## 1. Introduction



Aristotle's logical works contain the earliest formal study of
logic that we have. It is therefore all the more remarkable that
together they comprise a highly developed logical theory, one that was
able to command immense respect for many centuries: Kant, who was ten
times more distant from Aristotle than we are from him, even held that
nothing significant had been added to Aristotle's views in the
intervening two millennia.



In the last century, Aristotle's reputation as a logician has
undergone two remarkable reversals. The rise of modern formal logic
following the work of Frege and Russell brought with it a recognition
of the many serious limitations of Aristotle's logic; today,
very few would try to maintain that it is adequate as a basis for
understanding science, mathematics, or even everyday reasoning. At the
same time, scholars trained in modern formal techniques have come to
view Aristotle with new respect, not so much for the correctness of
his results as for the remarkable similarity in spirit between much of
his work and modern logic. As Jonathan Lear has put it,
"Aristotle shares with modern logicians a fundamental interest
in metatheory": his primary goal is not to offer a practical
guide to argumentation but to study the properties of inferential
systems themselves.


## 2. Aristotle's Logical Works: The *Organon*



The ancient commentators grouped together several of Aristotle's
treatises under the title *Organon* ("Instrument")
and regarded them as comprising his logical works:


1. *Categories*
2. *On Interpretation*
3. *Prior Analytics*
4. *Posterior Analytics*
5. *Topics*
6. *On Sophistical Refutations*



In fact, the title *Organon* reflects a much later controversy
about whether logic is a part of philosophy (as the Stoics maintained)
or merely a tool used by philosophy (as the later Peripatetics
thought); calling the logical works "The Instrument" is a
way of taking sides on this point. Aristotle himself never uses this
term, nor does he give much indication that these particular treatises
form some kind of group, though there are frequent cross-references
between the *Topics* and the *Analytics*. On the other
hand, Aristotle treats the *Prior* and *Posterior
Analytics* as one work, and *On Sophistical Refutations* is
a final section, or an appendix, to the *Topics*). To these
works should be added the *Rhetoric*, which explicitly declares
its reliance on the *Topics*.


## 3. The Subject of Logic: "Syllogisms"



All Aristotle's logic revolves around one notion: the
**deduction** (*sullogismos*). A thorough
explanation of what a deduction is, and what they are composed of,
will necessarily lead us through the whole of his theory. What, then,
is a deduction? Aristotle says:



> 
> A deduction is speech (*logos*) in which, certain things having
> been supposed, something different from those supposed results of
> necessity because of their being so. (*Prior Analytics* I.2,
> 24b18-20)
> 



Each of the "things supposed" is a
**premise** (*protasis*) of the argument, and what
"results of necessity" is the **conclusion**
(*sumperasma*).



The core of this definition is the notion of "resulting of
necessity" (*ex anankes sumbainein*). This
corresponds to a modern notion of logical consequence: \(X\) results
of necessity from \(Y\) and \(Z\) if it would be impossible for \(X\)
to be false when \(Y\) and \(Z\) are true. We could therefore take
this to be a general definition of "valid argument".


### 3.1 Induction and Deduction



Deductions are one of two species of argument recognized by Aristotle.
The other species is **induction**
(*epagoge*). He has far less to say about this than
deduction, doing little more than characterize it as "argument
from the particular to the universal". However, induction (or
something very much like it) plays a crucial role in the theory of
scientific knowledge in the *Posterior Analytics*: it is
induction, or at any rate a cognitive process that moves from
particulars to their generalizations, that is the basis of knowledge
of the indemonstrable first principles of sciences.


### 3.2 Aristotelian Deductions and Modern Valid Arguments



Despite its wide generality, Aristotle's definition of deduction
is not a precise match for a modern definition of validity. Some of
the differences may have important consequences:


1. Aristotle explicitly says that what results of necessity must be
different from what is supposed. This would rule out arguments in
which the conclusion is identical to one of the premises. Modern
notions of validity regard such arguments as valid, though trivially
so.
2. The plural "certain things having been supposed" was
taken by some ancient commentators to rule out by definition arguments
with only one premise, and Aristotle himself says in some places that
nothing new follows from just one premise.
3. The force of the qualification "because of their being
so" has sometimes been seen as ruling out arguments in which the
conclusion is not 'relevant' to the premises, e.g.,
arguments in which the premises are inconsistent, arguments with
conclusions that would follow from any premises whatsoever, or
arguments with superfluous premises.



Of these three possible restrictions, the most interesting would be
the third. This could be (and has been) interpreted as committing
Aristotle to something like a
 relevance logic.
 In fact, there are passages that appear to confirm this. However,
this is too complex a matter to discuss here.



However the definition is interpreted, it is clear that Aristotle does
not mean to restrict it only to a subset of the valid arguments. This
is why I have translated *sullogismos* with
'deduction' rather than its English cognate. In modern
usage, 'syllogism' means an argument of a very specific
form. Moreover, modern usage distinguishes between valid syllogisms
(the conclusions of which follow from their premises) and invalid
syllogisms (the conclusions of which do not follow from their
premises). The second of these is inconsistent with Aristotle's
use: since he defines a *sullogismos* as an argument in which
the conclusion results of necessity from the premises, "invalid
*sullogismos*" is a contradiction in terms. The first is
also at least highly misleading, since Aristotle does not appear to
think that the *sullogismoi* are simply an interesting subset
of the valid arguments. Moreover (see below), Aristotle expends great
efforts to argue that every valid argument, in a broad sense, can be
"reduced" to an argument, or series of arguments, in
something like one of the forms traditionally called a syllogism. If
we translate *sullogismos* as "syllogism", this
becomes the trivial claim "Every syllogism is a
syllogism",


## 4. Premises: The Structures of Assertions



Syllogisms are structures of sentences each of which can meaningfully
be called true or false: **assertions**
(*apophanseis*), in Aristotle's terminology. According to
Aristotle, every such sentence must have the same structure: it must
contain a **subject** (*hupokeimenon*) and a
**predicate** and must either affirm or deny the
predicate of the subject. Thus, every assertion is either the
**affirmation** *kataphasis* or the
**denial** (*apophasis*) of a single predicate of
a single subject.



In *On Interpretation*, Aristotle argues that a single
assertion must always either affirm or deny a single predicate of a
single subject. Thus, he does not recognize sentential compounds, such
as conjunctions and disjunctions, as single assertions. This appears
to be a deliberate choice on his part: he argues, for instance, that a
conjunction is simply a collection of assertions, with no more
intrinsic unity than the sequence of sentences in a lengthy account
(e.g. the entire *Iliad*, to take Aristotle's own
example). Since he also treats denials as one of the two basic species
of assertion, he does not view negations as sentential compounds. His
treatment of conditional sentences and disjunctions is more difficult
to appraise, but it is at any rate clear that Aristotle made no
efforts to develop a sentential logic. Some of the consequences of
this for his theory of demonstration are important.


### 4.1 Terms



Subjects and predicates of assertions are **terms**. A
term (*horos*) can be either individual, e.g.
*Socrates*, *Plato* or universal, e.g. *human*,
*horse*, *animal*, *white*. Subjects may be
either individual or universal, but predicates can only be universals:
*Socrates is human*, *Plato is not a horse*, *horses
are animals*, *humans are not horses*.



The word **universal** (*katholou*) appears to be
an Aristotelian coinage. Literally, it means "of a whole";
its opposite is therefore "of a particular"
(*kath' hekaston*). Universal terms are those which can
properly serve as predicates, while particular terms are those which
cannot.



This distinction is not simply a matter of grammatical function. We
can readily enough construct a sentence with "Socrates" as
its grammatical predicate: "The person sitting down is
Socrates". Aristotle, however, does not consider this a genuine
predication. He calls it instead a merely **accidental**
or **incidental** (*kata sumbebekos*)
predication. Such sentences are, for him, dependent for their truth
values on other genuine predications (in this case, "Socrates is
sitting down").



Consequently, predication for Aristotle is as much a matter of
metaphysics as a matter of grammar. The reason that the term
*Socrates* is an individual term and not a universal is that
the entity which it designates is an individual, not a universal. What
makes *white* and *human* universal terms is that they
designate universals.



Further discussion of these issues can be found in the entry on
 Aristotle's metaphysics.


### 4.2 Affirmations, Denials, and Contradictions



Aristotle takes some pains in *On Interpretation* to argue that
to every affirmation there corresponds exactly one denial such that
that denial denies exactly what that affirmation affirms. The pair
consisting of an affirmation and its corresponding denial is a
**contradiction** (*antiphasis*). In general,
Aristotle holds, exactly one member of any contradiction is true and
one false: they cannot both be true, and they cannot both be false.
However, he appears to make an exception for propositions about future
events, though interpreters have debated extensively what this
exception might be (see
 further discussion
 below). The principle that contradictories cannot both be true has
fundamental importance in Aristotle's metaphysics (see
 further discussion
 below).


