Aristotle’s logic, especially his theory of the syllogism, hashad an unparalleled influence on the history of Western thought. Itdid not always hold this position: in the Hellenistic period, Stoiclogic, and in particular the work of Chrysippus, took pride of place.However, in later antiquity, following the work of AristotelianCommentators, Aristotle’s logic became dominant, andAristotelian logic was what was transmitted to the Arabic and theLatin medieval traditions, while the works of Chrysippus have notsurvived.

This unique historical position has not always contributed to theunderstanding of Aristotle’s logical works. Kant thought thatAristotle had discovered everything there was to know about logic, andthe historian of logic Prantl drew the corollary that any logicianafter Aristotle who said anything new was confused, stupid, orperverse. During the rise of modern formal logic following Frege andPeirce, adherents of Traditional Logic (seen as the descendant ofAristotelian Logic) and the new mathematical logic tended to see oneanother as rivals, with incompatible notions of logic. More recentscholarship has often applied the very techniques of mathematicallogic to Aristotle’s theories, revealing (in the opinion ofmany) a number of similarities of approach and interest betweenAristotle and modern logicians.

This article is written from the latter perspective. As such, it isabout Aristotle’s logic, which is not always the same thing aswhat has been called “Aristotelian” logic.

• 1. Introduction
• 2. Aristotle’s Logical Works: TheOrganon
• 3. The Subject of Logic: “Syllogisms”
  • 3.1 Induction and Deduction
  • 3.2 Aristotelian Deductions and Modern Valid Arguments
• 4. Premises: The Structures of Assertions
  • 4.1 Terms
  • 4.2 Affirmations, Denials, and Contradictions
  • 4.3 All, Some, and None
• 5. The Syllogistic
  • 5.1 The Figures
  • 5.2 Methods of Proof: “Perfect” Deductions, Conversion, Reduction
  • 5.3 Disproof: Counterexamples and Terms
  • 5.4 The Deductions in the Figures (“Moods”)
  • 5.5 Metatheoretical Results
  • 5.6 Syllogisms with Modalities
• 6. Demonstrations and Demonstrative Sciences
  • 6.1 Aristotelian Sciences
  • 6.2 The Regress Problem
  • 6.3 Aristotle’s Solution: “It Eventually Comes to a Stop”
  • 6.4 Knowledge of First Principles:Nous
• 7. Definitions
  • 7.1 Definitions and Essences
  • 7.2 Species, Genus, and Differentia
  • 7.3 The Categories
  • 7.4 The Method of Division
  • 7.5 Definition and Demonstration
• 8. Dialectical Argument and the Art of Dialectic
  • 8.1 Dialectical Premises: The Meaning ofEndoxos
  • 8.2 The Two Elements of the Art of Dialectic
  • 8.3 The Uses of Dialectic and Dialectical Argument
• 9. Dialectic and Rhetoric
• 10. Sophistical Arguments
• 11. Non-Contradiction and Metaphysics
• 12. Time and Necessity: The Sea-Battle
• 13. Glossary of Aristotelian Terminology
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

Aristotle’s logical works contain the earliest formal study oflogic that we have. It is therefore all the more remarkable thattogether they comprise a highly developed logical theory, one that wasable to command immense respect for many centuries: Kant, who was tentimes more distant from Aristotle than we are from him, even held thatnothing significant had been added to Aristotle’s views in theintervening two millennia.

In the last century, Aristotle’s reputation as a logician hasundergone two remarkable reversals. The rise of modern formal logicfollowing the work of Frege and Russell brought with it a recognitionof the many serious limitations of Aristotle’s logic; today,very few would try to maintain that it is adequate as a basis forunderstanding science, mathematics, or even everyday reasoning. At thesame time, scholars trained in modern formal techniques have come toview Aristotle with new respect, not so much for the correctness ofhis results as for the remarkable similarity in spirit between much ofhis work and modern logic. As Jonathan Lear has put it,“Aristotle shares with modern logicians a fundamental interestin metatheory”: his primary goal is not to offer a practicalguide to argumentation but to study the properties of inferentialsystems themselves.

2. Aristotle’s Logical Works: TheOrganon

The ancient commentators grouped together several of Aristotle’streatises under the titleOrganon (“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 titleOrganon reflects a much later controversyabout whether logic is a part of philosophy (as the Stoics maintained)or merely a tool used by philosophy (as the later Peripateticsthought); calling the logical works “The Instrument” is away of taking sides on this point. Aristotle himself never uses thisterm, nor does he give much indication that these particular treatisesform some kind of group, though there are frequent cross-referencesbetween theTopics and theAnalytics. On the otherhand, Aristotle treats thePrior andPosteriorAnalytics as one work, andOn Sophistical Refutations isa final section, or an appendix, to theTopics). To theseworks should be added theRhetoric, which explicitly declaresits reliance on theTopics.

3. The Subject of Logic: “Syllogisms”

All Aristotle’s logic revolves around one notion: thededuction (sullogismos). A thoroughexplanation 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 havingbeen supposed, something different from those supposed results ofnecessity because of their being so. (Prior Analytics I.2,24b18–20)

Each of the “things supposed” is apremise (protasis) of the argument, and what“results of necessity” is theconclusion(sumperasma).

The core of this definition is the notion of “resulting ofnecessity” (ex anankês sumbainein). Thiscorresponds to a modern notion of logical consequence: \(X\) resultsof necessity from \(Y\) and \(Z\) if it would be impossible for \(X\)to be false when \(Y\) and \(Z\) are true. We could therefore takethis 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 isinduction(epagôgê). He has far less to say about this thandeduction, doing little more than characterize it as “argumentfrom the particular to the universal”. However, induction (orsomething very much like it) plays a crucial role in the theory ofscientific knowledge in thePosterior Analytics: it isinduction, or at any rate a cognitive process that moves fromparticulars to their generalizations, that is the basis of knowledgeof the indemonstrable first principles of sciences.

3.2 Aristotelian Deductions and Modern Valid Arguments

Despite its wide generality, Aristotle’s definition of deductionis not a precise match for a modern definition of validity. Some ofthe differences may have important consequences:

1. Aristotle explicitly says that what results of necessity must bedifferent from what is supposed. This would rule out arguments inwhich the conclusion is identical to one of the premises. Modernnotions of validity regard such arguments as valid, though triviallyso.
2. The plural “certain things having been supposed” wastaken by some ancient commentators to rule out by definition argumentswith only one premise, and Aristotle himself says in some places thatnothing new follows from just one premise.
3. The force of the qualification “because of their beingso” has sometimes been seen as ruling out arguments in which theconclusion is not ‘relevant’ to the premises, e.g.,arguments in which the premises are inconsistent, arguments withconclusions that would follow from any premises whatsoever, orarguments with superfluous premises.

