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. 8
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>>

the most common or widespread sense (because it is the ˜being™ involved in
every predication). The inconsistency between E 4 and 10 would then
be merely apparent.
Solution (i) is unconvincing: when in the course of a discussion of the
various senses of an expression Aristotle uses a sentence like ˜This is what
the expression means kuriÛtata™, ˜kuriÛtata™ normally means (not ˜in
the most common or widespread sense™, but) ˜in the most proper or strictest
sense™.6

Solution (ii). Heidegger7 tries to resolve the alleged inconsistency between
E 4 and 10 by assuming that they address different topics. In his view,
E 4 concerns the ˜being™ in the sense of being true which is a property of
thoughts, i.e. of mental items like beliefs; 10 instead concerns the ˜being™
in the sense of being true which is a property of objects. Aristotle could
consistently say, on the one hand, that the ˜being™ in the sense of being
true which is a property of thoughts is different from ˜being™ in the strict
sense, and, on the other, that the ˜being™ in the sense of being true which
is a property of objects is ˜being™ in the strictest sense.8
I think that some claims made by Heidegger in his attempt to resolve
the alleged inconsistency between E 4 and 10 are true and important: he
is right in claiming that E 4 and 10 are about different properties, and,
speci¬cally, that E 4 is about a property of thoughts while 10 is about a
property of objects.9 However, I also think that Heidegger™s solution cannot
stay as it stands. For it requires that this change of properties should not be
re¬‚ected in any change in terminology: Aristotle would be using the same
word, ˜true™, to express both the property that holds of thoughts and the
4 See Jaeger (1912), 51“2, cf. Wilpert (1940), 9; Owens (1951/78), 411“12; Tugendhat (1966b), 408; Berti
(1990), 111“13; Pritzl (1998), 178; Berti (2000), 6“7.
At Metaph. 1, 1045b 36 Aristotle perhaps uses ˜m†lista kur©wv™ to mean ˜in the most common or
5
widespread sense™.
6 Cf. Ross (1924), ii 275; Heidegger (1930), 83“5. Jaeger himself later abandoned solution (i): for in his
1957 edition of the Metaphysics he resorted to textual emendation (cf. the paragraph to which n. 1
above is appended).
7 See Heidegger (1926), 168, 305“6; (1930), 87“91.
8 Elsewhere (in his (1924/25), 187“8) Heidegger insists that the ˜being™ in the sense of being true
discussed in E 4 is a property of objects.
9 Cf. the subsection to which n. 70 of ch. 1 is appended.
10, 1051 b1: the text
236 Metaph.
property that holds of objects. This would be poor practice, particularly
in view of the fact that the passages exhibiting the two uses are supposed
to be reciprocally connected (for in E 4, at 1027b 28“9, Aristotle promises
to discuss later certain questions concerning what ˜is™ in the sense of being
true and what ˜is not™ in the sense of being false,10 and 10 seems his
ful¬lment of this promise).

Solution (iii). Other commentators11 have a better solution for the alleged
inconsistency between E 4 and 10. The words ˜kuriÛtata ¿n ˆlhq•v
£ ye“dov™ must constitute a single clause (˜being in the strictest sense
true or false™): the adverb ˜kuriÛtata™ can be construed only with ˜¿n™,
and ˜¿n™ must be construed with the phrase ˜ˆlhq•v £ ye“dov™ (which
would otherwise remain unconnected to its context). Since ˜t¼ m•n™ at
1051a 34 corresponds to ˜t¼ d•™ at 1051a 35 and to ˜t¼ d•™ at 1051b 1, the clause
˜kuriÛtata ¿n ˆlhq•v £ ye“dov™ explains the third sense in which what ˜is™
and what ˜is not™ are spoken of (in the immediately preceding portion of text
this role is played by the clauses ˜kat‡ t‡ scžmata t¤n kathgori¤n™ and
˜kat‡ d…namin £ –n”rgeian to…twn £ tˆnant©a™). So at 1051a 34“1051b 2
Aristotle is saying that what ˜is™ and what ˜is not™ are spoken of, in a ¬rst
sense, with reference to the categories, in a second sense, with reference to
the potentiality or the actuality of the categories or to their opposites, and,
in a third sense, ˜by being in the strictest sense true or false™. Hence in 10
there is no question of ˜being in the strictest sense™: the text only speaks of
˜being in the strictest sense true or false™. Since 10 does not contain the
claim that ˜being™ in the sense of being true is ˜being™ in the strictest sense,
the alleged inconsistency between E 4 and 10 evaporates.
Objection: since what is being offered is a classi¬cation of the uses of
˜to be™ or ˜being™, the adverb ˜kuriÛtata™ (˜in the strictest sense™) is most
naturally understood as introducing the strictest sense of ˜to be™ or ˜being™.12
Answer: since 10 discusses truth and falsehood, it is just as natural to
understand ˜kuriÛtata™ (˜in the strictest sense™) as introducing the strictest
sense in which something can be called true or false.13
Solution (iii) seems adequate. Hence I adopt the reading ˜kuriÛtata
¿n™ and understand the words ˜kuriÛtata ¿n ˆlhq•v £ ye“dov™ as a single
10 Cf. n. 67 of ch. 1.
11 Brandis (1835/66), ii.ii.i 519“20; Prantl (1855), 185; Grote (1880), 618; Ross (1908), ad loc.; (1924),
ii 275; Deninger (1961), 140“1; Tricot (1966), ii 522; Russo (1973/82), 273; Seidl (1989/91), ii 491;
Burnyeat et al. (1984), 156; Salmeri (1996), 243, 306.
12 Cf. Burnyeat et al. (1984), 156.
13 For an explanation of why Aristotle introduces the idea of the strictest sense in which something
can be called true or false cf. the subsection to which n. 42 of ch. 1 is appended.
Appendix 1 237
clause (˜by being in the strictest sense true or false™) explaining the third
sense in which what ˜is™ and what ˜is not™ are spoken of.

No reason for claiming that the notion of truth or the veridical sense of ˜to be™ have
a privileged status. Contrary to what Heidegger and other commentators
think,14 the reading ˜kuriÛtata ¿n™ constitutes no evidence for crediting
Aristotle with the idea that the notion of truth provides a privileged insight
into that of being. Similarly, pace Kahn,15 the reading in question provides
no support for the contention that in Greek the veridical sense of ˜to be™ is
fundamental.

14 Heidegger (1925/26), 174, 178, 190“3 (cf. (1930), 82“3); Volkmann-Schluck (1979), 281“2.
15 Kahn (1966), 250 (cf. Pritzl (1998), 178).
a pp e n d ix 2

10, 1051 b 2“3: the text
Metaph.




The evidence. At 1051b 2“3 various readings are available. E and J have
[a] to“to d¬ –pª t¤n pragm†twn –stª t¼ sugke±sqai £ dihr¦sqai.
€
This reading, also presupposed by William of Moerbeke™s translation, is
printed by some early editors (the Aldine edition, Brandis, and Weise).
Ab has instead
[b] to“to dŸ ›ti t¤n pragm†twn –stª t¤€ sugke±sqai £ dihr¦sqai.
€
No editor prints this.
Most modern editors (Bekker, Schwegler, Bonitz, D¨ bner, Christ, Ross,
u
Tredennick, and Jaeger) print
[c] to“to d¬ –pª t¤n pragm†twn –stª t¤€ sugke±sqai £ dihr¦sqai,
€
a text obtained by combining readings [a] and [b].

Evaluation. Reading [b] makes no sense and must be discarded. Reading
[a] has the edge on [c] because of the parallel with 1051b 11“13, 1051b 19“20,
and 1051b 33“5. I therefore opt for reading [a].




238
ap p en di x 3

Int. 7, 17 b 16“18: the text 1




The evidence. At 17b 16“20 Aristotle discusses contradictory pairs where one
member is a universal predicative assertion. Here is the text handed down
by the main manuscripts and printed by the majority of recent editors
(Bekker, Waitz, D¨ bner, Cooke, Minio-Paluello, and Zadro):
u
ˆntike±sqai m•n o”n kat†jasin ˆpo- 17b
T 60 16
j†sei l”gw ˆntijatik¤v tŸn t¼ kaq»lou shma©nousan t¤€
aÉt¤€ ‚ti oÉ kaq»lou, o³on pŽv Šnqrwpov leuk»v-oÉ pŽv
Šnqrwpov leuk»v, oÉdeªv Šnqrwpov leuk»v-›sti tiv Šnqrw-
pov leuk»v.2 20

Ammonius (in Int. 109, 24“5) reports that according to Porphyry at 17b 17
some manuscripts instead of ˜ˆntijatik¤v™ read ˜ˆpojantik¤v™ (some
manuscripts of Ammonius have ˜ˆpojatik¤v™, the reading we ¬nd also
in some manuscripts of de Interpretatione, e.g. in Laur. 72, 17).3 In n ˜t¤™ €
b b
(17 17) is added above the line. Bekker claims that at 17 17“18 B reads
˜tŸn t¼ kaq»lou shma©nousan t¦€ t¼ oÉ kaq»lou™, but he is wrong: B™s
reading is the same as that of the other main manuscripts.4 (Was this reading
attested in some other manuscript?) At 17b 18 Laur. 72, 4 omits the ¬rst ˜oÉ™.
In Vat. Palat. Gr. 74 a later hand added ˜t¦€ t¼ aÉt¼™ above ˜t¤€ aÉt¤™.5 €
In Oxon. New College C. 225 (fourteenth century, f. 29r) a variant on the
standard text at 17b 17“18 is reported: ˜tŸn t¼ kaq»lou shma©nousan ‚ti
kaq»lou t¦€ t¼ aÉt¼ ‚ti oÉ kaq»lou™ (˜tŸn t¼ kaq»lou shma©nousan
‚ti kaq»lou t¦™ is in the margin, ˜t¦€ t¼ aÉt¼™ is above ˜t¤€ aÉt¤™: the ˜t¦™
€ € €
at the end of the phrase in the margin is obviously picked up by the ˜t¦™ at
€
the beginning of the phrase above ˜t¤€ aÉt¤™). It remains unclear whether
€
this variant is a reader™s conjecture (the hand that wrote it seems different

1 2 Cf. Int. 14, 24b 1“6.
Cf. the subsection to which n. 46 of ch. 2 is appended.
3 Cf. the variants to ˜ˆntijatik¤v™ at Int. 14, 24b 5.
4 Cf. Waitz (1857), 726; Weidemann (1985), 52.
5 Cf. Weidemann (1985), 53; Weidemann (1994/2002), 214.

239
Int. 7, 17 b 16“18: the text
240
from that which copied the main text) or a record of some manuscript™s
reading.6
Some manuscripts of Boethius™ Latin translation of de Interpretatione
report the following rendering of 17b 16“18: ˜Opponi autem ad¬rmationem
negationi dico contradictorie quae universaliter signi¬cat ei quae non univer-
saliter™. The reading presupposed by this translation is ˜ˆntike±sqai m•n o”n
kat†jasin ˆpoj†sei l”gw ˆntijatik¤v tŸn t¼ kaq»lou shma©nousan
t¦€ t¼ oÉ kaq»lou™.7 (This is the reading which Bekker falsely claims to
¬nd in B.)

