Pons Asinorum in Logic

Pons Asinorum. Kneale & Kneale 186
Kneale and Kneale, p. 186-187 explain: "Aristotle suggests a technique for finding syllogisms to prove conclusions of the four different kinds. When we want to prove a universal affirmative conclusion with two given terms, we should consider characters which entail the predicate of the proposed conclusion and characters which are entailed by the subject of the proposed conclusion. When we want to prove a particular affirmative we should consider characters which entail the predicate and characters which entail the subject. When we want to prove a universal negative we should consider characters which are incompatible with the predicate and characters which are entailed by the subject or conversely characters which are entailed by the predicate and characters which are incompatible with the subject. And finally, when we want to prove a particular negative we should consider characters which are incompatible with the predicate and characters which entail the subject. ... The diagram which Alexander [of Aphrodisias] describes is not preserved in our text of his work, but it can easily be reconstructed as shown, According to Aristotle's lettering, A is the predicate of the conclusion, B the group of characters entailed by A, Γ the group of characters that entail A, Δ the group of characters incompatible with A, E the subject of the conclusion, and Z, H, Θ the groups of characters related to it in the same way as the groups opposite are related to A. We have inserted five continuous lines to represent the five cases noticed above (two for the proof of universal negative conclusions) and a dotted line to represent the case--{Baralipton) in which we can prove a particular affirmative conclusion from premisses that would suffice to prove the universal affirmative with transposed terms."

A nicer pons asinorum from the 17th century. Source unknown

Chrysippus and his indemonstrables modes of reasoning

From the Jeremy Kirby's entry on Chrysippus in the IEP.

"...Aristotelian arguments above made use of classes. The ... relata herein are propositions [not classes]—Stoics called these ‘sayables’—rather than classes. In Aristotelian logic, the key connectives are ‘all’, ‘some’, ‘is’, and ‘is not’. In Chrysippus’ logic, the key connectives are ‘if’, ‘or’, ‘and’, ‘not’. "

  • Modus ponens: If p, q; p; ergo, q. : p→q, p, ∴q
  • Modus tollens: If p, q; not q; ergo, not p. : p→q, ¬q, ∴¬p 
  • Modus ponendo tollens: Either p or q; p; ergo not q.: p∨q, p, ¬q
  • Modus tollendo ponens-1: Either p or q; not q; ergo p.: p∨q, ¬q,  
  • Modus tollendo ponens-2: Not both p and q; p; ergo, not q. ¬(p∧q), p, ¬q
  • The consequentia mirabilis

    If a proposition follows from its own negation, then it's true.
    Some instances in which this deductive method is used are:

    • Cogito: I think
    • <> all that thinks exist
    • ergo sum: therefore I exist.
    • Suppose that all my knowledge is false
    • but at least I know that I know something, although it's wrong
    • Therefore in knowing that I know something, I don't deceive myself
    Angelelli, I. () Here
    Mirabell, I. () Here
    Kneale, W. () Here

    Johann Kaspar Sulzer: De conceptibus

    Cap. I. De conceptibus et quomodo in species et genera ordinentur.
    §l Quicquid in rerum natura sensibus nostris observatur, singulare est.

    §2 Omne singulare ab alio singulari differt eo ipso, quod est aliud, et individuum vocatur.

    §3 Omnia itaque in rerum natura sunt diversa, et licet sint innumerabilis, tamen semper unum non est alterum: unum individuum non est alterum individuum: Jacobus non est Johannes.

    §4 Quamquam vero diversa sunt, in multis tamen convenire deprehendimus. In quibus conveniunt vel similia sunt, eo respectu ad eandem referuntur classem: in quibus non conveniunt, ad diversam classem referuntur.

    §5 Quae ad eandem classem pertinent, idem nomen habent: diversa classis diversum nomen obtinet, et nomen unius classis alteri non convenit. v.g. Jacobus et Henricus, quia similes sunt, pertinent ad unam eandemque classem, cui virorum nomen est. Hinc uterque est vir. Anna vero et Maria, quia multum differunt a Jacobo et Henrico, diversam classem constituunt, quae nomine feminarum gaudet. Est ergo et Anna et Maria femina, sed vir non est femina nec femina vir.

    §6 Ex collatione igitur rerum omnium plurimae classes rerum oriuntur, et semper diversae, quia classis una non est altera, §2, 3.

    §7 Classes, quae ex similitudine individuorum promanant, erunt classes primi ordinis, et dicuntur species, vgr. mares, feminae.

    §8 Classes primi ordinis, licet diversae, similitudinem quandam habent, quae similitudo exhibet classes secundi ordinis, seu classes specierum, et tales classes vocantur genera. Sic mares et feminae ex Jacobo, Henrico, Anna, Maria orti homines nominantur, hinc mares et feminae erunt species, homines vero genus.

    §9 Classes, quae ex similitudine classium secundi ordinis oriuntur, tertii ordinis classes producunt, et generum superiorum nomine veniunt. Tale genus superius est, si homines et bruta conferuntur, et ex eo, quod utrinque vivunt et sentiunt, in unam classem ordinantur, quae classis animalium nomen habet, animal ergo erit genus superius.

    §10 Ulterius sic procedendo classes quarti, quinti, cet. ordinis exhibentur, et sic superiora semper genera nobis sistuntur, donec tandem ad unam ultimam classem perveniamus, quae supremum genus dicitur, quod ens est [ ... ].
    §13 Quod itaque genus nomen habet, et quidquid de genere dicitur, id nomen quoque habent omnia, quae sub hoc genere comprehenduntur ad individua usque, et de iis quoque absolute dicitur. Exemplum sit animal, quia homines, bruta, mares, feminae, cet. comprehendit, dicetur homo est animal, brutum, vir, femina, Jacobus, Maria, cet., est animal. Et si dicitur: animal sentit, etiam homo, brutum, vir, femina, Jacobus, Maria, sentit.

    See: Angelleli, I. (1974) LA JERARQUIA DE CLASES DE JOHANN CASPAR SULZER (1755). Cuadernos de Filosofia. Universidad de Buenos Aires. XIV, 21 , 1974


    A philosophaster is a “A shallow or pseudo-philosopher; a smatterer or pretender in philosophy”, according to the Oxford English Dictionary.

