````http://cran.rstudio.com/bin/linux/ubuntu zesty/
``````

Then in the command line:

``````\$ sudo apt-get install r-base r-base-dev
``````
• We then installed RStudio using the package provided here. The R packages for analyzing Likert data and plotting them require `devtools`, which we installed following the instructions here:
``````\$ sudo apt-get install build-essential libcurl4-gnutls-dev libxml2-dev libssl-dev
\$ R
> install.packages('devtools')
``````
• Now you can install the `likert` package from Github:
``> install_github('likert', 'jbryer')``

## Preparing the data

The source data comes in CSV files, `teachingevals.csv` for the teaching evaluation responses, and `textbooksurvey.csv` for the textbook survey responses.

Since we entered the teaching evaluation responses manually, it was relatively simple to provide them in a format usable by R. Columns are `Respondent ID` for a unique identifier, `Gender` (`M` for male, `F` for female, `O` for other), `Major`, `Year`, `Q1` through `Q9` for the nine Likert questions. For each question, a response of one of `Strongly Agree`, `Agree`, `Neutral`, `Disagree`, or `Strongly Disagree` is recorded.

For the textbook survey we collected a whole lot of responses more, and the data SurveyMonkey provided came in a format not directly usable by R. We converted it to a more suitable format by hand.

• SurveyMonkey results have two header lines, the first being the question, the second being the possible responses in multiple-response questions. We have to delete the second line. For instance, a question may have five different possible responses, which correspond to five columns. If a box was checked, the corresponding cell in a response will contain the answer text, otherwise it will be empty. In single-choice and Likert responses, SurveyMonkey reports the text of the chosen answer. For analysis, we wanted a simple `1` for checked and `0` for unchecked, and a number from 1 to 5 for the Likert answers. This was done easily enough with some formulas and search-and-replacing.
• Since the question texts in the SurveyMonkey spreadsheet don’t make for good labels for importing from CSV, we replaced them all by generic labels such as `Q5` (or `Q6R2`, for Question 6, Response 2, for multiple-choice questions).
• We deleted data columns we don’t need such as timestamps and empty colums for data we didn’t collect such as names and IP addresses.
• We added columns so we can collate data more easily: `Section` to identify the individual course the data is from, `Course` for which course it is (`PHIL279` for Logic I, `PHIL379` for Logic II), `Term` for Fall or Winter term, `Open` to distinguish responses from sections using an open or a commercial text, and `Text` for the textbook used. Text is one of `SLC` (for Sets, Logic, Computation, `BBJ` (for Boolos, Burgess, and Jeffrey, Computability and Logic), `ForallX` (for forall x: Calgary Remix, `Chellas` (for Chellas, Elementary Formal Logic), or `Goldfarb` (for Goldfarb, Deductive Logic). This was done by combining multiple individual spreadsheets provided by SurveyMonkey into one. (One spreadsheet contained responses from three different “Email Collectors”, one for each section surveyed.) `Q27GPA` contains the answer to Question 27, “What grade do you expect to get?”, converted to a 4-point grade scale.
• Question 23, “Is the price of the textbook too high for the amount of learning support it provides?”, had the same answer scale as other questions (“Not at all” to “Very much so”), but the “Not at all” is now the positive answer, and “Very much so” the negative answer. To make it easier to produce a graph in line with the others, I added a `Q23Rev` column, where the values are reversed (i.e., `Q23Rev` = 6 – `Q23`).
• `Q26` is the 4-letter code of the major reported in the multiple-choice question 26, and `Q26R1` to `Q26R8` are responses to the checkboxes corresponding to options “Mathematics”, “Computer Science”, “Physics”, “Philosophy”, “Engineering”, “Neuroscience”, “Other”, and the write-in answer for Other. These responses don’t correspond to the questions asked: we offered “Lingustics” as an answer but noone selected it. A number of “Other” respondents indicated a Neuroscience major. So `Q26R6` is `NEUR` in `Q26`. Question 26 allowed multiple answers, `Q26` is the first answer only.

