Logic as a discipline starts with the transition from the more or lessunreflective use of logical methods and argument patterns to thereflection on and inquiry into these methods and patterns and theirelements, including the syntax and semantics of sentences. In Greekand Roman antiquity, discussions of some elements of logic and a focuson methods of inference can be traced back to the late 5^thcentury BCE. The Sophists, and later Plato (early 4^th c.)displayed an interest in sentence analysis, truth, and fallacies, andEubulides of Miletus (mid-4^th c.) is on record as theinventor of both the Liar and the Sorites paradox. But logic as afully systematic discipline begins with Aristotle, who systematizedmuch of the logical inquiry of his predecessors. His main achievementswere his theory of the logical interrelation of affirmative andnegative existential and universal statements and, based on thistheory, his syllogistic, which can be interpreted as a system ofdeductive inference. Aristotle’s logic is known as term-logic, sinceit is concerned with the logical relations between terms, such as‘human being’, ‘animal’,‘white’. It shares elements with both set theory andpredicate logic. Aristotle’s successors in his school, the Peripatos,notably Theophrastus and Eudemus, widened the scope of deductiveinference and improved some aspects of Aristotle’s logic.

In the Hellenistic period, and apparently independent of Aristotle’sachievements, the logician Diodorus Cronus and his pupil Philo (seethe entryDialectical school) worked out the beginnings of a logic that took propositions, ratherthan terms, as its basic elements. They influenced the second majortheorist of logic in antiquity, the Stoic Chrysippus(mid-3^rd c.), whose main achievement is the development ofa propositional logic, crowned by a deductive system. Regarded by manyin antiquity as the greatest logician, he was innovative in a largenumber of topics that are central to contemporary formal andphilosophical logic. The many close similarities between Chrysippus’philosophical logic and that of Gottlob Frege are especiallystriking. Chrysippus’ Stoic successors systematized his logic, andmade some additions.

The development of logic from c. 100 BCE to c. 250 CE remains mostlyin the dark, but there can be no doubt that logic was one of thetopics regularly studied and researched. At some point Peripateticsand Stoics began taking notice of each other’s logical systems, and wewitness some conflation of both terminologies and theories.Aristotelian syllogistic became known as ‘categoricalsyllogistic’ and the Peripatetic adaptation of Stoic syllogisticas ‘hypothetical syllogistic’. In the 2^ndcentury CE, Galen attempted to synthesize the two traditions; he alsoprofessed to have introduced a third kind of syllogism, the‘relational syllogism’, which apparently was meant to helpformalize mathematical reasoning. The attempt of some MiddlePlatonists (1^st c. BCE–2^nd c. CE) to claima specifically Platonic logic failed, and in its stead, theNeo-Platonists (3^rd–6^th c. CE) adopted ascholasticized version of Aristotelian logic as their own. In themonumental—if rarely creative—volumes of the Greekcommentators on Aristotle’s logical works we find elements of Stoicand later Peripatetic logic as well as Platonism, and ancientmathematics and rhetoric. Much the same holds for the Latin logicalwritings by Apuleius (2^nd c. CE) and Boethius(6^th c. CE), which pave the way for Aristotelian logic,thus supplemented, to enter the Medieval era.

• 1. Pre-Aristotelian Logic
  • 1.1 Syntax and Semantics
  • 1.2 Argument Patterns and Valid Inference
• 2. Aristotle
  • 2.1 Dialectics
  • 2.2 Sub-sentential Classifications
  • 2.3 Syntax and Semantics of Sentences
  • 2.4 Non-modal Syllogistic
  • 2.5 Modal Logic
• 3. The early Peripatetics: Theophrastus and Eudemus
  • 3.1 Improvements and Modifications of Aristotle’s Logic
  • 3.2 Prosleptic Syllogisms
  • 3.3 Forerunners ofModus Ponens andModus Tollens
  • 3.4 Wholly Hypothetical Syllogisms
• 4. Diodorus Cronus and Philo the Logician
• 5. The Stoics
  • 5.1 Logical Achievements Besides Propositional Logic
  • 5.2 Syntax and Semantics of Complex Propositions
  • 5.3 Arguments
  • 5.4 Stoic Syllogistic
  • 5.5 Logical Paradoxes
• 6. Epicurus and the Epicureans
• 7. Later Antiquity
• Bibliography
  • Greek and Latin Texts
  • Translations of Greek and Latin Texts
  • Secondary Literature
• Academic Tools
• Other Internet Resources
• Related Entries