### 4.3 All, Some, and None



One major difference between Aristotle's understanding of
predication and modern (i.e., post-Fregean) logic is that Aristotle
treats individual predications and general predications as similar in
logical form: he gives the same analysis to "Socrates is an
animal" and "Humans are animals". However, he notes
that when the subject is a universal, predication takes on two forms:
it can be either **universal** or
**particular**. These expressions are parallel to those
with which Aristotle distinguishes universal and particular terms, and
Aristotle is aware of that, explicitly distinguishing between a term
being a universal and a term being universally predicated of
another.




|  |  |  |
| --- | --- | --- |
|  | **Affirmations** | **Denials** |
| **Universal** | \(P\) affirmed of all of \(S\) | Every \(S\) is \(P\),

All \(S\) is (are) \(P\) | \(P\) denied of all of \(S\) | No \(S\) is \(P\) |
| **Particular** | \(P\) affirmed of some of \(S\) | Some \(S\) is (are) \(P\) | \(P\) denied of some of \(S\) | Some \(S\) is not \(P\),

Not every \(S\) is \(P\) |
| **Indefinite** | \(P\) affirmed of \(S\) | \(S\) is \(P\) | \(P\) denied of \(S\) | \(S\) is not \(P\) |



Whatever is affirmed or denied of a universal subject may be affirmed
or denied of it it **universally** (*katholou* or
"of all", *kata pantos*), **in part**
(*kata meros*, *en merei*), or
**indefinitely** (*adihoristos*).


#### 4.3.1 The "Square of Opposition"



In *On Interpretation*, Aristotle spells out the relationships
of contradiction for sentences with universal subjects as follows:




|  |  |  |
| --- | --- | --- |
|  | **Affirmation** | **Denial** |
| **Universal** | Every \(A\) is \(B\) | No \(A\) is \(B\) |
| **Particular** | Some \(A\) is \(B\) | Not every \(A\) is \(B\) |



Simple as it appears, this table raises important difficulties of
interpretation (for a thorough discussion, see the entry on the
 square of opposition).



In the *Prior Analytics*, Aristotle adopts a somewhat
artificial way of expressing predications: instead of saying
"\(X\) is predicated of \(Y\)" he says "\(X\)
belongs (*huparchei*) to \(Y\)". This should really be
regarded as a technical expression. The verb *huparchein*
usually means either "begin" or "exist, be
present", and Aristotle's usage appears to be a
development of this latter use.


#### 4.3.2 Some Convenient Abbreviations



For clarity and brevity, I will use the following semi-traditional
abbreviations for Aristotelian categorical sentences (note that the
predicate term comes *first* and the subject term
*second*):




|  |  |
| --- | --- |
| **Abbreviation** | **Sentence** |
| \(Aab\) | \(a\) belongs to all \(b\) (Every \(b\) is \(a\)) |
| \(Eab\) | \(a\) belongs to no \(b\) (No \(b\) is \(a\)) |
| \(Iab\) | \(a\) belongs to some \(b\) (Some \(b\) is \(a\)) |
| \(Oab\) | \(a\) does not belong to all \(b\) (Some \(b\) is not
\(a\)) |


## 5. The Syllogistic



Aristotle's most famous achievement as logician is his theory of
inference, traditionally called the **syllogistic**
(though not by Aristotle). That theory is in fact the theory of
inferences of a very specific sort: inferences with two premises, each
of which is a categorical sentence, having exactly one term in common,
and having as conclusion a categorical sentence the terms of which are
just those two terms not shared by the premises. Aristotle calls the
term shared by the premises the **middle term**
(*meson*) and each of the other two terms in the premises an
**extreme** (*akron*). The middle term must be
either subject or predicate of each premise, and this can occur in
three ways: the middle term can be the subject of one premise and the
predicate of the other, the predicate of both premises, or the subject
of both premises. Aristotle refers to these term arrangements as
**figures** (*schemata*):


### 5.1 The Figures




|  |  |  |  |
| --- | --- | --- | --- |
|  | **First Figure** | **Second Figure** | **Third Figure** |
|  | **Predicate** | **Subject** | **Predicate** | **Subject** | **Predicate** | **Subject** |
| **Premise** | \(a\) | \(b\) | \(a\) | \(b\) | \(a\) | \(c\) |
| **Premise** | \(b\) | \(c\) | \(a\) | \(c\) | \(b\) | \(c\) |
| **Conclusion** | \(a\) | \(c\) | \(b\) | \(c\) | \(a\) | \(b\) |



Aristotle calls the term which is the predicate of the conclusion the
**major** term and the term which is the subject of the
conclusion the **minor** term. The premise containing the
major term is the **major premise**, and the premise
containing the minor term is the **minor premise**.



Aristotle then systematically investigates all possible combinations
of two premises in each of the three figures. For each combination, he
either demonstrates that some conclusion necessarily follows or
demonstrates that no conclusion follows. The results he states are
correct.


### 5.2 Methods of Proof: "Perfect" Deductions, Conversion, Reduction



Aristotle's proofs can be divided into two categories, based on
a distinction he makes between "perfect" or
"complete" (*teleios*) deductions and
"imperfect" or "incomplete"
(*ateles*) deductions. A deduction is perfect if it
"needs no external term in order to show the necessary
result" (24b23-24), and it is imperfect if it "needs
one or several in addition that are necessary because of the terms
supposed but were not assumed through premises"
(24b24-25). The precise interpretation of this distinction is
debatable, but it is at any rate clear that Aristotle regards the
perfect deductions as not in need of proof in some sense. For
imperfect deductions, Aristotle does give proofs, which invariably
depend on the perfect deductions. Thus, with some reservations, we
might compare the perfect deductions to the axioms or primitive rules
of a deductive system.



In the proofs for imperfect deductions, Aristotle says that he
"reduces" (*anagein*) each case to one of the
perfect forms and that they are thereby "completed" or
"perfected". These completions are either
**probative** (*deiktikos*: a modern translation
might be "direct") or **through the
impossible** (*dia to adunaton*).



A direct deduction is a series of steps leading from the premises to
the conclusion, each of which is either a **conversion**
of a previous step or an inference from two previous steps relying on
a first-figure deduction. Conversion, in turn, is inferring from a
proposition another which has the subject and predicate interchanged.
Specifically, Aristotle argues that three such conversions are
sound:

 
\[\begin{align}
Eab &\rightarrow Eba \\
 Iab &\rightarrow Iba \\
 Aab &\rightarrow Iba 
\end{align}\]


He undertakes to justify these in *An. Pr.* I.2. From a modern
standpoint, the third is sometimes regarded with suspicion. Using it
we can get *Some monsters are chimeras* from the apparently
true *All chimeras are monsters*; but the former is often
construed as implying in turn *There is something which is a
monster and a chimera*, and thus that there are monsters and there
are chimeras. In fact, this simply points up something about
Aristotle's system: Aristotle in effect supposes that
*all* terms in syllogisms are non-empty. (For further
discussion of this point, see the entry on the
 square of opposition).



As an example of the procedure, we may take Aristotle's proof of
*Camestres*. He says:



> 
> If \(M\) belongs to every \(N\) but to no \(X\), then neither will
> \(N\) belong to any \(X\). For if \(M\) belongs to no \(X\), then
> neither does \(X\) belong to any \(M\); but \(M\) belonged to every
> \(N\); therefore, \(X\) will belong to no \(N\) (for the first figure
> has come about). And since the privative converts, neither will \(N\)
> belong to any \(X\). (*An. Pr.* I.5, 27a9-12)
> 



From this text, we can extract an exact formal proof, as follows:




|  |  |  |
| --- | --- | --- |
| Step       | Justification | Aristotle's Text |
| 1. \(MaN\) |  | *If \(M\) belongs to every \(N\)* |
| 2. \(MeX\) |  | *but to no \(X\),* |
| To prove:

\(NeX\) |  | *then neither will \(N\) belong to any \(X\).* |
| 3. \(MeX\) | (2, premise) | *For if \(M\) belongs to no \(X\),* |
| 4. \(XeM\) | (3, conversion of \(e\)) | *then neither does \(X\) belong to any \(M\);* |
| 5. \(MaN\) | (1, premise) | *but \(M\) belonged to every \(N\);* |
| 6. \(XeN\) | (4, 5, *Celarent*) | *therefore, \(X\) will belong to no \(N\) (for the first
figure has come about).* |
| 7. \(NeX\) | (6, conversion of \(e\)) | *And since the privative converts, neither will \(N\) belong
to any \(X\).* |



A completion or proof "through the impossible" shows that
a certain conclusion follows from a pair of premises by assuming as a
third premise the denial of that conclusion and giving a deduction,
from it and one of the original premises, the denial (or the contrary)
of the other premises. This is the deduction of an
"impossible", and Aristotle's proof ends at that
point. An example is his proof of *Baroco* in
27a36-b1:




|  |  |  |
| --- | --- | --- |
| Step | Justification | Aristotle's Text |
| 1. \(MaN\) |  | *Next, if \(M\) belongs to every \(N\),* |
| 2. \(MoX\) |  | *but does not belong to some \(X\),* |
| To prove: \(NoX\) |  | *then it is necessary for \(N\) not to belong to some
\(X\)* |
| 3. \(NaX\) | Contradictory of the desired conclusion | *For if it belongs to all,* |
| 4. \(MaN\) | Repetition of premise 1 | *and \(M\) is predicated of every \(N\),* |
| 5. \(MaX\) | (3, 4, Barbara) | *then it is necessary that \(M\) belongs to every
\(X\).* |
| 6. \(MoX\) | (5 is the contradictory of 2) | *But it was assumed not to belong to some.* |


### 5.3 Disproof: Counterexamples and Terms



Aristotle proves invalidity by constructing counterexamples. This is
very much in the spirit of modern logical theory: all that it takes to
show that a certain *form* is invalid is a single
*instance* of that form with true premises and a false
conclusion. However, Aristotle states his results not by saying that
certain premise-conclusion combinations are invalid but by saying that
certain premise pairs do not "syllogize": that is, that,
given the pair in question, examples can be constructed in which
premises of that form are true and a conclusion of any of the four
possible forms is false.