Of these three possible restrictions, the most interesting would bethe third. This could be (and has been) interpreted as committingAristotle to something like arelevance 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 doesnot mean to restrict it only to a subset of the valid arguments. Thisis why I have translatedsullogismos with‘deduction’ rather than its English cognate. In modernusage, ‘syllogism’ means an argument of a very specificform. Moreover, modern usage distinguishes between valid syllogisms(the conclusions of which follow from their premises) and invalidsyllogisms (the conclusions of which do not follow from theirpremises). The second of these is inconsistent with Aristotle’suse: since he defines asullogismos as an argument in whichthe conclusion results of necessity from the premises, “invalidsullogismos” is a contradiction in terms. The first isalso at least highly misleading, since Aristotle does not appear tothink that thesullogismoi are simply an interesting subsetof the valid arguments. Moreover (see below), Aristotle expends greatefforts to argue that every valid argument, in a broad sense, can be“reduced” to an argument, or series of arguments, insomething like one of the forms traditionally called a syllogism. Ifwe translatesullogismos as “syllogism”, thisbecomes the trivial claim “Every syllogism is asyllogism”,

4. Premises: The Structures of Assertions

Syllogisms are structures of sentences each of which can meaningfullybe called true or false:assertions(apophanseis), in Aristotle’s terminology. According toAristotle, every such sentence must have the same structure: it mustcontain asubject (hupokeimenon) and apredicate and must either affirm or deny thepredicate of the subject. Thus, every assertion is either theaffirmationkataphasis or thedenial (apophasis) of a single predicate ofa single subject.

InOn Interpretation, Aristotle argues that a singleassertion must always either affirm or deny a single predicate of asingle subject. Thus, he does not recognize sentential compounds, suchas conjunctions and disjunctions, as single assertions. This appearsto be a deliberate choice on his part: he argues, for instance, that aconjunction is simply a collection of assertions, with no moreintrinsic unity than the sequence of sentences in a lengthy account(e.g. the entireIliad, to take Aristotle’s ownexample). Since he also treats denials as one of the two basic speciesof assertion, he does not view negations as sentential compounds. Histreatment of conditional sentences and disjunctions is more difficultto appraise, but it is at any rate clear that Aristotle made noefforts to develop a sentential logic. Some of the consequences ofthis for his theory of demonstration are important.

4.1 Terms

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

The worduniversal (katholou) appears to bean Aristotelian coinage. Literally, it means “of a whole”;its opposite is therefore “of a particular”(kath’ hekaston). Universal terms are those which canproperly serve as predicates, while particular terms are those whichcannot.

This distinction is not simply a matter of grammatical function. Wecan readily enough construct a sentence with “Socrates” asits grammatical predicate: “The person sitting down isSocrates”. Aristotle, however, does not consider this a genuinepredication. He calls it instead a merelyaccidentalorincidental (kata sumbebêkos)predication. Such sentences are, for him, dependent for their truthvalues on other genuine predications (in this case, “Socrates issitting down”).

Consequently, predication for Aristotle is as much a matter ofmetaphysics as a matter of grammar. The reason that the termSocrates is an individual term and not a universal is thatthe entity which it designates is an individual, not a universal. Whatmakeswhite andhuman universal terms is that theydesignate universals.

Further discussion of these issues can be found in the entry onAristotle’s metaphysics.

4.2 Affirmations, Denials, and Contradictions

Aristotle takes some pains inOn Interpretation to argue thatto every affirmation there corresponds exactly one denial such thatthat denial denies exactly what that affirmation affirms. The pairconsisting of an affirmation and its corresponding denial is acontradiction (antiphasis). In general,Aristotle holds, exactly one member of any contradiction is true andone false: they cannot both be true, and they cannot both be false.However, he appears to make an exception for propositions about futureevents, though interpreters have debated extensively what thisexception might be (seefurther discussion below). The principle that contradictories cannot both be true hasfundamental importance in Aristotle’s metaphysics (seefurther discussion below).

4.3 All, Some, and None

One major difference between Aristotle’s understanding ofpredication and modern (i.e., post-Fregean) logic is that Aristotletreats individual predications and general predications as similar inlogical form: he gives the same analysis to “Socrates is ananimal” and “Humans are animals”. However, he notesthat when the subject is a universal, predication takes on two forms:it can be eitheruniversal orparticular. These expressions are parallel to thosewith which Aristotle distinguishes universal and particular terms, andAristotle is aware of that, explicitly distinguishing between a termbeing a universal and a term being universally predicated ofanother.

Whatever is affirmed or denied of a universal subject may be affirmedor denied of it ituniversally (katholou or“of all”,kata pantos),in part(kata meros,en merei), orindefinitely (adihoristos).

4.3.1 The “Square of Opposition”

InOn Interpretation, Aristotle spells out the relationshipsof contradiction for sentences with universal subjects as follows:

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

In thePrior Analytics, Aristotle adopts a somewhatartificial way of expressing predications: instead of saying“\(X\) is predicated of \(Y\)” he says “\(X\)belongs (huparchei) to \(Y\)”. This should really beregarded as a technical expression. The verbhuparcheinusually means either “begin” or “exist, bepresent”, and Aristotle’s usage appears to be adevelopment of this latter use.

4.3.2 Some Convenient Abbreviations

For clarity and brevity, I will use the following semi-traditionalabbreviations for Aristotelian categorical sentences (note that thepredicate term comesfirst and the subject termsecond):

5. The Syllogistic

Aristotle’s most famous achievement as logician is his theory ofinference, traditionally called thesyllogistic(though not by Aristotle). That theory is in fact the theory ofinferences of a very specific sort: inferences with two premises, eachof which is a categorical sentence, having exactly one term in common,and having as conclusion a categorical sentence the terms of which arejust those two terms not shared by the premises. Aristotle calls theterm shared by the premises themiddle term(meson) and each of the other two terms in the premises anextreme (akron). The middle term must beeither subject or predicate of each premise, and this can occur inthree ways: the middle term can be the subject of one premise and thepredicate of the other, the predicate of both premises, or the subjectof both premises. Aristotle refers to these term arrangements asfigures (schêmata):

5.1 The Figures

Aristotle calls the term which is the predicate of the conclusion themajor term and the term which is the subject of theconclusion theminor term. The premise containing themajor term is themajor premise, and the premisecontaining the minor term is theminor premise.

Aristotle then systematically investigates all possible combinationsof two premises in each of the three figures. For each combination, heeither demonstrates that some conclusion necessarily follows ordemonstrates that no conclusion follows. The results he states arecorrect.

5.2 Methods of Proof: “Perfect” Deductions, Conversion, Reduction

Aristotle’s proofs can be divided into two categories, based ona distinction he makes between “perfect” or“complete” (teleios) deductions and“imperfect” or “incomplete”(atelês) deductions. A deduction is perfect if it“needs no external term in order to show the necessaryresult” (24b23–24), and it is imperfect if it “needsone or several in addition that are necessary because of the termssupposed but were not assumed through premises”(24b24–25). The precise interpretation of this distinction isdebatable, but it is at any rate clear that Aristotle regards theperfect deductions as not in need of proof in some sense. Forimperfect deductions, Aristotle does give proofs, which invariablydepend on the perfect deductions. Thus, with some reservations, wemight compare the perfect deductions to the axioms or primitive rulesof a deductive system.

In the proofs for imperfect deductions, Aristotle says that he“reduces” (anagein) each case to one of theperfect forms and that they are thereby “completed” or“perfected”. These completions are eitherprobative (deiktikos: a modern translationmight be “direct”) orthrough theimpossible (dia to adunaton).

