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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

Appendix 2 – A fortiori discourse in the Mishna

If we wish to enumerate the use of a fortiori argument in the Talmud, we must first make a census of its use in the Mishna (closed ca. 200 CE) – and perhaps also the Tosefta (closed ca. 300 CE), if any – since the Mishna is a document found in both the Jerusalem and Babylonian Talmuds. Alternatively, when we list all the a fortiori arguments in each Talmud, we shall have to specify which of those arguments are Mishnaic and there will be overlap equal to that set. In any case, the two Talmuds do not cover the same Mishna divisions: the Jerusalem treats the first four and the Babylonian treats the second to fifth. Also, neither of them treats the last (sixth) division; so, the Mishna must in any case be researched separately (in this regard, see Neusner: Rabbinic Literature: An Essential Guide, p. 34).

In truth, even though the Mishna is habitually mostly looked at and seen through the prism of Gemara commentary, it is historically a separate document and should be treated as such. Three sorts of listing are possible. The simplest is just a list of locations where the argument can be found. A more thorough listing includes the relevant extract without comment. The best listing must include a full analysis of each case found (this of course requires acquaintance with traditional commentaries, notably the Gemara, but also Rashi, Tosafot and others). Ideally, this research should be done with reference to the original Hebrew and Aramaic texts; but English translations would be a good start. All this is of course a very big job, but it is feasible today with relative ease and speed thanks to the possibilities of computer assisted search for key words and phrases (although these should not be considered as yielding exhaustive results).

Alexander Samely, in about 2002, made a valiant and apparently original attempt to list and analyze all the a fortiori arguments in the Mishna, posting the results of his work on the Internet, in a database (which may be viewed at: mishnah.llc.manchester.ac.uk/search.aspx). As we saw in the main text, some of the cases he listed are not credible, and a few more cases need to be added to the list. On this basis, I propose the following tentative list of qal vachomer arguments in the Mishna, which includes 46 attempted cases of a fortiori argument, of which 42 are valid and 4 are invalid.

There is no guarantee that Samely’s collection of cases is exhaustive, as he has found additional cases over time; and moreover he apparently missed, or for some reason disregarded, 4 cases, viz.: 2 in Avot, 1 in Kelaim, and 1 in Pesahim (6:5). Some of his analyses are essentially retained, though rewritten by me in standard form, but some cases are partly or entirely reanalyzed by me. I here take the “texts” proposed by him for granted, although I have some doubts that these quotations are all really accurate (in view of spelling mistakes, unexplained brackets, etc.). Hence, the exposition below (alphabetically ordered) is only tentative, a model and starting point for future research based on more reliable source texts.

The following table lists the results obtained:

Reference

Qty

valid or not

Q-P, P-Q or pari

Mood

Possible a crescendo

Arakin 8:4

1

valid

P to Q

-s

Avot 1:5

1

valid

Q to P

+s

&

Avot 6:3

1

valid

Q to P

+a

&

Baba Qama 2:5 (a)

1

valid

Q to P

+a

&

Baba Qama 2:5 (b)

1

valid

Q to P

+a

Bekoroth 1:1

1

valid

pari

+s

Bekoroth 9:1

1

valid

P to Q

-s

Berakoth 9:5

1

valid

Q to P

+s

Demai 2:2

1

valid

Q to P

-p

Eduyyoth 6:2

1

valid

Q to P

+s

Eduyyoth 6:3 (a)

1

valid

pari

+s

Eduyyoth 6:3 (b)

1

valid

pari

+s

Hullin 2:7

1

valid

P to Q

-s

Hullin 10:1

1

valid

Q to P

+a

Hullin 12:4-5

1

valid

Q to P

+s

&

Kilaim 8:1

1

valid

Q to P

+s

Makkoth 1:7

1

valid

pari

+s

Makkoth 3:15 (a)

1

invalid

Q to P

+s

&

Makkoth 3:15 (b)

1

valid

Q to P

+s

&

Menahoth 8:5

1

valid

Q to P

+s

Nazir 7:4

1

valid

Q to P

+s

Nedarim 10:7

1

valid

P to Q

+p

Negaim 10:2

1

valid

Q to P

+s

Negaim 12:5 (a), (b), & (c)

3

valids

Q to P

+s, +s, +s

&&&

Pesahim 6:2 (a)

1

valid

Q to P

+s

Pesahim 6:2 (b) & (c)

2

valid rivals

Q-P, P-Q

+s, -s

Pesahim 6:5

1

valid

Q to P

+s

Sanhedrin 6:5

1

valid

Q to P

+s

&

Shebuoth 3:6

1

valid

Q to P

+s

Sotah 6:3 (a)

1

valid

Q to P

+s

Sotah 6:3 (b)

1

valid

Q to P

+s

Temurah 1:1 (a) & (b)

1 + 1

valid, invalid

pari, Q-P

-s, -s

Terumoth 5:4 (a) & (b)

2

invalids

Q to P

-s, -s

Yadayim 4:7

1

valid

Q to P

+s

Yadayim 4:8

1

valid

P to Q

-s

Yebamoth 8:3

1

valid

P to Q

-s

Yom Tov 5:2

1

valid

Q to P

+s

Zebahim 7:4

1

valid

Q to P

+s

Zebahim 7:6

1

valid

P to Q

-s

Zebahim 8:12

1

valid

P to Q

-a

Zebahim 12:3

1

valid

Q to P

+s

Table A2.1

The above list can be summarized as follows.

Mood of a fortiori argument

Number found

Of which, a pari

Possibly a crescendo

Positive subjectal {+s}

28

4

8

Negative subjectal (-s)

11

1

Positive predicatal {+p}

1

Negative predicatal (-p)

1

Copulative

41

5

8

Positive antecedental (+a)

4

2

Negative antecedental (-a)

1

Positive consequental (+c)

0

Negative consequental (-c)

0

Implicational

5

0

2

Total

46

5

10

Table A2.2

The detailed expositions that these tables are based on are given below.

Note that three arguments are invalid because they are negative subjectal, and yet go from minor to major; however, one of these invalids, viz. Terumoth 5:4 (b) is intentionally so, having been put forward only to illustrate the invalidity of its predecessor, viz. (a). A fourth argument, viz. Makkoth 3:15 (a), is invalid because it is a contrario. So there are, it seems (unless alternative explanations for them are put forward), two unintentional formal errors of a fortiori reasoning in the Mishna (2 out of 46 = 4.3%). But one of these errors is challenged by someone else within the text; so only one, viz. Temurah 1:1 (b), apparently passes unnoticed. Note also, in passing, that all the arguments listed involve normative judgment of some sort; none are purely factual in content.

Regarding the language used in these 46 arguments, a few observations are worth making. The main indicators or markers of a fortiori argument (in the English translation, which of course depends on the translator’s choices of words) are: inference, inferred, logical[1] (19 cases), how much more, how much more so, how much the more, how can, all the more, still more (16 cases), light-heavy and the like[2], more stringent (10 cases), a fortiori (2 cases), as well as various other expressions: so, also, so also, just as … so, to these… and yet, dayo, etc. (16 cases); these add up to more than 46 cases because there are overlaps, i.e. some a fortiori argument are indicated by more than one marker.

It seems that, in the Mishna at least, a crescendo arguments are always indicated by expressions with ‘more’, viz.: 5 how much more, 2 how much more so, 1 how much the more, 1 all the more and 1 more stringently. However, note well, expressions with ‘more’ are not exclusively used for a crescendo discourse, but may also signal purely a fortiori discourse. As regards the dayo objection, it only appears in relation to the two arguments in Baba Qama 2:5, of which the first is a crescendo, while the second is purely a fortiori. There is no mention of dayo in relation to the remaining nine (possibly) a crescendo arguments[3], which anyhow do not involve inference of a penalty from the Torah. Assuming our list is exhaustive, there are no other applications of dayo in the Mishna (though certainly many more in the Gemara). This is quite surprising![4]

Arakin 8:4

TEXT: “A man may devote part of his flock or of his herd or of his Canaanite slaves and bondwomen, or ‘of the field of his possession’; and if he devoted the whole of them they are not devoted – words of R. Eliezer. R. Eleazar ben Azariah said: And if Man does not even have authority to devote to the High One everything he owns, how much more is Man obliged to protect his possessions!”

MY READING: negative subjectal, major to minor.

Man’s use of his possessions for holy ends (P) is more religiously valuable (R) than man’s use of his possessions for profane ends [including the waste of his possessions without purpose] (Q) is.

If man’s use of his possessions for holy ends (P) is not religiously valuable (R) enough to be authorized without limit (S),

then man’s use of his possessions for profane ends (Q) is not religiously valuable (R) enough to be authorized without limit (S).



Avot 1:5

TEXT: “Yose ben Yohanan of Jerusalem was wont to say:… Engage not in overmuch converse with a woman. If he said this of his wife, how much more does it apply to the wife of another.”[5]

MY READING: positive subjectal, minor to major. Could be intended as a crescendo.

Speaking to another man’s wife (P) is more to be avoided (R) than speaking to one’s own wife (Q) is.

If speaking to one’s own wife (Q) is to be avoided (R) enough to be discommended by Yose (S),

then speaking to another man’s wife (P) is to be avoided (R) enough to be discommended by Yose (S).



An a crescendo reading would imply a stronger recommendation against speech in the latter case, in proportion to its moral unsuitability.