Two problems of literal interpretation. The literal interpretation of the initial
part of T 60 at 17b 16“18 poses two problems. The ¬rst concerns the phrase
˜t¤€ aÉt¤™ at 17b 17“18: it is at best an extremely compressed formula.8
€
The second problem concerns the phrase ˜‚ti oÉ kaq»lou™ at 17b 18: there
is nothing with which this phrase can be construed. (One is inclined to
construe ˜‚ti oÉ kaq»lou™ with ˜ˆpoj†sei™ at 17b 16“17, but there is not
enough of a connection between the two expressions for such a construction
to be possible. Things would pan out if, for instance, at an appropriate
point there were a ˜t¦€ [sc. shmaino…sh]™ to pick up ˜ˆpoj†sei™ just as
€
˜tŸn shma©nousan™ picks up ˜kat†jasin™.)9 These problems suggest that
the initial part of T 60 at 17b 16“18 is corrupt.

Emendations. The problems of literal interpretation posed by the initial
part of T 60 at 17b 16“18 instigate emendation.


In many manuscripts some gloss can be found above the words ˜t¤€ aÉt¤™ (17b 17“18): ˜Ëpokeim(”n)w€™
€
6
(B, Laur. 71, 35, and Vat. Barberin. Gr. 87, in all cases a later hand); ˜t¤€ Ëpokeim(”n)w€™ (Laur. 72,
4 and Laur. 87, 16, in both cases a later hand); ˜kathgoroum”(nw€)™ (Laur. 72, 12, a later hand); ˜£
t¤€ aÉt¤€ Ëpokeim(”n)w€ k(aª) kathgoroum(”n)w€™ (Marc. 211, the ¬rst hand); ˜Ëpokeim(”n)w€ k(aª)
kathgoroum(”n)w€™ (A, a later hand). In Laur. 72, 5 a later hand adds the words ˜kat†jasin kaª
ˆp»jasin tŸn pŽv £ tŸn oÉde©v™ above the phrase ˜t¼ kaq»lou shma©nousan™ (17b 17). In Vat.
Barberin. Gr. 87 a later hand adds ˜pr»tasin™ above ˜shma©nousan™ (17b 17).
7 Cf. Buhle (1792/1800), ii 71; Waitz (1844/46), i.xxii. Other manuscripts of Boethius™ Latin translation
report a different rendering of 17b 16“18: ˜Opponi autem ad¬rmationem negationi dico contradictorie quae
universale signi¬cat eidem, quoniam non universaliter™. This second rendering of 17b 16“18 presupposes
the reading of the main manuscripts. The two different renderings of 17b 16“18 are also attested by
different manuscripts for the lemma of Boethius™ First Commentary on Aristotle™s de Interpretatione,
but the second rendering only is attested for the lemma of Boethius™ Second Commentary on Aristotle™s
de Interpretatione. According to Conybeare (1892), 33, 98 some manuscripts of the Armenian version
of de Interpretatione presuppose a Greek text lacking ˜‚ti™, and perhaps presuppose ˜t¼ oÉ kaq»lou™.
8 Cf. Buhle (1792/1800), ii 71.
The word order does not allow ˜t¤€ aÉt¤€ ‚ti oÉ kaq»lou™ at 17b 17“18 to be understood as a single
9
phrase: ˜by virtue of the claim of non-universality itself ™ could translate ˜aÉt¤€ t¤€ ‚ti oÉ kaq»lou™,
not ˜t¤€ aÉt¤€ ‚ti oÉ kaq»lou™.
Appendix 3 241
At 17b 17“18 Pacius reads ˜tŸn t¼ kaq»lou shma©nousan <t¦> t¼ aÉt¼
€
‚ti oÉ kaq»lou™. 10

Du Val (recently followed by Weidemann) reads ˜tŸn t¼ kaq»lou <‚ti
kaq»lou> shma©nousan <t¦> t¼ aÉt¼ ‚ti oÉ kaq»lou™.11
€
Buhle (followed by Weise) adopts the reading presupposed by the ren-
dering of the passage handed down by some manuscripts of Boethius™ Latin
translation of de Interpretatione: ˜tŸn t¼ kaq»lou shma©nousan t¦€ t¼ oÉ
kaq»lou™.

Two apparent incongruities. There seem to be two incongruities in Aristotle™s
account of contradictory pairs where one member is a universal predicative
assertion. They both concern the relation between the general characteri-
sation (in the initial part of T 60 at 17b 16“18) of contradictory pairs where
one member is a universal predicative assertion, on the one hand, and the
two contradictory pairs introduced (in the ¬nal part of T 60 at 17b 18“20)
to clarify that general characterisation on the other:
(i) The general characterisation of contradictory pairs where one mem-
ber is a universal predicative assertion is formulated in the initial part of
T 60 at 17b 16“18. The literal interpretation of this portion of text poses
problems which suggest that what Aristotle wrote underwent some cor-
ruption (cf. the penultimate subsection). Therefore it is not clear what is
Aristotle™s general characterisation of contradictory pairs where one mem-
ber is a universal predicative assertion. According to one interpretation,
Aristotle is claiming that in every such contradictory pair one member
asserts something non-universally of the same universal of which the other
member asserts something universally. Suppose Aristotle claims this. In
T 20 Aristotle claimed, on the one hand, that something can be asserted
universally of a universal by producing a universal predicative assertion
(e.g. ˜Every man is white™ or ˜No man is white™), and, on the other, that
something can be asserted non-universally of a universal by producing an
indeterminate predicative assertion (e.g. ˜A man is white™ or ˜A man is not
white™).12 Therefore one expects that when he illustrates his general charac-
terisation of contradictory pairs where one member is a universal predicative
assertion, Aristotle should produce one or more contradictory pairs each

10 Pacius (1597b), 94; (1623), 107. This is just like the corrector of Vat. Palat. Gr. 74, on which Pacius
might depend.
11 Du Val (1629), 40; Weidemann (1985), 51“5; (1994/2002), 88, 214“15 (cf. Trendelenburg (1836/92),
76; Crivelli (1996), 152). Word order apart, this emendation is the same as the reading recorded by
Oxon. New College C. 225.
12 Cf. proposition [15] on p. 91 above.
Int. 7, 17 b 16“18: the text
242
of which consists of a universal predicative assertion and an indeterminate
predicative assertion (e.g. ˜Every man is white™ “ ˜A man is not white™ or ˜No
man is white™ “ ˜A man is white™). However, when he illustrates his general
characterisation (in the ¬nal part of T 60 at 17b 18“20), Aristotle produces
two contradictory pairs each of which consists of a universal predicative
assertion and a particular predicative assertion (˜Every man is white™ “ ˜Not
every man is white™ and ˜No man is white™ “ ˜Some man is white™).13
(ii) In his general characterisation (in the initial part of T 60 at 17b 16“18)
of contradictory pairs where one member is a universal predicative assertion,
Aristotle seems to say that a predicative assertion which is universal and
af¬rmative is contradictorily opposed to a negative assertion. When (in the
¬nal part of T 60 at 17b 18“20) he illustrates his general characterisation,
Aristotle produces not only a contradictory pair (˜Every man is white™ “ ˜Not
every man is white™) consisting of a universal af¬rmative and a particular
negative predicative assertion, but also one (˜No man is white™ “ ˜Some
man is white™) consisting of a universal negative and a particular af¬rmative
predicative assertion, i.e. a contradictory pair no member of which is a
universal af¬rmative predicative assertion.
There are at least three ways of rescuing the consistency of Aristotle™s
account of contradictory pairs where one member is a universal predicative
assertion. Each of these three rescues endeavours to show that the two
foregoing apparent incongruities in Aristotle™s account are merely apparent.

The ¬rst rescue dissolves the ¬rst apparent incongruity in Aristotle™s account
by attributing to him an extended conception of asserting non-universally:
not only an indeterminate predicative assertion, but also a particular pred-
icative assertion asserts something non-universally of a universal, and the
way in which a particular predicative assertion (e.g. ˜Some man is white™ or
˜Not every man is white™) asserts something non-universally of a universal is
different from the way in which any ˜coincident™ indeterminate predicative
assertion (˜A man is white™ or ˜A man is not white™)14 asserts something non-
universally of a universal (particular and indeterminate predicative asser-
tions all assert something non-universally of a universal merely because
they fail to assert something universally of it “ only universal predicative
assertions assert universally).15 If Aristotle holds this view, then there is no
incongruity if, on the one hand, in giving a general characterisation of con-
tradictory pairs where one member is a universal predicative assertion, he

13 14 For ˜coincident™ see n. 9 of the introduction.
Cf. Riondato (1957a), 55“6; Whitaker (1996), 88.
15 Cf. Maier (1896/1936), i 158“9; Riondato (1957b), 29; Zanatta (1992), 33“4.
Appendix 3 243
claims that in every such contradictory pair one member asserts something
non-universally of the same universal of which the other member asserts
something universally, and, on the other hand, in illustrating that general
characterisation, he produces two contradictory pairs each of which con-
sists of a universal predicative assertion and a particular (rather than an
indeterminate) predicative assertion.
The ¬rst rescue can dissolve the second apparent incongruity in Aristo-
tle™s account by assuming that Aristotle is expressing himself succinctly:16 he
formulates the general characterisation only for contradictory pairs of pred-
icative assertions where one member is universal af¬rmative and the other
particular negative, and ˜leaves it to the reader™ to formulate the general
characterisation for contradictory pairs of the other type.

The ¬rst rescue fails. In T 20, at 17b 8“10, Aristotle illustrates what it is to assert
something non-universally of a universal by mentioning two indeterminate
predicative assertions (˜A man is white™ and ˜A man is not white™): he does
not mention any particular predicative assertion, nor any assertion of any
other sort. This suggests that for Aristotle, there is an especially close link
between indeterminate predicative assertions and asserting something non-
universally of a universal, so close that it can be ruled out that a particular
predicative assertion could assert something non-universally of a universal
and do this in a way that is different from that in which any ˜coincident™
indeterminate predicative assertion asserts something non-universally of a
universal. This dif¬culty, combined with the uneasy treatment of the second
apparent incongruity, suggests that the ¬rst rescue of the consistency of
Aristotle™s account of contradictory pairs where one member is a universal
predicative assertion fails.