1. Pre-Aristotelian Logic

1.1 Syntax and Semantics

Some of the Sophists classified types of sentences (logoi)according to their force. So Protagoras (485–415 BCE), whoincluded wish, question, answer and command (Diels Kranz (DK) 80.A1,Diogenes Laertius (D. L.) 9.53–4), and Alcidamas (pupil ofGorgias, fl. 4^th BCE), who distinguished assertion(phasis), denial (apophasis), question and address(prosagoreusis) (D. L. 9.54). Antisthenes(mid-5^th–mid-4^th cent.) defined a sentenceas ‘that which indicates what a thing was or is’(D. L. 6.3, DK 45) and stated that someone who says what is speakstruly (DK49). Perhaps the earliest surviving passage on logic is foundin theDissoi Logoi orDouble Arguments (DK 90.4,c. 400 BCE). It is evidence for a debate over truth andfalsehood. Opposed were the views (i) that truth isa—temporal—property of sentences, and that a sentence istrue (when it is said), if and only if things are as the sentence saysthey are when it is said, and false if they aren’t; and (ii) thattruth is an atemporal property of what is said, and that what is saidis true if and only if the things are the case, false if they aren’tthe case. These are rudimentary formulations of two alternativecorrespondence theories of truth. The same passage displays awarenessof the fact that self-referential use of the truth-predicate can beproblematic—an insight also documented by the discovery of theLiar paradox by Eubulides of Miletus (mid-4^th c. BCE)shortly thereafter.

Some Platonic dialogues contain passages whose topic is indubitablylogic. In theSophist, Plato analyzes simple statements ascontaining a verb (rhêma), which indicates action, anda noun(onoma), which indicates the agent (Soph.261e–262a). Anticipating the modern distinction of logicaltypes, he argues that neither a series of nouns nor a series of verbscan combine into a statement (Soph. 262a–d). Plato alsodivorces syntax (‘what is a statement?’) from semantics(‘when is it true?’). Something (e.g. ‘Theaetetus issitting’) is a statement if it both succeeds in specifying asubject and says something about this subject. Plato thus determinessubject and predicate as relational elements in a statement andexcludes as statements subject-predicate combinations containing emptysubject expressions. Something is a true statement if with referenceto its subject (Theaetetus) it says of what is (e.g. sitting) that itis. Something is a false statement if with reference to its subject itsays of something other than what is (e.g. flying), that it is. HerePlato produces a sketch of a deflationist theory of truth(Soph. 262e–263d; cf.Crat. 385b). He alsodistinguished negations from affirmations and took the negationparticle to have narrow scope: it negates the predicate, not the wholesentence (Soph. 257b–c). There are many passages inPlato where he struggles to explain certain logical relations:for example his theory that things participate in Forms corresponds toa rudimentary theory of predication; in theSophist andelsewhere he grapples with the class relations of exclusion, union andco-extension; also with the difference between the ‘is’ ofpredication (being) and the ‘is’ of identity (sameness);and inRepublic 4, 436bff., he anticipates the law ofnon-contradiction. But his explications of these logical questions arecast in metaphysical terms, and so can at most be regarded asproto-logical.

1.2 Argument Patterns and Valid Inference

Pre-Aristotelian evidence for reflection on argument forms and validinference are harder to come by. Both Zeno of Elea (born c. 490 BCE)and Socrates (470–399) were famous for the ways in which theyrefuted an opponent’s view. Their methods display similarities withreductio ad absurdum, but neither of them seems to havetheorized about their logical procedures. Zeno produced arguments(logoi) that manifest variations of the pattern ‘this(i.e. the opponent’s view) only if that. But that is impossible. Sothis is impossible’. Socratic refutation was an exchange ofquestions and answers in which the opponents would be led, on thebasis of their answers, to a conclusion incompatible with theiroriginal claim. Plato institutionalized such disputations intostructured, rule-governed verbal contests that became known asdialectical argument. The development of a basic logical vocabularyfor such contests indicates some reflection upon the patterns ofargumentation.