A direct deduction is a series of steps leading from the premises tothe conclusion, each of which is either aconversionof a previous step or an inference from two previous steps relying ona first-figure deduction. Conversion, in turn, is inferring from aproposition another which has the subject and predicate interchanged.Specifically, Aristotle argues that three such conversions aresound:

He undertakes to justify these inAn. Pr. I.2. From a modernstandpoint, the third is sometimes regarded with suspicion. Using itwe can getSome monsters are chimeras from the apparentlytrueAll chimeras are monsters; but the former is oftenconstrued as implying in turnThere is something which is amonster and a chimera, and thus that there are monsters and thereare chimeras. In fact, this simply points up something aboutAristotle’s system: Aristotle in effect supposes thatall terms in syllogisms are non-empty. (For furtherdiscussion of this point, see the entry on thesquare of opposition).

As an example of the procedure, we may take Aristotle’s proof ofCamestres. 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 \(X\), thenneither does \(X\) belong to any \(M\); but \(M\) belonged to every\(N\); therefore, \(X\) will belong to no \(N\) (for the first figurehas 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:

A completion or proof “through the impossible” shows thata certain conclusion follows from a pair of premises by assuming as athird 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 thatpoint. An example is his proof ofBaroco in27a36–b1:

5.3 Disproof: Counterexamples and Terms

Aristotle proves invalidity by constructing counterexamples. This isvery much in the spirit of modern logical theory: all that it takes toshow that a certainform is invalid is a singleinstance of that form with true premises and a falseconclusion. However, Aristotle states his results not by saying thatcertain premise-conclusion combinations are invalid but by saying thatcertain premise pairs do not “syllogize”: that is, that,given the pair in question, examples can be constructed in whichpremises of that form are true and a conclusion of any of the fourpossible forms is false.

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

5.4 The Deductions in the Figures (“Moods”)

InPrior Analytics I.4–6, Aristotle shows that thepremise combinations given in the following table yield deductions andthat all other premise combinations fail to yield a deduction. In theterminology traditional since the middle ages, each of thesecombinations is known as amood Latinmodus,“way”, which in turn is a translation of Greektropos). Aristotle, however, does not use this expression andinstead refers to “the arguments in the figures”.

In this table, “\(\vdash\)” separates premises fromconclusion; it may be read “therefore”. The second columnlists the medieval mnemonic name associated with the inference (theseare still widely used, and each is actually a mnemonic forAristotle’s proof of the mood in question). The third columnbriefly summarizes Aristotle’s procedure for demonstrating thededuction.

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 twoaffirmative premises
4. A deduction with a negative conclusion must have one negativepremise.
5. A deduction with a universal conclusion must have two universalpremises

He also proves the following metatheorem:

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

His proof of this is elegant. First, he shows that the two particulardeductions of the first figure can be reduced, by proof throughimpossibility, 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 theparticular deductions in the other figures exceptBaroco andBocardo toDarii andFerio, thesedeductions can thus be reduced toBarbara andCelarent. This proof is strikingly similar both in structureand in subject to modern proofs of the redundancy of axioms in asystem.

Many more metatheoretical results, some of them quite sophisticated,are proved inPrior Analytics I.45 and inPriorAnalytics II. As noted below, some of Aristotle’smetatheoretical results are appealed to in the epistemologicalarguments of thePosterior Analytics.

5.6 Syllogisms with Modalities

Aristotle follows his treatment of “arguments in thefigures” with a much longer, and much more problematic,discussion of what happens to these figured arguments when we add thequalifications “necessarily” and “possibly” totheir premises in various ways. In contrast to the syllogistic itself(or, as commentators like to call it, theassertoricsyllogistic), thismodal syllogistic appears to be much lesssatisfactory and is certainly far more difficult to interpret. Here, Ionly outline Aristotle’s treatment of this subject and note someof 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 notP”, and “possibly P” to “not necessarily notP”. Aristotle gives these same equivalences inOnInterpretation. However, inPrior Analytics, he makes adistinction between two notions of possibility. On the first, which hetakes as his preferred notion, “possibly P” is equivalentto “not necessarily P and not necessarily not P”. He thenacknowledges an alternative definition of possibility according to themodern equivalence, but this plays only a secondary role in hissystem.

5.6.2 Aristotle’s General Approach

Aristotle builds his treatment of modal syllogisms on his account ofnon-modal (assertoric) syllogisms: he works his waythrough the syllogisms he has already proved and considers theconsequences of adding a modal qualification to one or both premises.Most often, then, the questions he explores have the form: “Hereis an assertoric syllogism; if I add these modal qualifications to thepremises, then what modally qualified form of the conclusion (if any)follows?”. A premise can have one of three modalities: it can benecessary, possible, or assertoric. Aristotle works through thecombinations 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 whichsyllogize in their assertoric forms, he does sometimes extend this;similarly, he sometimes considers conclusions in addition to thosewhich would follow from purely assertoric premises.

Since this is his procedure, it is convenient to describe modalsyllogisms in terms of the corresponding non-modal syllogism plus atriplet of letters indicating the modalities of premises andconclusion: \(N\) = “necessary”, \(P\) =“possible”, \(A\) = “assertoric”. Thus,“Barbara \(NAN\)” would mean “The formBarbara with necessary major premise, assertoric minorpremise, and necessary conclusion”. I use the letters“\(N\)” and “\(P\)” as prefixes for premisesas 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 ofconversion rules to prove validity. The conversion rules for necessarypremises are exactly analogous to those for assertoric premises:

Possible premises behave differently, however. Since he defines“possible” as “neither necessary norimpossible”, 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 asfollows:

In addition, Aristotle uses the intermodal principle \(N\rightarrowA\): that is, a necessary premise entails the corresponding assertoricone. 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 \(P\)” areactually inconsistent (“possibly \(P\)” entails“possibly not \(P\)”).

This leads to a further complication. The denial of “possibly\(P\)” for Aristotle is “either necessarily \(P\) ornecessarily not \(P\)”. The denial of “necessarily\(P\)” is still more difficult to express in terms of acombination of modalities: “either possibly \(P\) (and thuspossibly not \(P\)) or necessarily not \(P\)” This is importantbecause of Aristotle’s proof procedures, which include proofthrough impossibility. If we give a proof through impossibility inwhich we assume a necessary premise, then the conclusion we ultimatelyestablish is simply the denial of that necessary premise, not a“possible” conclusion in Aristotle’s sense. Suchpropositions do occur in his system, but only in exactly this way,i.e., as conclusions established by proof through impossiblity fromnecessary assumptions. Somewhat confusingly, Aristotle calls suchpropositions “possible” but immediately adds “ notin the sense defined”: in this sense, “possibly\(Oab\)” is simply the denial of “necessarily\(Aab\)”. Such propositions appear only as premises, never asconclusions.

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 atrivial consequence but instead offers proofs; in all but two cases,these are parallel to those offered for the assertoric case. Theexceptions areBaroco andBocardo, which he provedin the assertoric case through impossibility: attempting to use thatmethod here would require him to take the denial of a necessary \(O\)proposition as hypothesis, raising the complication noted above, andhe uses the procedure he callsecthesis instead (see Smith1982).