When possible, he does this by a clever and economical method: he
gives two triplets of terms, one of which makes the premises true and
a universal affirmative "conclusion" true, and the other
of which makes the premises true and a universal negative
"conclusion" true. The first is a counterexample for an
argument with either an \(E\) or an \(O\) conclusion, and the second
is a counterexample for an argument with either an \(A\) or an \(I\)
conclusion.


### 5.4 The Deductions in the Figures ("Moods")



In *Prior Analytics* I.4-6, Aristotle shows that the
premise combinations given in the following table yield deductions and
that all other premise combinations fail to yield a deduction. In the
terminology traditional since the middle ages, each of these
combinations is known as a **mood** Latin *modus*,
"way", which in turn is a translation of Greek
*tropos*). Aristotle, however, does not use this expression and
instead refers to "the arguments in the figures".



In this table, "\(\vdash\)" separates premises from
conclusion; it may be read "therefore". The second column
lists the medieval mnemonic name associated with the inference (these
are still widely used, and each is actually a mnemonic for
Aristotle's proof of the mood in question). The third column
briefly summarizes Aristotle's procedure for demonstrating the
deduction.





|  |  |  |
| --- | --- | --- |
| Form | Mnemonic | Proof |
| 
FIRST FIGURE |
| \(Aab, Abc \vdash Aac\) | *Barbara* | Perfect |
| \(Eab, Abc \vdash Eac\) | *Celarent* | Perfect |
| \(Aab, Ibc \vdash Iac\) | *Darii* | Perfect; also by impossibility, from *Camestres* |
| \(Eab, Ibc \vdash Oac\) | *Ferio* | Perfect; also by impossibility, from *Cesare* |
| 
SECOND FIGURE |
| \(Eab, Aac \vdash Ebc\) | *Cesare* | \((Eab, Aac)\rightarrow (Eba, Aac)\) \(\vdash\_{Cel} Ebc\) |
| \(Aab, Eac \vdash Ebc\) | *Camestres* | \((Aab, Eac) \rightarrow (Aab, Eca)= (Eca, Aab)\) \(\vdash\_{Cel}
Ecb \rightarrow Ebc\) |
| \(Eab, Iac \vdash Obc\) | *Festino* | \((Eab, Iac) \rightarrow (Eba,Iac)\) \(\vdash\_{Fer} Obc\) |
| \(Aab, Oac \vdash Obc\) | *Baroco* | \((Aab, Oac +Abc) \vdash\_{Bar} (Aac,Oac)\) \(\vdash\_{Imp}
Obc\) |
| 
THIRD FIGURE |
| \(Aac, Abc \vdash Iab\) | *Darapti* | \((Aac, Abc) \rightarrow (Aac,Icb)\) \(\vdash\_{Dar} Iab\) |
| \(Eac, Abc \vdash Oab\) | *Felapton* | \((Eac, Abc) \rightarrow (Eac,Icb)\) \(\vdash\_{Fer} Oab\) |
| \(Iac, Abc \vdash Iab\) | *Disamis* | \((Iac, Abc) \rightarrow (Ica, Abc)=(Abc,Ica)\) \(\vdash\_{Dar}
Iba \rightarrow Iab\) |
| \(Aac, Ibc \vdash Iab\) | *Datisi* | \((Aac, Ibc) \rightarrow (Aac,Icb)\) \(\vdash\_{Dar} Iab\) |
| \(Oac, Abc \vdash Oab\) | *Bocardo* | \((Oac, +Aab, Abc) \vdash\_{Bar} (Aac,Oac)\) \(\vdash\_{Imp}
Oab\) |
| \(Eac, Ibc \vdash Oab\) | *Ferison* | \((Eac, Ibc) \rightarrow (Eac, Icb)\) \(\vdash\_{Fer} Oab\) |



Table of the Deductions in the Figures



### 5.5 Metatheoretical Results



Having established which deductions in the figures are possible,
Aristotle draws a number of metatheoretical conclusions,
including:


1. No deduction has two negative premises
2. No deduction has two particular premises
3. A deduction with an affirmative conclusion must have two
affirmative premises
4. A deduction with a negative conclusion must have one negative
premise.
5. A deduction with a universal conclusion must have two universal
premises



He also proves the following metatheorem:



> 
> All deductions can be reduced to the two universal deductions in the
> first figure.
> 



His proof of this is elegant. First, he shows that the two particular
deductions of the first figure can be reduced, by proof through
impossibility, to the universal deductions in the second figure:



(*Darii*)
\((Aab, Ibc, +Eac)\vdash\_{Camestres} (Ebc, Ibc)\vdash\_{Imp}
Iac\)
(*Ferio*)
\((Eab, Ibc, +Aac)\vdash\_{Cesare} (Ebc, Ibc)\vdash\_{Imp}
Oac\)


He then observes that since he has already shown how to reduce all the
particular deductions in the other figures except *Baroco* and
*Bocardo* to *Darii* and *Ferio*, these
deductions can thus be reduced to *Barbara* and
*Celarent*. This proof is strikingly similar both in structure
and in subject to modern proofs of the redundancy of axioms in a
system.



Many more metatheoretical results, some of them quite sophisticated,
are proved in *Prior Analytics* I.45 and in *Prior
Analytics* II. As noted below, some of Aristotle's
metatheoretical results are appealed to in the epistemological
arguments of the *Posterior Analytics*.


### 5.6 Syllogisms with Modalities



Aristotle follows his treatment of "arguments in the
figures" with a much longer, and much more problematic,
discussion of what happens to these figured arguments when we add the
qualifications "necessarily" and "possibly" to
their premises in various ways. In contrast to the syllogistic itself
(or, as commentators like to call it, the *assertoric*
syllogistic), this *modal* syllogistic appears to be much less
satisfactory and is certainly far more difficult to interpret. Here, I
only outline Aristotle's treatment of this subject and note some
of the principal points of interpretive controversy.


#### 5.6.1 The Definitions of the Modalities



Modern modal logic treats necessity and possibility as interdefinable:
"necessarily P" is equivalent to "not possibly not
P", and "possibly P" to "not necessarily not
P". Aristotle gives these same equivalences in *On
Interpretation*. However, in *Prior Analytics*, he makes a
distinction between two notions of possibility. On the first, which he
takes as his preferred notion, "possibly P" is equivalent
to "not necessarily P and not necessarily not P". He then
acknowledges an alternative definition of possibility according to the
modern equivalence, but this plays only a secondary role in his
system.


#### 5.6.2 Aristotle's General Approach



Aristotle builds his treatment of modal syllogisms on his account of
non-modal (**assertoric**) syllogisms: he works his way
through the syllogisms he has already proved and considers the
consequences of adding a modal qualification to one or both premises.
Most often, then, the questions he explores have the form: "Here
is an assertoric syllogism; if I add these modal qualifications to the
premises, then what modally qualified form of the conclusion (if any)
follows?". A premise can have one of three modalities: it can be
necessary, possible, or assertoric. Aristotle works through the
combinations of these in order:


* Two necessary premises
* One necessary and one assertoric premise
* Two possible premises
* One assertoric and one possible premise
* One necessary and one possible premise



Though he generally considers only premise combinations which
syllogize in their assertoric forms, he does sometimes extend this;
similarly, he sometimes considers conclusions in addition to those
which would follow from purely assertoric premises.



Since this is his procedure, it is convenient to describe modal
syllogisms in terms of the corresponding non-modal syllogism plus a
triplet of letters indicating the modalities of premises and
conclusion: \(N\) = "necessary", \(P\) =
"possible", \(A\) = "assertoric". Thus,
"Barbara \(NAN\)" would mean "The form
*Barbara* with necessary major premise, assertoric minor
premise, and necessary conclusion". I use the letters
"\(N\)" and "\(P\)" as prefixes for premises
as well; a premise with no prefix is assertoric. Thus,
*Barbara* \(NAN\) would be \(NAab, Abc \vdash NAac\).


#### 5.6.3 Modal Conversions



As in the case of assertoric syllogisms, Aristotle makes use of
conversion rules to prove validity. The conversion rules for necessary
premises are exactly analogous to those for assertoric premises:



\[\begin{align}
NEab &\rightarrow NEba \\
 NIab &\rightarrow NIba \\
 NAab &\rightarrow NIba 
\end{align}\]


Possible premises behave differently, however. Since he defines
"possible" as "neither necessary nor
impossible", it turns out that \(x\) *is possibly* \(F\)
entails, and is entailed by, \(x\) *is possibly not* \(F\).
Aristotle generalizes this to the case of categorical sentences as
follows:

 
\[\begin{align}
PAab &\rightarrow PEab \\
 PEab &\rightarrow PAab \\
 PIab &\rightarrow POab \\
 POab &\rightarrow PIab 
\end{align}\]


In addition, Aristotle uses the intermodal principle \(N\rightarrow
A\): that is, a necessary premise entails the corresponding assertoric
one. However, because of his definition of possibility, the principle
\(A\rightarrow P\) does not generally hold: if it did, then
\(N\rightarrow P\) would hold, but on his definition
"necessarily \(P\)" and "possibly \(P\)" are
actually inconsistent ("possibly \(P\)" entails
"possibly not \(P\)").