The second rescue handles the ¬rst apparent incongruity in Aristotle™s account
by attributing to him a view similar to that attributed to him by the ¬rst
rescue: not only an indeterminate predicative assertion, but also a particular
predicative assertion asserts something non-universally of a universal, and
the way in which a particular predicative assertion asserts something non-
universally of a universal is the same as that in which any ˜coincident™
indeterminate predicative assertion asserts something non-universally of
a universal. (The second rescue is unlike the ¬rst in that, according to
the second rescue, Aristotle believes that a particular predicative assertion
asserts something non-universally of a universal in the same way as any

16 Cf. Weidemann (1985), 55“6.
Int. 7, 17 b 16“18: the text
244
˜coincident™ indeterminate predicative assertion, while, according to the
¬rst rescue, Aristotle believes that the way in which a particular predica-
tive assertion asserts something non-universally of a universal is different
from the way in which any ˜coincident™ indeterminate predicative assertion
performs such an operation.) If Aristotle holds this view, then there is no
incongruity if, on the one hand, in giving a general characterisation of con-
tradictory pairs where one member is a universal predicative assertion, he
claims that in every such contradictory pair one member asserts something
non-universally of the same universal of which the other member asserts
something universally, and, on the other hand, in illustrating that general
characterisation, he produces two contradictory pairs each of which con-
sists of a universal predicative assertion and a particular (rather than an
indeterminate) predicative assertion.
Here is some independent evidence supporting the second rescue. If in
the Prior Analytics Aristotle believes that every indeterminate predicative
assertion is logically equivalent to any ˜coincident™ particular predicative
assertion, then most passages from that work17 where Aristotle mentions or
uses indeterminate predicative assertions are coherent.18 Hence, in the Prior
Analytics Aristotle probably believes that every indeterminate predicative
assertion is logically equivalent to any ˜coincident™ particular predicative
assertion.19 But one would expect Aristotle to believe this if, as the sec-
ond rescue assumes, he thinks that a particular predicative assertion asserts
something non-universally of a universal in the same way as any ˜coincident™
indeterminate predicative assertion.
Note, however, that this independent evidence on behalf of the sec-
ond rescue is weak. For, if in the Prior Analytics Aristotle believes, on the
one hand, that every indeterminate predicative assertion logically entails
any ˜coincident™ particular predicative assertion, and, on the other, that
17 1.1, 24a 17; 24a 19“22; 2, 25a 5; 4, 26a 28“30; 26a 32“3; 26a 36“9; 26b 21“4; 5, 27b 36“8; 6, 29a 6“9; 7,
29a 19“29; 14, 33a 37“8; 15, 35b 14“15; 16, 36b 12“13; 17, 37b 13“16; 18, 38a 10“11; 19, 38b 35“7; 20, 39b 2“3;
21, 40a 1“2.
18 In one passage (APr. 1.27, 43b 11“15, cf. Int. 7, 17b 7“8) Aristotle seems to claim that indeterminate
predicative assertions can be understood as equivalent to universal predicative assertions. However,
elsewhere (Int. 7, 17b 34“7) he says that the indeterminate predicative assertion ˜A man is not white™
and the universal predicative assertion ˜No man is white™ appear to have the same meaning but really
˜neither mean the same nor hold necessarily together™. His view is probably that speakers sometimes
use an indeterminate predicative assertion to convey what would be appropriately expressed by
a universal predicative assertion (cf. Averroes in Int. apud Butterworth (1983), 139; Zadro (1974),
409“10).
19 Cf. Ammon. in Int. 100, 21“2; 110, 23“6; 111, 10“17; Boeth. in Int. Pr. Ed. 87, 2“88, 3; Steph. in Int. 31,
11“12; Waitz (1844/46), i 369; Ross (1923), 29; Kneale/Kneale (1962), 55; Ackrill (1963), 129; G. E. L.
Owen (1965), 86“7; Brunschwig (1967), 163; Rose (1968), 13; Sainati (1968), 228, 234; Brunschwig
(1969), 9, 13; Soreth (1973), 421; Thom (1981), 19; Cavini (1985), 30“2.
Appendix 3 245
not every particular predicative assertion logically entails any ˜coincident™
indeterminate predicative assertion “ on this alternative hypothesis, too,
most passages from the Prior Analytics where Aristotle mentions or uses
indeterminate predicative assertions turn out to be coherent.20 But, on this
alternative hypothesis, it is unlikely that Aristotle should believe that a par-
ticular predicative assertion asserts something non-universally of a universal
in the same way as a ˜coincident™ indeterminate predicative assertion.21
The second rescue raises two questions: (i) What is it, according to
Aristotle, for a universal p to be asserted non-universally to hold (not to
hold) of a universal s? (ii) Why does Aristotle initially characterise what it
is to assert non-universally by introducing indeterminate predicative asser-
tions and then unexpectedly introduce particular predicative assertions,
if he thinks that an indeterminate predicative assertion asserts something
non-universally of a universal in the same way as a ˜coincident™ particular
predicative assertion?
(i) In T 20 Aristotle, on the one hand, associates asserting universally with
predicative assertions whose subject is preceded by a universal quanti¬er
(˜every™ or ˜no™), and, on the other, associates asserting non-universally with
predicative assertions whose subject is preceded by no quanti¬er (the Greek
sentences which are translated in English by ˜A man is white™, ˜A man is
not white™, etc. contain no expression corresponding to the English inde¬-
nite pronoun ˜a™). The association of asserting universally with predicative
assertions whose subject is preceded by a universal quanti¬er indicates that
according to Aristotle for a universal p to be asserted universally to hold
(not to hold) of a universal s is for p to be asserted to hold (not to hold) of
items of which s holds with the additional speci¬cation that it is of all of
the items of which s holds that p holds (does not hold). Analogously, the
association of asserting non-universally with predicative assertions whose
subject is preceded by no quanti¬er strongly suggests that according to
Aristotle for a universal p to be asserted non-universally to hold (not to
hold) of a universal s is for p to be asserted to hold (not to hold) of items
of which s holds without any additional speci¬cation as to how many the
20 Cf. Whitaker (1996), 86.
21 Aristotle applies ˜ˆdi»ristov™ (˜indeterminate™) not only to indeterminate but also to particular
predicative assertions (see APr. 1.4, 26b 14“15; 6, 28b 28“9; Top. 3.6, 120a 6“31, reading ˜e« tin‡ ›jhsen™
at 120a 7 with Brunschwig). However, the sense in which ˜ˆdi»ristov™ applies to indeterminate pred-
icative assertions must be different from that in which it applies to particular predicative assertions
(cf. Brunschwig (1969), 13, 19). For ˜ˆdi»ristov™, when applied to indeterminate predicative asser-
tions, contrasts with ˜diwrism”nov™ (˜determinate™), which applies to particular predicative assertions
(see APr. 1.4, 26b 21“4). Hence ˜ˆdi»ristov™, when applied to indeterminate predicative assertions,
probably expresses the absence of quantifying particles. Obviously ˜ˆdi»ristov™ does not apply in
this sense to particular predicative assertions, which do contain quantifying particles.
Int. 7, 17 b 16“18: the text
246
items of which s holds are of which p holds (does not hold), i.e. without an
additional speci¬cation that it is of one or two or twenty or all of the items
of which s holds that p holds (does not hold). Therefore Aristotle probably
thinks that for a universal p to be asserted non-universally to hold (not to
hold) of a universal s is for p to be asserted to hold (not to hold) of at least
one of the items of which s holds, the possibility being left open that p
could hold of many or even of all of the items of which s holds.
(ii) Aristotle initially characterises asserting non-universally by introduc-
ing indeterminate predicative assertions because such a characterisation
clari¬es what asserting non-universally is, and he subsequently uses the
notion thus clari¬ed to explain particular predicative assertions, in which
he was most interested from the start. Perhaps Aristotle initially does not
associate asserting non-universally with particular predicative assertions,
whose subject is preceded by a particular quanti¬er (˜some™ or ˜not every™),
because the particular quanti¬er ˜some™ lends itself to be (mistakenly) inter-
preted as expressing unique instantiation, or even as being a vague proper
name (one might be inclined to make the mistake of taking ˜Some man is
white™ as meaning that there is exactly one man who is white, or even as
referring in a vague way to Socrates).

The second rescue fails. The second rescue shares one defect with the ¬rst:
it deals uneasily with the second apparent incongruity. But it encoun-
ters a further dif¬culty which speaks speci¬cally against it.22 If the second
rescue is correct, then Aristotle is committed to the claim that asserting
non-universally that white holds of man is contradictorily opposed both to
asserting universally that white does not hold of man (because the particu-
lar af¬rmative predicative assertion ˜Some man is white™ is contradictorily
opposed to the universal negative predicative assertion ˜No man is white™)
and to asserting non-universally that white does not hold of man (because
the indeterminate af¬rmative predicative assertion ˜A man is white™ is con-
tradictorily opposed to the indeterminate negative predicative assertion ˜A
man is not white™). This claim seems incompatible with Aristotle™s other
claim that ˜a single af¬rmation is contradictorily opposed to a single denial™
(7, 18a 8“9)23 “ note that according to the second rescue Aristotle believes
that a particular predicative assertion ˜Some man is white™ asserts some-
thing non-universally of the universal man in the same way as a ˜coincident™
indeterminate predicative assertion ˜A man is white™. Therefore the second
rescue also fails.

22 23 Cf. 17b 38“9; 10, 20b 3“4.
Cf. Whitaker (1996), 86“7.
Appendix 3 247
The third rescue differs radically from the ¬rst two with respect to the
reconstruction of Aristotle™s general characterisation of contradictory pairs
where one member is a universal predicative assertion. According to the
third rescue, in his general characterisation of contradictory pairs where
one member is a universal predicative assertion Aristotle makes two claims.
First, he claims that every universal predicative assertion af¬rms universality:
it af¬rms that it is universally that the universal signi¬ed by the predicate
holds, or does not hold, of the universal signi¬ed by the subject. Second,
he claims that the contradictory opposite of any universal predicative asser-
tion “ i.e. every particular predicative assertion “ denies universality: it denies
that it is universally that the universal signi¬ed by the predicate holds, or
does not hold, of the universal signi¬ed by the subject.24
How does the third rescue of the consistency of Aristotle™s account of
contradictory pairs where one member is a universal predicative asser-
tion dissolve the ¬rst apparent incongruity in that account? According
to the third rescue, in his general characterisation of contradictory pairs
where one member is a universal predicative assertion Aristotle does not
claim that in every such contradictory pair one member asserts something
non-universally of the same universal of which the other member asserts
something universally. Therefore, according to the third rescue, there is no
incongruity when (in the ¬nal part of T 60 at 17b 18“20) Aristotle illustrates
his general characterisation by producing two contradictory pairs each of
which consists of a universal predicative assertion and a particular predica-
tive assertion (rather than by producing one or more contradictory pairs
each of which consists of a universal predicative assertion and an indeter-
minate predicative assertion “ indeterminate predicative assertions having
been associated with asserting non-universally in T 20).
As for the second apparent incongruity, the third rescue handles it well.
For, according to the third rescue, Aristotle is claiming that every univer-
sal predicative assertion af¬rms universality (it af¬rms that it is universally
that the universal signi¬ed by the predicate holds, or does not hold, of
the universal signi¬ed by the subject): hence, for Aristotle, every universal
predicative assertion is in a way af¬rmative. Moreover, according to the
third rescue, Aristotle is also claiming that the contradictory opposite of
any universal predicative assertion “ i.e. every particular predicative asser-
tion “ denies universality (it denies that it is universally that the universal
signi¬ed by the predicate holds, or does not hold, of the universal signi¬ed
24 Cf. Viano (1955), 30“2; Riondato (1957a), 55; Morpurgo-Tagliabue (1971), 49; von Fritz (1972), 243;
Weidemann (1985), 55; (1994/2002), 214“15; Crivelli (1996), 152“3; Whitaker (1996), 86“9; Hafemann
(1999), 112, 114, 136; Pardey (2000), 21“4.
Int. 7, 17 b 16“18: the text
248
by the subject): hence, for Aristotle, every particular predicative assertion is
in a way negative. This is why in his general characterisation of contradic-
tory pairs where one member is a universal predicative assertion Aristotle
says that a predicative assertion which is universal and af¬rmative is con-
tradictorily opposed to a negative assertion, and then, when he comes to
illustrating this general characterisation, he produces not only a contra-
dictory pair consisting of a universal af¬rmative and a particular negative
predicative assertion (˜Every man is white™ “ ˜Not every man is white™), but
also a contradictory pair consisting of a universal negative and a particular
af¬rmative predicative assertion (˜No man is white™ “ ˜Some man is white™):
he regards the universal negative predicative assertion as af¬rmative because
it af¬rms universality (it af¬rms that it is universally that the universal white
fails to hold of the universal man), and he regards the particular af¬rmative
predicative assertion as negative because it denies universality (it denies
that it is universally that the universal white fails to hold of the universal
man).
Three further considerations suggest that the third rescue should be the
correct one.
First, the theory of quanti¬cation which the third rescue attributes to
Aristotle ¬ts with the account of contradictories and modality in chapters
12 and 13 of de Interpretatione. In these chapters Aristotle claims, on the
one hand, that the modal operators (˜it is possible™, ˜it is contingent™, etc.)
modify the copula of a predicative sentence, and, on the other, that in
the denial of a modally quali¬ed predicative sentence the negative particle
attaches (not to the copula, but) to the modal operator: e.g. the denial of
˜It is possible that Socrates is white™ is (not ˜It is possible that Socrates is
not white™, but) ˜It is not possible that Socrates is white™. The analysis of
contradictories and quanti¬cation which most closely corresponds to this
account of contradictories and modality is that according to which, on
the one hand, the universal quanti¬er modi¬es the copula of a predicative
sentence, and, on the other, in the denial of a universally quanti¬ed pred-
icative sentence the negative particle attaches (not to the copula, but) to the
universal quanti¬er: e.g. the denial of ˜It is universal that man is white™ is
(not ˜It is universal that man is not white™, but) ˜It is not universal that man
is white™. Clearly, such an account of contradictories and quanti¬cation is
close to views attributed to Aristotle by the third rescue.25
Second, the third rescue ¬ts well with the way in which Aristotle in
de Interpretatione speaks about contradictory pairs. For, several times in