The 5^th and early to mid-4^th centuries BCE alsosee great interest in fallacies and logical paradoxes. Besides theLiar, Eubulides is said to have been the originator of several otherlogical paradoxes, including the Sorites. Plato’sEuthydemuscontains a large collection of contemporary fallacies. In attempts tosolve such logical puzzles, a logical terminology develops here, too,and the focus on the difference between valid and invalid argumentssets the scene for the search for a criterion of validinference. Finally, it is possible that the shaping of deduction andproof in Greek mathematics that begins in the later 5^thcentury BCE served as an inspiration for Aristotle’s syllogistic.

2. Aristotle

(For a more detailed account see the entry onAristotle’s Logic in this encyclopedia.) Aristotle is the first great logician in thehistory of logic. His logic was taught by and large without rival fromthe 4^th to the 19^th centuries CE. Aristotle’slogical works were collected and put in a systematic order by laterPeripatetics, who entitled them theOrganon or‘tool’, because they considered logic not a part butrather an instrument of philosophy. TheOrganon contains, intraditional order, theCategories,DeInterpretatione,Prior Analytics,PosteriorAnalytics,Topics andSophistical Refutations.In addition,Metaphysics Γ is a logical treatise thatdiscusses the principle of non-contradiction, and some further logicalinsights are found scattered throughout Aristotle’s other works, suchas thePoetics,Rhetoric,De Anima,Metaphysics Δ and Θ, and some of the biologicalworks. Some parts of theCategories andPosteriorAnalytics would today be regarded as metaphysics, epistemology orphilosophy of science rather than logic. The traditional arrangementof works in theOrganon is neither chronological norAristotle’s own. The original chronology cannot be fully recoveredsince Aristotle seems often to have inserted supplements into earlierwritings at a later time. However, by using logical advances as a criterion, we can conjecture that most of theTopics,Sophistical Refutations,Categories andMetaphysics Γ predate theDe Interpretatione,which in turn predates thePrior Analytics and parts of thePosterior Analytics.

2.1 Dialectics

TheTopics provide a manual for participants in the contestsof dialectical argument as instituted in the Academy by Plato. Books2–7 provide general procedures or rules (topoi) abouthow to find an argument to establish or refute a given thesis. Thedescriptions of these procedures—some of which are so generalthat they resemble logical laws—clearly presuppose a notion oflogical form, and Aristotle’sTopics may thus count as theearliest surviving logical treatise. TheSophisticalRefutations are the first systematic classification of fallacies,sorted by what logical flaw each type manifests (e.g. equivocation,begging the question, affirming the consequent,secundumquid) and how to expose them.

2.2 Sub-sentential Classifications

Aristotle distinguishes things that have sentential unity through acombination of expressions (‘a horse runs’) from thosethat do not (‘horse’, ‘runs’); the latter aredealt with in theCategories (the title really means ‘predications’^[1]). They have no truth-value and signify one of the following: substance(ousia), quantity (poson), quality (poion),relation (pros ti), location (pou), time(pote), position (keisthai), possession(echein), doing (poiein) and undergoing(paschein). It is unclear whether Aristotle considers thisclassification to be one of linguistic expressions that can bepredicated of something else; or of kinds of predication; or ofhighest genera. InTopics 1 Aristotle distinguishes fourrelationships a predicate may have to the subject: it may give itsdefinition, genus, unique property, or accidental property. These areknown as predicables.

