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A FORTIORI LOGIC

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A FORTIORI LOGIC

Appendix 4 – A fortiori discourse by Plato and Aristotle

1. Plato

2. Aristotle

I found a number of instances of a fortiori discourse by Plato (15) and Aristotle (80). So it is evident that such discourse was relatively common in the Greece of the 5th to 4th century BCE. Note that Plato (or Socrates before him, since Plato attributes most of his to Socrates) uses a fortiori language, but apparently does not reflect upon it anywhere; whereas Aristotle, in a couple of instances, namely Topics 2:10 and 3:6 and Rhetoric 2:23, engages in a bit of reflection on the subject.

This research was primarily based on fifteen key phrases commonly indicative of a fortiori argument, namely: a fortiori, all the more/less, (how/so) much more/less, (how/so) much the more/less. Additionally, to be on the safe side, I looked for cases involving the less specific phrases: even more/less[1], more/less so (and later also for: still more/less, still the more/less). Another important source of discovery was the Kneales’s list of references in Aristotle’s works, which points to many a fortiori arguments that do not involve any of the key phrases. It is of course conceivable that there are still more a fortiori arguments, differently phrased, in both Plato’s and Aristotle’s works.

It must be said, however, that not all occurrences of a string of words that includes a key phrase of a fortiori are necessarily indicative of a fortiori argument. Consider for examples, in Plato’s LYSIS, the sentence “Suppose that I were to cover your auburn locks with white lead… They would only appear to be white… But that would not make them at all the more white, notwithstanding the presence of white in them;” and in his THE SYMPOSIUM, “Do you think that my head is so full of the theatre as not to know how much more formidable to a man of sense a few good judges are than many fools?” Although both these sentences contain word strings that usually signal a fortiori argument, neither of them intends such argument. In the first sentence, the string ‘all the more’ is accidental, part of the larger string ‘not at all the more’, which simply means ‘not more’. In the second sentence, although there is a comparative proposition, it is not used as the major premise of an argument; an argument could be formed with it, but none is.

Each instance of apparent a fortiori argument found is listed below, with an appropriate reference (as available in my sources). I interpret the text, so as to more clearly bring out its a fortiori form, proposing the minor premise and conclusion (given these, the major premise is obvious and so I left it out). The original text is in inverted commas, while my interpretation is without. Needless to say, some interpretations are open to debate; but the exact number of cases is not of great importance, here.

The arguments are classified into moods by means of the following symbols in {} brackets: + for positive, – for negative, s for subjectal, p for predicatal, a for antecedental, c for consequental, and & for a crescendo. If the predicate (or consequent) is the same in premise and conclusion, the mood is subjectal (or antecedental); whereas, if the subject (or antecedent) is the same in premise and conclusion, the mood is predicatal (or consequental).

1. Plato

Regarding Plato (Greece, ca. 428-347 BCE), I first looked for the fifteen key phrases of a fortiori discourse in an eBook called The Complete Plato[2], and only found 13 instances: a fortiori (1), all the more (4), much more (1), much less (6), so much the more (1). I then tried out some vaguer word strings, and found 2 additional instances: even more (1), more so (1). So, assuming my interpretations are correct, there are in Plato’s works at least 15 instances of a fortiori argument. There may, of course, be many more cases, which I have not discovered because I mistinterpreted the said phrases in them or because they do not involve these phrases.

Of the 15 cases found, 9 are positive subjectal (including 4 a crescendo), 1 is negative subjectal, 4 are negative predicatal, and 1 is negative consequental; no other forms are exemplified. Note that all 15 arguments are ontical; 4 of them might be characterized as ethical (Crito, Theaetetus, Gorgias, Laws Bk 12); 3 others concern cognitive abilities, but they should not be classed as logical-epistemic (Meno, Cratylus, Laws Bk 7). Also note, 9 of the instances are attributed to Socrates (Greece, ca. 469-399 BCE), while 4 were spoken by an unnamed Athenian, 1 by Alcibiades, and 1 by Cebes. The following are the cases found:

CRITO (p. 39). Socrates speaks to Crito: “our country is more to be valued and higher and holier far than mother or father or any ancestor… and if he may do no violence to his father or mother, much less may he do violence to his country.” {+s} If parents are to be valued etc. enough that no one may do them violence, then country is valued etc. enough that no one may do it violence. Notice in this example the explicit major premise that country is more to be valued than parents.[3]

GORGIAS (p. 195). Socrates speaks to Callicles: “If a man who is afflicted by great and incurable bodily diseases… is in no way benefited by him in having been saved from drowning, much less he who has great and incurable diseases, not of the body, but of the soul, which is the more valuable part of him.” {+s} If a man whose body is very sick has a life bitter enough to be not worth saving, then a man who is spiritually very sick has a life bitter enough to be not worth saving. Notice the statement that the soul is more valuable than the body, from which derives the major premise.

MENO (p. 252). Socrates speaks to Meno: “For I literally do not know what virtue is, and much less whether it is acquired by teaching or not.” {–p} If I am not wise enough to know what virtue is, then I am not wise enough to know how it is acquired.