Avot 6:3

TEXT: “If David king of Israel, who having learned from Ahitophel but a couple of worldly matters yet called him his master… how much the more honor must be shown by the one who learns from his associate but a single chapter, law, or verse or saying, or even a single letter of Torah.”

MY READING: positive antecedental, minor to major. Could be intended as a crescendo.

Learning holy matters from someone (P) entails honoring him (R) more than learning worldly matters from him (Q) does.

If learning worldly matters from someone (Q) entails honoring him (R) enough to justify calling him one’s master (S),

then learning holy matters from someone (P) entails honoring him (R) enough to justify calling him one’s master (S).



An a crescendo reading would imply still more demonstrations of gratitude in the latter case, in proportion to the greater honor due to the Torah.

Baba Qama 2:5 (a) and (b)

TEXT: “An ox which causes damage in the [private] domain to him that is injured – thus, if it gored, pushed, bit, lay down, or kicked in the public domain it pays only half-damages; but if in the private domain of him that was injured, R. Tarfon says: It pays full damages. But the Sages say: Half-damages. R. Tarfon said to them: What! If they have dealt leniently with damage caused by tooth or foot in the public domain, when no restitution is imposed, and stringently with like damage in the private domain of him that is injured, when full damages are imposed, then since they have dealt stringently with damage caused by the horn in the public domain, when half-damages are imposed, ought we not therefore to deal the more stringently with damage caused by the horn in the private domain of him that was injured, so that full damages shall be imposed? They answered: It is enough (dayo) if the inferred is as strict as that from which it is inferred: if [for damage] caused by the horn in the public domain half-damages [are imposed], so also [for like damage] in the private domain of him that was injured, half-damages [only are imposed].

MY READING: The first argument by R. Tarfon, in view of its ‘proportional’ conclusion, may be taken minimally as an argument pro rata. It could alternatively be interpreted more elaborately as an a crescendo argument, i.e. as an a fortiori cum pro rata argument. But in any case, note well, it cannot be considered as a purely a fortiori argument, unless we are ready to suggest (unnecessarily) that R. Tarfon here argues in a formally invalid manner. The Sages counter this argument by saying that the penalty prescribed in the conclusion must not be greater than that given in the minor premise; this has come to be known as the dayo (sufficiency) principle. R. Tarfon’s first argument, taken as pro rata, can be expressed as follows:

Just as, in one case (that of tooth & foot), damage in private domain (full payment) implies more legal liability than damage in public domain (no payment) –

so, in the other case (viz. horn), we can likewise say that damage in private domain implies more legal liability than damage in public domain (half payment).



Whence, R. Tarfon concludes with full payment for damage by horn in the private domain. The Sages may be construed to oppose this conclusion by means of a formally valid purely a fortiori argument, without pro rata extension, as follows:

Private domain damage (P) implies more legal liability (R) than public domain damage (Q) [as we know by extrapolation from the case of tooth & foot].

For horn, public domain damage (Q) implies legal liability (R) enough to make the payment half (S).

Therefore, for horn, private domain damage (P) implies legal liability (R) enough to make the payment half (S).



Note that the Sages do not explicitly formulate this a fortiori argument, but may reasonably be assumed to intend it in view of their preferred conclusion. The dayo principle here simply corresponds to the principle of deduction, which is that the conclusion cannot contain more information than is given in the premises. For this reason, the dayo principle has been equated by many commentators (myself included, in the past) to the principle of deduction; but as we shall see this is only a point of intersection between them – they are not the same.

FURTHER TEXT: “He said to them: My inference is not from one case of damage caused by the horn to another case of damage caused by the horn, but from what applies in a case of damage caused by the foot to what should apply in the case of damage caused by the horn: If they have dealt leniently with damage caused by the tooth or foot in the public domain and stringently with damage caused by the horn [also in the public domain], then since they have dealt stringently [more stringently] with damage caused by the tooth or foot in the private domain of him that was injured [than in the public domain], ought we not, therefore, to deal the more stringently with damage caused by the horn [in the private domain]?! They answered: It is enough (dayo) if the inferred is as strict as that from which it was inferred: [as in the case of damage caused by the horn] in the public domain half-damages [are imposed], so also [for damage caused by the horn] in the private domain of him that was injured, half-damages [only are imposed].”

MY READING: The second argument by R. Tarfon can be read both as a mere argument by analogy (pro rata) like before, and as a valid purely a fortiori (as against a crescendo) argument that is in accord with the Sages’ previous objection. The Sages nevertheless reject this restructured argument by saying that the penalty prescribed in the conclusion should not surpass that given in the raw data that was generalized into the major premise; this signifies an enlargement and complication of the dayo (sufficiency) principle, compared to its tenor in relation to the previous argument. R. Tarfon’s second argument can be expressed as an argument pro rata as follows:

Just as, in one case (that of public domain), damage by horn (full payment) implies more legal liability than damage by tooth & foot (no payment).

So, in the other case (viz. private domain), we can likewise say that damage by horn implies more legal liability than damage by tooth & foot (full payment).



Whence, R. Tarfon concludes with full payment for damage by horn in the private domain. This ‘proportional’ conclusion is all the more credible, since the exact same conclusion can this time be obtained by regular (i.e. not a crescendo) a fortiori argument, as follows:

Horn damage (P) implies more legal liability (R) than tooth & foot damage (Q) [as we know by extrapolation from the case of public domain].

For private domain, tooth & foot damage (Q) implies legal liability (R) enough to make the payment full (S).

Therefore, for private domain, horn damage (P) implies legal liability (R) enough to make the payment full (S).



The Sages remain unfazed by this new, double-barrel argument, repeating their previous “it is enough” objection, in exactly the same words. From this we must infer that their first objection did not consist merely in opposing a purely a fortiori argument to an argument pro rata or a crescendo, but referred to any source of information used in formulating the premises, limiting the conclusion drawn to such given information. In the present case, the limiting information is not apparent in the a fortiori argument as such, but is influential in the formation of its major premise. This interpretation is in accord with rabbinical commentary.

Note that both a fortiori arguments are positive antecedental in form.

Bekoroth 1:1

TEXT: “The priests and Levites are free (i.e. from the duty of the first-born, including unclean animals which are treated like human first-born in Num. 18:15), by reason of the light and heavy (mi-qal wahomer): If they exempted those (first-born) of the Israelites in the desert, it is an inference that they exempted their own (first-born).”

MY READING: positive subjectal, a pari (egalitarian)[6].

The (first-born) of the priests and Levites (P) were as much released in the desert from their duties (R) as the (first-born) of the Israelites (Q) were.

If the (first-born) of the Israelites (Q) were released in the desert from their duties as first-born (R) enough to be henceforth exempt from them (S),

then the (first-born) of the priests and Levites (P) were released from their duties as first-born (R) enough to be henceforth exempt from them (S).



Bekoroth 9:1

TEXT: “The tithe of cattle applies in the land (of Israel) and outside the land… It applies to the herd and to the flock, and they do not tithe from one to the other; to sheep and goats, and they do tithe from one to the other; to new and old, and they do not tithe from one to the other. For, it could be an inference: Just as for the new and old, which are not diverse kinds (cf. Lev. 19:19), they do not tithe from one to the other, is it not an inference that for sheep and goats, which are diverse kinds, they (also) do not tithe from one to the other?”

MY READING: negative subjectal, major to minor.

New and old (P) are more similar to each other (R) than sheep and goats (Q) are.

If new and old (P) are similar to each other (R) not enough to be interchangeable (S),

then sheep and goats (Q) are similar to each other (R) not enough to be interchangeable (S).



Berakoth 9:5

TEXT: “A man may not enter the temple mount with his staff, or his sandal, or his wallet; or with the dust upon his feet. Nor may he use it as a shortcut, let alone (miqal wa-homer) spit there. [Cambridge MS adds:] If it is forbidden to enter with shodden (sic) feet which implies lack of respect, how much more is spitting forbidden which implies contempt.”

MY READING: positive subjectal, minor to major.

Spitting there (P) is more disrespectful of Temple Mount (R) than entering it with staff, sandal, wallet or dust or using it as a shortcut (Q) is.

If entering it with staff, sandal, wallet or dust or using it as a shortcut (Q) is disrespectful (R) enough to be forbidden (S),

then spitting there (P) is disrespectful (R) enough to be forbidden (S).



Demai 2:2

TEXT: “He that takes it upon himself to be trustworthy must give tithe [for?] that which he eats and that which he sells and that which he buys [in order to sell again] and may not be the guest of an ‘am haarets’ (ignoramus). R. Yehudah says: The one who is a guest of an ‘am haarets’ is also trustworthy. They said to him: He is not trustworthy concerning himself, how can he be trustworthy concerning others?!”

MY READING: negative predicatal, minor to major.

More scrupulousness (R) is required to be trustworthy concerning others (P) than concerning oneself (Q).

If someone (S) is not scrupulous enough (R) (e.g. by abstaining to partake of the food of an ignoramus) to be trustworthy concerning himself (i.e. for the good of his own soul) (Q),

then he (S) is not scrupulous (R) enough (e.g. by tithing food before selling it) to be trustworthy concerning others (i.e. for the good of their souls) (P).