    There's also a book by Robert Burton, of Melancholic fame, by the same name. The character names are funny, all of them charlatans in philosophy and theology:

    Polypragmaticus, Pantomagus, Pedanus, Amphimacer, Theanus

    Orator: cultivated talent from a second ploughing

    For bring me a man as accomplished, as clear and acute in thinking, and as ready in delivery as you please; if, for all that, he is a stranger to social intercourse, precedent, tradition, and the manners and disposition of his fellow-countrymen, those commonplaces from which proofs are derived will avail him but little. I must have talent which has been cultivated, soil, as it were, not of a single ploughing, but both broken and given a second ploughing so as to be capable of bearing better and more abundant produce. And the cultivation is practice, listening, reading and written composition.

    Nam si tu mihi quamvis eruditum hominem adduxeris, quamvis acrem et acutum in cogitando, quamvis ad pronuntiandum expeditum, si erit idem in consuetudine civitatis, in exemplis, in institutis, in moribus ac voluntatibus civium suorum hospes, non multum ei loci proderunt illi, ex quibus argumenta promuntur. Subacto mihi ingenio opus est, ut agro non semel arato, sed et novato et iterato, quo meliores fetus possit et grandiores edere. Subactio autem est usus, auditio, lectio, litterae.

    [my free translation] If you bring me an erudite orator, intellectually clear and sharp, and able in delivery; his arguments will be of little avail to him if he's not familiar with the traditions, examples, institutions, morals, and tastes of his countrymen... 

    Cicero De Oratore, II, 131 (p. 293)

    Concepts of Evidence

    Image result for carnapFrom Carnap, Logical Foundations of Probability, p. 163. See Also Rabinovitch, p. 109
    1. Classificatory
      • e renders h probable
    2. Comparative
      • e1 renders h1 more probable than e2 renders h2
      • (e1 & e2) render h more probable than e1 alone does.
      • e1 renders h1 more probable than it does h2
    3. Quantitative
      • the probability of h on e is r

    the mark of an educated man...

    Our discussion will be adequate if it has as much clearness as the subject-matter admits of, for precision is not to be sought for alike in all discussions, any more than in all the products of the crafts. Now fine and just actions, which political science investigates, admit of much variety and fluctuation of opinion, so that they may be thought to exist only by convention, and not by nature. And goods also give rise to a similar fluctuation because they bring harm to many people; for before now men have been undone by reason of their wealth, and others by reason of their courage.

    We must be content, then, in speaking of such subjects and with such premisses to indicate the truth roughly and in outline, and in speaking about things which are only for the most part true and with premisses of the same kind to reach conclusions that are no better. In the same spirit, therefore, should each type of statement be received; for it is the mark of an educated man to look for precision in each class of things just so far as the nature of the subject admits; it is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician scientific proofs. Nicomachean Ethics, I, 3.

    and in Latin (secoind paragraph)

    In tradenda civili scientia, non agendum esse demonstrationibus accuratis, quia id subjectum eius non permittit

    (I don't like this Latin translation by William Wilkinson 1820 too much - it's very convoluted)

    Ergo satis erit, si, cum de rebus et ex rebus ejusmodi verba facimus, rudem quandam veri formam adumbremus: et cum de rebus, quae plurimum eveniunt, atque ex talibus disputationem instituimus,
    talia quoque concludamus. Eodemque modo quaecunque ab alio dicuntur accipi probarique debent; est enim hominis probe instituti tantam in unoquoque genere subtilitatem desiderare, quantam rei ipsius natura recipit; nihil enim videtur interesse, utrum mathematicum rationibus ad persuadendum ac commodatis utentem feras, an ab oratore demonstrationes postules.

    This one by Carolus Zell (better)

    Satis itaque erit, si, cum de ejusmodi rebus ex rationibus item ejusmodi explicatio instituatur, rudi quadam et crassa forma verum declaremus; satis item, si (cum et derebus, quae plerumque eveniunt, et vero etiam ex talibus rationibus disputationem instituamus) ad talem quoque modam concludamus. Εodem autem modo etiarti accipi oportebit, quae hic dicentur. Εst enim bene instituti hominis, tantam in unoquoque genere disputationis subtilitatem desiderare, quantam rei ipsius fert natura. Ut enim ridiculus sit, qui mathematicum probabilibus rationibus ntentem probet: ita is quoque, qui ab ora tore demonstrationes postulet.

    Structure follows Strategy

    The economic historian Alfred D. Chandler Jr. coined an aphorism that summarizes the characteristics of business structures: structure follows strategy.

    From the point of view of its origin, the general idea, not the useful input for business of course, already exists in Antiquity. In the Aristotle's system the structure embodies the formal and perhaps the efficient causes, to achieve certain final cause.

    Another way to put it is that that structure is purposeful, and that purpose pertains to some strategy inasmuch strategy is a chain of causes that are deployed to accomplish certain purpose(s). As such, the structure is not something different but a part of the strategy.

    Against Radical Empiricism - Experience needs Theory

    • Experience teaches nothing without theory. (W. Edwards Deming)
    • Without sensibility no object would be given to us, and without understanding none would be thought. Thoughts without content are empty, intuitions without concepts are blind (Kant, Critique Pure Reason.)
    • The Sophists then appeared, men of no system but surveying all, only to find a multitude of ineffectual predicates applied to the world. (W.A. Heidel)
    • A radical empiricism … denies the possibility of knowledge (Hans Reichenbach, The rise of scientific philosophy)
    • no matter their ‘depth’ and sophistication, machine learning algorithms merely fit model forms to data. (Coveney, Dougherty, Highfield)
    • Data have no meaning apart from their context (W. Shewhart)
    • Gelman and Loken warn about the dangers of Borgean "garden of forking paths" when analyzing data. here.



    Hogarth: Line of Beauty - serpentine

    The serpentine is the line of beauty.

    Gibbs vs Boltzmann Entropies


    Important observation: Entropy is a characteristic of a thermodynamic system, not of a physical system. Because a physical system admits of many thermodynamic systems. The thermodynamic state is defined via a set of parameters (or degrees of freedom).

    Paper here

    Plato's "Grand Lie" - a.k.a. "Noble Lie"

    Lie as a tool. Dangerous in the contents, and in the method of imposing it.