25 Cf. Rose (1968), 14; Cavini (1985), 15“16.
Appendix 3 249
de Interpretatione26 Aristotle (i) describes those contradictory pairs whose
members are a universal af¬rmative (negative) and a particular negative
(af¬rmative) predicative assertion as if in such contradictory pairs the
contradiction turned on the act of asserting something universally of a
universal, (ii) contrasts these contradictory pairs with those whose mem-
bers are indeterminate (af¬rmative and negative) predicative assertions, and
(iii) describes the latter contradictory pairs as if they were about universals
of which something is asserted non-universally. This manner of speaking
can be accommodated by the third rescue. For, according to the third res-
cue, when Aristotle describes those contradictory pairs whose members
are a universal af¬rmative (negative) and a particular negative (af¬rmative)
predicative assertion as if in such contradictory pairs the contradiction
turned on the act of asserting something universally of a universal, what
he has in mind is that in such contradictory pairs the contradiction turns
on whether it is universally that one universal holds (or does not hold) of
one universal: universality is af¬rmed by every universal predicative asser-
tion (af¬rmative as well as negative), denied by every particular predicative
assertion (negative as well as af¬rmative).27
Third, in Prior Analytics 1.1 (24b 28“30) Aristotle de¬nes the expressions
˜being predicated of every™ (˜kat‡ pant¼v kathgore±sqai™) and ˜being
predicated of no™ (˜kat‡ mhden¼v kathgore±sqai™).28 He does not de¬ne
˜being predicated of some™ and ˜not being predicated of every™. Why does
he not de¬ne these last two expressions? Perhaps one reason is that he avows
[53] The notion expressed by ˜being predicated of some™ (˜not being pred-
icated of every™) is nothing but the ˜denial™ of that expressed by ˜being
predicated of no™ (˜being predicated of every™).
Aristotle™s avowal of [53] would provide further support for the third rescue:
the conception of quanti¬ed predicative assertions presupposed by [53] is
essentially the same as the one attributed to Aristotle by the third rescue.29
In conclusion, the third rescue of the consistency of Aristotle™s account of
contradictory pairs where one member is a universal predicative assertion


26 7, 17b 26“37; 9, 18a 28“33; 18b 27“9; 10, 20a 3“14. 27 Cf. Whitaker (1996), 88.
28 a 14“15; 4, 25b 39“40; 26a 24; 26a 27; APo. 1.4, 73a 28“34.
Cf. 24
29 Aristotle de¬nes ˜being predicated of every™ and ˜being predicated of no™ to show that ¬rst-¬gure
syllogisms are ˜complete™, i.e. that no addition is needed for it to become manifest that the conclusion
follows necessarily from the premisses (cf. 1, 24b 22“4; 4, 25b 32“40; 26a 17“28). This, however, does
not explain why Aristotle should omit de¬ning ˜being predicated of some™ and ˜not being predicated
of every™: some ˜complete™ ¬rst-¬gure syllogisms (cf. 4, 26a 17“28) have premisses that involve being
predicated of some and not being predicated of every.
Int. 7, 17 b 16“18: the text
250
is more plausible than the ¬rst two. Can one read into 17b 16“18 the claims
attributed to Aristotle by the third rescue?

How do the possible readings ¬t with the third rescue?
(i) Consider the text of 17b 16“18 handed down by the main manuscripts:
[a] ˆntike±sqai m•n o”n kat†jasin ˆpoj†sei l”gw ˆntijatik¤v tŸn t¼
kaq»lou shma©nousan t¤€ aÉt¤€ ‚ti oÉ kaq»lou.
The claims which the third rescue attributes to Aristotle cannot be plausibly
read into [a].
(ii) Consider the text of 17b 16“18 presupposed by the rendering of the
passage handed down by some manuscripts of Boethius™ Latin translation
of de Interpretatione:
[b] ˆntike±sqai m•n o”n kat†jasin ˆpoj†sei l”gw ˆntijatik¤v tŸn t¼
kaq»lou shma©nousan t¦€ t¼ oÉ kaq»lou.
In de Interpretatione before 17b 16 we encounter two uses of the word
˜kaq»lou™: as a common noun denoting universals and as an adverb.30
It is therefore extremely likely that in [b] ˜kaq»lou™ should be used either
as a common noun denoting universals or as an adverb (I am assuming that
in its two occurrences in [b] ˜kaq»lou™ is used in the same way). The ¬rst
alternative is implausible because it commits Aristotle to the bizarre claim
that in a contradictory pair of the sort under discussion the af¬rmative
member signi¬es (refers to?) a universal while the negative member signi-
¬es (refers to?) something that is not a universal. Hence it can be plausibly
inferred that in [b] ˜kaq»lou™ is used as an adverb. Therefore in [b] the
¬rst occurrence of ˜kaq»lou™ and the occurrence of ˜oÉ kaq»lou™ are prob-
ably instances of adverbial phrases. It follows that instances of some verb-
phrase must be understood for them to modify. These understood instances
can be plausibly taken to be occurrences of ˜Ëp†rcein £ mŸ Ëp†rcein™.
For:
(ii.i) At 17b 2“3 Aristotle uses ˜ˆpoja©nesqai Þv Ëp†rcei ti £ mž
[. . .] tin©™. This suggests that [b] should be describing af¬rmations
and denials as signifying the Ëp†rcein £ mŸ Ëp†rcein of something to
something.
(ii.ii) In his discussion of contradictory opposition at 17a 26“30 (< T 58)
Aristotle says that an af¬rmative predicative assertion asserts something
as holding (˜Þv Ëp†rcon™) while a negative predicative assertion asserts
30 For ˜kaq»lou™ as a common noun denoting universals see 7, 17a 38; 17a 39; 17b 1; 17b 2; 17b 4; 17b 5;
17b 7; 17b 9; 17b 10; 17b 11; 17b 12; 17b 13; 17b 15. For ˜kaq»lou™ as an adverb see 7, 17b 3; 17b 5“6; 17b 7;
17b 9; 17b 12; 17b 13; 17b 15.
Appendix 3 251
something as not holding (˜Þv mŸ Ëp†rcon™).31 Moreover, at 17a 22“3 Aristo-
tle describes a simple assertion as ˜a signi¬cant spoken sound about whether
something holds or does not hold [Ëp†rcei ti £ mŸ Ëp†rcei]™. These
occurrences of forms of ˜Ëp†rcein™ and ˜mŸ Ëp†rcein™ parallel those I am
suggesting we understand in [b]™s elliptical formulation.
(ii.iii) Forms of ˜Ëp†rcein kaq»lou™ occur elsewhere in Aristotle™s
works.32 This provides a partial parallel for my suggested reconstruction
of [b]™s elliptical formulation.
If the suggestion that the understood instances of a verb-phrase in [b]™s
elliptical formulation should be occurrences of ˜Ëp†rcein £ mŸ Ëp†rcein™
is correct, reading [b] should be rendered by:
[b ] I say that an af¬rmation signifying the <holding, or failing to hold,>
universally is contradictorily opposed to a denial signifying the not
<holding, or failing to hold,> universally.
Translation [b ], in turn, should be understood as a compressed formulation
of
[b ] I say that an af¬rmation signifying that a universal p universally holds,
or fails to hold, of a universal s is contradictorily opposed to a denial
signifying that p does not universally hold, or fail to hold, of s.
The claims which the third rescue attributes to Aristotle can be read in [b ],
and therefore in [b] too.
(iii) Consider the reading of 17b 16“18 recorded by Oxon. New College
C. 225:
[c] ˆntike±sqai m•n o”n kat†jasin ˆpoj†sei l”gw ˆntijatik¤v tŸn t¼
kaq»lou shma©nousan ‚ti kaq»lou t¦€ t¼ aÉt¼ ‚ti oÉ kaq»lou.33
In [c] two occurrences of ˜Ëp†rcei £ mŸ Ëp†rcei™ must be understood
from the context. [c] should therefore be rendered by:
[c ] I say that an af¬rmation signifying that a universal <holds, or fails
to hold,> universally is contradictorily opposed to a denial signifying
that that same universal does not <hold, or fail to hold,> universally.
The claims which the third rescue attributes to Aristotle can be read in [c ],
and therefore in [c].
(iv) A possible emendation (which, to the best of my knowledge, has not
yet been suggested) is:

31 Cf. Top. 2.1, 109a 29“30. 32 See n. 2 of appendix 4.
33 Reading [c] is equivalent to the one favoured by Du Val and Weidemann (cf. n. 11 above).
Int. 7, 17 b 16“18: the text
252
[d] ˆntike±sqai m•n o”n kat†jasin ˆpoj†sei l”gw ˆntijatik¤v tŸn
t¼ kaq»lou shma©nousan t¤€ aÉt¤€ <Ëp†rcein £ mŸ Ëp†rcein t¼
aÉt¼ t¦> ‚ti oÉ kaq»lou.34
€
Perhaps [a] (i.e. the text of 17b 16“18 handed down by the main MSS)
came about because the words ˜Ëp†rcein £ mŸ Ëp†rcein t¼ aÉt¼ t¦™ €
dropped out from [d]. In support of the emendation at hand, consider
that these words might have easily dropped out from [d]: some scribe
after copying ˜t¤€ aÉt¤™ might have resumed copying after ˜t¼ aÉt¼ t¦™.
€ €
Moreover, ˜Ëp†rcein £ mŸ Ëp†rcein t¼ aÉt¼ t¦™ consists of 28 letters:
€
many lines of the papyrus PBerol inv. 5002 (¬fth century ad, containing
APo. 71b 19“72a 38) consist of exactly 28 letters. Thus, the words ˜Ëp†rcein
£ mŸ Ëp†rcein t¼ aÉt¼ t¦™ might have constituted a papyrus line which
€
was entirely skipped by a scribe. In the papyrus at hand the iota mutum
is written: this makes the palaeographical similarity between ˜t¤€ aÉt¤™ €
and ˜t¼ aÉt¼ t¦™ even greater. Note that on the suggested emendation [a]
€
was generated by only one mistake from Aristotle™s original text. [d] can be
rendered by:
[d ] I say that an af¬rmation signifying that one item universally holds,
or fails to hold, of one item is contradictorily opposed to a denial
signifying that that same item does not universally hold, or fail to
hold, of that same item.
The claims which the third rescue attributes to Aristotle can be read in [d ],
and therefore in [d].
(v) Alternatively, the original text at 17b 16“18 could have been
[e] ˆntike±sqai m•n o”n kat†jasin ˆpoj†sei l”gw ˆntijatik¤v tŸn t¼
kaq»lou shma©nousan t¤€ aÉt¤€ <t¼ aÉt¼ t¦> ‚ti oÉ kaq»lou,
€
and [a] might have come about because the words ˜t¼ aÉt¼ t¦™ dropped
€
out from [e]. The error this alternative emendation presupposes a scribe