EUTHYDEMUS (p. 298). Socrates speaks to Dionysorodus: “I am not a match for one of you; and a fortiori I must run away from two.” {–p} If I am not skilled enough for one, then I am not skilled enough for two.

CRATYLUS (p. 341). Socrates: “Well, but do you suppose that you will be able to analyse them in this way? for I am certain that I should not.” Hermogenes: “Much less am I likely to be able.” {+s} If Socrates is not intelligent enough to do the analysis, then Hermogenes is not intelligent enough to do it.

PHAEDO (p. 381). Cebes speaks to Socrates: “I would therefore rather not rely on the argument from superior strength to prove the continued existence of the soul after death. For granting even more than you affirm to be possible, and acknowledging not only that the soul existed before birth, but also that the souls of some exist, and will continue to exist after death, and will be born and die again and again, and that there is a natural strength in the soul which will hold out and be born many times—nevertheless, we may be still inclined to think that she will weary in the labours of successive births, and may at last succumb in one of her deaths and utterly perish; and this death and dissolution of the body which brings destruction to the soul may be unknown to any of us, for no one of us can have had any experience of it: and if so, then I maintain that he who is confident about death has but a foolish confidence, unless he is able to prove that the soul is altogether immortal and imperishable. {–c} If the various possibilities I grant are not conclusive enough to imply impossible the death of the soul, then the various possibilities you grant are not conclusive enough for that.

PHAEDRUS (p. 430). Socrates speaks to Phaedrus: “the qualities of their god they attribute to the beloved, wherefore they love him all the more.” {+s &} If someone ordinary has qualities enough to be loved, then someone godly has qualities enough to be loved – even more intensely.

THE SYMPOSIUM (p. 495). Alcibiades: “If Ajax could not be wounded by steel, much less he by money.” {–p} If Ajax is not vulnerable enough to be wounded by steel, then he is not vulnerable enough to be wounded by money.

THEAETETUS (p. 528). Socrates speaks to Theodorus: “Let us tell them that they [rogues] are all the more truly what they do not think they are because they do not know it.” {+s} If a rogue who is aware of his roguery is dishonest enough to be censured, then a rogue who is not aware of his roguery is dishonest enough to be censured.

THE REPUBLIC (Book 3, p. 858). Socrates to Adeimantus: “These [harmonies] then… must be banished; they are of no use, even to women who have a character to maintain, and much less to men.” {–p} If they are not becoming enough to be of use to women, then they are not becoming enough to be of use to men.[4]

THE REPUBLIC (Book 6, p. 933). Socrates speaks to Adeimantus: “Why, I said, we know that all germs or seeds, whether vegetable or animal, when they fail to meet with proper nutriment or climate or soil, in proportion to their vigour, are all the more sensitive to the want of a suitable environment, for evil is a greater enemy to what is good than to what is not…. There is reason in supposing that the finest natures, when under alien conditions, receive more injuries than the inferior, because the contrast is greater.” {+s &} If a coarse organism is sensitive enough to be negatively affected by an unsuitable environment, then a finer organism is sensitive enough to be negatively affected by an unsuitable environment – even more negatively.

THE LAWS (Book 1, p. 1058). “Athenian: In the first place, then, the revellers as well as the soldiers will require a ruler? Cleinias: To be sure; no men more so.” {+s} If other men are dependent enough to require a ruler, then the revellers as well as the soldiers are dependent enough to require a ruler.

THE LAWS (Book 3, p. 1087). Athenian speaks to Cleinias: “The fewness of the survivors at that time would have made them all the more desirous of seeing one another.” {+s &} If many survivors are lonely enough to desire seeing others, then few survivors are lonely enough to desire seeing others – even more intensely.

THE LAWS (Book 7, p. 1186). Athenian. “Any young man, and much more any old one, when he sees or hears anything strange or unaccustomed, does not at once run to embrace the paradox, but he stands considering….” {–s} If a young man encountering the unexpected is not sufficiently perplexed to refrain from reflection, then an old one encountering the unexpected is not sufficiently perplexed to refrain from reflection.

THE LAWS (Book 12, p. 1315). Athenian speaks to Cleinias: “And if he be seen to have come home neither better nor worse, let him be praised at any rate for his enthusiasm; and if he be much better, let him be praised so much the more.” {+s &} If a man who is neither better nor worse is worthy enough to be praised, then a man who has improved is worthy enough to praised – indeed praised more.

I found no cases in the following books: Apology, Charmides, Laches, Lysis, Euthyphro, Menexenus, Ion, Protagoras, Parmenides, Sophist, Statesman, Philebus, Timaeus, Critias.