In order to analyze the Likert data, we have to tell R which cells contain what, set the levels in the right order, and rename the columns so they are labelled with the question text instead of the generic `Q1` etc. We’ll begin with the teaching evaluation data. The code is in `teachingevals.R`. Open that file in RStudio. You can run individual lines from that file, or selections, by highlighting the commands you want to run and then clicking on the “run” button.

First we load the required packages. `likert` is needed for all the Likert stuff; `plyr` just so we have the `rename` function used later; and `reshape2` for the `melt` function.

``````require(likert)
require(plyr)
library(reshape2)``````

``````data <- read.csv("teachingeval.csv",
na.string="")``````

Now the table `data` contains everything in our CSV file, with empty cells having the `NA` value rather than an empty string. We want the responses to be labelled by the text of the question rather than just `Q1` etc.

``````data <- rename(data, c(
Q1 = "In-class work in groups has improved my understanding of the material",
Q2 = "Collaborative work with fellow students has made the class more enjoyable",
Q3 = "Being able to watch screen casts ahead of time has helped me prepare for class",
Q4 = "Having lecture slides available electronically is helpful",
Q5 = "I learned best when I watched a screencast ahead of material covered in class",
Q6 = "I learned best when I simply followed lectures without a screencast before",
Q7 = "I learned best studying material on my own in the textbook",
Q8 = "This course made me more likely to take another logic course",
Q9 = "This course made me more likely to take another philosophy course"))``````

The Likert responses are in colums 5-13, so let’s make a table with just those:

``responses <- data[c(5:13)]``

The `responses` table still contains just the answer strings; we want to tell R that these are levels, and have the labels in the right order (“Strongly Disagree” = 1, etc.)

``````mylevels <- c('Strongly Disagree', 'Disagree', 'Neutral', 'Agree', 'Strongly Agree')

for(i in seq_along(responses)) {
responses[,i] <- factor(responses[,i], levels=mylevels)
}``````

## Analyzing and Plotting

Now we can analyze the likert data.

``lresponses <- likert(responses)``

You can print the analyzed Likert data:

``````> lresponses
Item
1          In-class work in groups has improved my understanding of the material
2      Collaborative work with fellow students has made the class more enjoyable
3 Being able to watch screen casts ahead of time has helped me prepare for class
4                      Having lecture slides available electronically is helpful
5  I learned best when I watched a screencast ahead of material covered in class
6     I learned best when I simply followed lectures without a screencast before
7                     I learned best studying material on my own in the textbook
8                   This course made me more likely to take another logic course
9              This course made me more likely to take another philosophy course
Strongly Disagree  Disagree   Neutral    Agree Strongly Agree
1          1.785714  5.357143 10.714286 37.50000      44.642857
2          1.785714  0.000000 10.714286 37.50000      50.000000
3          8.928571 14.285714 26.785714 28.57143      21.428571
4          1.785714  1.785714  5.357143 37.50000      53.571429
5          7.142857 10.714286 37.500000 33.92857      10.714286
6          3.571429 19.642857 51.785714 21.42857       3.571429
7          3.571429 12.500000 23.214286 33.92857      26.785714
8         20.000000 10.909091 32.727273 27.27273       9.090909
9         16.363636 18.181818 38.181818 18.18182       9.090909``````

And now we plot it:

``````plot(lresponses,
ordered=FALSE,
group.order=names(responses),
colors=c('darkred','darkorange','palegoldenrod','greenyellow','darkgreen')) +
ggtitle("Teaching Evaluations")``````

The `group.order=names(responses)` makes the lines of the plot sorted in the order of the questions, you need `ordered=FALSE` or else it’ll be ordered alphabetically. Leave those out and you get it sorted by level of agreement. You can of course change the colors to suit.