This leads to a further complication. The denial of "possibly
\(P\)" for Aristotle is "either necessarily \(P\) or
necessarily not \(P\)". The denial of "necessarily
\(P\)" is still more difficult to express in terms of a
combination of modalities: "either possibly \(P\) (and thus
possibly not \(P\)) or necessarily not \(P\)" This is important
because of Aristotle's proof procedures, which include proof
through impossibility. If we give a proof through impossibility in
which we assume a necessary premise, then the conclusion we ultimately
establish is simply the denial of that necessary premise, not a
"possible" conclusion in Aristotle's sense. Such
propositions do occur in his system, but only in exactly this way,
i.e., as conclusions established by proof through impossiblity from
necessary assumptions. Somewhat confusingly, Aristotle calls such
propositions "possible" but immediately adds " not
in the sense defined": in this sense, "possibly
\(Oab\)" is simply the denial of "necessarily
\(Aab\)". Such propositions appear only as premises, never as
conclusions.


#### 5.6.4 Syllogisms with Necessary Premises



Aristotle holds that an assertoric syllogism remains valid if
"necessarily" is added to its premises and its conclusion:
the modal pattern \(NNN\) is always valid. He does not treat this as a
trivial consequence but instead offers proofs; in all but two cases,
these are parallel to those offered for the assertoric case. The
exceptions are *Baroco* and *Bocardo*, which he proved
in the assertoric case through impossibility: attempting to use that
method here would require him to take the denial of a necessary \(O\)
proposition as hypothesis, raising the complication noted above, and
he uses the procedure he calls *ecthesis* instead (see Smith
1982).


#### 5.6.5 The Problem of the "Two Barbaras" and Other Problems of Interpretation



Since a necessary premise entails an assertoric premise, every \(AN\)
or \(NA\) combination of premises will entail the corresponding \(AA\)
pair, and thus the corresponding \(A\) conclusion. Thus, \(ANA\) and
\(NAA\) syllogisms are always valid. However, Aristotle holds that
some, but not all, \(ANN\) and \(NAN\) combinations are valid.
Specifically, he accepts *Barbara* \(NAN\) but rejects
*Barbara* \(ANN\). Almost from Aristotle's own time,
interpreters have found his reasons for this distinction obscure, or
unpersuasive, or both, and often have not followed his view. His close
associated Theophrastus, for instance, adopted the simpler rule that
the modality of the conclusion of a syllogism was always the
"weakest" modality found in either premise, where \(N\) is
stronger than \(A\) and \(A\) is stronger than \(P\) (and where \(P\)
probably has to be defined as "not necessarily not").



Beginning with Albrecht Becker, interpreters using the methods of
modern formal logic to interpret Aristotle's modal logic have
seen the Two-Barbaras problem as only one of a series of difficulties
in giving a coherent interpretation of the modal syllogistic. A very
wide range of reconstructions has been proposed: see Becker 1933,
McCall 1963, Nortmann 1996, Van Rijen 1989, Patterson 1995, Thomason
1993, Thom 1996, Rini 2012, Malink 2013. The majority of
reconstructions do not attempt to reproduce every detail of
Aristotle's exposition but instead produce modified
reconstructions that abandon some of those results. Malink 2013,
however, offers a reconstruction that reproduces everything Aristotle
says, although the resulting model introduces a high degree of
complexity. (This subject quickly becomes too complex for summarizing
in this brief article.


## 6. Demonstrations and Demonstrative Sciences



A **demonstration** (*apodeixis*) is "a
deduction that produces knowledge". Aristotle's
*Posterior Analytics* contains his account of demonstrations
and their role in knowledge. From a modern perspective, we might think
that this subject moves outside of logic to epistemology. From
Aristotle's perspective, however, the connection of the theory
of *sullogismoi* with the theory of knowledge is especially
close.


### 6.1 Aristotelian Sciences



The subject of the *Posterior Analytics* is
*episteme*. This is one of several Greek words that
can reasonably be translated "knowledge", but Aristotle is
concerned only with knowledge of a certain type (as will be explained
below). There is a long tradition of translating
*episteme* in this technical sense as
**science**, and I shall follow that tradition here.
However, readers should not be misled by the use of that word. In
particular, Aristotle's theory of science cannot be considered a
counterpart to modern philosophy of science, at least not without
substantial qualifications.



We have scientific knowledge, according to Aristotle, when we
know:



> 
> the cause why the thing is, that it is the cause of this, and that
> this cannot be otherwise. (*Posterior Analytics* I.2)
> 



This implies two strong conditions on what can be the object of
scientific knowledge:


1. Only what is necessarily the case can be known scientifically
2. Scientific knowledge is knowledge of causes



He then proceeds to consider what science so defined will consist in,
beginning with the observation that at any rate one form of science
consists in the possession of a **demonstration**
(*apodeixis*), which he defines as a "scientific
deduction":



> 
> by "scientific" (*epistemonikon*), I mean
> that in virtue of possessing it, we have knowledge.
> 



The remainder of *Posterior Analytics* I is largely concerned
with two tasks: spelling out the nature of demonstration and
demonstrative science and answering an important challenge to its very
possibility. Aristotle first tells us that a demonstration is a
deduction in which the premises are:


1. true
2. **primary** (*prota*)
3. **immediate** (*amesa*, "without a
middle")
4. **better known** or **more familiar**
(*gnorimotera*) than the conclusion
5. **prior** to the conclusion
6. **causes** (*aitia*) of the conclusion



The interpretation of all these conditions except the first has been
the subject of much controversy. Aristotle clearly thinks that science
is knowledge of causes and that in a demonstration, knowledge of the
premises is what brings about knowledge of the conclusion. The fourth
condition shows that the knower of a demonstration must be in some
better epistemic condition towards them, and so modern interpreters
often suppose that Aristotle has defined a kind of epistemic
justification here. However, as noted above, Aristotle is defining a
special variety of knowledge. Comparisons with discussions of
justification in modern epistemology may therefore be misleading.



The same can be said of the terms "primary",
"immediate" and "better known". Modern
interpreters sometimes take "immediate" to mean
"self-evident"; Aristotle does say that an immediate
proposition is one "to which no other is prior", but (as I
suggest in the next section) the notion of priority involved is likely
a notion of logical priority that it is hard to detach from
Aristotle's own logical theories. "Better known" has
sometimes been interpreted simply as "previously known to the
knower of the demonstration" (i.e., already known in advance of
the demonstration). However, Aristotle explicitly distinguishes
between what is "better known for us" with what is
"better known in itself" or "in nature" and
says that he means the latter in his definition. In fact, he says that
the process of acquiring scientific knowledge is a process of
*changing* what is better known "for us", until we
arrive at that condition in which what is better known in itself is
also better known for us.


### 6.2 The Regress Problem



In *Posterior Analytics* I.2, Aristotle considers two
challenges to the possibility of science. One party (dubbed the
"agnostics" by Jonathan Barnes) began with the following
two premises:


1. Whatever is scientifically known must be demonstrated.
2. The premises of a demonstration must be scientifically known.



They then argued that demonstration is impossible with the following
dilemma:


1. If the premises of a demonstration are scientifically known, then
they must be demonstrated.
2. The premises from which each premise are demonstrated must be
scientifically known.
3. Either this process continues forever, creating an infinite
regress of premises, or it comes to a stop at some point.
4. If it continues forever, then there are no first premises from
which the subsequent ones are demonstrated, and so nothing is
demonstrated.
5. On the other hand, if it comes to a stop at some point, then the
premises at which it comes to a stop are undemonstrated and therefore
not scientifically known; consequently, neither are any of the others
deduced from them.
6. Therefore, nothing can be demonstrated.



A second group accepted the agnostics' view that scientific
knowledge comes only from demonstration but rejected their conclusion
by rejecting the dilemma. Instead, they maintained:


* Demonstration "in a circle" is possible, so that it is
possible for all premises also to be conclusions and therefore
demonstrated.



Aristotle does not give us much information about how circular
demonstration was supposed to work, but the most plausible
interpretation would be supposing that at least for some set of
fundamental principles, each principle could be deduced from the
others. (Some modern interpreters have compared this position to a
coherence theory of knowledge.) However their position worked, the
circular demonstrators claimed to have a third alternative avoiding
the agnostics' dilemma, since circular demonstration gives us a
regress that is both unending (in the sense that we never reach
premises at which it comes to a stop) and finite (because it works its
way round the finite circle of premises).


### 6.3 Aristotle's Solution: "It Eventually Comes to a Stop"



Aristotle rejects circular demonstration as an incoherent notion on
the grounds that the premises of any demonstration must be prior (in
an appropriate sense) to the conclusion, whereas a circular
demonstration would make the same premises both prior and posterior to
one another (and indeed every premise prior and posterior to itself).
He agrees with the agnostics' analysis of the regress problem:
the only plausible options are that it continues indefinitely or that
it "comes to a stop" at some point. However, he thinks
both the agnostics and the circular demonstrators are wrong in
maintaining that scientific knowledge is only possible by
demonstration from premises scientifically known: instead, he claims,
there is another form of knowledge possible for the first premises,
and this provides the starting points for demonstrations.