    "The rulers then of the city may, if anybody, fitly lie on account of enemies or citizens for the benefit of the state; no others may have anything to do with it but for a layman to lie to rulers of that kind we shall affirm to be as great a sin, nay a greater, than it is for a patient not to tell physician or an athlete his trainer the truth about his bodily condition, or for a man to deceive the pilot about the ship and the sailors as to the real condition of himself or a fellow-sailor, and how they fare." (Republic 389b-c)

    "...How then may we devise one of those needful falsehoods of which we lately spoke—just one Grand lie which may deceive the rulers, if that be possible, and at any rate the rest of the city? What sort of lie? he said. Nothing new, I replied; only an old Phoenician tale of what has often occurred before now in other places, (as the poets say, and have made the world believe,) though not in our time, and I do not know whether such an event could ever happen again, or could now even be made probable, if it did.

    ...I will speak ... the audacious fiction, which I propose to communicate gradually, first to the rulers, then to the soldiers, and lastly to the people. They are to be told that their youth was a dream, and the education and training which they received from us, an appearance only; in reality during all that time they were being formed and fed in the womb of the earth, where they themselves and their arms and appurtenances were manufactured; when they were completed, the earth, their mother, sent them up; and so, their country being their mother and also their nurse, they are bound to advise for her good, and to defend her against attacks, and her citizens they are to regard as children of the earth and their own brothers.

    ... Citizens, we shall say to them in our tale, you are brothers, yet God has framed you differently. Some of you have the power of command, and in the composition of these he has mingled gold, wherefore also they have the greatest honour; others he has made of silver, to be auxiliaries; others again who are to be husbandmen and craftsmen he has composed of brass and iron; and the species will generally be preserved in the children. But as all are of the same original stock, a golden parent will sometimes have a silver son, or a silver parent a golden son. And God proclaims as a first principle to the rulers, and above all else, that there is nothing which they should so anxiously guard, or of which they are to be such good guardians, as of the purity of the race. They should observe what elements mingle in their offspring; for if the son of a golden or silver parent has an admixture of brass and iron, then nature orders a transposition of ranks, and the eye of the ruler must not be pitiful towards the child because he has to descend in the scale and become a husbandman or artisan, just as there may be sons of artisans who having an admixture of gold or silver in them are raised to honour, and become guardians or auxiliaries. For an oracle says that when a man of brass or iron guards the State, it will be destroyed.

    Such is the tale; is there any possibility of making our citizens believe in it? Not in the present generation, he replied; there is no way of accomplishing this; but their sons may be made to believe in the tale, and their sons' sons, and posterity after them. 
    (Republic 414b ...)

    Collingwood + Rex Martin on what is Historical Explanation

    To start with it seems that the best summary of Collingwood thinking was elicited by Dray:
    "...for in so far as we say an action is purposive at all, no matter at what level of conscious deliberation, there is a calculation which could be constructed for it. .. And it is by eliciting some such calculation that we explain the action." LAE 123
    and C. Devanny,
    "...what we very often want is a reconstruction of the agent's calculation of means to be adopted toward his chosen end in the light of the circumstances in which he found himself." 116 (LAE 122)
    Next figure is adapted from Rex Martin's Historical Explanation p. 69

    Rex Martin expressed the meaning of each node and side of the triangle in a schema of seven propositions:
    1. The agent perceived himself to be in a certain situation and was disposed to act toward it in some definite way (e.g., as Caesar was disposed to curb the hostile incursions of the Britons). 
    2. There were a number of alternative courses of action (designated as A-e.g., invading-B, C, and so on) open to the agent who had the situational motivation described in (1). 
    3. The agent did want to achieve or accomplish such-and-so end (e.g., conquest), which he believed would satisfy his situational motivation.
    4. He believed that doing A was, in the circumstances already described, a means to accomplishing his stated purpose or a part of achieving it.
    5. There was no action other than A believed or seen by the agent to be a means to his goal which he preferred or even regarded as about equal.
    6. The agent had no other purpose which overrode that of accomplishing such-and-so. 
    7. And we might add, although Collingwood gave little attention to it, that the agent knew how to do A, was physically able to do it, would be able to do it in the situation as given, had the opportunity, etc
    I interpreted it graphically as (though not sure on 5 and 6):

    The statistical criteria for proving the relevance are taken from Wesley Salmon:

    1. Statistical relevance: (called simple relevance by Martin)
    2. Screening-off: (Outweighing or overruling)
    3. Homogeneity of the reference class (intrinsic relevance)

    Concept, Percept - Essence, Accident.

    William Arthur Heidel (1868-1941), a philosopher son of a preacher, makes these interesting observations. His booklet "The necessary and the contingent in the Aristotelian system (1896)" is a very lucid treatment of concepts (and percepts).
    • The Sophists then appeared, men of no system but surveying all, only to find a multitude of ineffectual predicates applied to the world.
    • It is quite true, Aristotle admits, to say that chance is irrational; for reason deals only with what occurs always or at least for the most part, whereas chance lies in the reverse of these.
    • Thus the contingent and the necessary, which possess a true meaning only within a limited scope defined by a particular end, are generalized and erected into absolute fact. But, really, absolute necessity is as unmeaning as absolute contingency. For both conceptions we shall do well to substitute that of less or greater completeness in the definition of fact.
    He advances the idea that concepts are made with a end in view
    • The concept, in other words, is to be gained by defining the particular: But just here we discover the bad influence of the Socratic induction, proceeding as it did by the elimination of the non-essential, without being fully conscious of the meaning of this exclusion...It was the nature of the particular, in fact, which constituted the concept.
    • Viewed from a practical standpoint the exclusion of the non-essential from the concept is not only justified, but it even indicates a truth which ought to lead to the destruction of the theoretical category of "things" and so of the "given." When we are engaged in realizing an end which we have set up after a preliminary review or examination of our means, we find in our experience as presented in memory certain clusters of qualities which \ve commonly denote as things. These clusters are the net results, so to speak, of innumerable previous experiences, in which these "things" did service as ends in themselves or as means toward further ends. We cannot too gratefully acknowledge the serviceableness of this our minds' economy, by which our experience and, therefore, our whole fund of materials or means for future action is definitely organized so as to obviate the fatality of depending on more or less chance suggestions.
    • The essential point, on the theoretical side, is to recognize that .. [he] readjust[s] these clusters of qualities, according as this or that content is peculiarly desirable for a particular end... 
    • The previously discarded qualities, now again seen in the "things," are classed as "accidents" as opposed to the "essence." This once done, the arena is prepared for all the fruitless battles that have been fought over substance and attribute and inherence. ... I shall hope to show later on that this psychological fallacy is at the base of the distinction between the necessary and the contingent.
    • ... the contingent and the necessary, which possess a true meaning only within a limited scope defined by a particular end, are generalized and erected into absolute fact. But, really, absolute necessity is as unmeaning as absolute contingency. For both conceptions we shall do well to substitute that of less or greater completeness in the definition of fact.