2.3 Syntax and Semantics of Sentences

When writing theDe Interpretatione, Aristotle had worked outthe following theory of simple sentences: a (declarative) sentence(apophantikos logos) or declaration (apophansis) isdelimited from other pieces of discourse like prayer, command andquestion by its having a truth-value. The truth-bearers that featurein Aristotle’s logic are thus linguistic items. They are spokensentences that directly signify thoughts (shared by all humans) andthrough these, indirectly, things. Written sentences in turn signifyspoken ones. (Simple) sentences are constructed from two signifyingexpressions which stand in subject-predicate relation to each other: aname and a verb (‘Callias walks’) or two names connectedby the copula ‘is’, which co-signifies the connection(‘Pleasure is good’) (Int. 3). Names are eithersingular terms or common nouns (An. Pr. I 27). Both can beempty (Cat. 10,Int. 1). Singular terms can onlytake subject position. Verbs co-signify time. A name-verb sentence canbe rephrased with the copula (‘Callias is (a) walking(thing)’) (Int. 12). As to their quality, a(declarative) sentence is either an affirmation or a negation,depending on whether it affirms or negates its predicate of itssubject. The negation particle in a negation has wide scope(Cat. 10). Aristotle defined truth separately foraffirmations and negations: An affirmation is true if it says of thatwhich is that it is; a negation is true if it says of that which isnot that it is not (Met. Γ.7 1011b25ff). Theseformulations, or in any case their Greek counterparts, can beinterpreted as expressing either a correspondence or a deflationistconception of truth. Either way, truth is a property that belongs to asentenceat a given time. As to their quantity, sentences aresingular, universal, particular or indefinite. Thus Aristotle obtainseight types of sentences, which are later dubbed ‘categoricalsentences’. The following are examples, paired by quality:

Universal and particular sentences contain a quantifier and bothuniversal and particular affirmatives were taken to have existentialimport. (See entryThe Traditional Square of Opposition). The logicalstatus of the indefinites is ambiguous and controversial(Int. 6–7).

Aristotle distinguishes between two types of sentential opposition:contraries and contradictories. A contradictory pair of sentences (anantiphasis) consists of an affirmation and its negation (i.e.the negation that negates of the subject what the affirmation affirmsof it). Aristotle assumes that—normally—one of these mustbe true, the other false. Contrary sentences are such that they cannotboth be true. The contradictory of a universal affirmative is thecorresponding particular negative; that of the universal negative thecorresponding particular affirmative. A universal affirmative and itscorresponding universal negative are contraries. Aristotle thus hascaptured the basic logical relations between monadic quantifiers(Int. 7).

Since Aristotle regards tense as part of the truth-bearer (as opposedto merely a grammatical feature), he detects a problem regardingfuture tense sentences about contingent matters: Does the principlethat of an affirmation and its negation one must be false, the othertrue, apply to these? What, for example, is the truth-value now of thesentence ‘There will be a sea-battle tomorrow’? Aristotlemay have suggested that the sentence has no truth-value now, and thatbivalence thus does not hold—despite the fact that it isnecessary for there either to be or not to be a sea-battle tomorrow,so that the principle of excluded middle is preserved (Int.9).

2.4 Non-modal Syllogistic

Aristotle’s non-modal syllogistic (Prior Analytics A1–7) is the pinnacle of his logic. Aristotle defines a syllogismas ‘an argument (logos) in which, certain things havingbeen laid down, something different from what has been laid downfollows of necessity because these things are so’. Thisdefinition appears to require (i) that a syllogism consists of atleast two premises and a conclusion, (ii) that the conclusion followsof necessity from the premises (so that all syllogisms arevalid arguments), and (iii) that the conclusion differs fromthe premises. Aristotle’s syllogistic covers only a small part of allarguments that satisfy these conditions.

Aristotle restricts and regiments the types of categorical sentencethat may feature in a syllogism. The admissible truth-bearers are nowdefined as each containing two different terms (horoi)conjoined by the copula, of which one (the predicate term) is said ofthe other (the subject term) either affirmatively or negatively.Aristotle never comes clear on the question whether terms are things(e.g., non-empty classes) or linguistic expressions for these things.Only universal and particular sentences are discussed. Singularsentences seem to be excluded and indefinite sentences are mostlyignored. AtAn. Pr. A 7 Aristotle mentions that by substituting anindefinite premise for a particular, one obtains a syllogism ofthe same kind.