To solve this problem, Aristotle needs to do something quite specific.
It will not be enough for him to establish that we can have knowledge
of *some propositions* without demonstrating them: unless it is
in turn possible to deduce all the other propositions of a science
from them, we shall not have solved the regress problem. Moreover (and
obviously), it is no solution to this problem for Aristotle simply to
*assert* that we have knowledge without demonstration of some
appropriate starting points. He does indeed say that it is his
position that we have such knowledge (*An. Post.* I.2,), but he
owes us an account of why that should be so.


### 6.4 Knowledge of First Principles: *Nous*



Aristotle's account of knowledge of the indemonstrable first
premises of sciences is found in *Posterior Analytics* II.19,
long regarded as a difficult text to interpret. Briefly, what he says
there is that it is another cognitive state, *nous* (translated
variously as "insight", "intuition",
"intelligence"), which knows them. There is wide
disagreement among commentators about the interpretation of his
account of how this state is reached; I will offer one possible
interpretation. First, Aristotle identifies his problem as explaining
how the principles can "become familiar to us", using the
same term "familiar" (*gnorimos*) that he
used in presenting the regress problem. What he is presenting, then,
is not a method of discovery but a process of becoming wise. Second,
he says that in order for knowledge of immediate premises to be
possible, we must have a kind of knowledge of them without having
learned it, but this knowledge must not be as "precise" as
the knowledge that a possessor of science must have. The kind of
knowledge in question turns out to be a capacity or power
(*dunamis*) which Aristotle compares to the capacity for
sense-perception: since our senses are innate, i.e., develop
naturally, it is in a way correct to say that we know what e.g. all
the colors look like before we have seen them: we have the capacity to
see them by nature, and when we first see a color we exercise this
capacity without having to learn how to do so first. Likewise,
Aristotle holds, our minds have by nature the capacity to recognize
the starting points of the sciences.



In the case of sensation, the capacity for perception in the sense
organ is actualized by the operation on it of the perceptible object.
Similarly, Aristotle holds that coming to know first premises is a
matter of a potentiality in the mind being actualized by experience of
its proper objects: "The soul is of such a nature as to be
capable of undergoing this". So, although we cannot come to know
the first premises without the necessary experience, just as we cannot
see colors without the presence of colored objects, our minds are
already so constituted as to be able to recognize the right objects,
just as our eyes are already so constituted as to be able to perceive
the colors that exist.



It is considerably less clear what these objects are and how it is
that experience actualizes the relevant potentialities in the soul.
Aristotle describes a series of stages of cognition. First is what is
common to all animals: perception of what is present. Next is memory,
which he regards as a retention of a sensation: only some animals have
this capacity. Even fewer have the next capacity, the capacity to form
a single experience (*empeiria*) from many repetitions of the
same memory. Finally, many experiences repeated give rise to knowledge
of a single universal (*katholou*). This last capacity is
present only in humans.



See Section 7 of the entry on
 Aristotle's psychology
 for more on his views about mind.


## 7. Definitions



The **definition** (*horos*, *horismos*)
was an important matter for Plato and for the Early Academy. Concern
with answering the question "What is so-and-so?" are at
the center of the majority of Plato's dialogues, some of which
(most elaborately the *Sophist*) propound methods for finding
definitions. External sources (sometimes the satirical remarks of
comedians) also reflect this Academic concern with definitions.
Aristotle himself traces the quest for definitions back to
Socrates.


### 7.1 Definitions and Essences



For Aristotle, a definition is "an account which signifies what
it is to be for something" (*logos ho to ti en einai
semainei*). The phrase "what it is to be" and
its variants are crucial: giving a definition is saying, of some
existent thing, what it is, not simply specifying the meaning of a
word (Aristotle does recognize definitions of the latter sort, but he
has little interest in them).



The notion of "what it is to be" for a thing is so
pervasive in Aristotle that it becomes formulaic: what a definition
expresses is "the what-it-is-to-be" (*to ti en
einai*), or in modern terminology, its essence.


### 7.2 Species, Genus, and Differentia



Since a definition defines an essence, only what has an essence can be
defined. What has an essence, then? That is one of the central
questions of Aristotle's metaphysics; once again, we must leave
the details to another article. In general, however, it is not
individuals but rather **species** (*eidos*: the
word is one of those Plato uses for "Form") that have
essences. A species is defined by giving its **genus**
(*genos*) and its **differentia**
(*diaphora*): the genus is the kind under which the species
falls, and the differentia tells what characterizes the species within
that genus. As an example, *human* might be defined as
*animal* (the genus) *having the capacity to reason*
(the differentia).


#### Essential Predication and the Predicables



Underlying Aristotle's concept of a definition is the concept of
**essential predication** (*kategoreisthai en
toi ti esti*, predication in the what it is). In any true
affirmative predication, the predicate either does or does not
"say what the subject is", i.e., the predicate either is
or is not an acceptable answer to the question "What is
it?" asked of the subject. Bucephalus is a horse, and a horse is
an animal; so, "Bucephalus is a horse" and
"Bucephalus is an animal" are essential predications.
However, "Bucephalus is brown", though true, does not
state what Bucephalus is but only says something about him.



Since a thing's definition says what it is, definitions are
essentially predicated. However, not everything essentially predicated
is a definition. Since Bucephalus is a horse, and horses are a kind of
mammal, and mammals are a kind of animal, "horse"
"mammal" and "animal" are all essential
predicates of Bucephalus. Moreover, since what a horse is is a kind of
mammal, "mammal" is an essential predicate of horse. When
predicate \(X\) is an essential predicate of \(Y\) but also of other
things, then \(X\) is a **genus** (*genos*) of
\(Y\).



A definition of \(X\) must not only be essentially predicated of it
but must also be predicated only of it: to use a term from
Aristotle's *Topics*, a definition and what it defines
must "counterpredicate"
(*antikategoreisthai*) with one another. \(X\)
counterpredicates with \(Y\) if \(X\) applies to what \(Y\) applies to
and conversely. Though X's definition must counterpredicate with
\(X\), not everything that counterpredicates with \(X\) is its
definition. "Capable of laughing", for example,
counterpredicates with "human" but fails to be its
definition. Such a predicate (non-essential but counterpredicating) is
a **peculiar property** or **proprium**
(*idion*).



Finally, if \(X\) is predicated of \(Y\) but is neither essential nor
counterpredicates, then \(X\) is an **accident**
(*sumbebekos*) of \(Y\).



Aristotle sometimes treats genus, peculiar property, definition, and
accident as including all possible predications (e.g. *Topics*
I). Later commentators listed these four and the differentia as the
five **predicables**, and as such they were of great
importance to late ancient and to medieval philosophy (e.g.,
Porphyry).


### 7.3 The Categories



The notion of essential predication is connected to what are
traditionally called the **categories**
(*kategoriai*). In a word, Aristotle is famous for having
held a "doctrine of categories". Just what that doctrine
was, and indeed just what a category is, are considerably more vexing
questions. They also quickly take us outside his logic and into his
metaphysics. Here, I will try to give a very general overview,
beginning with the somewhat simpler question "What categories
are there?"



We can answer this question by listing the categories. Here are two
passages containing such lists:



> 
> We should distinguish the kinds of predication (*ta gene
> ton kategorion*) in which the four predications
> mentioned are found. These are ten in number: what-it-is, quantity,
> quality, relative, where, when, being-in-a-position, having, doing,
> undergoing. An accident, a genus, a peculiar property and a definition
> will always be in one of these categories. (*Topics* I.9,
> 103b20-25)
> 
> 
> Of things said without any combination, each signifies either
> substance or quantity or quality or a relative or where or when or
> being-in-a-position or having or doing or undergoing. To give a rough
> idea, examples of substance are man, horse; of quantity: four-foot,
> five-foot; of quality: white, literate; of a relative: double, half,
> larger; of where: in the Lyceum, in the market-place; of when:
> yesterday, last year; of being-in-a-position: is-lying, is-sitting; of
> having: has-shoes-on, has-armor-on; of doing: cutting, burning; of
> undergoing: being-cut, being-burned. (*Categories* 4,
> 1b25-2a4, tr. Ackrill, slightly modified)
> 
> 
> 



These two passages give ten-item lists, identical except for their
first members. What are they lists \(of\)? Here are three ways they
might be interpreted:



The word "category" (*kategoria*) means
"predication". Aristotle holds that predications and
predicates can be grouped into several largest "kinds of
predication" (*gene ton
kategorion*). He refers to this classification
frequently, often calling the "kinds of predication"
simply "the predications", and this (by way of Latin)
leads to our word "category".