After writing the above, it occurred to me that I might find some more instances by searching for the four key phrases: ‘still (the) more/less’. Looking into the same document, I found no occurrences of ‘still the more’ or ‘still the less’. However, the string ‘still more’ occurred 38 times, and ‘still less’ occurred 8 times. Even so, I would not consider all these occurrences as pointing to a fortiori discourse. Very often, these expressions seem to intend no more than ‘some amount more’ and ‘some amount less’, respectively – i.e. their intent is only to signify some greater or lesser degree of something compared to something else, without any inference of the one from the other being claimed. Still, in some cases, inference does seem to be intended, and these may be considered as a fortiori arguments. The reader is encouraged to look for these and analyze them; I will give just one of them as an example, without further analysis:

THE APOLOGY (p. 28) Socrates: “Some one will say: Yes, Socrates, but cannot you hold your tongue, and then you may go into a foreign city, and no one will interfere with you? Now I have great difficulty in making you understand my answer to this. For if I tell you that to do as you say would be a disobedience to the God, and therefore that I cannot hold my tongue, you will not believe that I am serious; and if I say again that daily to discourse about virtue, and of those other things about which you hear me examining myself and others, is the greatest good of man, and that the unexamined life is not worth living, you are still less likely to believe me.”

2. Aristotle

For research into Aristotle (Greece, 384-322 BCE), I used a pdf copy of The Works of Aristotle[5]. I searched through that for the fifteen key phrases of a fortiori discourse, and found 40 instances: a fortiori (12), all the more (22), how much more (2), so much more (1), much more (2), much less (1). I then tried out some vaguer word strings, and found 3 additional instances: even more (2), more so (1). Additionally, I found many instances (37) of a fortiori argument not involving these key phrases, mostly thanks to the already mentioned Kneales’ list of references. So, assuming my interpretations are correct, there are in Aristotle’s works at least 80 instances of a fortiori argument. There may, of course, be many more cases, which I have not discovered because I mistinterpreted the said phrases in them or because they do not involve these phrases.

Of the 80 cases found, 48 are positive subjectal (including 7 a pari and 8 a crescendo), 22 are negative subjectal (including 4 a pari), 5 are positive predicatal, 3 are negative predicatal, and 2 are positive antecedental. Note that only 17 of these arguments are logical-epistemic, while the rest are ontical. All the relevant passages found are listed below[6].

POSTERIOR ANALYTICS (Tr. G. R. G. Mure). 3 instances, of which 1 is positive subjectal, 1 is positive predicatal and 1 is negative predicatal. All 3 arguments may be considered logical-epistemic.

1:1 (p. 222). “A man is asked, ‘Do you, or do you not, know that every pair is even?’ He says he does know it. The questioner then produces a particular pair, of the existence, and so a fortiori of the evenness, of which he was unaware.” {+p} If this particular pair is manifest enough to convince you of the existence of which you were unaware, then it is manifest enough to convince you of the evenness of which you were unaware. This argument may be considered logical-epistemic.

1:3 (p. 228). “Two premisses constitute the first and smallest foundation for drawing a conclusion at all and therefore a fortiori for the demonstrative syllogism of science.” {–p} If less than two premises are not informative enough to draw a conclusion at all, then less than two premises are not informative enough for the demonstrative syllogism of science. This argument may be considered logical-epistemic.

1:10 (pp. 242-3). “That which expresses necessary self-grounded fact, and which we must necessarily believe, is distinct both from the hypotheses of a science and from illegitimate postulate – I say ‘must believe’, because all syllogism, and therefore a fortiori [all] demonstration, is addressed not to the spoken word, but to the discourse within the soul, and though we can always raise objections to the spoken word, to the inward discourse we cannot always object.” {+s} If syllogism is inward discourse enough to be necessarily believed, then other forms of demonstration are inward discourse enough to be necessarily believed. This argument may be considered logical-epistemic.

TOPICS (Tr. W. A. Pickard-Cambridge). 24 instances, of which 15 are positive subjectal (including 7 a pari), 9 are negative subjectal (including 4 a pari). At least 8 of these arguments may be considered logical-epistemic.

2:10 (pp. 372-3). Moreover, argue from greater and less degrees…. If one predicate be attributed to two subjects; then supposing it does not belong to the subject to which it is the more likely to belong, neither does it belong where it is less likely to belong; while if it does belong where it is less likely to belong, then it belongs as well where it is more likely. Again: If two predicates be attributed to one subject, then if the one which is more generally thought to belong does not belong, neither does the one that is less generally thought to belong; or, if the one that is less generally thought to belong does belong, so also does the other. Moreover: If two predicates be attributed to two subjects, then if the one which is more usually thought to belong to the one subject does not belong, neither does the remaining predicate belong to the remaining subject; or, if the one which is less usually thought to belong to the one subject does belong, so too does the remaining predicate to the remaining subject. Moreover, you can argue from the fact that an attribute belongs, or is generally supposed to belong, in a like degree, in three ways, viz. those described in the last three rules given in regard to a greater degree. For supposing that one predicate belongs, or is supposed to belong, to two subjects in a like degree, then if it does not belong to the one, neither does it belong to the other; while if it belongs to the one, it belongs to the remaining one as well. Or, supposing two predicates to belong in a like degree to the same subject, then, if the one does not belong, neither does the remaining one; while if the one does belong, the remaining one belongs as well. The case is the same also if two predicates belong in a like degree to two subjects; for if the one predicate does not belong to the one subject, neither does the remaining predicate belong to the remaining subject, while if the one predicate does belong to the one subject, the remaining predicate belongs to the remaining subject as well.”