Another innovation in the syllogistic is Aristotle’s use of letters inplace of terms. The letters may originally have served simply asabbreviations for terms (e.g.An. Post. A 13); but in thesyllogistic they seem mostly to have the function either of schematicterm letters or of term variables with universal quantifiers assumedbut not stated. Where he uses letters, Aristotle tends to express thefour types of categorical sentences in the following way (with commonlater abbreviations in parentheses):

Instead of ‘holds’ he also uses ‘ispredicated’.

All basic syllogisms consist of three categorical sentences, in whichthe two premises share exactly one term, called the middle term, andthe conclusion contains the other two terms, sometimes called theextremes. Based on the position of the middle term, Aristotleclassified all possible premise combinations into three figures(schêmata): the first figure has the middle term (B)as subject in the first premise and predicated in the second; thesecond figure has it predicated in both premises, the third has it assubject in both premises:

A is also called the major term,C the minorterm. Each figure can further be classified according to whether ornot both premises are universal. Aristotle went systematically throughthe fifty-eight possible premise combinations and showed that fourteenhave a conclusion following of necessity from them, i.e. aresyllogisms. His procedure was this: He assumed that the syllogisms ofthe first figure are complete and not in need of proof, since they areevident. By contrast, the syllogisms of the second and third figuresare incomplete and in need of proof. He proves them by reducing themto syllogisms of the first figure and thereby ‘completing’them. For this he makes use of three methods:

1. conversion (antistrophê): a categorical sentence isconverted by interchanging its terms. Aristotle recognizes andestablishes three conversion rules: ‘fromAeBinferBeA’; ‘fromAiBinferBiA’ and ‘fromAaB inferBiA’. All but twosecond and third figure syllogisms can be proved by premiseconversion.
2. reductio ad impossibile (apagôgê):the remaining two are proved by reduction to the impossible, where thecontradictory of an assumed conclusion together with one of thepremises is used to deduce by a first figure syllogism a conclusionthat is incompatible with the other premise. Using the semanticrelations between opposites established earlier the assumed conclusionis thus established.
3. exposition or setting-out (ekthesis): this method, whichAristotle uses in addition to (i) and (ii), involves choosing or‘setting out’ some additional term, sayD, thatfalls in the non-empty intersection delimited by two premises,sayAxB andAxC,and usingD to justify the inference from the premises to aparticular conclusion,BxC. It is debatedwhether ‘D’ represents a singular or a generalterm and whether exposition constitutes proof.

For each of the thirty-four premise combinations that allow noconclusion Aristotle proves by counterexample that they allow noconclusion. As his overall result, he acknowledges four first figuresyllogisms (later named Barbara, Celarent, Darii, Ferio), four secondfigure syllogisms (Camestres, Cesare, Festino, Baroco) and six thirdfigure syllogisms (Darapti, Felapton, Disamis, Datisi, Bocardo,Ferison); these were later called the modes or moods of the figures.(The names are mnemonics: e.g. each vowel, or the first three in caseswhere the name has more than three, indicates in order whether thefirst and second premises and the conclusion were sentences of typea,e,i oro.) Aristotleimplicitly recognized that by using the conversion rules on theconclusions we obtain eight further syllogisms (An. Pr.53a3–14), and that of the premise combinations rejected asnon-syllogistic, some (five, in fact) will yield a conclusion in whichthe minor term is predicated of the major (An. Pr.29a19–27). Moreover, in theTopics Aristotle acceptedthe rules ‘fromAaB inferAiB’ and ‘fromAeBinferAoB’. By using these on the conclusionsfive further syllogisms could be proved, though Aristotle did notmention this.