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 thatsome, but not all, \(ANN\) and \(NAN\) combinations are valid.Specifically, he acceptsBarbara \(NAN\) but rejectsBarbara \(ANN\). Almost from Aristotle’s own time,interpreters have found his reasons for this distinction obscure, orunpersuasive, or both, and often have not followed his view. His closeassociated Theophrastus, for instance, adopted the simpler rule thatthe modality of the conclusion of a syllogism was always the“weakest” modality found in either premise, where \(N\) isstronger 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 ofmodern formal logic to interpret Aristotle’s modal logic haveseen the Two-Barbaras problem as only one of a series of difficultiesin giving a coherent interpretation of the modal syllogistic. A verywide range of reconstructions has been proposed: see Becker 1933,McCall 1963, Nortmann 1996, Van Rijen 1989, Patterson 1995, Thomason1993, Thom 1996, Rini 2012, Malink 2013. The majority ofreconstructions do not attempt to reproduce every detail ofAristotle’s exposition but instead produce modifiedreconstructions that abandon some of those results. Malink 2013,however, offers a reconstruction that reproduces everything Aristotlesays, although the resulting model introduces a high degree ofcomplexity. (This subject quickly becomes too complex for summarizingin this brief article.

6. Demonstrations and Demonstrative Sciences

Ademonstration (apodeixis) is “adeduction that produces knowledge”. Aristotle’sPosterior Analytics contains his account of demonstrationsand their role in knowledge. From a modern perspective, we might thinkthat this subject moves outside of logic to epistemology. FromAristotle’s perspective, however, the connection of the theoryofsullogismoi with the theory of knowledge is especiallyclose.

6.1 Aristotelian Sciences

The subject of thePosterior Analytics isepistêmê. This is one of several Greek words thatcan reasonably be translated “knowledge”, but Aristotle isconcerned only with knowledge of a certain type (as will be explainedbelow). There is a long tradition of translatingepistêmê in this technical sense asscience, and I shall follow that tradition here.However, readers should not be misled by the use of that word. Inparticular, Aristotle’s theory of science cannot be considered acounterpart to modern philosophy of science, at least not withoutsubstantial qualifications.

We have scientific knowledge, according to Aristotle, when weknow:

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

This implies two strong conditions on what can be the object ofscientific 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 scienceconsists in the possession of ademonstration(apodeixis), which he defines as a “scientificdeduction”:

by “scientific” (epistêmonikon), I meanthat in virtue of possessing it, we have knowledge.

The remainder ofPosterior Analytics I is largely concernedwith two tasks: spelling out the nature of demonstration anddemonstrative science and answering an important challenge to its verypossibility. Aristotle first tells us that a demonstration is adeduction in which the premises are:

1. true
2. primary (prota)
3. immediate (amesa, “without amiddle”)
4. better known ormore familiar(gnôrimôtera) than the conclusion
5. prior to the conclusion
6. causes (aitia) of the conclusion

The interpretation of all these conditions except the first has beenthe subject of much controversy. Aristotle clearly thinks that scienceis knowledge of causes and that in a demonstration, knowledge of thepremises is what brings about knowledge of the conclusion. The fourthcondition shows that the knower of a demonstration must be in somebetter epistemic condition towards them, and so modern interpretersoften suppose that Aristotle has defined a kind of epistemicjustification here. However, as noted above, Aristotle is defining aspecial variety of knowledge. Comparisons with discussions ofjustification in modern epistemology may therefore be misleading.

The same can be said of the terms “primary”,“immediate” and “better known”. Moderninterpreters sometimes take “immediate” to mean“self-evident”; Aristotle does say that an immediateproposition is one “to which no other is prior”, but (as Isuggest in the next section) the notion of priority involved is likelya notion of logical priority that it is hard to detach fromAristotle’s own logical theories. “Better known” hassometimes been interpreted simply as “previously known to theknower of the demonstration” (i.e., already known in advance ofthe demonstration). However, Aristotle explicitly distinguishesbetween what is “better known for us” with what is“better known in itself” or “in nature” andsays that he means the latter in his definition. In fact, he says thatthe process of acquiring scientific knowledge is a process ofchanging what is better known “for us”, until wearrive at that condition in which what is better known in itself isalso better known for us.

6.2 The Regress Problem

InPosterior Analytics I.2, Aristotle considers twochallenges to the possibility of science. One party (dubbed the“agnostics” by Jonathan Barnes) began with the followingtwo 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 followingdilemma:

1. If the premises of a demonstration are scientifically known, thenthey must be demonstrated.
2. The premises from which each premise are demonstrated must bescientifically known.
3. Either this process continues forever, creating an infiniteregress of premises, or it comes to a stop at some point.
4. If it continues forever, then there are no first premises fromwhich the subsequent ones are demonstrated, and so nothing isdemonstrated.
5. On the other hand, if it comes to a stop at some point, then thepremises at which it comes to a stop are undemonstrated and thereforenot scientifically known; consequently, neither are any of the othersdeduced from them.
6. Therefore, nothing can be demonstrated.

A second group accepted the agnostics’ view that scientificknowledge comes only from demonstration but rejected their conclusionby rejecting the dilemma. Instead, they maintained:

• Demonstration “in a circle” is possible, so that it ispossible for all premises also to be conclusions and thereforedemonstrated.

Aristotle does not give us much information about how circulardemonstration was supposed to work, but the most plausibleinterpretation would be supposing that at least for some set offundamental principles, each principle could be deduced from theothers. (Some modern interpreters have compared this position to acoherence theory of knowledge.) However their position worked, thecircular demonstrators claimed to have a third alternative avoidingthe agnostics’ dilemma, since circular demonstration gives us aregress that is both unending (in the sense that we never reachpremises at which it comes to a stop) and finite (because it works itsway 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 onthe grounds that the premises of any demonstration must be prior (inan appropriate sense) to the conclusion, whereas a circulardemonstration would make the same premises both prior and posterior toone 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 thatit “comes to a stop” at some point. However, he thinksboth the agnostics and the circular demonstrators are wrong inmaintaining that scientific knowledge is only possible bydemonstration 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 knowledgeofsome propositions without demonstrating them: unless it isin turn possible to deduce all the other propositions of a sciencefrom them, we shall not have solved the regress problem. Moreover (andobviously), it is no solution to this problem for Aristotle simply toassert that we have knowledge without demonstration of someappropriate starting points. He does indeed say that it is hisposition that we have such knowledge (An. Post. I.2,), but heowes us an account of why that should be so.

6.4 Knowledge of First Principles:Nous

Aristotle’s account of knowledge of the indemonstrable firstpremises of sciences is found inPosterior Analytics II.19,long regarded as a difficult text to interpret. Briefly, what he saysthere is that it is another cognitive state,nous (translatedvariously as “insight”, “intuition”,“intelligence”), which knows them. There is widedisagreement among commentators about the interpretation of hisaccount of how this state is reached; I will offer one possibleinterpretation. First, Aristotle identifies his problem as explaininghow the principles can “become familiar to us”, using thesame term “familiar” (gnôrimos) that heused 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 bepossible, we must have a kind of knowledge of them without havinglearned it, but this knowledge must not be as “precise” asthe knowledge that a possessor of science must have. The kind ofknowledge in question turns out to be a capacity or power(dunamis) which Aristotle compares to the capacity forsense-perception: since our senses are innate, i.e., developnaturally, it is in a way correct to say that we know what e.g. allthe colors look like before we have seen them: we have the capacity tosee them by nature, and when we first see a color we exercise thiscapacity without having to learn how to do so first. Likewise,Aristotle holds, our minds have by nature the capacity to recognizethe starting points of the sciences.