Here, Aristotle has two sets of subjectal a fortiori argument, one in which one term is superior to the other, and the other in which the terms are equal. In the each set, three examples are given, each of them in positive and negative forms. Thus, there are a total of 12 distinct subjectal arguments, including 6 positives (of which 3 are a pari) and 6 negatives (of which 3 are a pari). That is counting a pari arguments in only one direction, though his wording allows for two. For my more detailed interpretation of these arguments, see the main text. At least 6 of these arguments may be considered logical-epistemic.

3:6 (pp. 388-9). “Moreover you should judge by means of greater or smaller or like degrees: for if some member of another genus exhibit such and such a character in a more marked degree than your object, while no member of that genus exhibits that character at all, then you may take it that neither does the object in question exhibit it; e.g. if some form of knowledge be good in a greater degree than pleasure, while no form of knowledge is good, then you may take it that pleasure is not good either. Also, you should judge by a smaller or like degree in the same way: for so you will find it possible both to demolish and to establish a view, except that whereas both are possible by means of like degrees, by means of a smaller degree it is possible only to establish, not to overthrow. For if a certain form of capacity be good in a like degree to knowledge, and a certain form of capacity be good, then so also is knowledge; while if no form of capacity be good, then neither is knowledge. If, too, a certain form of capacity be good in a less degree than knowledge, and a certain form of capacity be good, then so also is knowledge; but if no form of capacity be good, there is no necessity that no form of knowledge either should be good. Clearly, then, it is only possible to establish a view by means of a less degree.”

Here, Aristotle first presents an abstract example of negative subjectal a fortiori argument with unequal terms, then a more concrete illustration of it. Next, he illustrates a pari subjectal argument in positive and negative forms; but his illustrations go only in one direction. Finally, he offers an example of positive subjectal argument with unequal terms. Thus, there are a total of 5 distinct subjectal arguments, including 2 positives (of which 1 is a pari) and 3 negatives (of which 1 is a pari). For further comments on these arguments, see the main text.

4:6 (pp. 415). “On the other hand, the comparison of the genera and of the species one with another is of use: e.g. supposing A and B to have a like claim to be genus, then if one be a genus, so also is the other. Likewise, also, if what has less claim be a genus, so also is what has more claim: e.g. if ‘capacity’ have more claim than ‘virtue’ to be the genus of self-control, and virtue be the genus, so also is capacity. The same observations will apply also in the case of the species. For instance, supposing A and B to have a like claim to be a species of the genus in question, then if the one be a species, so also is the other: and if that which is less generally thought to be so be a species, so also is that which is more generally thought to be so.”

Here, Aristotle first presents an example of positive subjectal a fortiori argument with equal terms. Next, with regard to positive subjectal argument with unequal terms, he offers us an abstract example followed by an illustration of it. Lastly, he presents two examples of positive subjectal argument, one a pari and one not so. Thus, there are a total of 5 distinct positive subjectal arguments (of which 2 are a pari). For further comments on these arguments, see the main text.

7:3 (pp. 496). “Moreover, look at it from the point of [sic] and like degrees, in all the ways in which it is possible to establish a result by comparing two and two together. Thus if A defines a better than B defines [b?] and B is a definition of [b?] so too is A of a. Further, if A’s claim to define a is like B’s to define b, and B defines b, then A too defines a. This examination from the point of view of greater degrees is of no use when a single definition is compared with two things, or two definitions with one thing; for there cannot possibly be one definition of two things or two of the same thing.”

Here, Aristotle presents 2 examples of positive subjectal a fortiori argument, 1 with unequal terms and 1 a pari. For further comments on these arguments, see the main text. These 2 arguments may be considered logical-epistemic.

PHYSICS (Tr. R. P. Hardie and R. K. Gaye). 1 instance, which is negative subjectal.

3:5 (p. 663). “Further, how can the infinite be itself any thing, unless both number and magnitude, of which it is an essential attribute, exist in that way? If they are not substances, a fortiori the infinite is not.” {–s} If number and magnitude are not concrete enough to be considered substances, then the infinite is not concrete enough to be considered a substance.

ON THE HEAVENS (Tr. J. L. Stocks). 3 instances, 2 being positive subjectal and 1 negative subjectal.

2:7 (p. 899). “Movement tends to create fire in wood, stone, and iron; and with even more reason should it have that effect on air, a substance which is closer to fire than these.” {+s} If wood, stone, and iron are close enough to fire that their movement tends to create fire in them, then air is close enough to fire that its movement tends to create fire in it.

4:6 (p. 951). “He [Democritus] says that the warm bodies moving up out of the water hold up heavy bodies which are broad, while the narrow ones fall through, because the bodies which offer this resistance are not numerous. But this would be even more likely to happen in air – an objection which he himself raises. His reply to the objection is feeble. In the air, he says, the ‘drive’ (meaning by drive the movement of the upward moving bodies) is not uniform in direction.” {–s} If air has not enough upward mobile warm bodies to prevent narrow bodies falling, then water has not enough upward mobile warm bodies to prevent narrow bodies sinking.[7]

4:6 (p. 951). “And these considerations apply with far greater force to air, since it is so much more easily divided than water.” {+s} If water is easily divided enough that these considerations apply to it, then air is divisible enough that these considerations apply to it.

