Ex impossibili sequitur quidlibet in the 12th C

At the AAL conference in Melbourne in July, one of the questions that I was asked was about medieval discussions of ex falso/impossibili/contradictione quidlibet sequiter (and it’s correlate, ad necessarium quodlibet sequiter).

The legitimacy of these inferences was significantly discussed during the 12th century, with the followers of Adam de Petit-Pont (called the Adamites or the Parvipontanians) accepting these principles, but others, in particular Abaelard and the so-called Nominales rejecting it. My point of contact with these discussions is via the Tractatus Emmeranus de Positio Impossibilis (1st half of the 13th C, edited by de Rijk in 1974), a treatise on the genre of obligationes which takes not a false statement but an impossible one as the positum. The author points out that the only way this genre can be sustained as a useful one is if the types of inferences used are restricted; in particular, the author singles out the “Adamite” thesis by name in order to identify it as unacceptable in the context of impossible positio:

And we should note that in this question everything does not follow from an impossible obligation. Thus, in this question one must not concede the consequence of the Adamites—namely that from the impossible anything follows [p. 218]

The principle is also attributed to Adam by John of Salisbury in the Metalogicon.

A couple of good modern discussions of this thesis can be found in Iwakuma 1993 (covering the 12th century), Spruyt 1993 (on the 13th century), and d’Ors 1993 (for the 14th century). In this post, I focus on Iwakuma’s paper and the 12th century.

The best-known 12th-century sources for the ex impossibili thesis are the Ars Meliduna and Alexander Neckam’s De Naturis Rerum (p. 125). The Ars Meliduna is written by a Melidunensis rather than a Parvipontanus, but the Parvipontani thesis is discussed in book IV, chapters 37-39. Neckam, on the other hand, accepts the thesis in a discussion in chapter 173, according to Iwakuma’s reinterpretation. In this section, Neckam starts off saying “Miror etiam quosdam damnare opinionem dicentium ex impossibili per se quodcumque sequi enuntiabile“, and gives six arguments. The first is a direct proof of ex impossibili, but Iwakuma says the remaining five “appear at first sight to have nothing to do with the thesis” (p. 126). The second argument is “one of the earliest presentations of the Liar’s paradox” (p. 126), and the other arguments “are very similar to the second in their structure” (p. 126). Iwakuma explains the relevance of the Liar paradox to questions of ex impossibili as follows:

Alexander’s second argument amounts to show that such an impossible antecedent is followed by a contradiction, namely Socrates says something both true and false. And once the contradiction holds, then it is easy to construct an argument to show that it is followed by any consequent, following the steps of the first argument…In just the same manner, one can interprete the other four arguments as proofs of the thesis. For in every argument of the four the antecedent of the first premise can be understood as impossible, and the conclusion is a contradiction (pp. 127-128).

But Iwakuma’s main aim in the paper is not discussion of Neckham and Ars Meliduna, but rather “to introduce two hitherto unknown texts and to comment on them, as well as on those already known, relevant to the thesis ex impossibili quidlibet sequiter” (p. 123). The two texts are MS Avranches 224, f. 3, from the late 12th century, and MS Munich clm 29520(2, also from the late 12th century.

The Avranches text discusses two counterarguments to the thesis, then gives two proofs of it (and a proof of the corollary ad necessarium) as well as responses to the purported counterarguments. Unfortunately, I neglected to scan the appendix in which this text is edited, and since Iwakuma doesn’t say anything further in the body of his article I can’t add anything more. (Guess I need to get that book via ILL again…or someday just purchase myself a copy!)

The Munich MS is incomplete, lacking the beginning so that it “leaves us uncertain as to what it discusses” (p. 130), though Iwakuma conjectures that it was probably written by a Parvipontanus on the ex impossibili thesis. As evidence, he points to a discussion of solutions to a problem set by Abelard wherein the solutions of the Nominales and the Melidunenses are rejected, but not the Parvipontani solution. In the same section, two more theses (ex inmodali sequitur modalis, et e converso and ex explicita sequitur inplicita, et e converso) are admitted; both of these are denied by the Porretani and Montani schools, but are consequences of ex impossibili, for:

a modal/immodal antecedent, if being impossible, is followed by any consequent, including immodal/modal ones. Similar considerations apply to inplicita/explicita propositions (p. 132).