    pseudo-Metaphysical claims of Positivism acc. to Collingwood

      Image result for collingwood robin
    • EM 143: Its central doctrine was that the only valid method of attaining knowledge is the method used in the natural sciences, and hence that no kind of knowledge is genuine unless it either is natural science or resembles natural science in method. 
    • EM 154: the first principle of positivistic metaphysics, the principle that all the presuppositions we can detect underlying our thought must be justified, and justified by an appeal to observed facts.
    • EM: 147: What is in fact a presupposition they misunderstood as a general proposition about matters of fact, advanced upon credit and awaiting verification

    Collingwood's examples of Absolute Presuppositions

    We do not acquire absolute presuppositions by arguing; on the contrary, unless we have them already arguing is impossible to us. Nor can we change them by arguing; unless they remained constant all our arguments would fall to pieces. We cannot confirm ourselves in them by ‘proving’ them; it is proof that depends on them, not they on proof (An Essay on Metaphysics 1998: 173).

    What is the difference between presuppositions and assumptions? C. Ribeiro states:
    Collingwood (1940) made a distinction between presuppositions and assumptions. Presuppositions are non-justified implicit implications. They differ from assumptions because the latter are stated openly, are explicit, not implicit. We assume by an act of free will: ‘To assume is to suppose by an act of free will. A person who ‘makes an assumption’ is making a supposition about which he is aware that he might if he chose make not that but another. (…)’ (Collingwood 1940: 27) Presuppositions, however, work in the darkness. But they establish logical connections with the statements formulated in our explicit thought. 
    Some examples of AP:
    • all events have a cause
    • the principle of the continuity of nature in time and space, 
    • the existence of God
    • the principle that mathematics is applicable to the natural world and hence that natural science is essentially an applied mathematics
    • all events happen according to law (EM, 150).
    • nature is uniform (Mill's EM, 152)


    Ribeiro, C. (2015) http://www.philosophica.ugent.be/fulltexts/90-3.pdf

    Principle of Calculated Risk

    Fleet Admiral Chester W. Nimitz portrait.jpgI wonder what is the originality, from a purely conceptual standpoint, of Nimitz' "Principle of Calculated Risk" (PCR). Aside from its indiscutible tactical value proven in WWII, this principle seems a commonsensical variation of a priciple that is ancient: Here's some possible ancestors:

    • Prov. 25:8: ... utter not hastily in a quarrel: lest afterward thou mayst not be able to make amends.
    • Lk. 14:31: "Or what king, going to make war against another king, sitteth not down first, and consulteth whether he be able with ten thousand to meet him that cometh against him with twenty thousand?"
    • "reasonable chance condition" [Cajetan says] that for a war to be just, the Prince ought to know he has enough power to be morally certain of victory: first because otherwise he exposes himself to the manifest danger of imposing on his state greater harm than is fair. Quaestio de bello (from Suarez in Franklin).
    • Thompson (1950), the user in this case should have available complete information concerning the cost of each operation as well as an estimate of the contingent gain or loss which will result if the forecast events do not occur. Then, in order to keep the cost of the series of operations at a minimum, decisions should be made by balancing the probability of occurrence of the foreoast event against the ratio of the cost to the contingent gain or loss.
    • Prudent avoidance principle in risk management
    • Precautionary principle in risk management

    Uncertainty, Veracity, Logicality

    According to Ortega y Gasset, veracity and logicality (logicidad) are two dimensions of the concept (The Idea of Principle in Leibniz)

    Keep reading!

    Proportio: Cusanus: Proportionabilia; Erigena: Comportionabilis; Vitruvius: Commodulatio

    Nicolas of Cusa offers an interesting argument which related number, proportion, substance, and accident:
    Iohannes Scotus EriugenaOmnis igitur inquisitio in comparativa proportione facili vel difficili existit; propter quod infinitum ut infinitum, cum omnem proportionem aufugiat, ignotum est. Proportio vero cum convenientiam in aliquo uno simul et alteritatem (otherness) dicat, absque numero intelligi nequit. Numerus ergo omnia proportionabilia includit. Non est igitur numerus in quantitate tantum, qui proportionem efficit, sed in omnibus, quae quovismodo substantialiter aut accidentaliter convenire possunt ac differre. (De Docta Ignorantia, iii).
    and the term pro-portion-abilia reveals its origin as a portion.

    And Scotus Erigena uses (coins?) another related term: comportionabilis (comportionalis) but related to equality whereas proportionabilis is related to number.
    Ut autem decurramus per media ad extrema, rursus dic
    imus, quodneque numerus est, quia non est ad paria (part) vel in paria proportionabilis,
    nec magnitudo, quia non est augmentabilis,
    nec parvitas, quia non est minorabilis,
    nec aequalitas, quia non est comportionabilis,
    nec similitudo, quia nulli rei comparabilis,
    nec dissimilitudo, quia a nullo est discrepabilis,
    nec stat, quia non est res immobilis,
    nec movetur, quia non est volubilis,
    nec silentium agit, quia non est verbo mentali vel vocali reprehensibilis. (Expositiones in Mysticam theologiam S. Dionysii, PL (auctor 810-877))
    and finally Vitruvius, the authority in symmetry explains the origin of proportione:
    Aedium compositio constat ex symmetria, cuius rationem diligentissime architecti tenere debent. Ea autem paritur a proportione, quae graece αναλογια dicitur. Proportio est ratae partis membrorum in omni opere totiusque commodulatio, ex qua ratio efficitur symmetriarum. Namque non potest aedis ulla sine symmetria atque proportione rationem habere compositionis, nisi uti hominis bene figurati membrorum habuerit exactam rationem. (De Architectura,,

    Adaequatio intellectus et rei

    Adaequatio intellectus et rei: adequation of the intellect to things (correspondence). Or to paraphrase Hermann Lotze (Logica #130, p. 156), "thought follows reality". A frequent assumption (and fallacy) in science.