METEOROLOGY (Tr. E. W. Webster). 1 instance, which is positive subjectal.

2:2 (p. 1076). “All of which is plainly impossible on the theory, and the more so as it derives the sea from Tartarus.” {+s} If the Phaedo theory, due to its inconsistency with various empirical facts (e.g. that rivers only flow downhill), even without mentioning that it derives the sea from Tartarus, is incredible enough to be rejected, then that theory, when its claim that the sea derives from Tartarus, is additionally taken into consideration, is incredible enough to be rejected. This argument may be considered logical-epistemic.

ON THE SOUL (Tr. J. A. Smith). 2 instances, 1 being negative subjectal and 1 negative predicatal.

1:3 (p. 1174). “Further, the cause of the revolution of the heavens is left obscure. It is not the essence of soul which is the cause of this circular movement – that movement is only incidental to soul – nor is, a fortiori, the body its cause.” {–s} If the essence of soul is not powerful enough to be the cause of the revolution of the heavens, then the body is not powerful enough to be the cause of the revolution of the heavens.

1:5 (p. 1183). “The problem might also be raised: What is that which unifies the elements into a soul? The elements correspond, it would appear, to the matter; what unites them, whatever it is, is the supremely important factor. But it is impossible that there should be something superior to, and dominant over, the soul (and a fortiori over the mind); it is reasonable to hold that mind is by nature most primordial and dominant, while their statement [is] that it is the elements which are first of all that is.” {–p} If something is not primordial enough to be dominant over the soul, then it is not primordial enough to be dominant over the mind.

ON SENSE AND THE SENSIBLE (Tr. J. I. Beare). 1 instance, which is negative subjectal.

5 (p. 1281-2). “But since even water by itself alone, that is, when unmixed, will not suffice for food – for anything which is to form a consistency must be corporeal – , it is still much less conceivable that air should be so corporealized [and thus fitted to be food].” {–s} If water is not corporeal enough to be food, then air is not corporeal enough to be food.

PARVA NATURALIA (Tr. J. I. Beare and G.R.T. Ross). 2 instances, 1 being positive subjectal and 1 negative predicatal.

7 (p. 1291). “If, then, the sensibles denominated co-ordinates though in different provinces of sense (e.g. I call Sweet and White coordinates though in different provinces) stand yet more aloof, and differ more, from one another than do any sensibles in the same province; while Sweet differs from White even more than Black does from White, it is still less conceivable that one should discern them [viz. sensibles in different sensory provinces whether co-ordinates or not] coinstantaneously than sensibles which are in the same province. Therefore, if coinstantaneous perception of the latter be impossible, that of the former is a fortiori impossible.” {+s} If sensibles in the same province are separate enough that their simultaneous perception is impossible, then sensibles in the different provinces are separate enough that their simultaneous perception is impossible.

7 (p. 1294). “But if the Soul does not, in the way suggested [i.e. with different parts of itself acting simultaneously], perceive in one and the same individual time sensibles of the same sense, a fortiori it is not thus that it perceives sensibles of different senses. For it is, as already stated, more conceivable that it should perceive a plurality of the former together in this way than a plurality of heterogeneous objects.” {–p} If the soul is not versatile enough to perceive simultaneously sensibles of the same sense, then the soul is not versatile enough to perceive simultaneously sensibles of different senses.

ON MEMORY AND REMINISCENCE (Tr. J. I. Beare). 1 instance, which is negative subjectal.

2 (p. 1307). “And since in the realm of nature occurrences take place which are even contrary to nature, or fortuitous, the same happens a fortiori in the sphere swayed by custom, since in this sphere natural law is not similarly established.” {–s} If natural law in the realm of nature is not sufficiently established to prevent occurrences (seemingly) taking place which are even contrary to nature or fortuitous, then natural law in the sphere swayed by custom is not sufficiently established to prevent occurrences (seemingly) taking place which are even contrary to nature or fortuitous.

HISTORY OF ANIMALS (Tr. D’Arcy Wentworth Thompson). 10 instances, of which 7 are positive subjectal (including a crescendo), 1 is positive predicatal and 2 are positive antecedental.

3:2 (p. 1463). “Now, as the nature of blood and the nature of the veins have all the appearance of being primitive, we must discuss their properties first of all, and all the more as some previous writers have treated them very unsatisfactorily.” {+a} If the primitiveness of the properties of blood and veins implies urgency enough for us to discuss them first, then their having been unsatisfactorily treated by past writers implies urgency enough for us to discuss them first.

5:14 (p. 1558). “If a sow be highly fed, it is all the more eager for sexual commerce, whether old or young.” {+s &} If a moderately-fed sow of any age is sated enough to be eager for sex, then a highly-fed one is sated enough to be eager for sex – even more eager.