If Iwakuma is right in his analysis and identification, then “this is one of the few valuable pieces of evidence on the Parvipontani‘s thesis from their own pens” (p. 133).


References

Anonymous. 2001. “The Emmeran Treatise on Impossible Positio“, in M. Yrjönsuuri, Medieval Formal Logic (Kluwer Academic Publishers): 217-223.

de Rijk, L. M. 1974. “Some Thirteenth Century Tracts on the Game of Obligation”, Vivarium 12: 94-123.

d’Ors, Angel. 1993. “Ex Impossibili Quodlibet Sequitur (John Buridan)”, in K. Jacobi, Argumentationstheorie (Brill): 195-212.

Iwakuma, Y. 1993. “Parvipontani‘s Thesis Ex Impossibili Quidlibet Sequitur: Comments on the Sources of the Thesis from the Twelfth Century”, in K. Jacobi, Argumentationstheorie (Brill): 123-133.

Spruyt, Joke. 1993. “Thirteenth-Century Positions on the Rule ‘Ex Impossibili Sequitur Quidlibet‘”, in K. Jacobi, Argumentationstheorie (Brill): 161-193.

Posted in Uncategorized | Tagged , , , , , , , , , , , , | Leave a comment

Avicenna in a Castle

Every August, I, a hundred of my friends, and all the medieval re-enactment gear you could want spend 10 days at Raglan Castle, in Wales. It is an amazing castle, and a very special place with lots of memories as I’ve been doing this since 2008. The length of the event lends itself to leisure, and one of the things I love to do is bring along medieval texts to read aloud and discuss. This year, I brought along Asad Q. Ahmed’s translation of the section on logic of Avicenna’s Deliverance — a book which has been sitting on my shelf for almost four years but which I have not yet read. The Deliverance was written in 1027, a turning point in Avicenna’s logical career. Ahmed says in the introduction: “Somewhere around 1027, Avicenna starts to show less patience with Aristotle himself, frequently pointing out the failure to implement one set of principles consistently throughout the Organon.” The Deliverance, however, was a compilation of work written prior to 1027, and so reflects a more orthodox approach.

Slightly over half a dozen of us spent more than an hour on the text. Not having read any of it, I didn’t do any preparation in advance of the discussion, so some of our discussion centered around “Well, I’m not entirely sure what he’s referring to here, possibly X”. We read all of sections 1-8, and then I glossed over and summarized sections 9-40. Here are some of the interesting points that came out of the discussion.

In section 1, “On Conceptualization and Assenting and the Method of Each”, Avicenna claims that “all primary cognition and scientific knowledge is either conceptualization or assenting”. Conceptualization is acquired by definition, while assenting comes via syllogism. Both definition and syllogism are divided into (1) the real, (2) the unreal but “beneficial to some extent in its own way”, and (3) the “false that resembles the real”. He makes the interesting point that humans are not naturally disposed to being able to distinguish these three; but since logic involves being able to make these distinctions, that is why it’s important to study logic. As evidence for the claim that people, by their nature, are not able to distinguish these three types is that if this weren’t the case, then “there would occur neither any disagreement among the wise nor any contradiction in the judgement of any single one of them”.

Both definitions and syllogisms are hylomorphic, being composed of matter and form. What is interesting here is that in order for a definition or a syllogism to be a good one, both the matter and the form must be good in combination with each other. For “just as corruption in the building of a house may occur on account of the matter even if the form is correct or on account of the form even if the matter is sound (or on both their accounts together)”, so may a definition or syllogism have be defective either because the form is good but the matter is not; the matter is good but the form is not; or because neither is. This is a distinctive view to me (if anyone knows of anyone else who holds such a view, please share in the comments!), because usually the importance of the form of a valid argument is stressed because if the form is good, then the argument will be good regardless of the matter that is inserted. Unfortunately, Avicenna doesn’t give an example of a syllogism or definition where the form is good but the matter is not.