Eduyyoth 6:2

TEXT: “[They said:] R. Yehoshua and R. Nechunya ben Elinathan of Kefar Ha-Bavli testified with regard to [the smallest] member of a corpse that it is unclean [in the sense of conveying uncleanness by overshadowing, mOhol 2:1], concerning which R. Eliezer says: They [i.e. the aforementioned two] have said this only of a member of living being [mOhol 1:7]. They [i.e. the first speakers] said to him: But is it not a [relationship between] lesser and greater: For the living being, which is clean, a limb severed from it is unclean – so is it not to be inferred that for a corpse, which is unclean, a limb severed from it is unclean?”

MY READING: positive subjectal, minor to major.

A limb severed from a corpse (a part severed from an unclean whole) (P) is more unclean (R) than a limb severed from a living body (a part severed from a clean whole) (Q).

If the limb severed from a living body (Q) is unclean (R) enough to be declared unclean (S),

then the limb severed from a corpse (P) is (R) enough to be declared unclean (S).



FURTHER TEXT: “R. Eliezer … said to them [i.e. the first speakers]: They [the aforementioned two] have said this only of a member of living being. Another answer is: The uncleanness of living beings is greater than the uncleanness of corpses, for the living being makes what it lies and sits on convey uncleanness to men and to garments, and [makes] what is above it maddaf-unclean so that it conveys uncleanness to food and liquids, which is not how the corpse makes unclean.”

MY READING: This is not an a fortiori argument, but an attempted refutation of the preceding a fortiori argument. The opponents correct the initial statement attributed to R. Yehoshua and R. Nechunya ben Elinathan of Kefar Ha-Bavli, saying they “testified with regard to [the smallest] member of a living being that it is unclean;” and they reject the major premise of the proposed a fortiori argument by reversing it, saying “The uncleanness of living beings is greater than the uncleanness of corpses.”

Eduyyoth 6:3 (a)

TEXT: “R. Eliezer … said to them: We find that a member from a living being is like a whole corpse (mOhol 2:1); as an olive’s bulk of flesh severed from a corpse is unclean, so an olive’s bulk of flesh severed from the member of a living being will be unclean.”

MY READING: positive subjectal, a pari (egalitarian).

An olive’s bulk of flesh severed from a living body (P) is as dead (R) as an olive’s bulk of flesh severed from a corpse (Q).

If an olive’s bulk of flesh severed from a corpse (Q) is dead (R) enough to be declared unclean (S),

then an olive’s bulk of flesh severed from the limb severed from a living body (P) is dead (R) enough to be declared unclean (S).



FURTHER TEXT: “R. Yehoshua and R. Nechunya … said to him: No. If you declare unclean an olive’s bulk of flesh severed from a corpse, of which you have declared unclean a barleycorn’s bulk of bone severed from it (mOhol 2:3), would you also declare unclean an olive’s bulk of flesh severed from the member of a living being, of which you have declared clean a barleycorn’s bulk of bone severed from it?”

MY READING: This is not an a fortiori argument, but an attempted refutation of the preceding a fortiori argument. The opponents argue:

Given that: if a barleycorn’s bulk of bone severed from a corpse is unclean, then an olive’s bulk of flesh severed from a corpse is unclean.

Does it follow that: if a barleycorn’s bulk of bone severed from a member of a living body is clean, then an olive’s bulk of flesh severed from the member of a living being is unclean? No!

What the opponents are saying is that the major premise (“a member from a living being is like a whole corpse”), is not as general as it is made to seem. A barleycorn’s bulk of bone is unclean severed from a corpse, yet clean severed from a member of a living being. Therefore, they say, we cannot infer an olive’s bulk of flesh severed from a member of a living being to be unclean from the fact that an olive’s bulk of flesh severed from a corpse is declared unclean.

Eduyyoth 6:3 (b)

TEXT: “R. Nechunya … We find that a member from a living being is like a whole corpse (mOhol 2:1); as a barleycorn’s bulk of bone that is severed from a corpse is unclean, so a barleycorn’s bulk of bone that is severed from a member from a living body will be unclean.”

MY READING: positive subjectal, a pari (egalitarian).

A barleycorn’s bulk of bone severed from a living body (P) is as dead (R) as a barleycorn’s bulk of bone severed from a corpse (Q) is.

If a barleycorn’s bulk of bone severed from a corpse (Q) is dead (R) enough to be declared unclean (S),

then a barleycorn’s bulk of bone severed from the limb severed from a living body (P) is dead (R) enough to be declared unclean (S).[7]



FURTHER TEXT: “They said to him: No. If you declare unclean a barleycorn’s bulk of bone severed from a corpse, of which you have declared unclean an olive’s bulk of flesh severed from it, would you also declare unclean a barleycorn’s bulk of bone severed from the member of a living being, of which you have declared clean an olive’s bulk of flesh that is severed from it?”

MY READING: This is not an a fortiori argument, but an attempted refutation of the preceding a fortiori argument. The opponents argue, as in the previous rebuttal:

Given that: if an olive’s bulk of flesh severed from a corpse is unclean, then a barleycorn’s bulk of bone severed from a corpse is unclean.

Does it follow that: if an olive’s bulk of flesh severed from the member of a living body is clean, then a barleycorn’s bulk of bone severed from the member of a living body is unclean? No![8]

Here again the major premise of the proposed a fortiori is put in doubt.

Hullin 2:7

TEXT: “R. Yose said: The things are lighter and weightier: Just as in the case (maqom) where the thought invalidates, regarding sacrifices (cf. mZeb 2:2), everything depends on the person officiating (i.e. not the owner), so in the case where thought does not invalidate, regarding profane slaughter, should not everything also depend on the slaughterer (i.e. not the owner, in this case a non-Jew)?”

MY READING: negative subjectal, major to minor.

Sacrificial slaughter (P) is more dependent on the right thought of the owner (R) than profane slaughter (Q) is.

If sacrificial slaughter (P) is dependent on the right thought of the owner (R) not enough to be invalidated (S),

then profane slaughter (Q) is dependent on the right thought of the owner (R) not enough to be invalidated (S).



Hullin 10:1

TEXT: “{“the shoulder and the two cheeks and the stomach” (Deut. 18:3) … applies to non-sacrificial slaughter but not to sacrificial slaughter.} For it might have been an inference [to say]: And if even non-sacrificial slaughter which is not liable to “breast and thigh” (Lev. 7:31) is liable to these [other] dues, is it not an inference to that consecrated animals which are liable to “breast and thigh” should be also liable to these [other] dues? {In this regard it is instructive that Scripture says: “[For I have taken the breast of elevation offering and the thigh of gift offering from the Israelites, from their sacrifices of well-being] and given them to Aaron the priest and his sons as a prescribed due for ever [from the Israelites]” (Lev. 7:34). You have there only what is said as stated.}”

MY READING: positive antecedental, minor to major.

Sacrificial offerings (P) are more liable to priestly dues (R) than non-sacrificial slaughter (Q) is (this being generalized from ‘breast and thigh’ dues to all dues).

If non-sacrificial slaughter (which has no priestly involvement) (Q) is liable to priestly dues (R) enough to necessitate payment of ‘the shoulder and the two cheeks and the stomach’ (S),

then sacrificial offerings (which have priestly involvement) (P) are liable to priestly dues (R) enough to necessitate payment of ‘the shoulder and the two cheeks and the stomach’ (S).



The last remark “You have there only what is said as stated” seems intended to deny the conclusion of the qal vachomer, saying that Lev. 7:34 is to be read as exclusive of ‘the shoulder and the two cheeks and the stomach’ (i.e. as davka).

Hullin 12:4-5

TEXT: “A man may not take ‘the dam together with the young’ even in order to effect with them the purification of the leper [where one of whose birds is let go at the end of the ritual, cf. Lev. 14:7]. And if (concerning) a ‘light’ commandment, involving an Issar, the Torah says: ‘So that it may be good with you and you have length of days’, how much more so (qal wahomer) concerning the weightier commandments which are in the Torah!”

MY READING: positive subjectal, minor to major. Could be intended as a crescendo.

Obeying weightier (more demanding) commandments (P) earns one more merit (R) than obeying lighter (less demanding) commandments (Q) does.

If someone obeying a light commandment (demanding merely one Issar) (Q) earns merit (R) enough to get rewarded with good and long life (S),

then someone weightier commandments (demanding more than one Issar) (P) earn merit (R) enough to get rewarded with good and long life (S).



An a crescendo reading would imply a greater reward (i.e. proportionately more good and longer life) in the latter case, in proportion to the greater merit of weightier commandments.

Kilaim 8:1

TEXT: “It is prohibited to sow kilaim [a certain mingling] of seeds, and to allow it to grow; but it is lawful to eat of it, and, a fortiori, to derive benefit therefrom.”

MY READING: This is a Mishnaic a fortiori argument not mentioned by Samely. It is a straightforward positive subjectal case:

Deriving benefit from something (P) is more innocuous (R) than eating of it (Q).

If eating of something (Q) is innocuous (R) enough to be lawful (S),

then, deriving benefit from it (P) is innocuous (R) enough to be lawful (S).



Makkoth 1:7

TEXT: “{‘On the evidence of two witnesses or three witnesses, shall he that is to die be put to death’ …] R. Aqiva says: The third witness is here mentioned only that the same stringency shall apply to him also, and that his condemnation shall be made like to that of the other two.} If thus Scripture punishes the (person) who is joined to those who commit transgressions in the same way as those who (actually) commit transgressions, how much more will it reward the (person) who is joined to those who fulfill commandments in the same way as those who (actually) fulfill commandments.”