7:1 (p. 1652). “And among men, the breasts grow more conspicuous and more like to those of women, both in young men and old, when the individual temperament is moist and sleek and the reverse of sinewy, and all the more among the dark-complexioned than the fair.” {+s} If the breasts of fair men are affected by the men’s temperament enough to grow, then the breasts of dark men are affected by the men’s temperament enough to grow.

7:4 (p. 1659). “Women who have connexion with their husbands shortly before childbirth are delivered all the more quickly.” {+s &} If women who do not have intercourse etc. are ready enough to give birth quickly, then women who do have intercourse etc. are ready enough to give birth quickly – even more so.

8:7 (p. 1690). “Cattle grow all the more in size when they are kept from sexual commerce over a number of years.” {+s &} If cattle which are allowed to have sex grow enough to reach a certain size, then cattle which are forbidden to have sex grow enough to reach a certain size – indeed, a greater size.

8:10 (p. 1692). “Sheep are fattened by twigs of the olive or of the oleaster, by vetch, and bran of every kind; and these articles of food fatten all the more if they be first sprinkled with brine.” {+s &} If certain specified feeds without brine are fattening enough that they fatten sheep to some extent, then these feeds with brine are fattening enough that they fatten sheep to some extent – a greater extent.

8:19 (p. 1707). “Whilst rain is wholesome for most fishes, it is, on the contrary, unwholesome for the mullet, the cephalus, and the so-called marinus, for rain superinduces blindness in most of these fishes, and all the more rapidly if the rainfall be superabundant.” {+p} If the eyesight of certain fishes (the mullet, etc.) is vulnerable enough to be affected slowly by a bit of rain, then their eyesight is vulnerable enough to be eventually affected rapidly by a lot of rain.

8:24 (p. 1714). “The bite is all the more dangerous if the mouse be pregnant when she bites, for the boils then burst, but do not burst otherwise.” {+s &} If the bite of a non-pregnant mouse (which does not burst boils) is dangerous enough to be avoided, then the bite of a pregnant mouse (which does burst boils) is dangerous enough to be avoided – even more so.

8:27 (p. 1717). “All insects, without exception, die if they be smeared over with oil; and they die all the more rapidly if you smear their head with the oil and lay them out in the sun.” {+s &} If any insect smeared with oil in ordinary ways is affected enough to soon die, then any insect whose head is smeared with oil and which is laid in the sun is affected enough to soon die – even sooner.

9:1 (p. 1726). “And so between the aegithus and the ass, owing to the fact that the ass, in passing a furze-bush, rubs its sore and itching parts against the prickles; by so doing, and all the more if it brays, it topples the eggs and the brood out of the nest, the young ones tumble out in fright, and the mother-bird, to avenge this wrong, flies at the beast and pecks at his sore places.” {+a} If the non-braying ass causes damage enough for the mother-bird to peck at it, then the braying ass causes damage enough for the mother-bird to peck at it.

METAPHYSICS (Tr. W. D. Ross). 4 instances, of which 3 are positive subjectal and 1 is positive predicatal. 1 of these arguments may be considered logical-epistemic.

1:8 (p. 2227-8). “But these thinkers are, after all, at home only in arguments about generation and destruction and movement; for it is practically only of this sort of substance that they seek the principles and the causes. But those who extend their vision to all things that exist, and of existing things suppose some to be perceptible and others not perceptible, evidently study both classes, which is all the more reason why one should devote some time to seeing what is good in their views and what bad from the standpoint of the inquiry we have now before us.” {+p} If certain views are interesting enough to deserve study, then they are interesting enough to justify our devoting some time to develop value judgments concerning them.

3:4 (p. 2255). “Further, since the matter exists, because it is ungenerated, it is a fortiori reasonable that the substance or essence, that which the matter is at any time coming to be, should exist; for if neither essence nor matter is to be, nothing will be at all, and since this is impossible there must be something besides the concrete thing, viz. the shape or form.” {+s} If the idea that matter (the concrete thing) exists without being generated is reasonable enough to be accepted, then the idea that the substance or essence (the shape or form) exists is reasonable enough to be accepted. This argument may be considered logical-epistemic.

4:4 (p. 2275). “The same account holds good with regard to ‘not being a man’, for ‘being a man’ and ‘being a not-man’ mean different things, since even ‘being white’ and ‘being a man’ are different; for the former terms are much more different so that they must a fortiori mean different things.” {+s} If ‘being white’ and ‘being a man’ are different enough to mean different things, then ‘being a not-man’ and ‘being a man’ are different enough to mean different things.

4:4 (p. 2279). “And if this is not knowledge but opinion, they should be all the more anxious about the truth, as a sick man should be more anxious about his health than one who is healthy.” {+s} If someone who has some apparent knowledge has enough doubt to incite him to verify it, then his having some apparent opinion should give rise to enough doubt to incite him to verify it.

NICOMACHEAN ETHICS (Tr. W. D. Ross). 4 instances, all positive subjectal (including 1 a crescendo).