In [d] ˜shma©nousan t¼™ + in¬nitive is followed by ˜shmaino…sh€ ‚ti™ (˜shmaino…sh™ is understood).
€
34
a 16“17, where only the second member of a pair of contrary
A similar asymmetry occurs at Int. 10, 20
predicative assertions is described by using ˜shma©nousa ‚ti™ + ¬nite verb form.
In [d] ˜Ëp†rcein £ mŸ Ëp†rcein™ follows the dative case (˜t¤€ aÉt¤™) with which it is construed,
€
and precedes its subject. At APr. 1.5, 27a 34 Aristotle uses the sentence ˜oÉdenª t¤€ M Ëp†rxei t¼ N™,
where ˜Ëp†rxei™ follows the dative case (˜oÉdenª t¤€ M™) with which it is construed, and precedes its
subject (cf. APo. 1.15, 79a 41“79b 1).
According to C (followed by Bekker, Weise, Waitz, D¨ bner, and Tredennick) at APr. 1.15, 35b 3“4
u
Aristotle describes a premiss of a syllogism by saying that it ˜shma©nh€ t¼ Ëp†rcein £ mŸ Ëp†rcein™.
(Ross excises ˜£ mŸ Ëp†rcein™.) This parallels [d]™s ˜t¼ [. . .] shma©nousan [. . .] Ëp†rcein £ mŸ
Ëp†rcein™.
Appendix 3 253
to have made is one that can be easily committed. If [e] was the original
text at 17b 16“18, then Aristotle expected the reader to supply ˜Ëp†rcein £
mŸ Ëp†rcein™ from the context.35 Reading [e] can be rendered in the same
way as [d].
Reading [b] is attested in the tradition (although it is not widely attested,
it is there). Moreover, the claims which the third rescue attributes to Aris-
totle can be read in [b]. Reading [b] is therefore the best reading for
17b 16“18.
35 In the Analytics (APr. 1.5, 27a 7“8; 27a 10“1; 2.10, 60b 20“2; APo. 1.15, 79a 41“79b 1) Aristotle employs
elliptical constructions for which some form of ˜Ëp†rcein™ must be supplied from the context.
a p p en d i x 4

The two-place relations in Aristotle™s
de¬nition of truth1



The two-place relations involved in the truth conditions for quanti¬ed asser-
tions. An important feature of [17], the de¬nition of truth for predica-
tive assertions I attribute to Aristotle, is that the truth conditions for
(those utterances which are) quanti¬ed predicative assertions involve two-
place relations (the combination of universally holding, the division of
universally failing to hold, the combination of not universally failing
to hold, and the division of not universally holding) obtaining between
the universals signi¬ed by the predicates and the subjects of the asser-
tions. These two-place relations are to some extent re¬‚ected in views for-
mulated in the logical writings of Aristotle himself and of some of his
pupils.

(i) The expression ˜to hold universally™. In some passages2 Aristotle uses the
expression ˜to hold universally™ (˜kaq»lou Ëp†rcein™) to express a relation
which is at least close to the combination of universally holding. This ¬ts
well with the part of [17] which concerns universal af¬rmative predicative
assertions.

(ii) Universal af¬rmative predicative assertions and being a part of a whole.
At the beginning of the Prior Analytics Aristotle says that he intends to
determine ˜what it is for this to be, or not to be, in this taken as a whole [–n
‚lw€ e²nai £ mŸ e²nai t»de t¤de]™ (1.1, 24a 13“14). Aristotle probably wants
€
to describe what universal af¬rmations and denials say. But the expression
he uses, i.e.
[a] –n ‚lw€ e²nai £ mŸ e²nai t»de t¤de,3
€

1 Cf. the subsection to which n. 52 of ch. 2 is appended.
2 APr. 1.24, 41b 6; 41b 22“3; 2.4, 57a 34; APo. 1.4, 74a 1; 5, 74a 5; 74a 24“5; 2.13, 96a 26. Cf. the use of ˜–pª
pl”on Ëp†rcein™ at APo. 2.13, 96a 25“7; 96a 29“30; 96a 33“4; 96b 8; 96b 9“10; 17, 99a 31“2.
3 Cf. APr. 1.4, 25b 33“4; APo. 2.15, 79a 40“1; 79b 6“7.

254
Appendix 4 255
is problematic. For in [a] ˜‚lw€™ seems to have to be understood differently
depending on whether it is taken with ˜e²nai™ or with ˜mŸ e²nai™. When
˜‚lw€™ is taken with ˜e²nai™, it yields the phrase
[b] –n ‚lw€ e²nai t»de t¤de,4
€
which, on its most natural (and traditional) interpretation, is paraphrased
by
[c] This is in this as in a whole of which it is a part.5
Interpretation [c] is a reasonable description of what universal af¬rmative
predicative assertions assert. However, when ˜‚lw€™ is taken with ˜mŸ e²nai™,
one gets
[d] –n ‚lw€ mŸ e²nai t»de t¤de.6
€
Now, if ˜‚lw€™ in [d] were understood in the same way as in [b], [d] ought
to be paraphrased by
[e] This is not in this as in a whole of which it is a part.
But [e] cannot be the correct paraphrase of [d] because [e] cannot describe
what is asserted by a universal negative predicative assertion “ [e] can at
most describe what is asserted by a particular negative predicative assertion.
The most plausible guess is that ˜‚lw€™ in [d] ought to be so understood as
to allow [d] to be paraphrased by
[f ] This is completely not in this.
Therefore [c] is probably Aristotle™s description of what is asserted by a uni-
versal af¬rmative predicative assertion: a universal af¬rmative predicative
assertion asserts that one universal (signi¬ed by the assertion™s predicate)
is related to another universal (signi¬ed by the assertion™s subject) as a
4 Cf. APr. 1.1, 24b 26“7; 4, 25b 32“3; 2.1, 53a 21“3; APo. 1.15, 79a 36“7; 79a 38; 79b 2; 79b 5; 79b 10; 79b 12“13;
79b 15; 79b 17; 16, 79b 38.
5 Cf. APr. 1.4, 25b 32“7; 2.1, 53a 21“4; Ph. 4.3, 210a 14“18; Alex. Aphr. in APr. 25, 2“9; Trendelenburg
(1836/92), 94“5; Waitz (1844/46), i 375; Ross (1949), 301; Patzig (1968), 91; Cosenza (1987), 28“31;
Patterson (1995), 19“21. According to Mignucci (1969), 182, [b] should be understood as equivalent
to
[c ] This subject holds of this predicate universally
with ˜–n ‚lw€™ taken as an adverb (parallel to ˜–n m”rei™ and used like ˜kaq»lou™ at APr. 1.23, 40b 23“
5; 32, 47a 13; 2.6, 58b 28“30; 20, 66b 16) and ˜t¤de™ as a dative of property. Such an interpretation
€
is implausible in the light of APr. 1.4, 25b 32“7 and 2.1, 53a 21“4, where Aristotle switches between
instances of ˜t¼ x –n ‚lw€ t¤€ y™ and the corresponding instances of ˜t¼ x –n t¤€ y™. Cosenza (1987),
28“31 also shows that Mignucci™s interpretation of [b] as equivalent to [c ] clashes with Aristotle™s
usage at APo. 1.15, 79a 36“40. Mignucci (1975a), 336“7 retracts this interpretation.
6 Cf. APo. 1.15, 79a 39.
256 Two-place relations
whole to one of its parts. This suggests that Aristotle associates universal
af¬rmative predicative assertions with something like the combination of
universally holding.7 Analogously, [f ] is probably Aristotle™s description of
what is asserted by a universal negative predicative assertion: a universal
negative predicative assertion asserts that one universal (signi¬ed by the
assertion™s predicate) is related to another universal (signi¬ed by the asser-
tion™s subject) in such a way as to be completely not in it, i.e. in such a way
as to be ˜outside™ it. This suggests that Aristotle associates universal negative
predicative assertions with something like the division of universally failing
to hold.8

(iii) Theophrastus and Eudemus on universal negative predicative assertions.
Aristotle™s pupils Theophrastus and Eudemus in their (now lost) Analytics
linked universal negative predicative assertions with something like the
division of universally failing to hold. Such an account of universal negative
predicative assertions allowed Theophrastus and Eudemus to construct a
proof of the conversion of universal negative predicative assertions which is
simpler than the one given by Aristotle in the Prior Analytics (1.2, 25a 14“17).9
Since Theophrastus™ work as a logician is in many respects an expansion
on Aristotle™s,10 Theophrastus perhaps knew that the account of universal
negative predicative assertions presupposed by his proof was approved by
Aristotle. Such an approval would ¬t well with the part of [17] which
concerns universal negative predicative assertions.

(iv) Particular predicative assertions and parts. Aristotle™s use of ˜–n m”rei™
and ˜kat‡ m”rov™ (˜particular™) to describe particular predicative assertions11
suggests that he regarded a particular af¬rmative predicative assertion as
asserting that the universal signi¬ed by its predicate includes at least part of
the universal signi¬ed by its subject (˜m”rov™ in Greek means ˜part™).12 Such
a view about particular af¬rmative predicative assertions is tantamount to
7 See also the use of ˜peri”cw™ at APr. 1.27, 43b 23; 43b 29“30. Cf. Miller (1971), 29“32; Cosenza (1983);
Patterson (1995), 19“22, 37.
8 Cf. Miller (1971), 29“32.
9 Cf. Alex. Aphr. in APr. 31, 4“10; 34, 13“15; 124, 18“21; 132, 24“32; 220, 12“16; On the Conversion
of Propositions apud Badawi (1971), 65; Ammon. in Int. 108, 29“30; 185, 14“18; in APr. 4, 12“14;
[Ammon.] in APr. 44, 1“4; 58, 19“23; 60, 21“2; Phlp. in APr. 48, 11“18; 74, 17“20; 74, 24“6; 124,
12“16; 129, 13“18; 198, 10“11; 201, 19“21; Bochenski (1947), 54“6; Graeser (1973), 75; Mignucci (1999a),
24“38. Authors who questioned the axiom that only syllogisms in the ¬rst ¬gure are complete also
linked universal negative predicative assertions with something like the division of universally failing
to hold: see Schol. in Arist. 157a 13“24; Them. apud Badawi (1987), 186.
10 Cf. Boeth. in Int. Sec. Ed. 12, 3“16; Barnes (1999b), 78; Mignucci (1999a), 21“4; Bobzien (2000), 102.
11 APr. 1.1, 24a 17; 2, 25a 10; 25a 20; etc.
12 At APr. 1.4, 26a 17 Aristotle says: ˜If one of the terms is related universally [kaq»lou] to another and
the other term is related partially [–n m”rei] [. . .]™. Cf. Cosenza (1987), 38“9; Mignucci (1997a), 69.
Appendix 4 257
the view that a particular af¬rmative predicative assertion asserts that the
universal signi¬ed by its predicate bears to the universal signi¬ed by its
subject something like the combination of not universally failing to hold.13
This ¬ts well with the part of [17] which concerns particular af¬rmative
predicative assertions.