    The idea was expressed by William of Auvergne (1190-1249) as 
    • adaequatio intellectus ad rem (adequation of the intellect to things). 
    Other versions of the apothegm are:
    • veritas est quod est, enuntiativus est natura veritatis et essentiae ejus (Isaac Israeli 823-932), 
    • veritas est dispositio in re exteriore cum est ei aequalitas (Avicenna), 
    • adequatio intellectus et rei (Aquinas following Israeli), 
    • adequatio rei cum intellectu (Albertus Magnus).
    • veritas transcendentalis significat entitatem rei, connotando cognitionem seu conceptum intellectus, cui talis entitas conformatur vel in quo talis res representatur (Suarez)
    • Ordo et connexio idearum idem est ac ordo et connexio rerum (Spinoza)
    Some wording of the schoolmen associated with this concept are:
    • conformitas, correspondentia, convenientia, adaequatio, representatio.


    • Veritas auterm enunciationis seu iudicii nihil aliud est quam conformitas ore factae aut iudicii mente peracto cum ipsa enuntiata seu iudicata (Gassendi)
    • ‘truth’, in the strict sense, refers to the conformity of a thought with its object (Descartes, Letter to Mersenne, 16.x.1639) 
    • Idea vera debet convenire cum suo ideato (Spinoza)
    • Let us be content with looking for truth in the correspondence between the propositions that are in the mind and the things they are about. (Leibniz New Essays, IV, v, 11)
    • Leibniz wrote: "Dialogus de Connexione inter res et verba, et veritatis realitate" in 1677

    20th Century

    Ortega y Gasset in Historia como Sistema argues that:
    Image result for ortega y gasset
    • "the world of reality and the world of thought are each a cosmos corresponding one to the other, each compact and continuous, wherein nothing is abrupt, isolated or inaccessible  ...Western man believes, then, that the world possesses a rational structure, that is to say, that reality possesses an organization coincident with the organization of the human intellect, taking this, of course, in its purest form, that of mathematical reason.” - "En los últimos años del siglo XVI y ... primeros del XVII ... cree, pues, el hombre de Occidente que el mundo posee una estructura racional, es decir, que la realidad tiene una organización coincidente con la del intelecto humano, se entiende, con aquella forma del humano intelecto que es le más pura: con la razón matemática."
    There's now the issue of whether the human intellect is really capable of grasping the infinite complexity of reality. John Duns Scoto phrases this as an objection:
    Atqui impossibilis omnino est talis adaequatio et commesuratio inter potentiam finitam et et obiectum infinitum. Ergo, pariter impossibile est infinitum ab intellecto finiti comprehendi.
    And White
    ...the finite mind is inadequate to grasp the infinite. SC108
    And Cusanus
    Quoniam ex se manifestum est infiniti ad finitum proportionem non esse (De Docta Ignorantia, Cap iii)
    and Trithemius also echoes this argument, having resort to an interesting term "incircumscriptibilem":
    Deum igitur nobis credere potius quae scire vel intelligere omnino necessarium fuit: propterea que penitus impossibile sit illam super divinam & incircumscriptibilem maiestatem comprehendi a nobis qui nihil intelligimus sine ministerio sensuum & discursu rationis. (Liber octo quaestionum ad Maximilianum Caesarem de fide et intellectu)
    and Lactantius:
    incogitabiles... inextricabiles...inaestimabilem potestatem. Dubitet vero aliquis, an quidquam difficile aut impossibile sit Deo, qui tanta tamque mirifica opera providentia excogitavit, virtute constituit, ratione perfecit; nunc autem spiritu sustentet, potestate moderetur, inexcogitabilis, ineffabilis, et nulli alii satis notus quam sibi? (Divinarum Inst., lib i)
    But Arthur Eddington introduces a radically different account:
    We have learnt that the exploration of the
    external world by the methods of physical science leads not to concrete reality, but to a world of symbols.
    Along the same lines James Jeans suggests,
    The essential fact is simply that all the pictures which science now draws of nature, and which alone seem capable of according with observational fact, are mathematical pictures...They are nothing more than pictures – fictions if you like, if by fiction you mean that science is not yet in contact with ultimate reality.” 

    I. Menocchio

    From Mennochio De Praesumptione, p. 6

    • Presumption: pre (before) sumptio (saying), that which we take for true before it has been legitimately proved.
    • Indicium: is born from plausibility.

    Baldus de Ubaldis on Components of Argument (praesumptio, fictio, indicium, coniectura, adminiculum)

    ¶Deinde debemus diffinire quædam alia: quia antequam perveniatur ad cognitionem rei alicuius: homo transit per multos gradus, & per multa media,& paulatim per partes devenit in notitiam veri, & ideo diffinio hæc,quæ sequuntur.

    Et primò quid est suspicio? Respondo quòd suspicio est aliqualis applicatio animi ad aliquid cum vehementi titubatione. Vel sic, Suspicio est motus quidam mentis ad aliquid cum vehementi titubatione: non tamen hgc diffinitio procedit loquendo de materia suspecti tutoris vel alterius administratoris, quia illa suspicio non est simplex suspicio, sed est suspicio permixta cum coniecturis, præsumptionibus, & aliis verisimilibus, ex quibus quis ab officio removet.

    ¶Quæro, quid est praesumptio? dico quòd praesumptio hominis est quidam conceptus causatus in mente ab aliqua probabili coniectura
    • I ask what is a presumption? I say that it is a concept formed in man's mind by a particular plausible conjecture.
    ¶Præsumptio vero iuris est duplex, una est presumptio simplex, que sic diffinitur: præsumptiò iuris simplex est similitudo quedam sufficiens ad rem dubiam, de qua creduntur credenda. ¶ Presumptio verò iuris & de iure sic diffinit, Præsumptio iuris, & de iure est status à iure, promulgatus ex indubia coniectura.

    ¶Fictio verò sic diffinitur, Fictio est falsitas pro veritate accepta, ex specialissima & iustissima causà in iure èxpressa.
    • Fiction is a falsity accepted as true for a very special and just cause expressed in Law.
    ¶Indicium vero est duplex, semiplenum seu dubitatum, & plenum seu indubitatum. indicium semiplenum est praesumptio fortiter movens animum ad aliquid credendum vel discredendum. Indicium vero plenum est demonstratio rei per signa sufficientia, per quæ animus in aliquo tanquàm in existente quiescit, & plus investigare non curat.

    Argumentum autem sic diffinitur, Argumentum est propositio ex aliquibus existentibus resultans, ad propositum ostendendum seu concludendum: unde quando plura indicia vel plures propositiones, vel plures præsumptiones, vel plures testes ad unam conclusionem ad aliquid probandum copulantur: istud sic copulare dicitur argumentatio, sive argumentum, quod est collatio plurium ad unam conclusionem.