1:13 (p. 2558). “But if this is so, clearly the student of politics must know somehow the facts about soul, as the man who is to heal the eyes or the body as a whole must know about the eyes or the body; and all the more since politics is more prized and better than medicine.” {+s} If medicine, which cares for the body, is prized enough to be studied, then politics, which cares for the soul, is prized enough to be studied.

3:9 (p. 2601). “Death and wounds will be painful to the brave man and against his will, but he will face them because it is noble to do so or because it is base not to do so. And the more he is possessed of virtue in its entirety and the happier he is, the more he will be pained at the thought of death; for life is best worth living for such a man, and he is knowingly losing the greatest goods, and this is painful. But he is none the less brave, and perhaps all the more so, because he chooses noble deeds of war at that cost.” {+s} If the man of low virtue, who has little to lose, is brave enough to choose battle, then the virtuous man, who has much to lose, is brave enough to choose battle.

7:12 (p. 2704) “Neither practical wisdom nor any state of being is impeded by the pleasure arising from it; it is foreign pleasures that impede, for the pleasures arising from thinking and learning will make us think and learn all the more.” {+s} If without intellectual pleasure we are interested enough to think and learn, then with intellectual pleasure we are interested enough to think and learn.

10:5 (p. 2771). “This happens, similarly, in all other cases, when one is active about two things at once; the more pleasant activity drives out the other, and if it is much more pleasant does so all the more, so that one even ceases from the other.” {+s &} If a moderately pleasant activity is absorbing enough to diminish conflicting activities, then an intensely pleasant activity is absorbing enough to diminish others, maybe even so much as to stop them.

POLITICS (Tr. Benjamin Jowett). 3 instances, all positive subjectal.

1:5 (p. 2796). “And if this is true of the body, how much more just that a similar distinction should exist in the soul?” {+s} If the distinctions between human bodies are significant enough to suggest different social roles, then the distinctions between human souls are significant enough to suggest different social roles.

2:5 (p. 2825). “The government, too, as constituted by Socrates, contains elements of danger; for he makes the same persons always rule. And if this is often a cause of disturbance among the meaner sort, how much more among high-spirited warriors?” {+s} If the meaner sort find government by the same persons unpleasant enough to cause disturbance, then high-spirited warriors will find government by the same persons unpleasant enough to cause disturbance.

7:15 (p. 3026). “Courage and endurance are required for business and philosophy for leisure, temperance and justice for both, and more especially in times of peace and leisure, for war compels men to be just and temperate, whereas the enjoyment of good fortune and the leisure which comes with peace tend to make them insolent. Those then who seem to be the best-off and to be in the possession of every good, have special need of justice and temperance – for example, those (if such there be, as the poets say) who dwell in the Islands of the Blest; they above all will need philosophy and temperance and justice, and all the more the more leisure they have, living in the midst of abundance.” {+s} If the better-off people in ordinary times need philosophy and temperance and justice enough to make them necessary, then they will in times of security and plenty need philosophy and temperance and justice enough to make them necessary.

RHETORIC (Tr. W. Rhys Roberts). 21 instances, of which 11 are positive subjectal (including 1 a crescendo), 8 are negative subjectal, and 2 are positive predicatal. Of these arguments, 4 may be considered logical-epistemic.

1:9 (p. 3168-9). “If a man runs into danger needlessly, much more will he do so in a noble cause; and if a man is open-handed to any one and every one, he will be so to his friends also.” {+s} If a man is brave enough to endanger himself needlessly, then he is brave enough to endanger himself in a noble cause. {+s} If a man is generous enough to be open-handed to strangers, then he will be generous enough to be open-handed to his friends also. (2 cases.)

2:3 (p. 3202). “The punishment of servants shows this: those who contradict us and deny their offence we punish all the more, but we cease to be incensed against those who agree that they deserved their punishment.” {+s &} If someone who admits having offended is enervating enough to be punished, then someone who denies having offended is enervating enough to be punished, perhaps more severely.

2:6 (p. 3211). “Once we are on a level with others, it is a disgrace to be, say, less well educated than they are; and so with other advantages: all the more so, in each case, if it is seen to be our own fault: wherever we are ourselves to blame for our present, past, or future circumstances, it follows at once that this is to a greater extent due to our moral badness.” {+s} If inferiority in any respect though not due to any fault is disgraceful enough to be despised, then such inferiority when due to a fault is disgraceful enough to be despised.

2:13 (p. 3226). “They love life; and all the more when their last day has come, because the object of all desire is something we have not got, and also because we desire most strongly that which we need most urgently.” {+s} If people who are not close to death love life enough to cling to it, then people who are close to death love life enough to cling to it.

2:19 (p. 3232). “If a thing can be produced without art or preparation, it can be produced still more certainly by the careful application of art to it.” {+p} If something is simple enough to be produced without art or preparation, then it is simple enough to produce with art and/or preparation.