In `textbooksurvey.R` we do much of the same stuff, except for the results of the textbook survey. Some sample differences:

Here’s how to group charts for multiple questions by textbook used:

``````lUseByText <- likert(items=survey[,27:31,drop=FALSE],
grouping=survey\$Text)
plot(lUseByText,
ordered=TRUE,
group.order=c('SLC','BBJ','ForallX','Chellas','Goldfarb'),
colors=c('darkred', 'darkorange', 'palegoldenrod','greenyellow','darkgreen')
) +
ggtitle("Textbook Use Patterns")``````

To plot a bar chart for a scaled question, but without centering the bars, use `centered=FALSE`:

``````lQ5byText <- likert(items=survey[,26,drop=FALSE],
grouping=survey\$Text)
plot(lQ5byText,
ordered=TRUE,
centered= FALSE,
group.order=c('SLC','BBJ','ForallX','Chellas','Goldfarb'),
colors=c('darkred','darkorange', 'gold', 'palegoldenrod','greenyellow','darkgreen')
) +
ggtitle("Textbook Use Frequency")``````

## Plotting Bar Charts for Multiple-Answer Questions

Some of the questions in the textbook survey allowed students to check multiple answers. We want those plotted with a simple bar chart, grouped by, say, the textbook used. To do this, we first have to the data for that. First, we extract the responses into a new table.

``Q1 <- survey[,c(6,7:13)]``

Now `Q1` is just the column `Text` and `Q1R1` through `Q1R7`. Next, we sum the answers (a checkmark is a 1, unchecked is 0, so number of mentions is the sum).

``Q1 <- ddply(Q1,.(Text),numcolwise(sum))``

Next, we convert this to “long form”:

``Q1 <- melt(sumQ1,id.var="Text")``

Now `Q1` has three columns: `Text`, `variable`, and `value`. Now we can plot it:

``````ggplot() +
geom_bar(
aes(x=Text,fill=variable,y=value),
data=Q1,
stat="identity") +
coord_flip() +
ggtitle("01. How do you access the textbook?") +
theme(legend.position = "bottom",
axis.title.x = element_blank()) +
guides(fill=guide_legend(title=NULL,ncol=1))``````

This makes a bar chart with `Text` on the x-axis, stacking `variable`, and using `values` for the value of each bar. `stat="identity"` means to just use `value` and not count. `coord_flip()` makes it into a horizontal chart. `ggtitle(...)` adds a title, `theme(...)` puts the legend on the bottom and removes the x axis label, and `guides(...)` formats the legend in one column.

## UPDATE: Better Visualization of Multiple-Answer Responses

I figured out a better way to visualize multiple-answer responses (thanks to Norbert Preining for the help!). You don’t want the number of respondents which checked a box, but the percentage of all respondents (in a category) who did, so instead of adding up a column you compute the mean for it. Also, `aggregate` is an easier way to do this, and it doesn’t make sense to stack the responses, so I’m going to graph them side-by-side.

Here’s the code:

``````# load responses for question 4 into df Q4
Q4 <- survey[,c(6,20:25)]