* First, the categories may be *kinds of predicate*:
predicates (or, more precisely, predicate expressions) can be divided
into ten separate classes, with each expression belonging to just one
class. This comports well with the root meaning of the word
*kategoria* ("predication"). On this
interpretation, the categories arise out of considering the most
general types of question that can be asked about something:
"*What* is it?"; "*How much* is
it?"; "*What sort* is it?";
"*Where* is it?"; "*What* is it
*doing*?" Answers appropriate to one of these questions
are nonsensical in response to another ("When is it?"
"A horse"). Thus, the categories may rule out certain
kinds of question as ill-formed or confused. This plays an important
role in Aristotle's metaphysics.
* Second, the categories may be seen as classifications of
*predications*, that is, kinds of relation that may hold
between the predicate and the subject of a predication. To say of
Socrates that he is human is to say what he \(is\), whereas to say
that he is literate is not to say what he is but rather to give a
quality that he *has*. For Aristotle, the relation of predicate
to subject in these two sentences is quite different (in this respect
he differs both from Plato and from modern logicians). The categories
may be interpreted as ten different ways in which a predicate may be
related to its subject. This last division has importance for
Aristotle's logic as well as his metaphysics.
* Third, the categories may be seen as *kinds of entity*, as
highest genera or kinds of thing that are. A given thing can be
classified under a series of progressively wider genera: Socrates is a
human, a mammal, an animal, a living being. The categories are the
highest such genera. Each falls under no other genus, and each is
completely separate from the others. This distinction is of critical
importance to Aristotle's metaphysics.



Which of these interpretations fits best with the two passages above?
The answer appears to be different in the two cases. This is most
evident if we take note of point in which they differ: the
*Categories* lists **substance** (*ousia*)
in first place, while the *Topics* list
**what-it-is** (*ti esti*). A substance, for
Aristotle, is a type of entity, suggesting that the
*Categories* list is a list of types of entity.



On the other hand, the expression "what-it-is" suggests
most strongly a type of predication. Indeed, the *Topics*
confirms this by telling us that we can "say what it is"
of an *entity* falling under any of the categories:



> 
> an expression signifying what-it-is will sometimes signify a
> substance, sometimes a quantity, sometimes a quality, and sometimes
> one of the other categories.
> 



As Aristotle explains, if I say that Socrates is a man, then I have
said what Socrates is and signified a substance; if I say that white
is a color, then I have said what white is and signified a quality; if
I say that some length is a foot long, then I have said what it is and
signified a quantity; and so on for the other categories. What-it-is,
then, here designates a kind of predication, not a kind of entity.



This might lead us to conclude that the categories in the
*Topics* are only to be interpreted as kinds of predicate or
predication, those in the *Categories* as kinds of being. Even
so, we would still want to ask what the relationship is between these
two nearly-identical lists of terms, given these distinct
interpretations. However, the situation is much more complicated.
First, there are dozens of other passages in which the categories
appear. Nowhere else do we find a list of ten, but we do find shorter
lists containing eight, or six, or five, or four of them (with
substance/what-it-is, quality, quantity, and relative the most
common). Aristotle describes what these lists are lists of in
different ways: they tell us "how being is divided", or
"how many ways being is said", or "the figures of
predication" (ta schemata tes kategorias). The
designation of the first category also varies: we find not only
"substance" and "what it is" but also the
expressions "this" or "the this" (*tode
ti*, *to tode*, *to ti*). These latter expressions
are closely associated with, but not synonymous with, substance. He
even combines the latter with "what-it-is"
(*Metaphysics* Z 1, 1028a10: "... one sense
signifies what it is and the this, one signifies quality
...").



Moreover, substances are for Aristotle fundamental for predication as
well as metaphysically fundamental. He tells us that everything that
exists exists because substances exist: if there were no substances,
there would not be anything else. He also conceives of predication as
reflecting a metaphysical relationship (or perhaps more than one,
depending on the type of predication). The sentence "Socrates is
pale" gets its truth from a state of affairs consisting of a
substance (Socrates) and a quality (whiteness) which is in that
substance. At this point we have gone far outside the realm of
Aristotle's logic into his metaphysics, the fundamental question
of which, according to Aristotle, is "What is a
substance?". (For further discussion of this topic, see the
entry on
 Aristotle's Categories
 and the entry on
 Aristotle's metaphysics,
 (Section 2).



See Frede 1981, Ebert 1985 for additional discussion of
Aristotle's lists of categories.



For convenience of reference, I include a table of the categories,
along with Aristotle's examples and the traditional names often
used for them. For reasons explained above, I have treated the first
item in the list quite differently, since an example of a substance
and an example of a what-it-is are necessarily (as one might put it)
in different categories.




|  |  |  |  |
| --- | --- | --- | --- |
| **Traditional name** | **Literally** | **Greek** | **Examples** |
| Substance | substance

"this"

what-it-is | *ousia*

*tode ti*

*ti esti* | man, horse

Socrates

"Socrates is a man" |
| Quantity | How much | *poson* | four-foot, five-foot |
| Quality | What sort | *poion* | white, literate |
| Relation | related to what | *pros ti* | double, half, greater |
| Location | Where | *pou* | in the Lyceum, in the marketplace |
| Time | when | *pote* | yesterday, last year |
| Position | being situated | *keisthai* | lies, sits |
| Habit | having, possession | *echein* | is shod, is armed |
| Action | doing | *poiein* | cuts, burns |
| Passion | undergoing | *paschein* | is cut, is burned |


### 7.4 The Method of Division



In the *Sophist*, Plato introduces a procedure of
"Division" as a method for discovering definitions. To
find a definition of \(X\), first locate the largest kind of thing
under which \(X\) falls; then, divide that kind into two parts, and
decide which of the two \(X\) falls into. Repeat this method with the
part until \(X\) has been fully located.



This method is part of Aristotle's Platonic legacy. His attitude
towards it, however, is complex. He adopts a view of the proper
structure of definitions that is closely allied to it: a correct
definition of \(X\) should give the **genus**
(*genos*: kind or family) of \(X\), which tells what kind of
thing \(X\) is, and the **differentia**
(*diaphora*: difference) which uniquely identifies \(X\) within
that genus. Something defined in this way is a
**species** (*eidos*: the term is one of
Plato's terms for "Form"), and the differentia is
thus the "difference that makes a species" (*eidopoios
diaphora*, "specific difference"). In *Posterior
Analytics* II.13, he gives his own account of the use of Division
in finding definitions.



However, Aristotle is strongly critical of the Platonic view of
Division as a method for *establishing* definitions. In
*Prior Analytics* I.31, he contrasts Division with the
syllogistic method he has just presented, arguing that Division cannot
actually prove anything but rather assumes the very thing it is
supposed to be proving. He also charges that the partisans of Division
failed to understand what their own method was capable of proving.


### 7.5 Definition and Demonstration



Closely related to this is the discussion, in *Posterior
Analytics* II.3-10, of the question whether there can be
both definition and demonstration of the same thing (that is, whether
the same result can be established either by definition or by
demonstration). Since the definitions Aristotle is interested in are
statements of essences, knowing a definition is knowing, of some
existing thing, what it is. Consequently, Aristotle's question
amounts to a question whether defining and demonstrating can be
alternative ways of acquiring the same knowledge. His reply is
complex:


1. Not everything demonstrable can be known by finding definitions,
since all definitions are universal and affirmative whereas some
demonstrable propositions are negative.
2. If a thing is demonstrable, then to know it just is to possess its
demonstration; therefore, it cannot be known just by definition.
3. Nevertheless, some definitions can be understood as demonstrations
differently arranged.



As an example of case 3, Aristotle considers the definition
"Thunder is the extinction of fire in the clouds". He sees
this as a compressed and rearranged form of this demonstration:


* Sound accompanies the extinguishing of fire.
* Fire is extinguished in the clouds.
* Therefore, a sound occurs in the clouds.



We can see the connection by considering the answers to two questions:
"What is thunder?" "The extinction of fire in the
clouds" (definition). "Why does it thunder?"
"Because fire is extinguished in the clouds"
(demonstration).



As with his criticisms of Division, Aristotle is arguing for the
superiority of his own concept of science to the Platonic concept.
Knowledge is composed of demonstrations, even if it may also include
definitions; the method of science is demonstrative, even if it may
also include the process of defining.


## 8. Dialectical Argument and the Art of Dialectic



Aristotle often contrasts *dialectical arguments* with
demonstrations. The difference, he tells us, is in the character of
their premises, not in their logical structure: whether an argument is
a *sullogismos* is only a matter of whether its conclusion
results of necessity from its premises. The premises of demonstrations
must be *true and primary*, that is, not only true but also
prior to their conclusions in the way explained in the *Posterior
Analytics*. The premises of dialectical deductions, by contrast,
must be **accepted** (*endoxos*).


### 8.1 Dialectical Premises: The Meaning of *Endoxos*



Recent scholars have proposed different interpretations of the term
*endoxos*. Aristotle often uses this adjective as a
substantive: *ta endoxa*, "accepted things",
"accepted opinions". On one understanding, descended from
the work of G. E. L. Owen and developed more fully by Jonathan Barnes
and especially Terence Irwin, the *endoxa* are a compilation of
views held by various people with some form or other of standing:
"the views of fairly reflective people after some
reflection", in Irwin's phrase. Dialectic is then simply
"a method of argument from [the] common beliefs [held by these
people]". For Irwin, then, *endoxa* are "common
beliefs". Jonathan Barnes, noting that *endoxa* are
opinions with a certain standing, translates with
"reputable".