Going beyond his basic syllogistic, Aristotle reduced the3^rd and 4^th first figure syllogisms to secondfigure syllogisms, thusde facto reducing all syllogisms toBarbara and Celarent; and later on in thePrior Analytics heinvokes a type of cut-rule by which a multi-premise syllogism can bereduced to two or more basic syllogisms. From a modern perspective,Aristotle’s system can be understood as a sequent logic in the styleof natural deduction and as a fragment of first-order logic. It hasbeen shown to be sound and complete if one interprets the relationsexpressed by the categorical sentences set-theoretically as a systemof non-empty classes as follows:AaB is true if andonly if the classA contains the classB.AeB is true if and only if the classesA andB are disjoint.AiB is trueif and only if the classesA andB are notdisjoint.AoB is true if and only if the classA does not contain the classB. It is generallyagreed, though, that Aristotle’s syllogistic is a kind of relevancelogic rather than classical. The vexing textual question what exactlyAristotle meant by ‘syllogisms’ has received several rivalinterpretations, including one that they are a certain type ofconditional propositional form. Most plausibly, perhaps, Aristotle’scomplete andincomplete syllogisms taken together should beunderstood as formally valid premise-conclusion arguments; and hiscomplete andcompleted syllogisms taken together as (sound)deductions.

2.5 Modal Logic

Aristotle is also the originator of modal logic. In addition toquality (as affirmation or negation) and quantity (as singular,universal, particular, or indefinite), he takes categorical sentencesto have a mode; this consists of the fact that the predicate is saidto hold of the subject either actually or necessarily or possibly orcontingently or impossibly. The latter four are expressed by modaloperators that modify the predicate, e.g. ‘It is possible forA to hold of someB’; ‘Anecessarily holds of everyB’.

InDe Interpretatione 12–13, Aristotle (i) concludesthat modal operators modify the whole predicate (or the copula, as heputs it), not just the predicate term of a sentence. (ii) He statesthe logical relations that hold between modal operators, such as that‘it is not possible forA not to hold ofB’ implies ‘it is necessary forA tohold ofB’. (iii) He investigates what thecontradictories of modalized sentences are, and decides that they areobtained by placing the negator in front of the modal operator. (iv)He equates the expressions ‘possible’ and‘contingent’, but wavers between a one-sidedinterpretation (where necessity implies possibility) and a two-sidedinterpretation (where possibility implies non-necessity).

Aristotle develops his modal syllogistic inPrior Analytics1.8–22. He settles on two-sided possibility (contingency) andtests for syllogismhood all possible combinations of premise pairs ofsentences with necessity (N), contingency (C) or no (U) modaloperator: NN, CC, NU/UN, CU/UC and NC/CN. Syllogisms with the lastthree types of premise combinations are called mixed modalsyllogisms. Apart from the NN category, which mirrors unmodalizedsyllogisms, all categories contain dubious cases. For instance,Aristotle accepts:

A necessarily holds of allB.
B holds of allC.
ThereforeA necessarily holds of allC.

This and other problematic cases were already disputed in antiquity,and more recently have sparked a host of complex formalizedreconstructions of Aristotle’s modal syllogistic. As Aristotle’stheory is conceivably internally inconsistent, the formal models thathave been suggested may all be unsuccessful.

3. The early Peripatetics: Theophrastus and Eudemus

Aristotle’s pupil and successor Theophrastus of Eresus(c. 371–c. 287 BCE) wrote more logical treatises than histeacher, with a large overlap in topics. Eudemus of Rhodes (later4^th cent. BCE) wrote books entitledCategories,Analytics andOn Speech. Of all these works only anumber of fragments and later testimonies survive, mostly in commentators on Aristotle. Theophrastus and Eudemus simplified some aspects ofAristotle’s logic, and developed others where Aristotle left us onlyhints.

3.1 Improvements on and Modifications of Aristotle’s Logic

The two Peripatetics seem to have redefined Aristotle’s first figure,so that it includes every syllogism in which the middle term issubject of one premise and predicate of the other. In this way, fivetypes of non-modal syllogisms only intimated by Aristotle later in hisPrior Analytics (Baralipton, Celantes, Dabitis, Fapesmo andFrisesomorum) are included, but Aristotle’s criterion that firstfigure syllogisms are evident is given up (Theophrastus fr. 91,Fortenbaugh). Theophrastus and Eudemus also improved Aristotle’s modaltheory. Theophrastus replaced Aristotle’s two-sided contingency with one-sided possibility, so that possibility no longer entailsnon-necessity. Both recognized that the problematic universal negative(‘A possibly holds of noB’) is simplyconvertible (Theophrastus fr. 102A Fortenbaugh). Moreover, theyintroduced the principle that in mixed modal syllogisms the conclusionalways has the same modal character as the weaker of the premises(Theophrastus frs. 106 and 107 Fortenbaugh), where possibility isweaker than actuality, and actuality than necessity. In this wayAristotle’s modal syllogistic is notably simplified and manyunsatisfactory theses, like the one mentioned above (that from‘NecessarilyAaB’ and‘BaC’ one can infer ‘NecessarilyAaC’) disappear.