(v) Aristotle™s jargon for particular negative predicative assertions. Aristotle™s
use of ˜Ëperte©nein™ (˜to exceed™),14 ˜›xw –pekte©nein™ (˜to extend outside™),15
and ˜›xw Ëp†rcein™ (˜to hold outside™)16 suggests that in his view a partic-
ular negative predicative assertion says that one universal (signi¬ed by the
assertion™s subject) extends outside, or exceeds, another universal (signi¬ed
by the assertion™s predicate). Thus, Aristotle probably associates particu-
lar negative predicative assertions with something like the division of not
universally holding. This ¬ts well with the part of [17] which concerns
particular negative predicative assertions.

(vi) Particular negative predicative assertions and proofs of invalidity by
counter-example. Two of the proofs of invalidity by counter-example17 sug-
gest that Aristotle thinks that a particular negative predicative assertion is
true just in case some part of the universal signi¬ed by its subject holds of
none of the items of which the universal signi¬ed by its predicate holds.
This view ¬ts well with the idea that particular negative predicative asser-
tions are linked with the division of not universally holding.

13 At APr. 1.28, 43b 43“44a 2 Aristotle says that to establish that A holds of some B one must ¬nd an
X of all of which both A and B hold. This suggests that Aristotle might be associating particular
af¬rmative predicative assertions with something like the relation of not universally failing to hold
(cf. Thom (1981), 73“5). Cf. the use of ˜–pall†ttw™ at APo. 1.15, 79b 7“8; 79b 11; GA 2.1, 733a 27; 4.4,
770b 6; 6, 774b 17.
14 APr. 1.14, 33a 39“40; 2.23, 68b 24; 27, 70b 34; APo. 1.22, 84a 25. 15 APo. 2.13, 96a 24“5.
16 a 30“1. 17 APr. 1.4, 26a 39“26b 14.
APo. 2.13, 96
a p p en d i x 5

Aristotle™s theory of truth for predicative assertions:
formal presentation 1


Alphabet. Let me ¬rst give the alphabet of the formal language in which
the formal presentation of Aristotle™s theory of truth is formulated.
[54] The logical symbols of the alphabet are:
two parentheses: ˜(™ and ˜)™;
one propositional connective of one argument: ˜¬™;
four propositional connectives of two arguments: ˜&™, ˜∨™, ˜’™, and ˜”™;
two quanti¬ers: ˜∀™ and ˜∃™;
one two-place relation constant: ˜=™.
These logical symbols are understood in the usual way.
[55] The descriptive symbols of the alphabet include:
eleven one-place relation constants: ˜V ™, ˜Sn ™, ˜Sa ™, ˜Se ™, ˜Si ™, ˜So ™, ˜Sy ™,
˜Sd ™, ˜U ™, ˜I ™, and ˜X ™;
six two-place relation constants: ˜T ™, ˜H ™, ˜K ™, ˜D™, ˜Ch ™, and ˜Ck ™;
thirteen three-place relation constants: ˜Cu ™, ˜Du ™, ˜Cp ™, ˜Dp ™, ˜Cs ™, ˜Ds ™,
˜P ™, ˜Ju ™, ˜Su ™, ˜Jp ™, ˜Sp ™, ˜Js ™, and ˜Ss ™;
in¬nitely many individual variables: ˜x™, ˜y™, ˜z™, ˜u™, ˜v™, ˜w™, ˜t™, ˜s™, ˜x0 ™,
˜x1 ™, ˜x2 ™, . . .;
in¬nitely many individual parameters: ˜a™, ˜b™, ˜c™, ˜d ™, . . . , ˜p™, ˜q™, ˜a0 ™,
˜a1 ™, ˜a2 ™, . . .;
in¬nitely many one-place relation parameters: ˜R 0 ™, ˜R 11 ™, ˜R 2 ™, . . .;
1 1

in¬nitely many two-place relation parameters: ˜R 0 ™, ˜R 12 ™, ˜R 2 ™, . . .;
2 2
.
.
.
The meanings of the relation constants are best explained by considering
them in context:
[56] Every instance of every schema in the following left-hand side column
has the same meaning as the corresponding instance of the schema on
the same line in the following right-hand side column:
1 Cf. the subsection to which n. 52 of ch. 2 is appended.

258
Appendix 5 259
˜a is an utterance™2
˜Va™
˜Sn a™ ˜a is a present-tense predicative assertion™3
˜Sa a™ ˜a is a universal af¬rmative present-tense predicative assertion™
˜Se a™ ˜a is a universal negative present-tense predicative assertion™
˜Si a™ ˜a is a particular af¬rmative present-tense predicative asser-
tion™
˜So a™ ˜a is a particular negative present-tense predicative assertion™4
˜Sy a™ ˜a is a singular af¬rmative present-tense predicative assertion™
˜Sd a™ ˜a is a singular negative present-tense predicative assertion™5
˜Ua™ ˜a is a universal™
˜Ia™ ˜a is an individual™
˜a is a time™6
˜Xa™
˜a is true at b™
˜Tab™
˜a is signi¬ed by the part of b that constitutes its subject™7
˜Hab™
˜a is signi¬ed by the part of b that constitutes its predicate™8
˜Kab™
˜a is other than b™9
˜Dab™
˜Ch ab™ ˜the part of a that constitutes its subject and the part of b that
constitutes its subject are tokens of the same type™
˜Ck ab™ ˜the part of a that constitutes its predicate and the part of b
that constitutes its predicate are tokens of the same type™10
˜Cu abg™ ˜at g a is combined with b in such a way as universally to
hold of it™
˜Du abg™ ˜at g a is divided from b in such a way as universally to fail
to hold of it™
˜Cp abg™ ˜at g a is combined with b in such a way as not universally
to fail to hold of it™
˜Dp abg™ ˜at g a is divided from b in such a way as not universally to
hold of it™
˜Cs abg™ ˜at g a is combined with b in such a way as to hold of it™

2 ˜V ™ is an abbreviation of the Latin ˜vox™, which in the present context should be taken to mean
˜utterance™.
3 The subscript ˜n™ is an abbreviation of ˜now™.
4 The subscripts ˜a™, ˜e™, ˜i ™, and ˜o™ are the traditional symbols for (respectively) universal af¬rmative,
universal negative, particular af¬rmative, and particular negative predicative assertions.
5 The subscripts ˜y™ and ˜d ™ are abbreviations of ˜yes™ and ˜denial™.
6 ˜X ™ is an abbreviation of the Greek ˜cr»nov™ (˜time™).
7 ˜H ™ is an abbreviation of the Greek ˜Ëpoke©menon™, which in the present context should be taken to
mean ˜ontological subject™, i.e. ˜object signi¬ed by the grammatical subject™.
8 ˜K ™ is an abbreviation of the Greek ˜kathgoro…menon™, which in the present context should be
taken to mean ˜ontological predicate™, i.e. ˜object signi¬ed by the grammatical predicate™.
9 ˜D™ is an abbreviation of ˜distinct™.
10 ˜Ch ™ and ˜Ck ™ stand for ˜common subject™ and ˜common predicate™.
260 Formal theory of truth for predicative assertions
˜Ds abg™ ˜at g a is divided from b in such a way as to hold outside it™
˜at g a is predicated of b™
˜Pabg™
˜Ju abg™ ˜a is asserted by g to be combined with b in such a way as
universally to hold of it™
˜Su abg™ ˜a is asserted by g to be divided from b in such a way as
universally to fail to hold of it™
˜Jp abg™ ˜a is asserted by g to be combined with b in such a way as
not universally to fail to hold of it™
˜Sp abg™ ˜a is asserted by g to be divided from b in such a way as not
universally to hold of it™
˜Js abg™ ˜a is asserted by g to be combined with b in such a way as to
hold of it™
˜Ss abg™ ˜a is asserted by g to be divided from b in such a way as to
hold outside it™

Formulae. Formulae are formed in the standard way. The usual conven-
tions for economising on parentheses are adopted (the conjunctive and the
disjunctive connective, ˜&™ and ˜∨™, are taken to have more binding power
than the conditional and the biconditional connective, ˜’™ and ˜”™).

The de¬nition of truth relies on nine assumptions:
[57] ∀x (Sn x ’ Vx).
[58] ∀x (Sn x ” Sa x ∨ Se x ∨ Si x ∨ So x ∨ Sy x ∨ Sd x).
[59] ∀x ((Sa x ’ ¬(Se x ∨ Si x ∨ So x ∨ Sy x ∨ Sd x)) & (Se x ’ ¬(Si x ∨ So x
∨ Sy x ∨ Sd x)) & (Si x ’ ¬(So x ∨ Sy x ∨ Sd x)) & (So x ’ ¬(Sy x ∨
Sd x)) & (Sy x ’ ¬Sd x)).
[60] ∀x (Sn x ’ ∃y (Kyx & ∀z (Kzx ’ z = y) & Uy)).
[61] ∀x (Sa x ∨ Se x ∨ Si x ∨ So x ’ ∃y (Hyx & ∀z (Hzx ’ z = y) & Uy)).
[62] ∀x (Sy x ∨ Sd x ’ ∃y (Hyx & ∀z (Hzx ’ z = y) & Iy)).
[63] ∀x∀y (Sn x & Sn y & Ch xy ’ ∃z (Hzx & Hzy)).
[64] ∀x∀y (Sn x & Sn y & Ck xy ’ ∃z (Kzx & Kzy)).
[65] ∀x (Sn x ’ (Sa x ” ∃y ∃z (Kyx & Hzx & Ju yzx)) & (Se x ” ∃y ∃z (Kyx
& Hzx & Su yzx)) & (Si x ” ∃y ∃z (Kyx & Hzx & Jp yzx)) & (So x ”
∃y ∃z (Kyx & Hzx & Sp yzx)) & (Sy x ” ∃y ∃z (Kyx & Hzx & Js yzx))
& (Sd x ” ∃y ∃z (Kyx & Hzx & Ss yzx))).
Appendix 5 261
Here is the de¬nition:
[66] ∀x (Sa x ’ ∀t (Xt ’ (Txt ” ∃y ∃z (Kyx & Hzx & Cu yzt)))).
[67] ∀x (Se x ’ ∀t (Xt ’ (Txt ” ∃y ∃z (Kyx & Hzx & Du yzt)))).
[68] ∀x (Si x ’ ∀t (Xt ’ (Txt ” ∃y ∃z (Kyx & Hzx & Cp yzt)))).
[69] ∀x (So x ’ ∀t (Xt ’ (Txt ” ∃y ∃z (Kyx & Hzx & Dp yzt)))).
[70] ∀x (Sy x ’ ∀t (Xt ’ (Txt ” ∃y ∃z (Kyx & Hzx & Cs yzt)))).
[71] ∀x (Sd x ’ ∀t (Xt ’ (Txt ” ∃y ∃z (Kyx & Hzx & Ds yzt)))).