In the case of sensation, the capacity for perception in the senseorgan is actualized by the operation on it of the perceptible object.Similarly, Aristotle holds that coming to know first premises is amatter of a potentiality in the mind being actualized by experience ofits proper objects: “The soul is of such a nature as to becapable of undergoing this”. So, although we cannot come to knowthe first premises without the necessary experience, just as we cannotsee colors without the presence of colored objects, our minds arealready so constituted as to be able to recognize the right objects,just as our eyes are already so constituted as to be able to perceivethe colors that exist.

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

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

7. Definitions

Thedefinition (horos,horismos)was an important matter for Plato and for the Early Academy. Concernwith answering the question “What is so-and-so?” are atthe center of the majority of Plato’s dialogues, some of which(most elaborately theSophist) propound methods for findingdefinitions. External sources (sometimes the satirical remarks ofcomedians) also reflect this Academic concern with definitions.Aristotle himself traces the quest for definitions back toSocrates.

7.1 Definitions and Essences

For Aristotle, a definition is “an account which signifies whatit is to be for something” (logos ho to ti ên einaisêmainei). The phrase “what it is to be” andits variants are crucial: giving a definition is saying, of someexistent thing, what it is, not simply specifying the meaning of aword (Aristotle does recognize definitions of the latter sort, but hehas little interest in them).

The notion of “what it is to be” for a thing is sopervasive in Aristotle that it becomes formulaic: what a definitionexpresses is “the what-it-is-to-be” (to ti êneinai), or in modern terminology, its essence.

7.2 Species, Genus, and Differentia

Since a definition defines an essence, only what has an essence can bedefined. What has an essence, then? That is one of the centralquestions of Aristotle’s metaphysics; once again, we must leavethe details to another article. In general, however, it is notindividuals but ratherspecies (eidos: theword is one of those Plato uses for “Form”) that haveessences. A species is defined by giving itsgenus(genos) and itsdifferentia(diaphora): the genus is the kind under which the speciesfalls, and the differentia tells what characterizes the species withinthat genus. As an example,human might be defined asanimal (the genus)having the capacity to reason(the differentia).

Essential Predication and the Predicables

Underlying Aristotle’s concept of a definition is the concept ofessential predication (katêgoreisthai entôi ti esti, predication in the what it is). In any trueaffirmative predication, the predicate either does or does not“say what the subject is”, i.e., the predicate either isor is not an acceptable answer to the question “What isit?” asked of the subject. Bucephalus is a horse, and a horse isan animal; so, “Bucephalus is a horse” and“Bucephalus is an animal” are essential predications.However, “Bucephalus is brown”, though true, does notstate what Bucephalus is but only says something about him.

Since a thing’s definition says what it is, definitions areessentially predicated. However, not everything essentially predicatedis a definition. Since Bucephalus is a horse, and horses are a kind ofmammal, and mammals are a kind of animal, “horse”“mammal” and “animal” are all essentialpredicates of Bucephalus. Moreover, since what a horse is is a kind ofmammal, “mammal” is an essential predicate of horse. Whenpredicate \(X\) is an essential predicate of \(Y\) but also of otherthings, then \(X\) is agenus (genos) of\(Y\).

A definition of \(X\) must not only be essentially predicated of itbut must also be predicated only of it: to use a term fromAristotle’sTopics, a definition and what it definesmust “counterpredicate”(antikatêgoreisthai) with one another. \(X\)counterpredicates with \(Y\) if \(X\) applies to what \(Y\) applies toand conversely. Though X’s definition must counterpredicate with\(X\), not everything that counterpredicates with \(X\) is itsdefinition. “Capable of laughing”, for example,counterpredicates with “human” but fails to be itsdefinition. Such a predicate (non-essential but counterpredicating) isapeculiar property orproprium(idion).

Finally, if \(X\) is predicated of \(Y\) but is neither essential norcounterpredicates, then \(X\) is anaccident(sumbebêkos) of \(Y\).

Aristotle sometimes treats genus, peculiar property, definition, andaccident as including all possible predications (e.g.TopicsI). Later commentators listed these four and the differentia as thefivepredicables, and as such they were of greatimportance to late ancient and to medieval philosophy (e.g.,Porphyry).

7.3 The Categories

The notion of essential predication is connected to what aretraditionally called thecategories(katêgoriai). In a word, Aristotle is famous for havingheld a “doctrine of categories”. Just what that doctrinewas, and indeed just what a category is, are considerably more vexingquestions. They also quickly take us outside his logic and into hismetaphysics. Here, I will try to give a very general overview,beginning with the somewhat simpler question “What categoriesare there?”

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

We should distinguish the kinds of predication (ta genêtôn katêgoriôn) in which the four predicationsmentioned 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 definitionwill always be in one of these categories. (Topics I.9,103b20–25)

Of things said without any combination, each signifies eithersubstance or quantity or quality or a relative or where or when orbeing-in-a-position or having or doing or undergoing. To give a roughidea, 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; ofhaving: has-shoes-on, has-armor-on; of doing: cutting, burning; ofundergoing: being-cut, being-burned. (Categories 4,1b25–2a4, tr. Ackrill, slightly modified)

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

The word “category” (katêgoria) means“predication”. Aristotle holds that predications andpredicates can be grouped into several largest “kinds ofpredication” (genê tônkatêgoriôn). He refers to this classificationfrequently, often calling the “kinds of predication”simply “the predications”, and this (by way of Latin)leads to our word “category”.

• First, the categories may bekinds of predicate:predicates (or, more precisely, predicate expressions) can be dividedinto ten separate classes, with each expression belonging to just oneclass. This comports well with the root meaning of the wordkatêgoria (“predication”). On thisinterpretation, the categories arise out of considering the mostgeneral types of question that can be asked about something:“What is it?”; “How much isit?”; “What sort is it?”;“Where is it?”; “What is itdoing?” Answers appropriate to one of these questionsare nonsensical in response to another (“When is it?”“A horse”). Thus, the categories may rule out certainkinds of question as ill-formed or confused. This plays an importantrole in Aristotle’s metaphysics.
• Second, the categories may be seen as classifications ofpredications, that is, kinds of relation that may holdbetween the predicate and the subject of a predication. To say ofSocrates that he is human is to say what he \(is\), whereas to saythat he is literate is not to say what he is but rather to give aquality that hehas. For Aristotle, the relation of predicateto subject in these two sentences is quite different (in this respecthe differs both from Plato and from modern logicians). The categoriesmay be interpreted as ten different ways in which a predicate may berelated to its subject. This last division has importance forAristotle’s logic as well as his metaphysics.
• Third, the categories may be seen askinds of entity, ashighest genera or kinds of thing that are. A given thing can beclassified under a series of progressively wider genera: Socrates is ahuman, a mammal, an animal, a living being. The categories are thehighest such genera. Each falls under no other genus, and each iscompletely separate from the others. This distinction is of criticalimportance 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 mostevident if we take note of point in which they differ: theCategories listssubstance (ousia)in first place, while theTopics listwhat-it-is (ti esti). A substance, forAristotle, is a type of entity, suggesting that theCategories list is a list of types of entity.