3.2 Prosleptic Syllogisms

Theophrastus introduced the so-called prosleptic premises andsyllogisms (Theophrastus fr. 110 Fortenbaugh). A prosleptic premise isof the form:

For allX, if Φ(X), then Ψ(X)

where Φ(X) and Ψ(X) stand for categoricalsentences in which the variableX occurs in place of one ofthe terms. For example:

1. A [holds] of all of that of all ofwhichB [holds].
2. A [holds] of none of that which[holds] of allB.

Theophrastus considered such premises to contain three terms, two ofwhich are definite (A,B), one indefinite(‘that’, or the bound variableX). We canrepresent (1) and (2) as

∀X (BaX →AaX)

∀X (XaB →AeX)

Prosleptic syllogisms then come about as follows: They are composed ofa prosleptic premise and the categorical premise obtained byinstantiating a term (C) in the antecedent ‘opencategorical sentence’ as premises, and the categorical sentencesone obtains by putting in the same term (C) in the consequent‘open categorical sentence’ as conclusion. Forexample:

A [holds] of all of that of all of whichB[holds].
B holds of allC.
Therefore,A holds of allC.

Theophrastus distinguished three figures of these syllogisms,depending on the position of the indefinite term (also called‘middle term’) in the prosleptic premise; for example (1)produces a third figure syllogism, (2) a first figure syllogism. Thenumber of prosleptic syllogisms was presumably equal to that of typesof prosleptic sentences: with Theophrastus’ concept of the firstfigure these would be sixty-four (i.e. 32 + 16 + 16). Theophrastusheld that certain prosleptic premises were equivalent to certaincategorical sentences, e.g. (1) to ‘A is predicated ofallB’. However, for many, including (2), no suchequivalent can be found, and prosleptic syllogisms thus increased theinferential power of Peripatetic logic.

3.3 Forerunners ofModus Ponens andModus Tollens

Theophrastus and Eudemus considered complex premises which they called‘hypothetical premises’ and which had one of the followingtwo (or similar) forms:

If something isF, it isG

Either something isF or it isG    (withexclusive ‘or’)

They developed arguments with them which they called ‘mixed froma hypothetical premise and a probative premise’ (Theophrastusfr. 112A Fortenbaugh). These arguments were inspired by Aristotle’ssyllogisms ‘from a hypothesis’ (An. Pr. 1.44);they were forerunners ofmodus ponens andmodustollens and had the following forms (Theophrastus frs. 111 and112 Fortenbaugh), employing the exclusive ‘or’:

Theophrastus also recognized that the connective particle‘or’ can be inclusive (Theophrastus fr. 82A Fortenbaugh);and he considered relative quantified sentences such as thosecontaining ‘more’, ‘fewer’, and ‘thesame’ (Theophrastus fr. 89 Fortenbaugh), and seems to havediscussed syllogisms built from such sentences, again following upupon what Aristotle said about syllogisms from a hypothesis(Theophrastus fr. 111E Fortenbaugh).

3.4 Wholly Hypothetical Syllogisms

Theophrastus is further credited with the invention of a system of thelater so-called ‘wholly hypothetical syllogisms’(Theophrastus fr. 113 Fortenbaugh). These syllogisms were originallyabbreviated term-logical arguments of the kind

If [something is]A, [it is]B.
If [something is]B, [it is]C.
Therefore, if [something is]A, [it is]C.

and at least some of them were regarded as reducible to Aristotle’scategorical syllogisms, presumably by way of the equivalences to‘EveryA isB’, etc. In parallel toAristotle’s syllogistic, Theophrastus distinguished three figures;each had sixteen modes. The first eight modes of the first figure areobtained by going through all permutations with ‘notX’ instead of ‘X’ (withXforA,B,C); the second eight modes areobtained by using a rule of contraposition on the conclusion:

(CR)

From ‘ifX,Y’ infer‘if the contradictory ofY then the contradictory ofX’

The sixteen modes of the second figure were obtained by using (CR) onthe schema of the first premise of the first figure arguments,e.g.