    ¶Adminiculum est duplex, scilicet vehemens, & non vehemens. Adminiculum vehemens sic diffinitur, Adminiculum vehemens et suppletio defectus probationis, per sé folum non operans, fed cum alia probatione cooperans, probatur hoc extra. Adminiculum non vehemens,est aliqualis confirmatio rei probabilis, vel est aliqualis confirmatio veri tendens ad aliquid suspendendum de defectu probationis: &talia adminicula per se sola etiam plura non sufficiunt.

    ¶Coniecturas vero sic diffinitur, Coniectura est acceptio seu reputatio veri, ex aliquo alio sic verisimiliter ordinato: sicut per circulum coniecturamur tabernam, per habitum meretricem.

    His vero omnibus opposita sunt nescientiam & error. Nescientia autem sic diffinitur, Nescientia seu ignorantia est intellectus non determinans se ad verum,vel falsum.

    ¶Error autem est privatio intellecto, determinans se ad falfum: vel est priuatio causata ad intellectum per falsam imaginationem. Ideo autem dixi, privatio: quia omne de quo non est verum, est quædam privatio, unde dictum est de qualibet probatione, quaedam sunt manifesta,quædam sunt,quæ sunt notoria, de quibus oportet videre verisimile sic diffinitur.

    ¶Verisimile est id, quod vero videtur simile, non verisimile, sic diffinitur, Non verisimile est quod à vero dissimile videtur.

    Baldus ad liber ix Codicis (p. 212-213)

    Josiah W. Gibbs, a dignified gentleman

    File:JWGibbs.jpgI was interested by the depictions that students made of the character of Josiah Gibbs, arguably one of the best American scientists. It seems as if they were the exact opposite of today's scholars in the US who love to christen their group of students with their last name: "Smith Lab" or much worse "Smith Gang". Here's some of the observations (thanks Wikipedia):
    Gibbs was not an advertiser for personal renown nor a propagandist for science; he was a scholar, scion of an old scholarly family, living before the days when research had become résearch ... Gibbs was not a freak, he had no striking ways, he was a kindly dignified gentleman.
    — E. B. Wilson, 1931
    and also:
    was always neatly dressed, usually wore a felt hat on the street, and never exhibited any of the physical mannerisms or eccentricities sometimes thought to be inseparable from genius ... His manner was cordial without being effusive and conveyed clearly the innate simplicity and sincerity of his nature.
    — Lynde Wheeler, 1951
    Unassuming in manner, genial and kindly in his intercourse with his fellow-men, never showing impatience or irritation, devoid of personal ambition of the baser sort or of the slightest desire to exalt himself, he went far toward realizing the ideal of the unselfish, Christian gentleman. In the minds of those who knew him, the greatness of his intellectual achievements will never overshadow the beauty and dignity of his life.
    — H. A. Bumstead, 1903
    It's difficult for today's scholars to be described in that manner many of whom happily assume the role of salesmen and celebrities air to impress the NSF, NIH, private donors,.... and get their bucks.

     Gibbs free energy. (Right) Thermodynamic Surface done by Maxwell from Gibbs functions.


    Futuritionis is a latin term which was related to the concept of probability and also of potentiality. Was it related to the future prediction of something?

    The full spectrum of terms is: pastness (praeteritionis), presentness (praesentalitatis), and futurity (futuritionis).

    1. Mathematical Meaning 

    Here's some texts (J. Bernoulli Ars Conjectandi):

    Certitudo rei cujusvis spectatur vel objective & in se; nec aliud significat, quam ipsam veritatem existentiae aut futuritionis illius rei: vel subjective & in ordine ad nos; & consistit in mensura cognitionis nostræ circa hanc veritatem. . . . Probabilitas enim est gradus certitudinis, & ab hac differt ut pars a toto.
    Translation Sylla p. 315
    The certainty of anything is considered either objectively and in itself or subjectively and in relation to us. Objectively, certainty means nothing else than the truth of the present or future existence of the thing. Subjectively, certainty is the measure of our knowledge concerning this truth. . . . Probability, indeed, is degree of certainty, and differs from the latter as a part from the whole.
    Translation mine:
    The certitude of things obtains either as objective, i.e. essentially, meaning nothing but the reality of the thing's existence or futurity, or subjective, i.e. arranged to us, consisting in the extent of our knowledge about that reality.
    Everything under the Sun that is or has been, past, present, or future, in itself and objectively, always has total certitude. It is apparent what is present and past, for in their very existing or having existed, cannot be otherwise. Nor should be argued of future things.

    2. Theological Meaning

    For one thing, the locution Futurition is almost always found in theological treatises. One instance of such discussion is afforded by a lengthy philosophical consideration by Richard Baxter (1615-1691) in An ANSWER TO Mr. Polehill's Exceptions about Futurition.

    In Reformed circles: Leidecker, Burman, 

    Another example is the Franciscan Bonaventura: ratione futuritionis. (Sent., Bk. I, d. 41, a. 2, q. 1, 4 arg. ad opp.).

    Or Aquinas ratio futuritionis futurorum.

    Or Tomasz Młodzianowski (1666)
    Si autem nulla futura fuissent futura, processisset verbum ex cognitione non futurae futuritionis futurorum.

    3. Philosophical Meaning

    Kant following Leibniz said:

    The events which occur in the world have been determined with such certainty, that divine foreknowledge, which is incapable of being mistaken, apprehends both their futurition (futuritio) and the impossibility of their opposite. New Elucidation 1:400.


    Jacob Bernoulli. The Art of Conjecturing, together with Letter to a Friend on Sets in Court Tennis. Johns Hopkins University Press, Baltimore, 2006. Translation of [22] and commentary by Edith Sylla. 9, 99, 100

    Funes and the Shas Pollak

    Wiki dixit: "Shas Pollak were Jewish mnemonists who ... memorized the exact layout of words in more than 5,000 pages of the 12 books of the standard edition of the Babylonian Talmud."

    Funes the Memorious remembered absolutely everything, not just what he read, but also each second of his life. In order to recount what had happened to him the day before, he'd spend 24 hours.

    Procrustean Bed

    A problem is stretched to fit in a preconceived framework. (Procrustes either cut the limbs of opponents or stretched them hammering to fit in on one of his beds).

    Presumptions, Presuppositions, Assumption

    Presumptions, Presuppositions, Assumption, Assertion, Postulate differences.

    Presumption and Probability in Talmud

    Hazakah indicates presumption and sometimes coincides with rabbah, or rubba, or rov which is the rule of the majority, or a type of frequentist probability.