On the other hand, the expression “what-it-is” suggestsmost strongly a type of predication. Indeed, theTopicsconfirms this by telling us that we can “say what it is”of anentity falling under any of the categories:

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

As Aristotle explains, if I say that Socrates is a man, then I havesaid what Socrates is and signified a substance; if I say that whiteis a color, then I have said what white is and signified a quality; ifI say that some length is a foot long, then I have said what it is andsignified 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 theTopics are only to be interpreted as kinds of predicate orpredication, those in theCategories as kinds of being. Evenso, we would still want to ask what the relationship is between thesetwo nearly-identical lists of terms, given these distinctinterpretations. However, the situation is much more complicated.First, there are dozens of other passages in which the categoriesappear. Nowhere else do we find a list of ten, but we do find shorterlists containing eight, or six, or five, or four of them (withsubstance/what-it-is, quality, quantity, and relative the mostcommon). Aristotle describes what these lists are lists of indifferent ways: they tell us “how being is divided”, or“how many ways being is said”, or “the figures ofpredication” (ta schêmata tês katêgorias). Thedesignation of the first category also varies: we find not only“substance” and “what it is” but also theexpressions “this” or “the this” (todeti,to tode,to ti). These latter expressionsare closely associated with, but not synonymous with, substance. Heeven combines the latter with “what-it-is”(Metaphysics Z 1, 1028a10: “… one sensesignifies what it is and the this, one signifies quality…”).

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

See Frede 1981, Ebert 1985 for additional discussion ofAristotle’s lists of categories.

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

7.4 The Method of Division

In theSophist, Plato introduces a procedure of“Division” as a method for discovering definitions. Tofind a definition of \(X\), first locate the largest kind of thingunder which \(X\) falls; then, divide that kind into two parts, anddecide which of the two \(X\) falls into. Repeat this method with thepart until \(X\) has been fully located.

This method is part of Aristotle’s Platonic legacy. His attitudetowards it, however, is complex. He adopts a view of the properstructure of definitions that is closely allied to it: a correctdefinition of \(X\) should give thegenus(genos: kind or family) of \(X\), which tells what kind ofthing \(X\) is, and thedifferentia(diaphora: difference) which uniquely identifies \(X\) withinthat genus. Something defined in this way is aspecies (eidos: the term is one ofPlato’s terms for “Form”), and the differentia isthus the “difference that makes a species” (eidopoiosdiaphora, “specific difference”). InPosteriorAnalytics II.13, he gives his own account of the use of Divisionin finding definitions.

However, Aristotle is strongly critical of the Platonic view ofDivision as a method forestablishing definitions. InPrior Analytics I.31, he contrasts Division with thesyllogistic method he has just presented, arguing that Division cannotactually prove anything but rather assumes the very thing it issupposed to be proving. He also charges that the partisans of Divisionfailed to understand what their own method was capable of proving.

7.5 Definition and Demonstration

Closely related to this is the discussion, inPosteriorAnalytics II.3–10, of the question whether there can beboth definition and demonstration of the same thing (that is, whetherthe same result can be established either by definition or bydemonstration). Since the definitions Aristotle is interested in arestatements of essences, knowing a definition is knowing, of someexisting thing, what it is. Consequently, Aristotle’s questionamounts to a question whether defining and demonstrating can bealternative ways of acquiring the same knowledge. His reply iscomplex:

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

As an example of case 3, Aristotle considers the definition“Thunder is the extinction of fire in the clouds”. He seesthis 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 theclouds” (definition). “Why does it thunder?”“Because fire is extinguished in the clouds”(demonstration).

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

8. Dialectical Argument and the Art of Dialectic

Aristotle often contrastsdialectical arguments withdemonstrations. The difference, he tells us, is in the character oftheir premises, not in their logical structure: whether an argument isasullogismos is only a matter of whether its conclusionresults of necessity from its premises. The premises of demonstrationsmust betrue and primary, that is, not only true but alsoprior to their conclusions in the way explained in thePosteriorAnalytics. The premises of dialectical deductions, by contrast,must beaccepted (endoxos).

8.1 Dialectical Premises: The Meaning ofEndoxos

Recent scholars have proposed different interpretations of the termendoxos. Aristotle often uses this adjective as asubstantive:ta endoxa, “accepted things”,“accepted opinions”. On one understanding, descended fromthe work of G. E. L. Owen and developed more fully by Jonathan Barnesand especially Terence Irwin, theendoxa are a compilation ofviews held by various people with some form or other of standing:“the views of fairly reflective people after somereflection”, in Irwin’s phrase. Dialectic is then simply“a method of argument from [the] common beliefs [held by thesepeople]”. For Irwin, then,endoxa are “commonbeliefs”. Jonathan Barnes, noting thatendoxa areopinions with a certain standing, translates with“reputable”.

My own view is that Aristotle’s texts support a somewhatdifferent understanding. He also tells us that dialectical premisesdiffer from demonstrative ones in that the former arequestions, whereas the latter areassumptions orassertions: “the demonstrator does not ask, buttakes”, he says. This fits most naturally with a view ofdialectic as argument directed at another person by question andanswer and consequently taking as premises that other person’sconcessions. Anyone arguing in this manner will, in order to besuccessful, have to ask for premises which the interlocutor is liableto accept, and the best way to be successful at that is to have aninventory of acceptable premises, i.e., premises that are in factacceptable to people of different types.

In fact, we can discern in theTopics (and theRhetoric, which Aristotle says depends on the art explainedin theTopics) an art of dialectic for use in such arguments.My reconstruction of this art (which would not be accepted by allscholars) is as follows.

8.2 The Two Elements of the Art of Dialectic

Given the above picture of dialectical argument, the dialectical artwill consist of two elements. One will be a method for discoveringpremises from which a given conclusion follows, while the other willbe a method for determining which premises a given interlocutor willbe likely to concede. The first task is accomplished by developing asystem 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 lesssystematic and rests on various “common” terms. The secondtask is accomplished by developing lists of the premises which areacceptable to various types of interlocutor. Then, once one knows whatsort of person one is dealing with, one can choose premisesaccordingly. Aristotle stresses that, as in all arts, the dialecticianmust study, not what is acceptable to this or that specific person,but what is acceptable to this or that type of person, just as thedoctor studies what is healthful for different types of person:“art is of the universal”.