MY READING: positive subjectal, a pari (egalitarian).

Joining those who fulfill commandments (P) earns one as much merit (R) as actual fulfilling of commandments (Q) does.[9]

If someone actually fulfilling commandments (Q) earns merit (R) enough to be rewarded in a certain way (S),

then someone joining those who fulfill commandments (P) earns merit (R) enough to be rewarded in that way (S).



Makkoth 3:15 (a)

TEXT: “R. Hananiah ben Gamliel (also) said: If the one who commits one transgression has his life taken away, all the more will the one who performs one commandment be given [or restored] his life!”

MY READING: positive subjectal, minor to major. Strictly speaking, this attempted a fortiori argument is invalid, because the subsidiary term (S) is not the same in the minor premise and conclusion.

The one who performs one commandment (P) deserves more credit (R) than the one who commits one transgression (who deserves not credit, but debit) (Q) does.

If the one who commits one transgression (Q) deserves credit (R) enough to have his life taken away (S1),

then the one who performs one commandment (P) deserves credit (R) enough to be given [or restored] his life! (S2).



We could regard this argument more generously as valid, if we look upon it as a crescendo, i.e. if we assume an unstated additional premise about proportionality to be tacitly intended. In that case, S1 (life taken away) and S2 (life given or restored) are viewed as two sides of a continuum S (life), the former being negative and the latter positive. This continuum being parallel to the continuum R (‘credit’ in a broad senses, ranging from actual debit to actual credit), we can reason proportionately. Just as Rq implies S1, so Rp implies S2.

However, even then the argument is of very doubtful validity, because it is essentially a contrario. Notice that not only the predicates (life forfeited, life restored) are contrary, but also the subjects (commits transgression, performs commandment) are contrary. This is not per se something inconceivable; however, the difficulty lies in the coupling of these two pairs of contraries. By what formal means does the speaker know that the switchover from the first predicate to the second is tied precisely to the switchover from first subject to the second? Obviously, he perceives a causative relation between the subjects and predicates, i.e. he believes that transgression causes life to be forfeited and performing commandments causes life to be restored. Fair enough; this may well be true as a pair of observations (or through further inductive and deductive arguments). But the problem is in the inference from the premises to the conclusion. If the conclusion is already known by observation (or however), and is appealed to in order to justify the said coupling of switchovers, then the proposed a fortiori argument constitutes circular reasoning. It begs the question, since its putative conclusion can only really be drawn if it is previously given.

Makkoth 3:15 (b)

TEXT: “R. Shimon ben Rabbi says: Behold it says: ‘Only be firm not to eat the blood, for the blood is the life…’ And if the person who separates from the blood, from which man recoils, receives a reward; then the person who separates from robbery and forbidden sexual relations, which man covets and desires, how much more so will he acquire merit for himself and his generations and the generations of his generations until the end of all the generations!”

MY READING: positive subjectal, minor to major. Could be intended as a crescendo.

A person who separates from robbery and forbidden sexual relations (P) has more inner resistance to overcome (R) than a person who separates from the blood (Q).

If a person who separates from the blood (Q), from which (many a) man recoils, has enough inner resistance to overcome (R) that he merits to receive a reward (S),

then a person who separates from robbery and forbidden sexual relations (P), which (many a) man covets and desires, has enough inner resistance to overcome (R) that he merits to receive a reward (S).



An a crescendo reading would imply a greater reward in the latter case, in proportion to the inner obstacles that had to be overcome. Although at first sight, R. Shimon’s argument appears a crescendo, it can definitely also be interpreted as purely a fortiori. The latter is possible in two ways, in both of which the minor premise and conclusion have the same predicate (the subsidiary term, S): either (a) both propositions state that the subject vaguely “receives a reward” (for self and perhaps children), or (b) both propositions state that the subject “will he acquire merit for himself and his generations and the generations of his generations until the end of all the generations!” In the event of (a), R. Shimon’s conclusion must be taken as mere hyperbole, and cannot be accepted literally as the logical conclusion of the a fortiori argument as such. In the event of (b), R. Shimon’s conclusion must be taken as having been tacitly intended also in his more vaguely put minor premise. But, equally well, we may consider R. Shimon’s conclusion as occurring after the a fortiori argument, the product of a subsequent pro rata argument, i.e. as an extrapolation (in time) based on other considerations (e.g. the principle of measure for measure), i.e. as an a crescendo conclusion.

To be sure, underlying this positive subjectal argument, with the middle term “having some resistance to overcome,” is a positive predicatal argument, with the middle term “having some self-control.” The latter may be formulated as follows:

More self-control (R) is required to avoid robbery and incest (P), which arouse desire, than to avoid shedding blood (Q), which arouses aversion.

If a person (S) has self-control (R) enough to avoid robbery and incest (P),

Then that person (S) has self-control (R) enough to avoid shedding blood (Q).



However, this argument lacks the information about reward, and moreover proceeds from major to minor. For the inference of reward, the subjectal form used by R. Shimon seems more appropriate.

Menahoth 8:5

TEXT: “Also, for the meal offerings should be inferred that they require ‘pure olive oil’: Just as the Menorah which is not to do with eating, requires ‘pure olive oil’, so meal offerings, which are to do with eating, is it not an inference that they should require ‘pure olive oil’? Scripture instructs by saying: ‘[olive oil] pure, beaten, for lighting…’ – and not pure, beaten for the meal offerings.”

MY READING: positive subjectal, minor to major.

Oil for the meal offering (P) has more to do with eating (R) than oil for lighting the Menorah (Q).

If oil for lighting the Menorah (Q), though not intended as food, has enough to do with eating (R) (R = 0) to need to be pure olive oil (S),

then meal offerings (P), which are intended as food, have enough to do with eating (R) (R > 0) to need to be pure olive oil (S).



The final sentence (“Scripture instructs, etc.”), read davka, i.e. as exclusive of anything not explicitly mentioned therein, is intended as a rebuttal of the putative conclusion. In this case, the problem lies not with the major premise (which seems credible enough), but with the minor premise (which seems forced anyway).

Nazir 7:4

TEXT: “R. Eleazar said in the name of R. Yehoshua: For whatsoever uncleanness from a corpse a Nazirite must cut off his hair [mishnah 2], for that too is a man culpable if he enters into the Temple; and for whatsoever uncleanness from a corpse a Nazirite need not cut off his hair [mishnah 3], for that too is a man not culpable if he enters into the Temple. R. Meir said: Would there not thus be less stringency than [when uncleanness is contracted from] a creeping thing! [Lev. 5:2, Num. 19:20] R. Aqiva said: I argued before R. Eliezer: If because of the contact or carrying of a barleycorn’s bulk of bone which does not render a man unclean by overshadowing a Nazirite must cut off his hair, how much more, then, ought he to cut off his hair because of the contact or carrying of a quarter-log of blood [cf. mishnah 3] which renders a man unclean by overshadowing! He said to me: What is this, Aqiva? We cannot here argue from the lesser to the greater [since it is the accepted ruling]. But when I came and declared these words before R. Yehoshua, he said to me: You have spoken well; but thus have they said as Halakhah.”

MY READING: R. Akiva’s argument is positive subjectal, minor to major.

The contact or carrying of a quarter-log of blood (P) renders a man more unclean by overshadowing (R) than the contact or carrying of a barleycorn’s bulk of bone (Q) does.

If a Nazirite’s contact or carrying of a barleycorn’s bulk of bone (Q) renders him unclean by overshadowing (R) (R = 0) enough to make him have to cut off his hair (S),

then a Nazirite’s contact or carrying of a quarter-log of blood (P) renders him unclean by overshadowing (R) (R > 0) enough to make him have to cut off his hair (S).



R. Akiva seems to sustain R. Eleazar[10], by arguing a fortiori as described. Then R. Akiva explains that R. Eliezer objected to this a fortiori argument (as against Halakha), to this, whereas R. Yehoshua approved of it (i.e. as formally valid) but suggested the Halakha goes the other way anyway (like R. Eliezer).

Nedarim 10:7 I (2) = A4.2

TEXT: “If a man said to his wife ‘Let every vow be established that you shall vow from this time forth until I return from such a place’, he has said nothing; but if he said, ‘Let them be void’, R. Eliezer says: They are cancelled. But the Sages say: They are not cancelled. R. Eliezer said: If he can cancel vows which have already had [for a time, before he cancelled them] the force of a ‘prohibition’ (cf. Num. 30:3) [as any vow of his wife that he cancels], can he not also cancel vows which have not yet the force of a ‘prohibition’? {They answered: Behold, it is written. ‘Her husband may establish it and her husband may cancel it’ – that which comes under the category of ‘establishing’ also comes under the category of ‘cancelling’, and that which does not come under the category of ‘establishing’ [also] does not come under the category of ‘cancelling’.}”

MY READING: R. Eliezer’ argument is positive predicatal, major to minor:

More authority (R) is required to cancel vows which already had the force of prohibition (P) than vows which do not yet have such force (Q).

If a husband (S) has authority (R) enough to cancel his wife’s vows which already have the force of a prohibition (P),

then a husband (S) has authority (R) enough to cancel his wife’s vows which do not yet have the force of a prohibition (Q).



The Sages reject this conclusion, effectively by denying the major premise. By saying that the husband can only cancel vows that he can establish, they mean (if I understand correctly) that since he is away and not able to establish his wife’s vows individually, he has no authority to cancel them collectively in advance.