My own view is that Aristotle's texts support a somewhat
different understanding. He also tells us that dialectical premises
differ from demonstrative ones in that the former are
*questions*, whereas the latter are *assumptions* or
*assertions*: "the demonstrator does not ask, but
takes", he says. This fits most naturally with a view of
dialectic as argument directed at another person by question and
answer and consequently taking as premises that other person's
concessions. Anyone arguing in this manner will, in order to be
successful, have to ask for premises which the interlocutor is liable
to accept, and the best way to be successful at that is to have an
inventory of acceptable premises, i.e., premises that are in fact
acceptable to people of different types.



In fact, we can discern in the *Topics* (and the
*Rhetoric*, which Aristotle says depends on the art explained
in the *Topics*) an art of dialectic for use in such arguments.
My reconstruction of this art (which would not be accepted by all
scholars) is as follows.


### 8.2 The Two Elements of the Art of Dialectic



Given the above picture of dialectical argument, the dialectical art
will consist of two elements. One will be a method for discovering
premises from which a given conclusion follows, while the other will
be a method for determining which premises a given interlocutor will
be likely to concede. The first task is accomplished by developing a
system for classifying premises according to their logical structure.
We might expect Aristotle to avail himself here of the syllogistic,
but in fact he develops quite another approach, one that seems less
systematic and rests on various "common" terms. The second
task is accomplished by developing lists of the premises which are
acceptable to various types of interlocutor. Then, once one knows what
sort of person one is dealing with, one can choose premises
accordingly. Aristotle stresses that, as in all arts, the dialectician
must study, not what is acceptable to this or that specific person,
but what is acceptable to this or that type of person, just as the
doctor studies what is healthful for different types of person:
"art is of the universal".


#### 8.2.1 The "Logical System" of the *Topics*



The method presented in the *Topics* for classifying arguments
relies on the presence in the conclusion of certain
"common" terms (*koina*) -- common in the
sense that they are not peculiar to any subject matter but may play a
role in arguments about anything whatever. We find enumerations of
arguments involving these terms in a similar order several times.
Typically, they include:


1. Opposites (*antikeimena*, *antitheseis*)
	1. Contraries (*enantia*)
	2. Contradictories (*apophaseis*)
	3. Possession and Privation (*hexis kai steresis*)
	4. Relatives (*pros ti*)
2. Cases (*ptoseis*)
3. "More and Less and Likewise"



The four types of **opposites** are the best represented.
Each designates a type of term pair, i.e., a way two terms can be
opposed to one another. **Contraries** are polar
opposites or opposed extremes such as hot and cold, dry and wet, good
and bad. A pair of **contradictories** consists of a term
and its negation: good, not good. A **possession** (or
condition) and **privation** are illustrated by sight and
blindness. **Relatives** are relative terms in the modern
sense: a pair consists of a term and its correlative, e.g. large and
small, parent and child.



The argumentative patterns Aristotle associated with
**cases** generally involve inferring a sentence
containing adverbial or declined forms from another sentence
containing different forms of the same word stem: "if what is
useful is good, then what is done usefully is done well and the useful
person is good". In Hellenistic grammatical usage,
*ptosis* meant "case" (e.g. nominative,
dative, accusative); Aristotle's use here is obviously an early
form of that.



Under the heading **more and less and likewise**,
Aristotle groups a somewhat motley assortment of argument patterns all
involving, in some way or other, the terms "more",
"less", and "likewise". Examples: "If
whatever is \(A\) is \(B\), then whatever is more (less) \(A\) is more
(less) \(B\)"; "If \(A\) is more likely \(B\) than \(C\)
is, and \(A\) is not \(B\), then neither is \(C\)"; "If
\(A\) is more likely than \(B\) and \(B\) is the case, then \(A\) is
the case".


#### 8.2.2 The *Topoi*



At the heart of the *Topics* is a collection of what Aristotle
calls *topoi*, "places" or "locations".
Unfortunately, though it is clear that he intends most of the
*Topics* (Books II-VI) as a collection of these, he never
explicitly defines this term. Interpreters have consequently disagreed
considerably about just what a *topos* is. Discussions may be
found in Brunschwig 1967, Slomkowski 1996, Primavesi 1997, and Smith
1997.


### 8.3 The Uses of Dialectic and Dialectical Argument



An *art* of dialectic will be useful wherever dialectical
argument is useful. Aristotle mentions three such uses; each merits
some comment.


#### 8.3.1 Gymnastic Dialectic



First, there appears to have been a form of stylized argumentative
exchange practiced in the Academy in Aristotle's time. The main
evidence for this is simply Aristotle's *Topics*,
especially Book VIII, which makes frequent reference to rule-governed
procedures, apparently taking it for granted that the audience will
understand them. In these exchanges, one participant took the role of
answerer, the other the role of questioner. The answerer began by
asserting some proposition (a *thesis*: "position"
or "acceptance"). The questioner then asked questions of
the answerer in an attempt to secure concessions from which a
contradiction could be deduced: that is, to **refute**
(*elenchein*) the answerer's position. The questioner was
limited to questions that could be answered by yes or no; generally,
the answerer could only respond with yes or no, though in some cases
answerers could object to the form of a question. Answerers might
undertake to answer in accordance with the views of a particular type
of person or a particular person (e.g. a famous philosopher), or they
might answer according to their own beliefs. There appear to have been
judges or scorekeepers for the process. Gymnastic dialectical contests
were sometimes, as the name suggests, for the sake of exercise in
developing argumentative skill, but they may also have been pursued as
a part of a process of inquiry.


#### 8.3.2 Dialectic That Puts to the Test



Aristotle also mentions an "art of making trial", or a
variety of dialectical argument that "puts to the test"
(the Greek word is the adjective *peirastike*, in the
feminine: such expressions often designate arts or skills, e.g.
*rhetorike*, "the art of rhetoric").
Its function is to examine the claims of those who say they have some
knowledge, and it can be practiced by someone who does not possess the
knowledge in question. The examination is a matter of refutation,
based on the principle that whoever knows a subject must have
consistent beliefs about it: so, if you can show me that my beliefs
about something lead to a contradiction, then you have shown that I do
not have knowledge about it.



This is strongly reminiscent of Socrates' style of
interrogation, from which it is almost certainly descended. In fact,
Aristotle often indicates that dialectical argument is by nature
refutative.


#### 8.3.3 Dialectic and Philosophy



Dialectical refutation cannot of itself establish any proposition
(except perhaps the proposition that some set of propositions is
inconsistent). More to the point, though deducing a contradiction from
my beliefs may show that they do not constitute knowledge, failure to
deduce a contradiction from them is no proof that they are true. Not
surprisingly, then, Aristotle often insists that "dialectic does
not prove anything" and that the dialectical art is not some
sort of universal knowledge.



In *Topics* I.2, however, Aristotle says that the art of
dialectic is useful in connection with "the philosophical
sciences". One reason he gives for this follows closely on the
refutative function: if we have subjected our opinions (and the
opinions of our fellows, and of the wise) to a thorough refutative
examination, we will be in a much better position to judge what is
most likely true and false. In fact, we find just such a procedure at
the start of many of Aristotle's treatises: an enumeration of
the opinions current about the subject together with a compilation of
"puzzles" raised by these opinions. Aristotle has a
special term for this kind of review: a *diaporia*, a
"puzzling through".



He adds a second use that is both more difficult to understand and
more intriguing. The *Posterior Analytics* argues that if
anything can be proved, then not everything that is known is known as
a result of proof. What alternative means is there whereby the first
principles of sciences are known? Aristotle's own answer as
found in *Posterior Analytics* II.19 is difficult to interpret,
and recent philosophers have often found it unsatisfying since (as
often construed) it appears to commit Aristotle to a form of apriorism
or rationalism both indefensible in itself and not consonant with his
own insistence on the indispensability of empirical inquiry in natural
science.



Against this background, the following passage in *Topics* I.2
may have special importance:



> 
> It is also useful in connection with the first things concerning each
> of the sciences. For it is impossible to say anything about the
> science under consideration on the basis of its own principles, since
> the principles are first of all, and we must work our way through
> about these by means of what is generally accepted about each. But
> this is peculiar, or most proper, to dialectic: for since it is
> examinative with respect to the principles of all the sciences, it has
> a way to proceed.
> 



A number of interpreters (beginning with Owen 1961) have built on this
passage and others to find dialectic at the heart of Aristotle's
philosophical method. Further discussion of this issue would take us
far beyond the subject of this article (the fullest development is in
Irwin 1988; see also Nussbaum 1986 and Bolton 1990; for criticism,
Hamlyn 1990, Smith 1997).


## 9. Dialectic and Rhetoric



Aristotle says that rhetoric, i.e., the study of persuasive speech, is
a "counterpart" (*antistrophos*) of dialectic and
that the rhetorical art is a kind of "outgrowth"
(*paraphues ti*) of dialectic and the study of character types.
The correspondence with dialectical method is straightforward:
rhetorical speeches, like dialectical arguments, seek to persuade
others to accept certain conclusions on the basis of premises they
already accept. Therefore, the same measures useful in dialectical
contexts will, mutatis mutandis, be useful here: knowing what premises
an audience of a given type is likely to believe, and knowing how to
find premises from which the desired conclusion follows.