Having introduced the subject matter of logic, in section 2, Avicenna explains what the benefits of logic are. (Chapters like this are always my favorite, because they justify how I spend my life.) He first explains the different types of good definition and syllogism. There are two types of good definitions, true definitions and descriptions (which are merely convincing rather than true), while there are three types of good syllogism. The first is the correct type, and is called demonstration. The second is a convincing syllogism which “generates a kind of assent that resembles certainty”, and these are the dialectical ones. Then there is a weak type which “generates overwhelming belief”, and these are the rhetorical ones. Finally, there is the false definition, which is called misleading, and the false syllogism, which is called sophistical. A sophistical syllogism “presents itself as a demonstrative or dialectical syllogism, while not being so”. Finally, there is a fifth type of syllogism which does not generate any type of assent but rather effects the imagination, and this is called the poetic syllogism. (Side note: I’ve never heard of the poetic syllogism before. Is this in Aristotle? Who else discusses these?)

Avicenna then notes that the relation of logic to deliberation is the same as grammar to speech and prosody to poetry, but while “a sound nature and innate faculty of discernment can perhaps dispense with the study of grammar and prosody”, there is no substitute for the study of logic.

Sections 3 and 4 cover Simple Utterances and Complex Utterances. The former are those utterances which are significative on their own, no part of which is significative, while the latter are those utterances which are significative but which have significative parts.

Sections 5 and 6 cover Universal and Particular Simple Utterances, which are distinguished on the basis of whether they signify “the many by way of one coinciding meaning” or not. Here, “the many” can either be many in existence, such as “man” which signifies many men, or in the imagination, such as “sun” which signifies one existing sun but nothing prevents it from signifying in the imagination many other suns. Particular simple utterances are those “whose unique meaning cannot possibly be anything more than a unique thing”. Examples of particular simple utterances include proper names and deictic descriptions such as “this sun” or “this man”.

This is pretty much all Avicenna has to say about simple utterances at this point, and the next section, 7, is a long one dedicated to what counts as essential, because every universal utterance is either essential or accidental. An essential utterance “sets down the quiddity of that of which it is said”, which, you have to admit, is not an entirely helpful definition. A further gloss is provided that

the essential is such that, if the meaning [of the subject] is understood and occurs in the mind and if the meaning of what is essential to it is understood and occurs in the mind at the same time, it would be impossible for the essence of the subject to be understood unless first the meaning [of that which is essential to it] is already understood to belong to it.

From this it is clear that the essential is not that which is merely inseparable from its substance. For it is inseparable of a triangle that the sum of its angles equal two right angles, but this is not an essential property of a triangle, because one can understand “triangle” without necessarily first understanding “sum of its angles equally two right angles”. This can be contrasted with the example of “man” and “animal”; “animal” is essential to “man” because you cannot understand “man” without understanding “animal”.

Given all this, section 8 is a short discussion of the Accidental, which is that is not essential. After this point, we stopped reading aloud entire sections and rather I skimmed and summarized.

Sections 9 and 10 are on answers to the questions “What is it?” and “Which thing is it?” We skipped these sections and went on ahead to the classification of the five types of universal utterance (section 11), genus, specific difference, species, property, and accident, each of which is given their own section (12, 14, 13, 15, 16, respectively). Sections 18-20 cover nouns, verbs, and particles, and then we have definitions of statements (21), which are merely complex utterances; propositions (22), which are statements in which “there is a relationship between two things such that the judgment ‘true’ or ‘false’ follows from it”; and attributive propositions (23), where the two things being related can both be picked out by simple utterances.

Statements themselves can also be related to each other, and the result is a conditional proposition (24). There are two types of conditional propositions, conjunctive and disjunctive. A conjunctive conditional proposition is one like “if the sun rises, the morning exists” (25), while a disjunctive conditional proposition is one like “Either this number is even or this number is odd” (26).

After this we are introduced to concepts of affirmation and negation, subject and predicate, singular and indefinite propositions, and then into everything that goes into the Square of Oppositions. At this point we stopped reading from Avicenna and I grabbed a stick and began drawing in the sand. It was not very easy to photograph, but here you go:

square of oppositions drawn in sand

Avicenna in a castle

Posted in Uncategorized | Tagged , , , , | 3 Comments

Call for proposals/interest/suggestions/ideas: British Society for the History of Philosophy

The British Society for the History of Philosophy is having a three-day conference at the University of Sheffield, April 6-8 2017, and has just sent out a call for proposals.

Would any of our readers be interested in clubbing together with me to put together a proposal or two on logic?


Call for Papers

The British Society for the History of Philosophy invites scholars to submit symposium and individual paper proposals for its general conference. Symposia and individual papers are invited on any topic and any period of the history of philosophy.