2:23 (p. 3245). “4. Another line of proof is the ‘a fortiori’. Thus it may be argued that if even the gods are not omniscient, certainly human beings are not. The principle here is that, if a quality does not in fact exist where it is more likely to exist, it clearly does not exist where it is less likely. Again, the argument that a man who strikes his father also strikes his neighbours follows from the principle that, if the less likely thing is true, the more likely thing is true also; for a man is less likely to strike his father than to strike his neighbours. The argument, then, may run thus. Or it may be urged that, if a thing is not true where it is more likely, it is not true where it is less likely; or that, if it is true where it is less likely, it is true where it is more likely: according as we have to show that a thing is or is not true. This argument might also be used in a case of parity, as in the lines: Thou hast pity for thy sire, who has lost his sons: Hast none for Oeneus, whose brave son is dead? And, again, ‘if Theseus did no wrong, neither did Paris’; or ‘the sons of Tyndareus did no wrong, neither did Paris’; or ‘if Hector did well to slay Patroclus, Paris did well to slay Achilles’. And ‘if other followers of an art are not bad men, neither are philosophers’. And ‘if generals are not bad men because it often happens that they are condemned to death, neither are sophists’. And the remark that ‘if each individual among you ought to think of his own city’s reputation, you ought all to think of the reputation of Greece as a whole’.”

Here, Aristotle starts with a concrete example of negative subjectal argument, which he repeats in more abstract form. Next, he gives us a concrete example of positive subjectal argument, which he likewise repeats in more abstract form. Then he formulates in relatively abstract terms a negative subjectal and a positive subjectal. Finally, he points out that a fortiori argument can be used with equal terms, and offers us 7 subjectal examples (of which 5 are negative and 2 are positive) whose “parity” is, however, unclear. Thus, we may count a total of 13 subjectal arguments, including 5 positive ones and 8 negative ones (leaving open the number of a pari). Note that, although the whole passage is explicitly referred to as concerning a fortiori argument, none of these examples contain a key word or phrase signaling a fortiori argument. 4 of the arguments here may be considered logical-epistemic.

2:23 (p. 3252). “Thus Thrasybulus accused Leodamas of having had his name recorded as a criminal on the slab in the Acropolis, and of erasing the record in the time of the Thirty Tyrants: to which Leodamas replied, ‘Impossible: for the Thirty would have trusted me all the more if my quarrel with the commons had been inscribed on the slab.’” {+p} If Leodamas was trustworthy to the Thirty enough with his name erased from the slab in the Acropolis, then he was trustworthy enough to them with his name recorded on it as an opponent of the commons.

3:11 (p. 3284). “Liveliness is specially conveyed by metaphor, and by the further power of surprising the hearer; because the hearer expected something different, his acquisition of the new idea impresses him all the more.” {+s} If someone hearing an unsurprising new idea may be impressed enough to adopt it, then when hearing a surprising new idea is impressed enough to adopt it.

After writing the above, it occurred to me that I might find some more instances by searching for the four key phrases: ‘still (the) more/less’. Looking into the same document, I found no occurrences of ‘still the more’ or ‘still the less’. However, the string ‘still more’ occurred 35 times, and ‘still less’ occurred 7 times. Even so, I would not consider all these occurrences as pointing to a fortiori discourse. Very often, these expressions seem to intend no more than ‘some amount more’ and ‘some amount less’, respectively – i.e. their intent is only to signify some greater or lesser degree of something compared to something else, without any inference of the one from the other being claimed. Still, in some cases, inference does seem to be intended, and these may be considered as a fortiori arguments. The reader is encouraged to look for these and analyze them; I will give just one of them as an example, without further analysis:

NICOMACHEAN ETHICS 4:7 (p. 1676). “For the man who loves truth, and is truthful where nothing is at stake, will still more be truthful where something is at stake.”



[1] ‘X is even more than Y’ means ‘Y is a lot (or sufficiently much), and X > Y’. Likewise, ‘X is even less than Y’ means ‘Y is a little (or sufficiently little), and X < Y’.

[2] Feedbooks, from ebooks.adelaide.edu.au/p/plato/. Based on the 1891 translation by Benjamin Jowett (of The Dialogues of Plato?). I assume this edition is really complete.

[3] It is thanks to Wiseman’s mention of this case that I was moved to look for cases with the vaguer key phrases ‘much more/less’, ‘more/less so’.

[4] I am not sure what the intended middle term is here; so I have opted for ‘becoming’.

[5] Ed. William David Ross. Chicago: Encyclopædia Britannica, 1952.

[6] It should be noted that often, when Aristotle makes observations of proportionality, using words indicative of relative quantity (like “more” or “less”), his intent is a pro rata statement rather than an a fortiori argument. For examples: “The greater [prosperity] is, the more exposed is it to risk” (Nicomachean Ethics, 8:1), “The more virtuous a city is, the more happy it is” (Politics, 7:2).

[7] Democritus is apparently saying that, although one might object that in air the upward mobile warm bodies, being more numerous, would be more likely to uphold things (i.e. narrow bodies as well as broad ones) than those in water do, this does not in fact occur because in air the upward motion is more scattered than in water. Whence the major premise of the a fortiori argument involved is: Air has more upward mobile warm bodies than water does, and the minor premise and conclusion are as stated (major to minor).

2016-08-04T12:14:29+00:00