    There's an 11 volume work on hazakah by R. Jacob L. Kroch.


    Kroch, Jacob Leib ben Shemaiah. Encyclopaedia Judaica. . Encyclopedia.com. (February 19, 2017).
    J.L. Kroch, Hazakah Rabbah, 1 (1927), introduction by P.J. Kohn; J.L. Kroch, Halakhah Rabbah, 2 (1966), introduction by A. Krauss.

    Argument moves in Babylonian Talmud

    Image result for talmud arguments
    The Talmud provides interesting techniques of logical analysis and debate. For all its logical value, Talmudists devised and applied conceptual instruments to unravel tacit implications or ramifications in arguments. The arguments put forward are subjected to these conceptual instruments in order to test their cogency or validity to the extreme. These instruments frequently are casuistic hypothetical situations, applications of Mishna, Baraitot, etc. To the superficial eye, Talmudic arguments seem artificial and not a few times bordering on the absurd. However, these conceptual instruments have a surgical function which they perform well and are very helpful for the patient student.

    R. L. Jacobs (The Talmudic Argument, CUP, 13f) asserts that the following argumentative moves are found in Talmud:
    • Arguments based on pure reason
      1. Argument from authority
      2. Argument from comparison
      3. Argument by differentiation
      4. on the contrary argument
      5. acceptance of an argument in part
      6. Argument based on an opponent's position
      7. Argument exposing the flaws in an opponent's argument
    • Arguments based on the facts or interpretation of facts
      1. Argument based on geographical or historical conditions
      2. Argument based on the analysis of states of mind
    • Other types of arguments
      1. Readmission of an argument that has been previously rejected
      2. Argument against a statement of the obvious
    • Arguments from Texts
      1. Argument to resolve a contradiction between two sources
      2. Argument by textual emendation (Mishnah)
      3. Argument from the principle of literary economy
    • Versions of the same argument
      1. argument presented by different teachers
      2. consequences of different arguments
      3. limited application of an argument

    Majority, Multitude, and Probability in Biblical and Talmudic writers

    R. Louis Jacobs observes that:
    Thus probability in Hebrew is rab (רָב, abundance). It seems an interesting vindication of the frequentist version of probability! But specifically, it reinforces the intuitive idea that probability are statements about aggregates.

    Interestingly enough, there is the the Justinian Digest a rule of thumb to overcome evidential deadlocks enunciated by Paulus that hinges on the concept of the majority (plerumque):
    In obscuris, inspici solere quod, verisimilius est, aut quod plerumque fieri solet. Corpus Iuris Civilis, Justinian Digest 50.17.114.
    (when there is obscurity, we usually regard that what has appearance of truth or what is mostly done) 

    Jacobs, L. (1984) The Talmudic Argument. Cambridge UP, p. 50f.

    De ars ignorandi

    File:SebastianCastellio.jpgSebastian Castellio included curious but challenging aspects in the title to his treatise: De Arte dubitandi & confidendi, ignorandi, & sciendi published in Basel in 1563. I call attention to:

    ars ignorandi: 'De ignorando hoc dico: ignorare nobis ea licet, quae homini non sunt ad salutem necessaria, quae multa esse nemo sapiens negabit' (p.50) and also 'Ignorare autem ea licet, quae nec a Deo praecepta, nec homini ad Deum cognoscendum officumve suum discendum aut faciendum iusticiaque fungendum, sun necessaria' (p.51).

    Of course, these observations remind one of  1 Cor 2:2: Non enim judicavi me scire aliquid inter vos, nisi Jesum Christum, et hunc crucifixum.

    • Castellion, S. (1981). De arte dubitandi et confidendi, ignorandi et sciendi. Brill.
    • Castellion, S. (1953). De l'art de douter et de croire, d'ignorer et de savoir. Traduit de l'original Latin par Charles Baudouin. Genéve, Jeheber.
    • Calvetti, C. G. (2005). Il testamento dottrinale di Sebastiano Castellion e l'evoluzione razionalistica del suo pensiero. Vita e Pensiero.
    • For the disquisitions of Castellion on hazard, or chance, see Bru, B. (2006) The Bernoulli Code. J. Electronique d'Histoire des Probabilites et de la Statistique. 2, 1. Here 

    Recessional by RUDYARD KIPLING


    Related Poem Content Details

    God of our fathers, known of old, 
       Lord of our far-flung battle-line, 
    Beneath whose awful Hand we hold 
       Dominion over palm and pine— 
    Lord God of Hosts, be with us yet, 
    Lest we forget—lest we forget! 

    The tumult and the shouting dies; 
       The Captains and the Kings depart: 
    Still stands Thine ancient sacrifice, 
       An humble and a contrite heart. 
    Lord God of Hosts, be with us yet, 
    Lest we forget—lest we forget! 

    Far-called, our navies melt away; 
       On dune and headland sinks the fire: 
    Lo, all our pomp of yesterday 
       Is one with Nineveh and Tyre! 
    Judge of the Nations, spare us yet, 
    Lest we forget—lest we forget! 

    If, drunk with sight of power, we loose 
       Wild tongues that have not Thee in awe, 
    Such boastings as the Gentiles use, 
       Or lesser breeds without the Law— 
    Lord God of Hosts, be with us yet, 
    Lest we forget—lest we forget! 

    For heathen heart that puts her trust 
       In reeking tube and iron shard, 
    All valiant dust that builds on dust, 
       And guarding, calls not Thee to guard, 
    For frantic boast and foolish word— 
    Thy mercy on Thy People, Lord! 

    Source: A Choice of Kipling's Verse (1943)

    Marchetti constant: Anthropological Invariants in Travel Behavior, and Zahavi's UMOT

    "Marchetti's constant is the average time spent by a person for commuting each day, which is approximately one hour." [Source Wiki].

    Zahavi "fundamental equation of travel demand": 

    M / T = v . c

    where M is travel-money budget, T is daily travel-time budget, v is mean daily speed, and c is average cost per unit distance traveled. In words:

    What individuals are willing to spend in money (M) and time (T) = product individual would like to purchase from system supply in its performance (v) and price (c) of using it.

    Explanation: if travel system is slow, then, people exhaust travel-time budget long before the travel-money budget. This disequilibrium is solved by buying faster travel (e.g. a car, or more cars)

    Marchetti paper Here
    Zahavi's report here and here.