Proposals for either symposia (3-4 thematically related presentations) or individual presentations (approximately 25-30 minutes) are welcome. Symposium submissions are especially encouraged.

Proposal Submission Deadline: 1 October 2016

Decision by: 1 December 2016

Submissions should be sent as an email attachment (in Word) to: bshp@sheffield.ac.uk

Proposals for symposia should include:

– Title of symposium
– Symposium summary statement (maximum 500 words)
– Titles and abstracts of papers (maximum 500 words for each paper)
– Address of each participant, including e-mail, phone, and institution
– Name and email of symposium organizer, who will serve as contact person


If you are interested, comment below; depending on who is, we can decide on number of sessions/specific topics, and then think about specific abstracts, etc.

Posted in Uncategorized | Tagged , | Leave a comment

Non-reflexive consequence relations

Among the many notes that I scribbled on handouts at the AAL a few weeks ago, one of them (on the handout for Colin Caret’s talk “Prospects for Non-Reflexivity” reads “Medieval non-reflexivity? Examples where A\nvdash A“. I had a vague feeling that because the Aristotelian syllogistic requires that the major, minor, and middle terms all be distinct, one could argue that it is an example of a non-reflexive consequence relation simply because something of the form A,B\vdash A would be malformed. It turns out, a friend has recently published on exactly this topic: Matthew Duncombe, “Irreflexivity and Aristotle’s Syllogismos”, Philosophical Quarterly 64, no. 256 (2014): 434-452:

Abstract: Aristotle’s definition of syllogismos at Prior Analytics 24b18–20 specifies syllogistic consequence as an irreflexive relation: the conclusion must be different from each premise and any conjunction of the premises. Typically, commentators explain this irreflexivity condition as Aristotle’s attempt to brand
question-begging syllogismoi illegitimate in argumentative contexts. However, this explanation faces two problems. First, it fails to consider all the contexts in which Aristotle thinks syllogismoi are deployed. Secondly, irreflexivity rules out only some arguments that Aristotle considers question begging. Here I address these problems. First, I examine all the contexts in which Aristotle thinks syllogismoi can be
used. Secondly, I argue that, for each context, irreflexivity makes sense as a condition, but for different reasons. Assuming that a condition which holds in each context is a condition on syllogistic consequence tout court, this explains why Aristotle holds syllogistic consequence to be an irreflexive relation.

Here I want to highlight a couple of important points that Duncombe makes in this article.

First: “irreflexivity may seem strange, because if truth preservation is necessary and sufficient for a consequence relation, then that relation ought to be reflexive” (p. 434): Thus, if a consequence relation is irreflexive, this entails that there is more to that notion of consequence than mere truth preservation. If one things of consequence as “truth preservation” plus something more (which of course needs to be fleshed out, then one needn’t take the route that Beall and Restall take, denying logics with irreflexive consequence relations the status of being real logics, instead calling them “logics by courtesy and by family resemblance” (p. 437).

Second: irreflexivity cannot be (merely) a way to block question-begging, since there are examples which Aristotle considers question-begging which are not reflexive (p. 435; also, sec. II.2). Duncombe notes that five different types of question beginning can be identified in Aristotle, of which “the second and third types of question begging do not seem to be violations of irreflexivity” (p. 441).

Duncombe’s conclusion is that there is no uniform explanation for irreflexivity, but rather that irreflexivity is a consequence of different factors in different types of argumentative contexts: “Although in different contexts irreflexivity is a condition on syllogismoi for different reasons, in any pragmatic context in which Aristotle envisions syllogismoi being used, it is a plausible condition
for giving a syllogismos” (p. 435), and “it is not necessary that there is one single notion of validity across all contexts, but, as it turns out, the same notion applies across all contexts, for diverse reasons” (p. 441). A crucial feature of the various pragmatic contexts is that they are all ones which involve “logic in action”, in demonstrative, dialectic, peirastic or eristic contexts (p. 443). In each of these contexts, there is a reason why irreflexivity is desirable, even if it is a different reason in each context. Such a view of logic, with a focus on the application and use of logic, can be considered either surprisingly modern — or it can be considered a case where modern logicians are (unknowingly) returning to the original roots of logic as a dialectical tool. Thus, it should come as no surprise that just as modern interest in irreflexive logics tracks a growing interest in “logic in action”, so too irreflexivity comes hand-in-hand with Aristotle’s view of the application(s) of logic.