The *Rhetoric* does fit this general description: Aristotle
includes both discussions of types of person or audience (with
generalizations about what each type tends to believe) and a summary
version (in II.23) of the argument patterns discussed in the
*Topics*. For further discussion of his rhetoric see
 Aristotle's rhetoric.


## 10. Sophistical Arguments



Demonstrations and dialectical arguments are both forms of valid
argument, for Aristotle. However, he also studies what he calls
**contentious** (*eristikos*) or
**sophistical** arguments: these he defines as arguments
which only apparently establish their conclusions. In fact, Aristotle
defines these as apparent (but not genuine) *dialectical*
*sullogismoi*. They may have this appearance in either of two
ways:


1. Arguments in which the conclusion only appears to follow of
necessity from the premises (apparent, but not genuine,
*sullogismoi*).
2. Genuine *sullogismois* the premises of which are merely
apparently, but not genuinely, acceptable.



Arguments of the first type in modern terms, appear to be valid but
are really invalid. Arguments of the second type are at first more
perplexing: given that acceptability is a matter of what people
believe, it might seem that whatever appears to be *endoxos*
must actually be *endoxos*. However, Aristotle probably has in
mind arguments with premises that may at first glance *seem* to
be acceptable but which, upon a moment's reflection, we
immediately realize we do not actually accept. Consider this example
from Aristotle's time:


* Whatever you have not lost, you still have.
* You have not lost horns.
* Therefore, you still have horns



This is transparently bad, but the problem is not that it is invalid:
the problem is rather that the first premise, though superficially
plausible, is false. In fact, anyone with a little ability to follow
an argument will realize that at once upon seeing this very
argument.



Aristotle's study of sophistical arguments is contained in
*On Sophistical Refutations*, which is actually a sort of
appendix to the *Topics*.



To a remarkable extent, contemporary discussions of fallacies
reproduce Aristotle's own classifications. See Dorion 1995 for
further discussion.


## 11. Non-Contradiction and Metaphysics



Two frequent themes of Aristotle's account of science are (1)
that the first principles of sciences are not demonstrable and (2)
that there is no single universal science including all other sciences
as its parts. "All things are not in a single genus", he
says, "and even if they were, all beings could not fall under
the same principles" (*On Sophistical Refutations* 11).
Thus, it is exactly the universal applicability of dialectic that
leads him to deny it the status of a science.



In *Metaphysics* IV (G), however, Aristotle takes what
appears to be a different view. First, he argues that there is, in a
way, a science that takes being as its genus (his name for it is
"first philosophy"). Second, he argues that the principles
of this science will be, in a way, the first principles of all (though
he does not claim that the principles of other sciences can be
demonstrated from them). Third, he identifies one of its first
principles as the "most secure" of all principles: the
principle of non-contradiction. As he states it,



> 
> It is impossible for the same thing to belong and not belong
> simultaneously to the same thing in the same respect (*Met.* )
> 



This is the most secure of all principles, Aristotle tells us, because
"it is impossible to be in error about it". Since it is a
first principle, it cannot be demonstrated; those who think otherwise
are "uneducated in analytics". However, Aristotle then
proceeds to give what he calls a "refutative
demonstration" (*apodeixai elenktikos*) of this
principle.



Further discussion of this principle and Aristotle's arguments
concerning it belong to a treatment of his metaphysics (see
 Aristotle: Metaphysics).
 However, it should be noted that: (1) these arguments draw on
Aristotle's views about logic to a greater extent than any
treatise outside the logical works themselves; (2) in the logical
works, the principle of non-contradiction is one of Aristotle's
favorite illustrations of the "common principles"
(*koinai archai*) that underlie the art of dialectic.



See
 Aristotle's Metaphysics,
 Aristotle on non-contradiction, Dancy 1975, and Code 1986 for further discussion.


## 12. Time and Necessity: The Sea-Battle



The passage in Aristotle's logical works which has received
perhaps the most intense discussion in recent decades is *On
Interpretation* 9, where Aristotle discusses the question whether
every proposition about the future must be either true or false.
Though something of a side issue in its context, the passage raises a
problem of great importance to Aristotle's near contemporaries
(and perhaps contemporaries).



A **contradiction** (*antiphasis*) is a pair of
propositions one of which asserts what the other denies. A major goal
of *On Interpretation* is to discuss the thesis that, of every
such contradiction, one member must be true and the other false. In
the course of his discussion, Aristotle allows for some exceptions.
One case is what he calls **indefinite** propositions
such as "A man is walking": nothing prevents both this
proposition and "A man is not walking" being
simultaneously true. This exception can be explained on relatively
simple grounds.



A different exception arises for more complex reasons. Consider these
two propositions:


1. There will be a sea-battle tomorrow
2. There will not be a sea-battle tomorrow



It seems that exactly one of these must be true and the other false.
But if (1) is *now* true, then there *must* be a
sea-battle tomorrow, and there *cannot* fail to be a sea-battle
tomorrow. The result, according to this puzzle, is that nothing is
possible except what actually happens: there are no unactualized
possibilities.



Such a conclusion is, as Aristotle is quick to note, a problem both
for his own metaphysical views about potentialities and for the
commonsense notion that some things are up to us. He therefore
proposes another exception to the general thesis concerning
contradictory pairs.



This much would probably be accepted by most interpreters. What the
restriction is, however, and just what motivates it are matters of
wide disagreement. It has been proposed, for instance, that Aristotle
adopted, or at least flirted with, a three-valued logic for future
propositions, or that he countenanced truth-value gaps, or that his
solution includes still more abstruse reasoning. The literature is
much too complex to summarize: see Anscombe, Hintikka, D. Frede,
Whitaker, Waterlow.



Historically, at least, it is likely that Aristotle is responding to
an argument originating with the Megarian philosophers. He ascribes
the view that only that which happens is possible to the Megarians in
*Metaphysics* IX (Th). The puzzle with which he is
concerned strongly recalls the "Master Argument" of
 Diodorus Cronus
 especially in certain further details. For instance, Aristotle
imagines the statement about tomorrow's sea battle having been
uttered ten thousand years ago. If it was true, then its truth was a
fact about the past; if the past is now unchangeable, then so is the
truth value of that past utterance. This recalls the Master
Argument's premise that "what is past is necessary".
Diodorus Cronus was active a little after Aristotle, and he was
certainly influenced by Megarian views, whether or not it is correct
to call him a Megarian (David Sedley 1977 argues that he was instead a
member of the
 Dialectical School
 which was in any event an offshoot of the Megarians; see Dorion 1995
and Doring 1989, Ebert 2008 and the article Dialectical School).
It is therefore likely that Aristotle's target here is some
Megarian argument, perhaps a forerunner of Diodorus' Master
Argument.


## 13. Glossary of Aristotelian Terminology


* Accept: *tithenai* (in a dialectical argument)
* Accepted: *endoxos* (also 'reputable'
'common belief')
* Accident: *sumbebekos* (see *incidental*)
* Accidental: *kata sumbebekos*
* Affirmation: *kataphasis*
* Affirmative: *kataphatikos*
* Assertion: *apophansis* (sentence with a truth value,
declarative sentence)
* Assumption: *hupothesis*
* Belong: *huparchein*
* Category: *kategoria* (see the discussion in Section
7.3).
* Contradict: *antiphanai*
* Contradiction: *antiphasis* (in the sense
"contradictory pair of propositions" and also in the sense
"denial of a proposition")
* Contrary: *enantion*
* Deduction: *sullogismos*
* Definition: *horos*, *horismos*
* Demonstration: *apodeixis*
* Denial (of a proposition): *apophasis*
* Dialectic: *dialektike* (the *art* of
dialectic)
* Differentia: *diaphora*; specific difference, *eidopoios
diaphora*
* Direct: *deiktikos* (of proofs; opposed to "through
the impossible")
* Essence: *to ti esti*, *to ti en einai*
* Essential: *en toi ti esti* (of predications)
* Extreme: *akron* (of the major and minor terms of a
deduction)
* Figure: *schema*
* Form: *eidos* (see also Species)
* Genus: *genos*
* Immediate: *amesos* ("without a middle")
* Impossible: *adunaton*; "through the
impossible" (*dia tou adunatou*), of some proofs.
* Incidental: see Accidental
* Induction: *epagoge*
* Middle, middle term (of a deduction): *meson*
* Negation (of a term): *apophasis*
* Objection: *enstasis*
* Particular: *en merei*, *epi meros* (of a
proposition); *kath'hekaston* (of individuals)
* Peculiar, Peculiar Property: *idios*, *idion*
* Possible: *dunaton*, *endechomenon*;
*endechesthai* (verb: "be possible")
* Predicate: *kategorein* (verb);
*kateegoroumenon* ("what is predicated")
* Predication: *kategoria* (act or instance of
predicating, type of predication)
* Primary: *proton*
* Principle: *arche* (starting point of a
demonstration)
* Quality: *poion*
* Reduce, Reduction: *anagein*,
*anagoge*
* Refute: *elenchein*; refutation, *elenchos*
* Science: *episteme*
* Species: *eidos*
* Specific: *eidopoios* (of a differentia that "makes a
species", *eidopoios diaphora*)
* Subject: *hupokeimenon*
* Substance: *ousia*
* Term: *horos*
* Universal: *katholou* (both of propositions and of
individuals)