8.2.1 The “Logical System” of theTopics

The method presented in theTopics for classifying argumentsrelies on the presence in the conclusion of certain“common” terms (koina) — common in thesense that they are not peculiar to any subject matter but may play arole in arguments about anything whatever. We find enumerations ofarguments 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 sterêsis)
   4. Relatives (pros ti)
2. Cases (ptôseis)
3. “More and Less and Likewise”

The four types ofopposites are the best represented.Each designates a type of term pair, i.e., a way two terms can beopposed to one another.Contraries are polaropposites or opposed extremes such as hot and cold, dry and wet, goodand bad. A pair ofcontradictories consists of a termand its negation: good, not good. Apossession (orcondition) andprivation are illustrated by sight andblindness.Relatives are relative terms in the modernsense: a pair consists of a term and its correlative, e.g. large andsmall, parent and child.

The argumentative patterns Aristotle associated withcases generally involve inferring a sentencecontaining adverbial or declined forms from another sentencecontaining different forms of the same word stem: “if what isuseful is good, then what is done usefully is done well and the usefulperson is good”. In Hellenistic grammatical usage,ptôsis meant “case” (e.g. nominative,dative, accusative); Aristotle’s use here is obviously an earlyform of that.

Under the headingmore and less and likewise,Aristotle groups a somewhat motley assortment of argument patterns allinvolving, in some way or other, the terms “more”,“less”, and “likewise”. Examples: “Ifwhatever 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 \(A\) isthe case”.

8.2.2 TheTopoi

At the heart of theTopics is a collection of what Aristotlecallstopoi, “places” or “locations”.Unfortunately, though it is clear that he intends most of theTopics (Books II–VI) as a collection of these, he neverexplicitly defines this term. Interpreters have consequently disagreedconsiderably about just what atopos is. Discussions may befound in Brunschwig 1967, Slomkowski 1996, Primavesi 1997, and Smith1997.

8.3 The Uses of Dialectic and Dialectical Argument

Anart of dialectic will be useful wherever dialecticalargument is useful. Aristotle mentions three such uses; each meritssome comment.

8.3.1 Gymnastic Dialectic

First, there appears to have been a form of stylized argumentativeexchange practiced in the Academy in Aristotle’s time. The mainevidence for this is simply Aristotle’sTopics,especially Book VIII, which makes frequent reference to rule-governedprocedures, apparently taking it for granted that the audience willunderstand them. In these exchanges, one participant took the role ofanswerer, the other the role of questioner. The answerer began byasserting some proposition (athesis: “position”or “acceptance”). The questioner then asked questions ofthe answerer in an attempt to secure concessions from which acontradiction could be deduced: that is, torefute(elenchein) the answerer’s position. The questioner waslimited to questions that could be answered by yes or no; generally,the answerer could only respond with yes or no, though in some casesanswerers could object to the form of a question. Answerers mightundertake to answer in accordance with the views of a particular typeof person or a particular person (e.g. a famous philosopher), or theymight answer according to their own beliefs. There appear to have beenjudges or scorekeepers for the process. Gymnastic dialectical contestswere sometimes, as the name suggests, for the sake of exercise indeveloping argumentative skill, but they may also have been pursued asa part of a process of inquiry.

8.3.2 Dialectic That Puts to the Test

Aristotle also mentions an “art of making trial”, or avariety of dialectical argument that “puts to the test”(the Greek word is the adjectivepeirastikê, in thefeminine: such expressions often designate arts or skills, e.g.rhêtorikê, “the art of rhetoric”).Its function is to examine the claims of those who say they have someknowledge, and it can be practiced by someone who does not possess theknowledge in question. The examination is a matter of refutation,based on the principle that whoever knows a subject must haveconsistent beliefs about it: so, if you can show me that my beliefsabout something lead to a contradiction, then you have shown that I donot have knowledge about it.

This is strongly reminiscent of Socrates’ style ofinterrogation, from which it is almost certainly descended. In fact,Aristotle often indicates that dialectical argument is by naturerefutative.

8.3.3 Dialectic and Philosophy

Dialectical refutation cannot of itself establish any proposition(except perhaps the proposition that some set of propositions isinconsistent). More to the point, though deducing a contradiction frommy beliefs may show that they do not constitute knowledge, failure todeduce a contradiction from them is no proof that they are true. Notsurprisingly, then, Aristotle often insists that “dialectic doesnot prove anything” and that the dialectical art is not somesort of universal knowledge.

InTopics I.2, however, Aristotle says that the art ofdialectic is useful in connection with “the philosophicalsciences”. One reason he gives for this follows closely on therefutative function: if we have subjected our opinions (and theopinions of our fellows, and of the wise) to a thorough refutativeexamination, we will be in a much better position to judge what ismost likely true and false. In fact, we find just such a procedure atthe start of many of Aristotle’s treatises: an enumeration ofthe opinions current about the subject together with a compilation of“puzzles” raised by these opinions. Aristotle has aspecial term for this kind of review: adiaporia, a“puzzling through”.

He adds a second use that is both more difficult to understand andmore intriguing. ThePosterior Analytics argues that ifanything can be proved, then not everything that is known is known asa result of proof. What alternative means is there whereby the firstprinciples of sciences are known? Aristotle’s own answer asfound inPosterior Analytics II.19 is difficult to interpret,and recent philosophers have often found it unsatisfying since (asoften construed) it appears to commit Aristotle to a form of apriorismor rationalism both indefensible in itself and not consonant with hisown insistence on the indispensability of empirical inquiry in naturalscience.

Against this background, the following passage inTopics I.2may have special importance:

It is also useful in connection with the first things concerning eachof the sciences. For it is impossible to say anything about thescience under consideration on the basis of its own principles, sincethe principles are first of all, and we must work our way throughabout these by means of what is generally accepted about each. Butthis is peculiar, or most proper, to dialectic: for since it isexaminative with respect to the principles of all the sciences, it hasa way to proceed.

A number of interpreters (beginning with Owen 1961) have built on thispassage and others to find dialectic at the heart of Aristotle’sphilosophical method. Further discussion of this issue would take usfar beyond the subject of this article (the fullest development is inIrwin 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, isa “counterpart” (antistrophos) of dialectic andthat 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 persuadeothers to accept certain conclusions on the basis of premises theyalready accept. Therefore, the same measures useful in dialecticalcontexts will, mutatis mutandis, be useful here: knowing what premisesan audience of a given type is likely to believe, and knowing how tofind premises from which the desired conclusion follows.

TheRhetoric does fit this general description: Aristotleincludes both discussions of types of person or audience (withgeneralizations about what each type tends to believe) and a summaryversion (in II.23) of the argument patterns discussed in theTopics. For further discussion of his rhetoric seeAristotle’s rhetoric.

10. Sophistical Arguments

Demonstrations and dialectical arguments are both forms of validargument, for Aristotle. However, he also studies what he callscontentious (eristikos) orsophistical arguments: these he defines as argumentswhich only apparently establish their conclusions. In fact, Aristotledefines these as apparent (but not genuine)dialecticalsullogismoi. They may have this appearance in either of twoways:

1. Arguments in which the conclusion only appears to follow ofnecessity from the premises (apparent, but not genuine,sullogismoi).
2. Genuinesullogismois the premises of which are merelyapparently, but not genuinely, acceptable.