    Talmudic reasoning

    There are principles of Talmudic hermeneutics which constitute an interesting method alternative, although related in some instances, to the syllogism. Three groups are listed: the 7 rules of Hillel, the 13 rules of R. Ishmael, and the 32 rules of R. Eliezer b. Jose Ha-Gelili (all come from the Jewish Encyclopedia 1906).

    7 rules of Hillel 
    1. Ḳal (ḳol) wa-ḥomer (קל וחומר): "Argumentum a minori ad majus" or "a majori ad minus"; corresponding to the scholastic proof a fortiori.
    2. Gezerah shawah (גזירה שוה): Argument from analogy. Biblical passages containing synonyms or homonyms are subject, however much they differ in other respects, to identical definitions and applications.
    3. Binyan ab mi-katub eḥad (בנין אב מכתוב אחד): Application of a provision found in one passage only to passages which are related to the first in content but do not contain the provision in question.
    4. Binyan ab mi-shene ketubim (בנין אב מכתוב אחד): The same as the preceding, except that the provision is generalized from two Biblical passages.
    5. Kelal u-Peraṭ and Peraṭ u-kelal (כלל ופרט ופרט וכלל): Definition of the general by the particular, and of the particular by the general.
    6. Ka-yoẓe bo mi-maḳom aḥer (כיוצא בו ממקום אחר): Similarity in content to another Scriptural passage.
    7. Dabar ha-lamed me-'inyano (דבר הלמד מעניינו): Interpretation deduced from the context.
    13 rules of R. Ishmael b. Elisha: (from Jewish Encyclopedia 1906)
    1. Ḳal wa-ḥomer: Identical with the first rule of Hillel.
    2. Gezerah shawah: Identical with the second rule of Hillel.
    3. Binyan ab: Rules deduced from a single passage of Scripture and rules deduced from two passages. This rule is a combination of the third and fourth rules of Hillel.
    4. Kelal u-Peraṭ: The general and the particular.
    5. u-Peraṭ u-kelal: The particular and the general.
    6. Kelal u-Peraṭ u-kelal: The general, the particular, and the general.
    7. The general which requires elucidation by the particular, and the particular which requires elucidation by the general.
    8. The particular implied in the general and excepted from it for pedagogic purposes elucidates the general as well as the particular.
    9. The particular implied in the general and excepted from it on account of the special regulation which corresponds in concept to the general, is thus isolated to decrease rather than to increase the rigidity of its application.
    10. The particular implied in the general and excepted from it on account of some other special regulation which does not correspond in concept to the general, is thus isolated either to decrease or to increase the rigidity of its application.
    11. The particular implied in the general and excepted from it on account of a new and reversed decision can be referred to the general only in case the passage under consideration makes an explicit reference to it.
    12. Deduction from the context.
    13. When two Biblical passages contradict each other the contradiction in question must be solved by reference to a third passage.
    Rules seven to eleven are formed by a subdivision of the fifth rule of Hillel; rule twelve corresponds to the seventh rule of Hillel, but is amplified in certain particulars; rule thirteen does not occur in Hillel, while, on the other hand, the sixth rule of Hillel is omitted by Ishmael.

    32 Rules of Eliezer b. Jose Ha-Gelili
    1. Ribbuy (extension): The particles 'et," "gam," and "af," which are superfluous, indicate that something which is not explicitly stated must be regarded as included in the passage under consideration, or that some teaching is implied thereby.
    2. Mi'uṭ (limitation): The particles "ak," "raḳ" and "min" indicate that something implied by the concept under consideration must be excluded in a specific case.
    3. Ribbuy aḥar ribbuy (extension after extension): When one extension follows another it indicates that more must be regarded as implied.
    4. Mi'uṭ aḥar mi'uṭ (limitation after limitation): A double limitation indicates that more is to be omitted.
    5. Ḳal wa-ḥomer meforash: "Argumentum a minori ad majus," or vice versa, and expressly so characterized in the text.
    6. 6. Ḳal wa-ḥomer satum: "Argumentum a minori ad majus," or vice versa, but only implied, not explicitly declared to be one in the text. This and the preceding rule are contained in the Rules of Hillel, No. 1. 
    7. identical with Rule 2 of Hillel.
    8. identical with Rule 3 of Hillel 
    9. Derek ḳeẓarah: Abbreviation is sometimes used in the text when the subject of discussion is self-explanatory.
    10. Dabar shehu shanuy (repeated expression): Repetition implies a special meaning.
    11. Siddur she-neḥlaḳ: Where in the text a clause or sentence not logically divisible is divided by the punctuation, the proper order and the division of the verses must be restored according to the logical connection.
    12. Anything introduced as a comparison to illustrate and explain something else, itself receives in this way a better explanation and elucidation.
    13. When the general is followed by the particular, the latter is specific to the former and merely defines it more exactly (comp. Rules of Hillel, No. 5).
    14. Something important is compared with something unimportant to elucidate it and render it more readily intelligible.
    15. Same as Rule 13 of R. Ishmael.
    16. Dabar meyuḥad bi-meḳomo: An expression which occurs in only one passage can be explained only by the context. This must have been the original meaning of the rule, although another explanation is given in the examples cited in the baraita.
    17. A point which is not clearly explained in the main passage may be better elucidated in another passage.
    18. A statement with regard to a part may imply the whole.
    19. A statement concerning one thing may hold good with regard to another as well.
    20. A statement concerning one thing may apply only to something else.
    21. If one object is compared to two other objects, the best part of both the latter forms the tertium quid of comparison.
    22. A passage may be supplemented and explained by a parallel passage.
    23. A passage serves to elucidate and supplement its parallel passage.
    24. When the specific implied in the general is especially excepted from the general, it serves to emphasize some property characterizing the specific.
    25. The specific implied in the general is frequently excepted from the general to elucidate some other specific property, and to develop some special teaching concerning it.
    26. Mashal (parable).
    27. Mi-ma'al: Interpretation through the preceding.
    28. Mi-neged: Interpretation through the opposite.
    29. Gemaṭria: Interpretation according to the numerical value of the letters.
    30. Noṭariḳon: Interpretation by dividing a word into two or more parts.
    31. Postposition of the precedent. Many phrases which follow must be regarded as properly preceding, and must be interpreted accordingly in exegesis.
    32. Many portions of the Bible refer to an earlier period than do the sections which precede them, and vice versa.
    • 7 Rules of Hillel: here
    • 13 Rules of R. Ishmael: here
    • 32 Rules of R. Eliezer: here