Negaim 10:2

TEXT: “‘Thin yellow hair’ means uncleanness: clustered together or dispersed, surrounded or not surrounded, turned [yellow by the scall] or not turned – words of R. Yehudah. R. Shimon says: It only means uncleanness if turned. R. Shimon said: And it is a an inference: If the white hair, against which another hair does not afford protection, does not render unclean except when turned, then the ‘yellow thin hair’, against which another hair does afford protection (cf. Lev. 13:31), is it not an inference that it also does not shall render unclean except when turned?’

MY READING: R. Shimon’s argument is positive subjectal, minor to major.

‘Yellow thin hair’ (P) is afforded more protection against uncleanness by another hair (R) than white hair (Q) is.

If someone with ‘white hair’ (Q) is afforded enough protection against uncleanness by another hair (R) (R = 0) to not-render him unclean except when it is turned (S),

then someone with ‘yellow thin hair’ (P) is afforded enough protection against uncleanness by another hair (R) (R > 0) to not-render him unclean except when it is turned (S).



TEXT: “R. Yehudah said: In every place where it was necessary to say ‘turned’, it [i.e. Scripture] said ‘turned’ (e.g. Lev. 13:3). But the scall, about which it is said: ‘And there is no yellow hair in it’, renders unclean [whether the hair] turned [yellow[11]] or whether it did not turn.”

MY READING: R. Yehudah proposes a case which apparently belies or gives an exception to the conclusion of the previous a fortiori argument, i.e. he says that: whether it turned or did not turn, scall renders unclean. How is this a rebuttal? I am not sure. I would rather look at the above conclusion clause ‘except when turned’ and suggest that R. Yehudah is saying: the above conclusion says that unturned implies not unclean, whereas scall is a case where albeit unturned, nevertheless unclean is implied.

Negaim 12:5

TEXT: “About what then does the Torah take care? About his earthenware utensils, and about his flask and his [oil] vessels. If the Torah thus cares for his humble possession, how much more for his beloved possession! If [it thus cares] for his possession, how much more for the life of his sons and daughters! If [it thus cares] for those of the wicked, how much more for those of the righteous!”

MY READING: There are here three distinct a fortiori arguments, all of positive subjectal (minor to major) form. These could be intended as a crescendo. The first is:

Beloved possessions (P) are more valuable (R) than humble possessions (Q) are.

If humble possessions (Q) are valuable (R) enough to be taken care of by the Torah (S).

then beloved possessions (P) are valuable (R) enough to be taken care of by the Torah (S).



Similarly the other two. Note that there is a progression in value, from humble to beloved possessions, from material possessions to life of children, from life of children of wicked to those of righteous. The conclusion of first is a springboard for the next, which is in turn a springboard for the third. A crescendo readings would imply more care taken by the Torah in each succeeding case, in proportion to the value of the possessions.

Pesahim 6:2 (a)

TEXT: “These acts pertaining to the Pesah offering override the Sabbath: slaughtering it, tossing its blood, scraping its entrails and burning its fat pieces. But the roasting of it and rising its entrails do not override the Sabbath. Carrying it [to the Temple] and bringing it from the outside to within the Sabbath limit and cutting off a wen [from the carcass] do not override the Sabbath. R. Eliezer says: They do override it. And is it not an inference: Just as the slaughtering which comes under [Sabbath] work overrides the Sabbath, those [activities] which come [only] under [Sabbath] rest – should they not [also] override the Sabbath?”

MY READING: This is positive subjectal, minor to major.

Activities classed under Sabbath rest (P) are more leniently regulated (R) than activities classed under Sabbath work (Q).

If activities classed under Sabbath work (Q) are leniently regulated (R) enough to be permitted on the Sabbath (S),

then, activities classed under Sabbath rest (P) are leniently regulated (R) enough to be permitted on the Sabbath (S).



FURTHER TEXT: “R. Joshua said to him: The festival day proves [it], for on it they have allowed [activities] under the category of work and [activities] under the category of rest are forbidden. R. Eliezer said to him: What is this, Joshua! What is a proof from that which is allowed to that which is commanded?”

MY READING: R. Joshua objects to R. Eliezer’s argument by pointing out that on a festival some Sabbath work activities are permitted and some Sabbath rest activities are forbidden. This means that R. Eliezer’s major premise about relative leniency of regulation is not universally true – and so the conclusion he draws cannot be drawn. R. Eliezer replies by claiming that R. Joshua is inferring something commanded from something allowed. I do not know to what he is referring specifically.

FURTHER TEXT: “R. Aqiva replied and said: The sprinkling [of the sin offering water on day 3 and 7 after attracting corpse-uncleanness] proves [it], for it is commanded and it comes under [Sabbath] rest, but it does not override the Sabbath. Thus also do not be astounded at those [other] ones, for they [too], despite being commanded and [only] under the category of rest, do not override the Sabbath.”

MY READING: R. Akiva is saying: Sprinkling is commanded Sabbath rest, yet is forbidden on a Festival. Therefore, conceivably, other things may be commanded Sabbath rest, yet be forbidden on a Festival. This like the preceding objection is designed to neutralize R. Eliezer’s a fortiori argument.

Pesahim 6:2 (b) and (c)

TEXT: “R. Eliezer said to him: And on this [itself] do I base an inference: (And) if the slaughtering which is under the category of work overrides the Sabbath, the sprinkling which is under the category of rest – should it not be inferred that it overrides the Sabbath [also]? R. Aqiva said to him: Or the reverse! If the sprinkling which is [only under the category] of rest does not override the Sabbath, the slaughtering which is [under the category] of work – should it not be inferred that it [also] does not override the Sabbath?”

MY READING: Here we have for once two rival a fortiori arguments! This is worth mentioning as an example of such rivalry.

R. Eliezer’s is almost the same argument as already seen, except that here ‘slaughtering’ and ‘sprinkling’ are specifically mentioned instead of the vaguer minor and major term. It is positive subjectal, going from minor to major.

Activities classed under Sabbath rest (P) are more leniently regulated (R) than activities classed under Sabbath work (Q).

If the slaughtering which comes under Sabbath work (Q) is leniently regulated (R) enough to be permitted on the Sabbath (S),

then, sprinkling which comes [only] under Sabbath rest (P) is leniently regulated (R) enough to be permitted on the Sabbath (S).



R. Aqiva’s retort is negative subjectal, going from major to minor, as follows:

Activities classed under Sabbath rest (P) are more leniently regulated (R) than activities classed under Sabbath work (Q).

If sprinkling which comes [only] under Sabbath rest (P) is leniently regulated (R) not enough to be permitted on the Sabbath (S),

then, the slaughtering which comes under Sabbath work (Q) is leniently regulated (R) not enough to be permitted on the Sabbath (S).



What is the status of this controversy? The two arguments in fact formally imply each other, since they have the same major premise. What puts them in opposition to each other is that each speaker assumes himself to have a true minor premise, and therefore his opponent to have a false conclusion. Presented with the two arguments, and no other information, we have no way to choose between them. Though contrary, they are both equally cogent hypothetical scenarios, given their common major premise. It is a standoff. The answer is presumably given further on in the text.

Pesahim 6:5

TEXT: “R. Eliezer argues: If a person, when he has changed the name of the paschal sacrifice, which sacrifice he may slaughter on the Sabbath, is deemed to be guilty; does it not follow that when he had changed the names of other sacrifices which are already prohibited to be offered thereon as such, that he must a fortiori, be considered guilty?”

MY READING: This is a Mishnaic a fortiori argument not mentioned by Samely. It can be put in positive subjectal form:

Changing the purpose of a sacrifice that must not be slaughtered on the Sabbath (P) is more culpable (R) than changing the purpose of a sacrifice that may be slaughtered on the Sabbath (Q).

If a person who changes the purpose of a sacrifice (such as the paschal sacrifice) which may be slaughtered on the Sabbath (Q) is culpable (R) enough to be liable to a sin-offering (S),

then, a person who changes the purpose of a sacrifice which (though eligible on Pesach) must not be slaughtered on the Sabbath (P) is culpable (R) enough to be liable to a sin-offering (S).



FURTHER TEXT: “To this R. Joshua answered: You cannot apply what is affirmed in respect to the sacrifice, when it was changed to that which it is unlawful to offer on the Sabbath, to other sacrifices where the name has been changed to what is lawful. R. Eliezer replied: The offerings brought for the whole congregation [of Israel] shall prove [my assertion,] for it is lawful to offer them on the Sabbath under their proper name; yet whoever brings other offerings under their denomination is declared to be guilty. Then R. Joshua answered: You cannot apply what is affirmed in respect to the offerings of the whole congregation which have a determinate number, to the paschal sacrifice which has no determinate number.”

MY READING: R. Joshua apparently denies the major premise, saying that relabeling a sacrifice as equivalent to one unlawful on the Sabbath (e.g. changing the purpose of a paschal offering to some other) is not comparable to relabeling a sacrifice as equivalent to one lawful on the Sabbath (e.g. changing the purpose of some other offering to paschal). To defend his major premise, R. Eliezer retorts that whereas sacrifices for the whole congregation may be offered on the Sabbath under their name (i.e. as public offerings), other sacrifices cannot be offered on the Sabbath under that name (i.e. as public offerings); that is, the latter name change does not make them Sabbath compatible (just as the name change to paschal sacrifice does not make an offering Sabbath compatible). But R. Joshua rejects that defense, saying that whereas the offerings of the whole congregation are limited in number, the paschal sacrifice is not (so no comparison between them is possible).