Propositions [66]“[71] appeal to six relations of combination and division:
Cu (the combination of universally holding ), Du (the division of universally
failing to hold ), Cp (the combination of not universally failing to hold ), Dp
(the division of not universally holding ), Cs (the combination of holding ),
and Ds (the division of holding outside). They are de¬ned as follows:
[72] ∀x ∀y ∀t (Cu xyt ” ∀z (Iz & Pyzt ’ Pxzt)).
[73] ∀x ∀y ∀t (Du xyt ” ∀z (Iz & Pyzt ’ ∀u (Iu & Pxut ’ Dzu))).
[74] ∀x ∀y ∀t (Cp xyt ” ∃z (Iz & Pyzt & Pxzt)).
[75] ∀x ∀y ∀t (Dp xyt ” ∃z (Iz & Pyzt & ∀u (Iu & Pxut ’ Dzu))).
[76] ∀x ∀y ∀t (Cs xyt ” Pxyt).
[77] ∀x ∀y ∀t (Ds xyt ” ∀z (Iz & Pxzt ’ Dyz)).
Some additional clauses are necessary to pin down the foregoing:
[78] ∀x ∀y ∀z ∀t (Pxyt & Pyzt ’ Pxzt).
[79] ∃t Xt.
[80] ∀x (Ux ” ∃y ∃z ∃t ∃s (Pxyt & Pxzs & Dyz)).11
[81] ∀x (Ix ” ¬Ux).12
[82] ∀x (Ux ’ ∀t (Xt ’ ∃y (Iy & Pxyt))).13
[83] ∀x ∀y (Sn x & Sn y ’ (Ch xy ’ Ch yx) & (Ck xy ’ Ck yx)).
[84] ∀x ∀y (Dxy ” ¬x = y).

11 Cf. [10] on p. 80 above and the subsection to which it belongs.
12 Cf. [11] on p. 80 above and the subsection to which it belongs.
13 Cf. the subsection to which n. 6 of ch. 2 is appended.
262 Formal theory of truth for predicative assertions
A few remarks are in order.
Remark (i): alternative truth conditions for quanti¬ed predicative assertions.
The truth conditions for quanti¬ed present-tense predicative assertions
given by [66]“[69] are such that some quanti¬ed present-tense predicative
assertion can be true at one time and false at another. These truth conditions
correspond to a temporally ˜speci¬c™ reading of quanti¬ed present-tense
predicative assertions. Alternative truth conditions are available which do
not allow quanti¬ed present-tense predicative assertions to have different
truth-values at different times, and correspond to a temporally ˜unquali¬ed™
reading of these assertions.14 I shall not offer a formal presentation of these
alternative truth conditions “ the task would not present any particular
dif¬culties.

Remark (ii): the Square of Opposition. Here are some trivial consequences
of the foregoing de¬nitions and assumptions:
[85] ∀x ∀y ∀t (Cu xyt ” ¬Dp xyt).
[86] ∀x ∀y ∀t (Du xyt ” ¬Cp xyt).
[87] ∀x ∀y ∀t (Ux & Uy & Xt ’ ¬(Cu xyt & Du xyt)).
[88] ∀x ∀y ∀t (Ux & Uy & Xt ’ (Cu xyt ’ Cp xyt) & (Du xyt ’ Dp xyt)).
[89] ∀x ∀y ∀t (Ux & Uy & Xt ’ Cp xyt ∨ Dp xyt).
The proofs of [85] and [86] rely only on de¬nitions [72]“[75] and [84].
For instance, consider the proof of [85]. Assume Cu abc. Then (by [72]) ∀z
(Iz & Pbzc ’ Pazc). Suppose Dp abc. Then (by [75]) ∃z (Iz & Pbzc & ∀u
(Iu & Pauc ’ Dzu)). Let then d be such that Id & Pbdc & ∀u (Iu & Pauc ’
Ddu). Then Padc. Then Ddd. Then (by [84]) ¬d = d, which is absurd.
Then ¬Dp abc. So Cu abc ’ ¬Dp abc. Vice versa, assume ¬Dp abc. Let e
be such that Ie & Pbec. Suppose that ¬Paec. Let f be such that If & Pafc.
Suppose e = f. Then Paec, contrary to the hypothesis. Then ¬e = f. Then
(by [84]) Def. But f was arbitrary. Then ∀u (Iu & Pauc ’ Deu). Then ∃z (Iz
& Pbzc & ∀u (Iu & Pauc ’ Dzu)). Then (by [75]) Dp abc, contrary to the
assumption. Then Paec. But e was arbitrary. Then ∀z (Iz & Pbzc ’ Pazc).
Then (by [72]) Cu abc. So ¬Dp abc ’ Cu abc. Then Cu abc ” ¬Dp abc. But
a, b, and c were arbitrary. Then ∀x ∀y ∀t (Cu xyt ” ¬Dp xyt), i.e. [85] holds.
The proof of [86] is equally simple.
14 For Aristotle™s distinction between temporally ˜speci¬c™ and ˜unquali¬ed™ readings of quanti¬ed
present-tense predicative assertions see APr. 1.15, 34b 7“18 (cf. APo. 1.4, 73a 28“34; 8, 75b 21“30; 2.12,
96a 15; GC 1.3, 317b 5“13; Hintikka (1957), 68“9).
Appendix 5 263
The proofs of [87], [88], and [89] rely (not only on de¬nitions [72]“[75]
and [84], but also) on assumption [82]: [87], [88], and [89] hold because
every universal is always predicated of at least one individual. I omit them
because they are trivial.
From [58], [60], [61], [63], [64], [66]“[69], and [85]“[89] it follows (triv-
ially) that the laws of the Square of Opposition are valid:
[90] Law of Contradictories. A universal af¬rmative (negative) present-tense
predicative assertion is true when and only when any ˜coincident™
particular negative (af¬rmative) present-tense predicative assertion is
not true:15 ∀x ∀y (((Sa x & So y) ∨ (Se x & Si y)) & Ck xy & Ch xy ’ ∀t
(Xt ’ (Txt ” ¬Tyt))).16
[91] Law of Contraries. A universal af¬rmative and a ˜coincident™ universal
negative present-tense predicative assertion are never both true:17 ∀x
∀y (Sa x & Se y & Ck xy & Ch xy ’ ∀t (Xt ’ ¬(Txt & Tyt))).
[92] Law of Subalternation. When a universal af¬rmative (negative) present-
tense predicative assertion is true, any ˜coincident™ particular af¬rma-
tive (negative) present-tense predicative assertion is also true:18 ∀x ∀y
(((Sa x & Si y) ∨ (Se x & So y)) & Ck xy & Ch xy & ’ ∀t (Xt ’ (Txt ’
Tyt))).
[93] Law of Subcontraries. For any particular af¬rmative and any ˜coincident™
particular negative present-tense predicative assertion, it is always the
case that at least one of them is true:19 ∀x ∀y (Si x & So y & Ck xy &
Ch xy ’ ∀t (Xt ’ (Txt ∨ Tyt)).

Remark (iii): contradictory opposition of singular predicative assertions. Here
is the principle governing the ontological relations of combination and
division relevant to singular present-tense predicative assertions:
[94] ∀x ∀y ∀t (Iy ’ (Cs xyt ” ¬Ds xyt)).


15 Int. 7, 17b 26“7; 9, 18a 28“33; Top. 2.3, 110a 32“7; 3.6, 120a 8“14; 120a 20“4.
16 For ˜coincident™ see n. 9 of the introduction.
17 Int. 7, 17b 20“4; 10, 20a 16“18; 14, 24b 6“7; APr. 2.15, 63b 23“8; Top. 2.1, 109a 5“6; 2, 109b 23“5.
18 Top. 2.1, 108b 34“109a 6; 3.6, 119a 34“6; 120a 14“20 (cf. Slomkowski (1997), 121“4). The Laws of Conver-
sion for universal and particular af¬rmations (see APr. 1.2, 25a 7“12), joined with some straightforward
existential assumptions about predicative assertions, entail the Law of Subalternation for af¬rmations
(cf. Pozzi (1974), 46; Crivelli (1989), 74).
19 Aristotle never formulates the Law of Subcontraries (cf. Blanch´/Dubucs (1970/96), 41): he only says
e
that a particular af¬rmative and a ˜coincident™ particular negative predicative assertion can both be
true (see Int. 7, 17b 24“6; 10, 20a 16“20; Top. 3.6, 120a 12“14).
264 Formal theory of truth for predicative assertions
Proposition [94] yields the principle that governs contradictory singular
present-tense predicative assertions:
[95] ∀x ∀y (Sy x & Sd y & Ck xy & Ch xy ’ ∀t (Xt ’ (Txt ” ¬Tyt))).
The proofs of [94] and [95] are trivial.

Remark (iv): predicative assertions classi¬ed. The thesis only partly captured
by [58] and [59] is that the genus present-tense-predicative-assertion (a
universal) divides into the six species (universals) universal-af¬rmative-
present-tense-predicative-assertion, universal-negative-present-tense-pre-
dicative-assertion, particular-af¬rmative-present-tense-predicative-asser-
tion, particular-negative-present-tense-predicative-assertion, singular-
af¬rmative-present-tense-predicative-assertion, and singular-negative-
present-tense-predicative-assertion (each of these species has many
individuals as members, and these individuals are present-tense pred-
icative assertions “ utterances, expression-tokens). One might add
one layer by postulating that each of the six species divides into
many subspecies (universals) which are sentence-types (e.g. the species
universal-af¬rmative-present-tense-predicative-assertion would divide
into the many subspecies which are the sentence-types ˜Every man is an
animal™, ˜Every man is tall™, ˜Every dog is barking™, etc.). (Propositions [58]
and [59] do not attempt to formulate this.)

Remark (v): what is the domain of quanti¬cation? Throughout the present
account of Aristotle™s theory of truth I have assumed that every quanti¬ed
present-tense predicative assertion quanti¬es over the individuals (rather
than over both the universals and the individuals) which the universal sig-
ni¬ed by its subject is predicated of.20 There are three reasons for attributing
this thesis to Aristotle.
(i) The thesis agrees best with the most natural reading of quanti¬ed
present-tense predicative assertions: e.g. ˜Some animal is hiding in that
corner™ is most naturally understood as saying that some individual which
the universal animal is predicated of is hiding in that corner (it can hardly be
understood as saying that some individual or universal which the universal
animal is predicated of is hiding in that corner).
(ii) A remark in Posterior Analytics 1.4 suggests that universal predicative
assertions quantify over individuals: ˜[. . .] if animal holds of every man,
then if it is true to call this one a man, it is true to call him an animal

20 Cf. Apelt (1891b), 138; Sainati (1968), 227; Miller (1971), 30“1.
Appendix 5 265
too™ (73a 30“1). (The reason why this remark merely suggests that universal
predicative assertions quantify over individuals is that it would be true even
if universal predicative assertions were to quantify over universals as well as
individuals.)21
However, Mario Mignucci makes a good case for attributing to Aristotle
alternative truth conditions, according to which every quanti¬ed predica-
tive assertion quanti¬es over both the universals and the individuals of
which the universal signi¬ed by its subject is predicated.22 It would not
be dif¬cult to implement these alternative truth conditions, but I will not
undertake the task here.
21 Cf. Mignucci (1997b), 138.
22 See Mignucci (1997a), 68“70; (1997b), 138“9; (2000), 13“15 (cf. Church (1965), 418).
ap p e n d ix 6

The failure of Bivalence for future-tense assertions:
formal presentation



Is Aristotle coherent? Aristotle™s position in Int. 9 has been taken to be inco-
herent.1 For Aristotle accepts Excluded Middle (which involves endorsing
every instance of ˜a ∨ ¬a™) but denies Bivalence (which involves denying
that at every time every assertion is either true or false). It is hard to see how
this could be coherent “ but the following formal semantic theory shows
that it is.2

Expanded semantic theory. Appendix 5 contained a formal presentation of
a semantic theory for a fragment of natural language that includes some
present-tense predicative assertions. Here, in appendix 6, this semantic
theory is expanded. The expanded semantic theory covers a slightly larger
fragment of natural language, a fragment including not only some present-
tense predicative assertions, but also some past- and future-tense predicative
assertions, some ˜tomorrow™-assertions (i.e. predicative assertions that begin
with an utterance of the phrase ˜tomorrow it will be the case that™), and
some negative and disjunctive assertions.