Though Duncombe identifies different reasons why irreflexivity is desirable in each of the different contexts, he argues that there may in fact be a general principle which underlies these different reasons:

But there may be interesting reasons why syllogismoi in different contexts converge on irreflexivity. For example, in mono-agent demonstrations the premises must be more acceptable than the conclusion (Smith 1989: 64b31–2; cf. Smith 1997: 159b8–9). When Aristotle comes to discuss multi-agent dialectic, particularly the use of dialectic for training (Smith 1997: 159a25–6) and inquiry (Smith 1997: 101a34–7), he formulates a corresponding condition for multi-agent contexts: the premises the answerer grants should be more acceptable than the conclusion (Smith 1997: 131)…it makes sense to specify that such an answerer should not admit a premise, unless it is more acceptable than the conclusion, since this condition corresponds to how to give a good demonstration…Such training dialectics would need irreflexivity as a condition (p. 451).

So not only is irreflexivity justifiable in this context, it may even be necessary!

Does anyone have any other examples of ancient or medieval irreflexive consequence relations? Please share in the comments!

Posted in Uncategorized | Tagged , , , , | Leave a comment

Why should we care about history of logic? (expanded)

The goal of this post is to expand on the slides, linked in the previous post, for the talk I gave at the Australasian Association of Logic two weeks ago, on “Why should we care about history of logic?” This isn’t by any means a transcription of my talk, but it is a general summary of the thoughts that I talked about.

I have been studying, researching, and presenting on topics in medieval logic for more than a decade now. Medieval logic sits in an uncomfortable position at the intersection of mathematical/philosophical logic and medieval studies, and my conference schedule takes me to events from both sides of the spectrum. One thing I have found very interesting is that the logicians are generally much more interested in the medieval stuff than the medievalists are in the logical stuff. I have to spend much more time motivating to the medievalists why this material is interesting to look at, and why I am using modern logical tools to do so, while the logicians generally seem simply curious about the history of their field — and if I throw in a manuscript image or two, it’s a nice break from, say, dualities in Stone algebras. For the first few years, particularly while I was still a PhD student, I was happy to focus on convincing others that what I was doing was interesting and they should accept my abstracts and papers for conferences and journals. But recently I’ve had the desire to move from convincing others that what I am doing is important to convincing others that they should be doing it too.

This is a big step: Many people may be interested in the history of logic as a historical curiosity — happy that someone else is doing it but not interested in doing it themselves. My aim in the talk was to show that logic differs relevantly from other scientific disciplines, and that these differences mean that the study of the history of logic is more than just historical or mere curiosity, but in fact should be seen as the study of logic, no “history of” modifier needed (slide 2).

In order to motivate why contemporary logicians should care about the history of logic, it’s worth first looking at other scientific disciplines where the history of the subject is a mere curiosity. Consider disciplines such as medicine, biology, chemistry, and astronomy. We do not ask medical doctors to read medieval treatises on the four humors, or to integrate medieval cures for headaches (sawing open the skull) or venereal diseases (inject mercury into the offending organ) into their modern arsenal (slides 3-5). No one takes seriously Aristotle’s claim that women have fewer teeth than men (slides 6-7). Chemists need not know about the roles of Mercury and Venus in alchemical processes (slides 8-9). Astronomy students are not taught how to calculate epicycles (slide 10). Certainly if a scientist showed an interest in any of these aspects of the history of their field, this would be encouraged, but not because we think studying the history of these disciplines will provide material relevant to the modern practice of these disciplines — something that I want to argue is different when we consider logic.

What makes logic different from these other disciplines? One answer might be that what we thought we knew then about medicine, biology, chemistry, astronomy was simply false. Now we know better, so now we don’t look to the medieval texts as our guides (slide 11).

But then consider the case of mathematics. Modern mathematicians do not generally study any history of mathematics. Try an experiment: Next time you run into a practicing mathematician, ask them to calculate the subdouble subsesquitertius of 10. Even if you manage to pronounce this correctly, they will almost certainly have no idea what you mean, nor any idea of any of the other complicated names of proportions offered by Roger Bacon in the 13th century (slide 12). And yet, it is nevertheless true that 3 is a subsuperbipartient of 5. Perhaps another alternative is that historical mathematics lacks a necessary clarity (slide 13), a clarity that modern approaches to mathematics have. This explanation seems to have more bite; for it helps us understand why it is that some historical mathematics is still read and used, namely Euclid’s Elements (slides 14-15). In fact, one might argue that it is precisely its clarity that has afforded it its long and fruitful usage.