Sanhedrin 6:5

TEXT: ‘R. Meir said: … says God (var. Scripture), I am pained at the blood of the wicked, how much more at the blood of the righteous!’

MY READING: positive subjectal, minor to major. Could be intended as a crescendo.

God has for the righteous (P) more concern (R) than He has for the wicked (Q).

If God has for the wicked (Q) concern (R) enough to be pained at their blood (S),

then God has for the righteous (P) concern (R) enough to be pained at their blood (S).



An a crescendo reading would imply God’s greater unhappiness in the latter case, in proportion to His greater love for the righteous.

Shebuoth 3:6

TEXT: “If he swore to cancel the commandment and did not cancel it, he is free (but see mShebu 3:8); [if he swore] to fulfill the commandment and did not fulfill it, he is free. Yet, it might be inferred that he was culpable, as according to the words of R. Yehudah ben Batyra. R. Yehudah ben Batyra said: If he is liable for [broken] oaths concerning that which is discretionary, for which no oath was imposed from Mount Sinai, is it not logical that he should be liable for [broken] oaths concerning commandments, for which an oath was imposed from Mount Sinai?”

MY READING: R. Yehuda’s argument is a positive subjectal, minor to major.

Broken oath about commandment (P) is more binding (R) than broken oath about discretionary item (Q) is.

If broken oath about discretionary item (Q) is binding (R) enough to make one liable (S),

then broken oath about commandment (P) is binding (R) enough to make one liable (S).



FURTHER TEXT: “They said to him: No; if you speak of an oath concerning what is discretionary, in which a No is as [valid as] a Yes, would you say the same for an oath concerning a [positive] commandment, where the No is not as [valid as] the Yes? (some mss add: So that a person taking an oath to cancel it, and did not cancel it, is not liable).”

MY READING: The rebuttal apparently denies the truth of the major premise – i.e. not all broken oaths about a commandment are taken that seriously. I gather from Samely that this refers to oaths against the commandment which are exempt from liability if not fulfilled.

Sotah 6:3 (a)

TEXT: “[mishnah 2] If [even] one witness said: I have seen her that she was defiled, she does not drink [the Sotah waters], and not only this, but even a slave, even a female slave, behold these are believed … Her mother-in-law, the daughter of her mother-in-law … behold these are believed … [3] It could have been a [correct] inference [to say]: If the initial testimony, which renders her not forbidden forever [to her husband], cannot be established by less than two witnesses, should not that which does render her forbidden forever, [also] be established by a minimum of two witnesses? {In this regard it is instructive that Scripture says, ‘and there is no witness against her’ – any testimony regarding her.}”

MY READING: This is positive subjectal, minor to major.

The later testimony (P) forbids wife to husband for longer time (R) than the initial testimony (Q) does.

If the initial testimony (Q) forbids wife to husband (not forever) for long (R) enough to require at least two witnesses (S),

then the later testimony (P) forbids wife to husband (forever) for long (R) enough to require at least two witnesses (S).



Sotah 6:3 (b)

TEXT: “There is an inference to be drawn from the less to the more stringent concerning the first testimony from this very fact [that only one witness is necessary]: Just as the last testimony which renders her forbidden forever, behold, is established by one witness [only], should not the first testimony which does not render her forbidden for ever also be capable of being established by one witness [only]?!”

MY READING: This is positive subjectal, minor to major.

The initial testimony (P) forbids wife to husband for shorter time (R) than the later testimony (Q) does.

If the later testimony (Q) forbids wife to husband (forever) briefly (R) enough to require only one witness (S),

then the initial testimony (P) forbids wife to husband (not forever) briefly (R) enough to require only one witness (S).



This argument, take note, is intended to rebut – or at least to rival (being apparently an equally cogent alternative) – the preceding one. Notice that though the terms here are labeled by me similarly to those there (i.e. P, Q, R, S), the meanings are different. Here, the middle term is the relative of the previous middle term (referring to shortness of time instead of length of time), and consequently the roles of the initial and final testimony are reverse; moreover, the subsidiary terms has changed from “at least two witnesses” to “only one witness.”

FURTHER TEXT: “{In this regard it is instructive that Scripture says: ‘For he has found in her the indecency of a matter [and he writes for her a bill of divorce]’, and above it says: ‘According to two witnesses [or according to three witnesses] shall the matter be established’. Just as the ‘matter’ enunciated above is [established] according to two witnesses, so the ‘matter’ enunciated here is according to two witnesses also.}”

MY READING: The above counterargument (b) is rejected by reference to Scripture, which specifies two or more witnesses for the initial testimony. It does not follow, however, that the previous argument (a) is established. I suspect (though this needs verification) that the Mishna which advocates only one witness for the final testimony is maintained, somehow.

Temurah 1:1 (a) and (b)

TEXT: “{The priests may substitute what is theirs and Israelites substitute what is theirs. The priests do not substitute the sin offering, and not the guilt offering and not the firstling. R. Yohanan ben Nuri said: And why do they not substitute the firstling? R. Aqiva said to him: The sin offering and the guilt offering are a gift to the priest; and the firstling is a gift to the priest. Just as they may not substitute the sin offering and the guilt offering, so they may not substitute the firstling.}”

MY READING: R. Akiva’s initial argument seems to be a negative subjectal, a pari (egalitarian).

The sin/guilt offerings (P) are as much a gift to the priest (R) as the firstling (Q) is.

If the sin/guilt offerings (P) are gifts (R) not enough to be substitutable (S),

then the firstling (Q) is a gift (R) not enough to be substitutable (S).



FURTHER TEXT: “R. Yohanan ben Nuri said to him: What do I have [knowing that] there is no substitution of sin offering and guilt offering, for to these they have no right while they [the animals] are alive, and yet you are telling me regarding the firstling to which they do have a right while it is alive? {R. Aqiva said to him: And is it not already said: ‘And it will be that both it and its substitute will be holy’? Where does its holiness take effect for it? In the house of the owner. So also substitution, in the house of the owner.}”

MY READING: R. Yohanan’s argument is intended to rival the preceding one by R. Akiva. It appeals to an additional distinction between live and dead offerings, which makes the attempted a fortiori argument not egalitarian, and therefore invalid, because though it is negative subjectal, it is yet minor to major.

What becomes priestly property while alive (P) is more fully owned (R) than what becomes priestly property only after slaughter (Q) is.

If the sin/guilt offering, which becomes priestly property only after slaughter (Q) is fully owned (R) not enough to be substitutable (S),

then the firstling, which becomes priestly property while alive (P) is fully owned (R) not enough to be substitutable (S).



R. Akiva apparently counters this invalid argument, if I understand correctly, with a claim that both offerings are equally holy and that holiness takes effect as soon as it comes into the owner’s home, so that substitution can take effect at once. This is not a third a fortiori argument, but an attempt to neutralize R. Yohanan’s rival a fortiori argument by denying his minor premise and conclusion. It is noteworthy that R. Akiva does not here (apparently) challenge R. Yohanan on more formal ground, i.e. by pointing out that his reasoning process is invalid.

Terumoth 5:4 (a) and (b)

TEXT: “If one seah of unclean heave offering fell into a hundred seahs of clean heave offering, the House of Shammai forbid the whole, but the School of Hillel permit it. The House of Hillel said to the House of Shammai: Since clean [heave offering] is forbidden to non-priests and unclean is forbidden to priests, if the clean can be outweighed cannot the unclean be outweighed too? The House of Shammai answered: No! If the ‘light’ common produce, which is permitted to non-priests, neutralizes what is clean (cf. mTer 5:3), should the ‘weighty’ heave offering, which is forbidden to non-priests, neutralize what is unclean?! After they agreed, R. Eliezer said: It should be taken up and burnt. But the Sages say: It is lost through its scantiness.”

MY READING: There are in fact two a fortiori arguments here, both of them negative subjectal, and both invalid because minor to major (instead of major to minor). The first argument, by the House of Hillel, is intended as valid; the second argument, by the House of Shammai, is put forward as invalid: it is formulated in order to show up the invalidity of the first argument.

The Hillel House argument seems to be the following:

What is forbidden [even] to priests (unclean heave offerings) (P) is more restricted (R) than what is forbidden to non-priests [but not to priests] (clean heave offerings) (Q).

If the clean heave offerings, which are forbidden to non-priests [but not to priests], (Q) are restricted (R) not enough to be prevented from being outweighed by clean common food (= effectively turned into clean common food by mixture in 100 times more of it) (S),

then the unclean heave offerings, which are forbidden [even] to priests (P) are restricted (R) not enough to be prevented from being outweighed by clean common food (= effectively turned into clean common food by mixture in 100 times more of it) (S).



This argument is fallacious: one can well imagine the clean being outweighed but the unclean not being outweighed. To say that the latter follows the former is a non-sequitur. This is apparently the intent of the objection by the Shammai House. They are not so much proposing a counter a fortiori argument as denying the process of the Hillel House proposal. Nevertheless, they modify the wording of the a fortiori argument as follows, presumably so as to show more clearly its absurdity: “If the ‘light’ common produce, which is permitted to non-priests, neutralizes what is clean (cf. mTer 5:3), should the ‘weighty’ heave offering, which is forbidden to non-priests, neutralize what is unclean?!”