1 Cf. Cic. Fat. 16. 37“8 (an attack on some Epicureans whose position resembles Aristotle™s); D. C.
Williams (1951), 289“90; Quine (1953), 19; D. C. Williams (1954), 253; Bradley (1959/68), 237“8;
Strang (1960), 454; Kneale/Kneale (1962), 46“8; Hintikka (1964/73a), 148, 163, 166“7; von Kutschera
(1986), 215.
2 A formal model for Int. 9™s theory of truth was offered at the beginning of the twentieth century
by Jan Lukasiewicz (see his (1918); (1922), 125“6; (1930), 175“6 “ cf. Prior (1953), 317; (1955/62),
240“4; (1957), 25“6, 85“6; (1967d), 1“3; Patzig (1973), 921, 927). Lukasiewicz postulated three truth-
values for assertions: truth, falsehood, and an ˜indeterminate™ value. He then described truth-tables
for the standard connectives. Lukasiewicz™s three-valued logic, whatever its logical merits, cannot
work as an exegesis of Int. 9 because it fails to validate Excluded Middle (cf. Sugihara (1954); King-
Farlow (1958/59), 36; Haack (1974/96), 84“5; Knabenschuh de Porta (1997), 203“4, 208). Lukasiewicz™s
truth-tables can be so modi¬ed as to validate Excluded Middle, but these modi¬cations have other
unacceptable consequences. Late in his life, Lukasiewicz abandoned his interpretation in terms of
three-valued logic in favour of one in terms of four-valued logic (see his (1957), 166“80). A four-valued
logic is implausible as an interpretation of Int. 9, which does not distinguish two ways of being neither
true nor false.

266
Appendix 6 267
Expansion of the alphabet. The alphabet of the semantic theory presented
in appendix 5, described in [54] and [55], must be expanded. Here are the
symbols to be added:
[96] The descriptive symbols of the alphabet include:
eight one-place relation constants: ˜S ™, ˜Sb ™, ˜Sf ™, ˜St ™, ˜Sc ™, ˜Sv ™, ˜M ™, and
˜B ™;
ten two-place relation constants: ˜∈™, ˜Ib ™, ˜If ™, ˜It ™, ˜Ic ™, ˜O™, ˜W ™, ˜<™,
˜T *™, and ˜F *™;
two three-place relation constants: ˜Iv ™ and ˜C ™.
As before, the meanings of the relation constants are explained by consid-
ering them in context:
[97] Every instance of every schema in the following left-hand side column
has the same meaning as the corresponding instance of the schema on
the same line in the following right-hand side column:
˜Sa™ ˜a is an assertion™
˜Sb a™ ˜a is a past-tense predicative assertion™
˜Sf a™ ˜a is a future-tense predicative assertion™
˜St a™ ˜a is a “tomorrow”-assertion™
˜Sc a™ ˜a is a negative assertion™
˜Sv a™ ˜a is a disjunctive assertion™3
˜a is a set™4
˜Ma™
˜Ba™ ˜a is a branch™
˜a is an element of b™
˜∈ab™
˜Ib ab™ ˜a consists of an utterance of “it has been the case that”
followed by b™
˜If ab™ ˜a consists of an utterance of “it will be the case that” followed
by b™
˜It ab™ ˜a consists of an utterance of “tomorrow it will be the case
that” followed by b™
˜Ic ab™ ˜a consists of an utterance of “it is not the case that” followed
by b™
˜a is a token of the same type as b™5
˜Oab™
˜b lies within the day after the day within which a lies™6
˜Wab™
˜a precedes b™
˜<ab™
3 The subscripts ˜b™, ˜f ™, ˜t™, ˜c™, and ˜v™ stand for ˜before™, ˜future™, ˜tomorrow™, ˜contradictory™, and ˜vel ™
(the Latin word for ˜or™).
4 ˜M ™ is an abbreviation of the German ˜Menge™ (˜set™).
5 ˜O ™ stands for ˜occurrence of the same type as™.
6 ˜W ™ stands for ˜within the day after the day within which™.
268 The failure of Bivalence
˜T *ab™ ˜a is true at b™
˜F *ab™ ˜a is false at b™
˜Iv abg™ ˜a consists of an utterance of “either”, followed by b, followed
by an utterance of “or”, followed by g™7
˜Cabg™ ˜a is correct at b on g™
For the sake of readability, expressions of set-theoretical membership
and temporal order are written in the usual way: every instance of the
schema ˜a ∈ b™ stands for the corresponding instance of the schema
˜∈a b™, and every instance of the schema ˜a < b™ stands for the corre-
sponding instance of the schema ˜<ab™.
Note the new symbol, ˜T *™, for truth. We already have a symbol, ˜T ™, for the
property of truth that applies to present-tense predicative assertions.8 The
present expanded semantic theory concerns not only present-tense predica-
tive assertions, but also assertions of other kinds. The property of truth that
applies to all assertions (not only to present-tense predicative assertions) is
different from the property of truth that applies only to present-tense pred-
icative assertions, and must be de¬ned differently. For this reason a new
symbol for truth is needed. However, as far as present-tense predicative
assertions are concerned, the property of truth signi¬ed by the new symbol
˜T *™ has the same extension as the one signi¬ed by the old symbol ˜T ™.9

Kinds of assertions. There are six kinds of assertions: present-tense, past-
tense, future-tense, ˜tomorrow™-, negative, and disjunctive assertions. Their
main characteristics are spelled out by the following principles:
[98] ∀x (Sx ’ Vx).
[99] ∀x (Sb x ” ∃y (Sn y & Ib xy)).
[100] ∀x (Sf x ” ∃y (Sn y & If xy)).
[101] ∀x (St x ” ∃y (Sn y & It xy)).
[102] ∀x (Sc x ” ∃y ((Sn y ∨ Sb y ∨ Sf y ∨ St y) & Ic xy)).
[103] ∀x (Sv x ” ∃y ∃z ((Sn y ∨ Sb y ∨ Sf y ∨ St y ∨ Sc y) & (Sn z ∨ Sb z ∨ Sf z
∨ St z ∨ Sc z) & Iv xyz)).
[104] ∀x ((Sn x ’ ¬(Sb x ∨ Sf x ∨ St x ∨ Sc x ∨ Sv x)) & (Sb x ’ ¬(Sf x ∨
St x ∨ Sc x ∨ Sv x)) & (Sf x ’ ¬(St x ∨ Sc x ∨ Sv x)) & (St x ’ ¬(Sc x
∨ Sv x)) & (Sc x ’ ¬Sv x)).
7 The ˜I ™ in ˜Ib ™, ˜If ™, ˜It ™, ˜Ic ™, and ˜Iv ™ is an abbreviation of ˜in¬‚ected from™.
8 9 Cf. [140] below.
Cf. [55] and [56] above.
Appendix 6 269
[105] ∀x (Sx ” Sn x ∨ Sb x ∨ Sf x ∨ St x ∨ Sc x ∨ Sv x).
The foregoing principles allow only a restricted range of assertions: e.g. the
only disjunctive assertions permitted are those whose disjuncts are either
present- or past- or future-tense or ˜tomorrow™- or negative assertions. The
principles could be so modi¬ed as to allow assertions of other kinds (e.g.
disjunctive assertions whose disjuncts are further disjunctive assertions),
but I am not presently interested in constructing a semantic theory that
covers a large fraction of natural language.

In¬‚ection of predicates vs in¬‚ection of whole assertions. Propositions [99] and
[100] take tense in¬‚ection to be performed on complete assertions. For
instance, the past-tense assertion ˜It has been the case that some man is
pale™ is obtained by adding an utterance of ˜it has been the case that . . .™
to the whole present-tense assertion ˜Some man is pale™. This is not the
only way one can conceive of tense in¬‚ection. An alternative is to regard
tense in¬‚ection as performed on verb-phrases. According to this alternative
conception, a past-tense assertion like ˜Some man was pale™ is composed of
utterances of ˜some man™ and of the past-tense verb-phrase ˜was pale™, which
in turn is obtained by in¬‚ecting the present-tense verb-phrase ˜is pale™. In
fact, some remarks from de Interpretatione 3 suggest that this alternative
conception of tense in¬‚ection is closer to Aristotle™s view of the matter:
T 61 Similarly, ˜recovered™ and ˜will recover™ are not verbs but cases of verbs. They
differ from the verb in that it additionally signi¬es the present time, they the
time outside the present. (16b 16“18)
The difference between the two conceptions of tense in¬‚ection re¬‚ects
on truth conditions. On the truth conditions associated with the ¬rst con-
ception, the past-tense assertion ˜It has been the case that some dark man is
pale™ never is true (because no time is preceded by a time when ˜Some dark
man is pale™ is true). On the truth conditions associated with the second
conception, the past-tense assertion ˜Some dark man was pale™ is sometimes
true because its predicate, an utterance of the past-tense verb-phrase ˜was
pale™, is true of some dark man (pale men do tan).10
I opt for the ¬rst conception of tense in¬‚ection because it is more
amenable to logical treatment. Moreover, as far as singular past- or future-
tense assertions whose subjects are proper names are concerned, the truth
conditions associated with the two conceptions are equivalent: ˜It has been
the case that Socrates is pale™ is true when and only when ˜Socrates was
pale™ is. But Int. 9 focuses on singular future-tense predicative assertions.
10 Cf. Ockham in Int. i.vi 10, 23“4; 10, 26“8.
270 The failure of Bivalence
Tokens of the same type. A part of the expanded semantic theory to be
presented relies on the two-place relation of being-a-token-of-the-same-
type-as. This relation is de¬ned case by case:
[106] ∀x ∀y (Sn x & Vy ’ (Oxy ” ((Sa x & Sa y) ∨ (Se x & Se y) ∨ (Si x & Si y)
∨ (So x & So y) ∨ (Sy x & Sy y) ∨ (Sd x & Sd y)) & Ck xy & Ch xy)).
[107] ∀x ∀y ∀z (Sn y & Ib xy & Vz ’ (Oxz ” ∃u (Sn u & Ib zu & Oyu))).
[108] ∀x ∀y ∀z (Sn y & If xy & Vz ’ (Oxz ” ∃u (Sn u & If zu & Oyu))).
[109] ∀x ∀y ∀z (Sn y & It xy & Vz ’ (Oxz ” ∃u (Sn u & It zu & Oyu))).
[110] ∀x ∀y ∀z ((Sn y ∨ Sb y ∨ Sf y ∨ St y) & Ic xy & Vz ’ (Oxz ” ∃u ((Sn u
∨ Sb u ∨ Sf u ∨ St u) & Ic zu & Oyu))).
[111] ∀x ∀y ∀z ∀u ((Sn y ∨ Sb y ∨ Sf y ∨ St y ∨ Sc y) & (Sn z ∨ Sb z ∨ Sf z ∨ St z
∨ Sc z) & Iv xyz & Vu ’ (Oxu ” ∃v ∃w ((Sn v ∨ Sb v ∨ Sf v ∨ St v ∨
Sc v) & (Sn w ∨ Sb w ∨ Sf w ∨ St w ∨ Sc w) & Iv uvw & Oyv & Ozw))).

The tree of time. The central ontological idea of the semantic theory out-
lined in these pages is that of a tree. A tree (which represents the total-
ity of time) is based on the two-place relation < (which represents the
relation of preceding) holding between times. < arranges times in a tree-
like structure. One can thereby distinguish two directions: the ˜backwards™
direction (towards the root), the direction of the past by moving in which
one encounters earlier times, and the ˜forward™ direction (away from the
root), the direction of the future by moving in which one encounters later
times.

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