Still, one shouldn’t draw the conclusion of “medieval = unclear” and “modern = clear”; for even modern logic can lack that necessary clarity (slide 16-17)!

Before I go on to say something about why it is that logic is different from some of these other scientific disciplines, I want to say something about why it matters that we study the history of logic. The most straightforward answer is the oft-repeated adage that “Those who cannot remember the past are condemned to repeat it” (slide 18). I have lost track of the number of times that I’ve been at a talk where someone has said “no one has ever done X before Y”, where Y is maybe Leibniz at the oldest, but is most often someone much more modern, and I have raised my hand during the Q&A to say “Uh, Z did this seven hundred years earlier”. People who do not know the history of their field are far more comfortable making sweeping generalisations about what was done in that history, and in many cases, falsely. Even if one is not interested in learning what was done in logic in the Middle Ages, one should still be sensitive to issues of priority and attribution (a well-known example being the so-called “DeMorgan’s Law”, articulated by a number of 14th C logicians including Jean Buridan and William of Ockham (slide 19)).

So why is logic different from chemistry, medicine, biology, astronomy? The next section of the talk was devoted to going through examples of developments in medieval logic that are of relevance and interest to contemporary logicians, because they provide us with alternative or complementary accounts of some of the same phenomena (slides 20-27). But this way of answer the “why” question still leaves open another “why” question: What is it about logic as a discipline that makes the medieval developments still relevant? Here I do not have a concrete answer to give. One answer that you might wish to offer is that the reason why medieval developments in logic have just as much use and application today as they did centuries ago is that they reflect some timeless truth, some underlying fact about the real nature of logic, that is context and culture independent and necessarily true. I am hesitant to take up such a view, mostly for hubristic reasons: Such a view can have an easy consequence that developments in argumentation and reasoning outside of the Aristotelian-European tradition fall short of reflecting that “real nature” of logic. Another possibility, and one which I think is a more likely explanation, is that what we find important in the use and application of logic changes over time — sometimes being closely linked to linguistic usage and practice, sometimes instead being connected with mathematics — and that what we have seen over the last few decades is a change in emphasis which has lead us back to a point where we have many of the same goals and motivations that the medieval authors did — a focus on dynamic reasoning, multi-agent settings, problems of truth and possibility, questions of how language is actually used, etc. Because our goals align, the developments and insights can still be relevant to our pursuits nowadays (slide 28).

By now I hope I’ve convinced you of the utility of becoming more familiar with the developments in the history of logic, as a practicing modern logician. Now what? (slide 29) Not everyone is going to go out and learn Latin; but that isn’t necessary as in recent years more and more reliable and high-quality translations into English have been published, as well as secondary sources such as the forthcoming Cambridge Companion to Medieval Logic edited by Stephen Read and Catarina Dutilh Novaes. There is also ample resources for support via social media, including this blog and the Medieval Logic FB group

Posted in Uncategorized | Tagged , , , , , , | Leave a comment

Why should we care about history of logic?

That was the talk of the keynote I gave at the Australasian Association for Logic conference in Melbourne a few hours ago. Of course, I naturally care quite a bit about history of logic, but tonight I wanted to spend some time explaining to other people why they too should care (judging from the enthusiastic nods in the audience, I convinced them. Either that, or they just enjoyed all the pictures in my slides).

I’d love to devote a post here to the topic — and intend to do so in the future — but today is not the day as it’s far too late in the evening, Melbourne time, to do it justice, so instead, I’ll link to my slides, for those who’d like to see the pictures: slides for AAL talk.

Posted in Uncategorized | Tagged | 2 Comments

Pre-1600 logic on googlebooks

I’ve been spending most of my time thinking about modern stuff rather than medieval stuff in the last few weeks, but there’s nothing like a commitment to have a post on this blog every Thursday to make sure I think about medieval stuff every week, at least for a little bit.

Today, you get a brief list of books published before 1600 that deal with “logica” and are available on googlebooks.

Have fun!

Posted in Uncategorized | Tagged , , , , , , , , , , , , | Leave a comment