# aggregate by Text, computing means = percent respondents who checked box
Q4 <- aggregate( . ~ Text, data=Q4, mean)

# make table long form for ggplot
Q4 <- melt(Q4,id.var="Text")

ggplot() +
geom_bar(
aes(x=variable,fill=Text,y=value),
data=Q4,
stat="identity", position="dodge") +
coord_flip() +
ggtitle("04. When using the text in electronic form, do you....") +
theme(legend.position = "bottom",
axis.title.x = element_blank()) +
guides(fill=guide_legend(title=NULL,ncol=1)) +
scale_fill_brewer(palette="Dark2") +
scale_y_continuous(labels = scales::percent)
``````

I drank the Koolaid and set up my CV so it’s generated automatically from a YAML file with a Pandoc template. The basic functionality is copied from `bmschmidt/CV-pandoc-healy`. My version generates the bibliography from a BibTeX file however, using `biblatex`.  The `biblatex` code is tweaked to include links to PhilPapers and Google Scholar citation counts.

The whole thing is on Github.

cv-zach

# A Fistful of Commits

I just checked in a whole bunch of changes to the part on first-order logic. Most of it is in preparation for a new version of the Logic II textbook Sets, Logic, Computation, which Nicole is planning to use in the Fall term.  Also important, and that’s why I put it right here at the top:

PDFs now live on builds.openlogicproject.org. The builds site has a nice index page now rather than a plain file list. If you link to a PDF on my ucalgary site, please update your link; that site will no longer be updated and will probably disappear sometime soon.

Here’s a list of changes:
• I’ve revised the completeness theorem thoroughly. (This was issue 38.) The main change is that instead of constructing a maximally consistent set, we construct a complete and consistent set. Of course, those are extensionally the same; but both the reason for why we need them and the way we construct them is directly related to completeness and only indirectly to maximal consistency: We want a set that contains exactly one of A or ¬A for every sentence so we can define a term model and prove the truth lemma. And we construct that set by systematically adding either A or ¬A (whichever one is consistent) to the original set. So it makes pedagogical sense to say that’s what we’re doing rather than confuse students with the notion of maximal consistency, prove that the Lindenbaum construction yields a maximally consistent set, and show that maximally consistent sets are complete so we can define the term model from it. Credit for the idea goes to Greg Restall, who does this in his course on Advanced Logic.  (I kind of wonder why standard textbooks mention maximally consistent sets. I’m guessing it’s because if you consider uncountable languages you have to use Zorn’s lemma to prove Lindenbaum’s theorem, and then using maximality is more natural. Is that right?) A bonus effect of this change is that a direct proof of the compactness theorem is now a tedious but relatively easy adaptation of the completeness proof; and I’ve added a section on this (leaving most of the details as exercises).
• I’ve revised the soundness proofs for sequent calculus and natural deduction, where the individual cases are now more clearly discussed. (This was issue 74 and issue 125.)
• In the process I also simplified a bunch of things, filled in some details, and corrected some errors. This includes fixing issue 110, and cleaning up the whole stuff about extensionality. There is a new section on assignments which you may need to add to your driver file unless you include `fol/syn/syntax-and-semantics.tex` in its entirety.
• The natural deduction system now uses Prawitz’s standard rules, i.e., the double negation elimination rule has been replaced with the classical absurdity rule, and the negation rules are now the special cases of the conditional rules with ⊥ as consequent. This was issue 144. Comparing the system to other treatments in the literature is now easier, and the chapter will integrate more seamlessly with the part on intuitionistic logic that’s in the works.
• The sequent calculus chapter now uses sequents that are made up of sequences, not sets, of formulas. This was issue 145. This is the standard way of doing it, and will make it easier to add material on substructural logics. It also makes the soundness proof a lot easier to understand.
• In both the sequent calculus and natural deduction chapters, the material on quantifiers is now separated from that on propositional connectives. Eventually it should be possible to present propositional logic separately (or only), and now you can reuse only the material on propositional logic. (This was issue 77.)
• There is a new chapter on proof systems, and the intro sections from both the natural deduction and sequent calculus chapters have moved there. So if you only want one of the proof systems, you’ll have to include the relevant intro section in the chapter on the proof system “by hand.” But if you include both, you now have an additional chapter that introduces and compares them.  (This was issue 61.)
• Sets, Logic, Computation (the textbook for Logic II) now includes both sequent calculus and natural deduction!
Note: The formatting of the rules in both systems now uses a `defish` (“definition-ish”) environment. If your remix uses a custom `-envs.sty` file, you will need to add a definition for that at the end (see `open-logic-envs.sty`). The textbooks for Logic II and Logic III have been updated accordingly.

# 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.”