What is forbidden to non-priests (‘weighty’ heave offerings) (P) is more restricted (R) than what is permitted to non-priests (‘light’ common produce) (Q).

If the ‘light’ common produce, which is permitted to non-priests, (Q) is restricted (R) not enough to be prevented from being neutralized by clean common food (= effectively turned into clean common food by mixture in 100 times more of it) (S),

then the ‘weighty’ heave offerings, which are forbidden to non-priests, (P) are restricted (R) not enough to be prevented from being neutralized by clean common food (= effectively turned into clean common food by mixture in 100 times more of it) (S).



This argument differs from the preceding in that it concerns only non-priests, ranging from what is forbidden to them to what is permitted to them. This clarifies the logical issue a bit, removing complications in the terms. If what is permitted to them can be neutralized, then surely what is forbidden to them can be neutralized too? The logical answer is of course: no – one can conceive the former being true without the latter being true. So this is an illicit process – i.e. the argument is invalid, going from minor to major whereas it should have gone from major to minor (i.e. if the forbidden can be neutralized then yes, surely the permitted can be so too). So this second a fortiori argument is invalid too – but intentionally so, so as to emphasize the invalidity of the first a fortiori argument.

In my opinion, Shammai here beats Hillel; i.e. Hillel House has not proven its point and Shammai House has demonstrated that absence of proof (though that does not mean it proves the opposite point). The last sentence in this passage, “After they agreed, R. Eliezer said: It should be taken up and burnt. But the Sages say: It is lost through its scantiness.” seems to say that the two sides agreed that Shammai House was right in its critique of Hillel House.

ADDITIONAL NOTE. Moreover that the conclusion of Hillel House is about outweighing by clean common food – but a further argument is tacitly implied, that if unclean heave offerings are outweighed by clean common food, then they are also a fortiori outweighed by clean heave offerings (which is the desired final conclusion), since the latter are more holy than the former. Similarly, the (ad absurdum) conclusion of Shammai House is about outweighing by clean common food – but a further argument is tacitly implied, that if ‘weighty’ heave offerings are outweighed by clean common food, then they are also a fortiori outweighed by clean heave offerings (which is the desired final conclusion), since the latter are more holy than the former. So we may say that we in fact have four a fortiori arguments here! The first two (explicit) are invalid, but the latter two (implicit) would be valid. I do not count the latter, since no one in the text has actually stated them.

Yadayim 4:7

TEXT: “The Sadducees say: We raise a complaint against you, o Pharisees, (for you say: If my ox and my donkey have caused damage they are culpable [making me liable]; but if my slave and female slave have caused damage, they are free [causing no liability for me]). Just as with regard to my ox and my donkey, concerning which I am not liable through commandments, behold I am culpable for damage, is it not logical that with regard to my slave and my female slave, concerning whom I am liable through commandments, I should be liable for damage?”

MY READING: The Sadducees propose the following positive subjectal, minor to major:

The owner of a male or female slave (P) is more liable through commandments (R) than the owner of an ox or donkey (Q) is.

If the owner of an ox or donkey (Q) is liable through commandments (R) (R = 0) enough to be culpable for damage (S),

Then the owner of a male or female slave (P) is liable through commandments (R) (R > 0) enough to be culpable for damage (S).



FURTHER TEXT: “They said to them: No. If you say this about my ox and my donkey that have no understanding, will you also say it about my slave and female slave who have understanding? So that if I provoke him he goes and sets fire to someone’s stack of corn and I am liable to compensate?”

MY READING: The Pharisees object to the above a fortiori argument of the Sadducees by denying its major premise, saying: the owner is responsible for his animals because they cannot understand laws, but the owner is not responsible for his slaves because they can understand laws.

Yadayim 4:8

TEXT: ‘A Galilean heretic (var.: Sadducee) said: I raise a complaint against you, O Pharisees, for you write the [name of the] ruler together with [the name of] Moses in a bill of divorce. The Pharisees say: We [raise a complaint] against you, O Galilean heretic, for you write the Name [of the] God together with the [name of] the ruler on [one] page, and not only that, but you write the ruler above and the Name beneath, (var. as it is said:) ‘And Pharaoh said: Who is the Lord that I shall listen to his voice to let go Israel? [I do not know the Lord and also Israel I shall not let go]’. {And when he was smitten, what does he say? ‘The Lord is righteous [and I and my people are the wicked ones].’}’

MY READING: The Pharisee argument is best expressed as negative subjectal, since the argument goes from major to minor.

God (P) is more worthy of being dissociated from earthly rulers (R) than Moses (Q) is.

If God (P) is worthy of being dissociated from earthly rulers (R) not enough to have his name excluded from a document with an earthly ruler’s name in it (S) (specifically, in the Torah, with Pharaoh),

then Moses (Q) is worthy of being dissociated from earthly rulers (R) not enough to have his name excluded from a document with an earthly ruler’s name in it (S) (specifically, in a bill of divorce, with any current ruler).



This a fortiori argument is put forward by the Pharisees, in order to arrive at a conclusion which contradicts the Sadducee’s (or Galilean’s) assertion (which is not an argument, notice) that we cannot write the name of Moses together with that of the ruler in a bill of divorce. They say, citing an instance from Scripture: not only can such names appear together, but the more honorable one can even appear beneath the less honorable one, and not only in bill of divorce but in any document.

Yebamoth 8:3

TEXT: “{‘An Ammonite and a Moabite’ is prohibited [to marry an Israelite] and their prohibition is an everlasting prohibition (cf. verse). But their females are allowed right away. An Egyptian and Edomite are only prohibited for three generations, males as well as and females.} R. Shimon allows the females right away. R. Shimon said: The things are lighter and heavier: If in a place for which it [Scripture] forbids the males with an everlasting prohibition, it allows the females right away, then in a place for which it forbids the males only for three generations, is it not logical that the females are allowed right away?! They answered: If this is Halakhah [which you have received] we receive it. But if it is but an inference [of your own] a counter-interference may rebut it. He answered: Not so, but I declare what is Halakhah.”

MY READING: This argument is negative subjectal, since it goes from major to minor.

The females of peoples whose males are forbidden forever (namely, Ammonites and Moabites) (P) are more liable to exclusion (R) than the females of peoples whose males are forbidden for three generations (namely, Egyptians and Edomites) (Q) are.

If the females of peoples whose males are forbidden forever (P) are liable to exclusion (R) not enough to be prevented from inclusion forthwith (S),

then the females of peoples whose males are forbidden for three generations (Q) are liable to exclusion (R) not enough to be prevented from inclusion forthwith (S).



Yom Tov 5:2

TEXT: “Any act that is culpable on the Sabbath, whether by virtue of the rules concerning Sabbath rest (cf. Erub. 10:3, 15) or concerning acts of choice or concerning pious duties, is culpable also on a festival day. And these by virtue of the rules concerning Sabbath rest: no one may climb a tree or ride a beast or swim on water or clap the hands or slap the thighs or stamp the feet [or dance]. And these by virtue of the rules concerning acts of choice: no one may sit in judgment or conclude a betrothal or perform Halitsah or contract levirate marriage. And these by virtue of the rules concerning pious duties: no-one may dedicate anything or make a vow of valuation or devote anything or set apart heave offering or tithes (Deut. 14:22-29). All these things have they prescribed [as culpable] on a festival day: still more so [are they culpable] on the Sabbath. A Festival-day differs from the Sabbath in nothing but the preparing of necessary food (cf. mMeg 1:5).”

MY READING: This is a positive subjectal, minor to major.

The Sabbath (P) is more restrictive (R) than any Festival day (Q) is.

If a Festival day (Q) is restrictive (R) enough to prescribe certain listed actions (S),

then the Sabbath (P) is restrictive (R) enough to prescribe the same listed actions (S).



We are given that everything prescribed on Sabbath is so on Festival day, except food preparation; and everything prescribed on Festival day is all the more so on Sabbath[, without exception]. The first proposition is almost general but exceptive. The second, which is the reverse if-then, is fully general. Both propositions are needed to fully express the relation between the two situations. As regards the a fortiori argument, it is an apparent redundancy, since we anyway know its conclusion before and independently of its premises. Nevertheless, it can be presented as a useful rule of thumb. That is, it may not be of hermeneutic/theoretical value, but it is heuristic/practical utility. In any case, the argument is formally valid and that is what concerns us here.

Zebahim 7:4

TEXT: “If the whole offering of a bird was offered below (the red line) after the manner of a sin offering and under the name sin offering, R. Eliezer says: The law of sacrilege still applies to it. R. Yehoshua says: The law of sacrilege no longer applies to it. R. Eliezer said: If the sin offering, which is not subject to the law of sacrilege when it is offered under that name, becomes subject to the law of sacrilege if it is offered under another name, how much more must the whole offering, which is subject to the law of sacrilege when it is offered under that name, be subject to the law of sacrilege when it is offered under another name.”

MY READING: R. Eliezer’s argument is positive subjectal, minor to major.

The whole-offering (P), being subject to law of sacrilege under its own name, is more susceptible to sacrilege (R) than the sin-offering (Q), which is not subject to law of sacrilege under its own name.