Arguments of the first type in modern terms, appear to be valid butare really invalid. Arguments of the second type are at first moreperplexing: given that acceptability is a matter of what peoplebelieve, it might seem that whatever appears to beendoxosmust actually beendoxos. However, Aristotle probably has inmind arguments with premises that may at first glanceseem tobe acceptable but which, upon a moment’s reflection, weimmediately realize we do not actually accept. Consider this examplefrom 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 superficiallyplausible, is false. In fact, anyone with a little ability to followan argument will realize that at once upon seeing this veryargument.

Aristotle’s study of sophistical arguments is contained inOn Sophistical Refutations, which is actually a sort ofappendix to theTopics.

To a remarkable extent, contemporary discussions of fallaciesreproduce Aristotle’s own classifications. See Dorion 1995 forfurther 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 sciencesas its parts. “All things are not in a single genus”, hesays, “and even if they were, all beings could not fall underthe same principles” (On Sophistical Refutations 11).Thus, it is exactly the universal applicability of dialectic thatleads him to deny it the status of a science.

InMetaphysics IV (Γ), however, Aristotle takes whatappears to be a different view. First, he argues that there is, in away, a science that takes being as its genus (his name for it is“first philosophy”). Second, he argues that the principlesof this science will be, in a way, the first principles of all (thoughhe does not claim that the principles of other sciences can bedemonstrated from them). Third, he identifies one of its firstprinciples as the “most secure” of all principles: theprinciple of non-contradiction. As he states it,

It is impossible for the same thing to belong and not belongsimultaneously 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 afirst principle, it cannot be demonstrated; those who think otherwiseare “uneducated in analytics”. However, Aristotle thenproceeds to give what he calls a “refutativedemonstration” (apodeixai elenktikôs) of thisprinciple.

Further discussion of this principle and Aristotle’s argumentsconcerning it belong to a treatment of his metaphysics (seeAristotle: Metaphysics). However, it should be noted that: (1) these arguments draw onAristotle’s views about logic to a greater extent than anytreatise outside the logical works themselves; (2) in the logicalworks, the principle of non-contradiction is one of Aristotle’sfavorite illustrations of the “common principles”(koinai archai) that underlie the art of dialectic.

SeeAristotle’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 receivedperhaps the most intense discussion in recent decades isOnInterpretation 9, where Aristotle discusses the question whetherevery proposition about the future must be either true or false.Though something of a side issue in its context, the passage raises aproblem of great importance to Aristotle’s near contemporaries(and perhaps contemporaries).

Acontradiction (antiphasis) is a pair ofpropositions one of which asserts what the other denies. A major goalofOn Interpretation is to discuss the thesis that, of everysuch contradiction, one member must be true and the other false. Inthe course of his discussion, Aristotle allows for some exceptions.One case is what he callsindefinite propositionssuch as “A man is walking”: nothing prevents both thisproposition and “A man is not walking” beingsimultaneously true. This exception can be explained on relativelysimple grounds.

A different exception arises for more complex reasons. Consider thesetwo 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) isnow true, then theremust be asea-battle tomorrow, and therecannot fail to be a sea-battletomorrow. The result, according to this puzzle, is that nothing ispossible except what actually happens: there are no unactualizedpossibilities.

Such a conclusion is, as Aristotle is quick to note, a problem bothfor his own metaphysical views about potentialities and for thecommonsense notion that some things are up to us. He thereforeproposes another exception to the general thesis concerningcontradictory pairs.

This much would probably be accepted by most interpreters. What therestriction is, however, and just what motivates it are matters ofwide disagreement. It has been proposed, for instance, that Aristotleadopted, or at least flirted with, a three-valued logic for futurepropositions, or that he countenanced truth-value gaps, or that hissolution includes still more abstruse reasoning. The literature ismuch too complex to summarize: see Anscombe, Hintikka, D. Frede,Whitaker, Waterlow.

Historically, at least, it is likely that Aristotle is responding toan argument originating with the Megarian philosophers. He ascribesthe view that only that which happens is possible to the Megarians inMetaphysics IX (Θ). The puzzle with which he isconcerned strongly recalls the “Master Argument” ofDiodorus Cronus especially in certain further details. For instance, Aristotleimagines the statement about tomorrow’s sea battle having beenuttered ten thousand years ago. If it was true, then its truth was afact about the past; if the past is now unchangeable, then so is thetruth value of that past utterance. This recalls the MasterArgument’s premise that “what is past is necessary”.Diodorus Cronus was active a little after Aristotle, and he wascertainly influenced by Megarian views, whether or not it is correctto call him a Megarian (David Sedley 1977 argues that he was instead amember of theDialectical School which was in any event an offshoot of the Megarians; see Dorion 1995and Döring 1989, Ebert 2008 and the article Dialectical School).It is therefore likely that Aristotle’s target here is someMegarian argument, perhaps a forerunner of Diodorus’ MasterArgument.

13. Glossary of Aristotelian Terminology

• Accept:tithenai (in a dialectical argument)
• Accepted:endoxos (also ‘reputable’‘common belief’)
• Accident:sumbebêkos (seeincidental)
• Accidental:kata sumbebêkos
• Affirmation:kataphasis
• Affirmative:kataphatikos
• Assertion:apophansis (sentence with a truth value,declarative sentence)
• Assumption:hupothesis
• Belong:huparchein
• Category:katêgoria (see the discussion in Section7.3).
• 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:dialektikê (theart ofdialectic)
• Differentia:diaphora; specific difference,eidopoiosdiaphora
• Direct:deiktikos (of proofs; opposed to “throughthe impossible”)
• Essence:to ti esti,to ti ên einai
• Essential:en tôi ti esti (of predications)
• Extreme:akron (of the major and minor terms of adeduction)
• Figure:schêma
• Form:eidos (see also Species)
• Genus:genos
• Immediate:amesos (“without a middle”)
• Impossible:adunaton; “through theimpossible” (dia tou adunatou), of some proofs.
• Incidental: see Accidental
• Induction:epagôgê
• Middle, middle term (of a deduction):meson
• Negation (of a term):apophasis
• Objection:enstasis
• Particular:en merei,epi meros (of aproposition);kath’hekaston (of individuals)
• Peculiar, Peculiar Property:idios,idion
• Possible:dunaton,endechomenon;endechesthai (verb: “be possible”)
• Predicate:katêgorein (verb);katêegoroumenon (“what is predicated”)
• Predication:katêgoria (act or instance ofpredicating, type of predication)
• Primary:prôton
• Principle:archê (starting point of ademonstration)
• Quality:poion
• Reduce, Reduction:anagein,anagôgê
• Refute:elenchein; refutation,elenchos
• Science:epistêmê
• Species:eidos
• Specific:eidopoios (of a differentia that “makes aspecies”,eidopoios diaphora)
• Subject:hupokeimenon
• Substance:ousia
• Term:horos
• Universal:katholou (both of propositions and ofindividuals)

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Acknowledgments

I am indebted to Alan Code, Marc Cohen, and Theodor Ebert for helpfulcriticisms of earlier versions of this article. I thank FranzFritsche, Nikolai Biryukov, Ralph E. Kenyon, Johann Dirry, BenGreenberg, Hasan Masoud, Marc Michael Hämmerling, James Whitely,and edward@logicmuseum.com for calling my attention to errors.