If the sin-offering (Q) is susceptible to sacrilege (R) (R = 0) enough to be subject to the law of sacrilege under another name (S),

then the whole-offering (P) is susceptible to sacrilege (R) (R > 0) enough to be subject to the law of sacrilege under another name (S).



FURTHER TEXT: “R. Yehoshua said to him: No, as you argue of the sin offering, which when its name is changed to that of a whole offering thereby becomes changed to a thing subject to the law of sacrilege, would you also argue of a whole offering, which when its name is changed to that of a sin offering thereby becomes changed to a thing not subject to the law of sacrilege)?”

MY READING: R. Yehoshua objects to R. Eliezer’s a fortiori argument, by considering changes of status from sin offering to whole offering and vice versa. He points out that in the former case, the change makes the offering become subject to the law of sacrilege; whereas in the latter case, the change makes the offering cease to be subject to the law of sacrilege. This denies the conclusion of the a fortiori argument, and thus puts in doubt the process of inference. What is formally wrong with that process? The answer to this question is that although superficially the subsidiary term is the same in the minor premise and conclusion, if we examine it more closely we realize that it is not really so. The words used are the same, but their underlying meaning is quite different. In the minor premise, the offering becomes truly subject to the law of sacrilege, whereas in the conclusion the offering becomes not subject to it. In the subsidiary term, we cannot use the same relative language as we use in the middle term. Whereas (R) = 0 and (R) > 0 can both count as (R), (S) = 0 and (S) > 0 cannot both count as (S). This is precisely the meaning of R. Yehoshua’s objection.[12]

Zebahim 7:6

TEXT: “If he had nipped off the head (of the bird) and it was found to be terefah, R. Meir says: It does not convey uncleanness of the gullet. R. Yehudah says: It conveys uncleanness of the gullet. R. Meir said: If in the case of a beast which as carrion would convey uncleanness by contact or carrying, slaughtering renders clean the uncleanness of an animal that is terefah, is it not an inference that, in the case of a bird which as carrion would not convey uncleanness by contact or carrying, slaughtering should render clean the uncleanness of an animal that is terefah? Just as we find with slaughtering that it renders fit for eating and renders clean the terefah from its uncleanness, so nipping off (the head) which renders fit for eating, renders clean the terefah from its uncleanness.”

MY READING: R. Meir’s argument negative subjectal, major to minor.

A beast (which as carrion would so convey) (P) is more able as carrion to convey uncleanness by contact or carrying (R) than a bird (which as carrion would not so convey) (Q) is.

If a beast (P) is able as carrion to convey uncleanness by contact or carrying (R) not enough to prevent its slaughtering from rendering clean the uncleanness of its terefah (S),

then a bird (Q) is able as carrion to convey uncleanness by contact or carrying (R) not enough to prevent its slaughtering from rendering clean the uncleanness of its terefah (S).



Thereafter, R. Meir argues by analogy from slaughtering to nipping off head. It goes apparently: “Since slaughtering and nipping-off both render fit for eating, then just as the former renders clean the terefah from its uncleanness, so does the latter.”

FURTHER TEXT: “R. Yose says: It is enough (dayo) to compare the carrion of a beast: slaughtering renders clean, nipping off does not.”

MY READING: R. Yose objects to R. Meir’s arguments. Though he uses the language of dayo, saying “it is enough,” I do not think he is really invoking the principle of sufficiency, since there is no quantitative or other difference in the subsidiary term of R. Meir’s proposed conclusion. R. Yose denies that the conclusion of the a fortiori argument follows from its minor premise (which he accepts), saying: “[Even though] slaughtering the carrion of a beast renders clean, [still] nipping off [head of bird] does not [render clean].” This implies that R. Yose doubts R. Meir’s major premise, for some reason.

Zebahim 8:12

TEXT: “If the blood of a sin offering was received into two cups and one of them was brought to the outside [of the temple court], the one that remained inside is fit. If one of them was brought inside [the sanctuary], R. Yose Ha-Gelili declares the outside one fit [i.e. the one that is in the temple court], and the Sages declare it unfit. R. Yose Ha-Gelili said: If in the case where the thought renders unfit, [as when there is an intention to sprinkle] outside [the temple court, cf. mZeb 2:2], this [outside] does not render unfit the remainder [still inside the temple area] like the one that was brought out, is it not an inference that in a case where thought does not render unfit, [as when there is an intention to sprinkle] inside [the sanctuary], the one that remains outside is [also] not made like the one brought in [to the sanctuary, namely invalid, mZeb 8:11] ?”

MY READING: R. Yose’s argument is negative antecedental, major to minor.

Wrongly sprinkling blood outside the temple (which renders it unfit) (P) is more ritually problematic (R) than wrongly sprinkling blood inside the sanctuary (which does not render it unfit) (Q).

If wrongly sprinkling blood outside the temple (P) is ritually problematic (R) not enough to render the remaining blood unfit (S),

then wrongly sprinkling blood inside the sanctuary (Q) is ritually problematic (R) not enough to render the remaining blood unfit (S).



Zebahim 12:3

TEXT: “The hides of the lesser holy offerings belong to the owners; the hides of the most holy offerings belong to the priests. Light and heavy: Just as when in the case of a burnt offering for which they do not have the right to its flesh, they have the right to its hide, for the most holy offerings, for which they have a right to their flesh, is it not an inference that they also have the right to their hides?”

MY READING: This is a positive subjectal, minor to major.

The most holy offerings (whose flesh does belong to the priests) (P) belong to the priests (R) more than the burnt offerings (whose flesh does not belong to the priests) (Q) do.

If the priests have in relation to the burnt offerings (Q) ownership rights (R) enough to have the right to the hides (S),

then the priests have in relation to the most holy offerings (P) ownership rights (R) enough to have the right to the hides (S).



FURTHER TEXT: “The (case of the) altar cannot serve as standard (countering the inference), for it does not have the hide in any case.”

MY READING: This is an objection: a denial of the conclusion, which puts in doubt a premise or the process.

DISCLAIMER: I would like to emphasize that I am not a Talmudist. Being but an amateur, it is quite possible that I have partly or wholly misunderstood some of the texts. I do not pretend here to have fully and accurately explicated the Mishna passages listed – I am not knowledgeable in Jewish law enough to do that. All I have tried to do is to briefly interpret the a fortiori aspect of these discourses in standard form, as they appear without looking at the wider context. Of course, I should have devoted more study to this field, and even consulted an expert in it in order to confirm or correct my interpretations, but I chose not to do so, considering that I had my hands full already with more pressing matters. I would be very grateful to anyone who, finding errors in my treatment, tells me about them. I would certainly encourage anyone who can improve on my work to do so. If the latter wishes me to publish his or her commentary, please submit it to www.logicforum.org.



[1] These are formulated as negative questions: e.g. “Is it not an inference that?” I presume that such expressions refer to the phrase “eino din” in the original Hebrew. I do not at this time have the resources to verify translations.

[2] These are all, I presume, literal translations of expressions like qal vachomer or miqal lechomer.

[3] The expression “dayo” is used in Zebahim 7:6, but as I point out there this use is rhetorically and not literally intended.

[4] This may explain why the lists of Hillel and R. Ishmael do not mention a dayo principle, though they mention a fortiori argument. Also, it suggests that the two dayo objections in the said Mishna may have been intended as merely ad hoc solutions, which were only later elaborated as guiding principles in the Gemara.

[5] Louis Jacobs, in his Studies (in a footnote on p. 4), suggests that this a fortiori argument, being by Jose ben Johanan (c. 160 BCE), “may be the earliest reference in Rabbinic literature to the qal wa-homer.” Then he adds that “the highly plausible suggestion has been made that the qal wa-homer does not belong to Jose’s saying but is a later addition;” in support of which he mentions Schwarz and Daube. But I would not regard a suggestion by the latter two writers as highly plausible.

[6] The reason I think it is a pari is that I see no reason offhand to treat the two parties differently, i.e. I assume all Israelites at once and equally (not just the non-Levites) were removed from old the “first-born” religious régime and placed in the new “priestly caste” religious régime. That some Levites thereby lost and found the same duties does not really affect this change of overall framework.

[7] Note that this resembles the earlier a fortiori argument by R. Eliezer, except that the subsumed part is “a barleycorn’s bulk of bone” instead of “an olive’s bulk of flesh”.

[8] Notice that the “given” here is the reverse of the previous “given,” and likewise the “does it follow that” antecedent and consequent are reversed! This seems to suggests that these rebuttals are quite hypothetical, and apparently not intended as factual.

[9] The major premise is obtained by generalization from potential and actual transgressions (bad) to all relations to commandments (bad and good), including fulfillment of commandments (good).

[10] R. Eleazar’s argument looks like mere analogy (actually, an argument by inversion, i.e. a contrario): if uncleanness of Nazirite from corpse is sufficient to impose haircut then it is sufficient to forbid Temple entry; likewise, if uncleanness of Nazirite from corpse is insufficient to impose haircut then it is insufficient to forbid Temple entry. R. Meir’s reply to this argument is intended to put it in doubt.

[11] The [yellow] interpolation may be Samely’s.

[12] After this, R. Eliezer counters R. Yehoshua’s objection, by pointing out that in some cases name change of an offering subject to the law of sacrilege does not cause that offering to cease to be subject to said law – so this constancy may well apply to the whole offering. But R. Yehoshua rejects this analogy, pointing out certain differences between the proposed analogues.

2016-08-05T04:41:58+00:00