If [something is] notB, [it is] notA.
If [something is]B, [it is]C.
Therefore, if [something is]A, [it is]C.

The sixteen modes of the third figure were obtained by using (CR) onthe schema of the second premise of the first figure arguments,e.g.

If [something is]A, [it is]B.
If [something is] notC, [it is] notB.
Therefore, if [something is]A, [it is]C.

Theophrastus claimed that all second and third figure syllogisms couldbe reduced to first figure syllogisms. If Alexander of Aphrodisias(2^nd c. CE Peripatetic) reports faithfully, any use of (CR)which transforms a syllogism into a first figure syllogism was such areduction. The large number of modes and reductions can be explainedby the fact that Theophrastus did not have the logical means forsubstituting negative for positive components in an argument. In laterantiquity, after some intermediate stages, and possibly under Stoicinfluence, the wholly hypothetical syllogisms were interpreted aspropositional-logical arguments of the kind

Ifp, thenq.
Ifq, thenr.
Therefore, ifp, thenr.

4. Diodorus Cronus and Philo the Logician

In the later 4^th to mid 3^rd centuries BCE,contemporary with Theophrastus and Eudemus, a loosely connected groupof philosophers, sometimes referred to as dialecticians (see entry‘Dialectical School’) and possibly influenced byEubulides, conceived of logic as a logic of propositions. Their bestknown exponents were Diodorus Cronus and his pupil Philo (sometimescalled ‘Philo of Megara’). Although no writings of theirsare preserved, there are a number of later reports of theirdoctrines. They each made groundbreaking contributions to thedevelopment of propositional logic, in particular to the theories ofconditionals and modalities.

A conditional (sunêmmenon) was considered anon-simple proposition composed of two propositions and the connectingparticle ‘if’. Philo, who may be credited with introducingtruth-functionality into logic, provided the following criterion fortheir truth: A conditional is falsewhen andonlywhen its antecedent is true and its consequent is false, andit is true in the three remaining truth-value combinations. ThePhilonian conditional resembles material implication, exceptthat—since propositions were conceived of as functions of timethat can have different truth-values at different times—it maychange its truth-value over time. For Diodorus, a conditionalproposition is true if it neither was nor is possible that itsantecedent is true and its consequent false. The temporal elements inthis account suggest that the possibility of a truth-value change inPhilo’s conditionals was meant to be improved on. With his own modalnotions (see below) applied, a conditional is Diodorean-true now ifand only if it is Philonian-true at all times. Diodorus’ conditionalis thus reminiscent of strict implication. Philo’s and Diodorus’conceptions of conditionals lead to variants of the‘paradoxes’ of material and strict implication—afact the ancients were aware of (Sextus Empiricus [S. E.]M.8.109–117).

Philo and Diodorus each considered the four modalities possibility,impossibility, necessity and non-necessity. These were conceived of asmodal properties or modal values of propositions, not as modaloperators. Philo defined them as follows: ‘Possible is thatwhich is capable of being true by the proposition’s own nature… necessary is that which is true, and which, as far as it isin itself, is not capable of being false. Non-necessary is that whichas far as it is in itself, is capable of being false, and impossibleis that which by its own nature is not capable of being true.’Diodorus’ definitions were these: ‘Possible is that which eitheris or will be [true]; impossible that which is false and willnot be true; necessary that which is true and will not be false;non-necessary that which either is false already or will befalse.’ Both sets of definitions satisfy the following standardrequirements of modal logic: (i) necessity entails truth and truthentails possibility; (ii) possibility and impossibility arecontradictories, and so are necessity and non-necessity; (iii)necessity and possibility are interdefinable; (iv) every propositionis either necessary or impossible or both possible andnon-necessary. Philo’s definitions appear to introduce mere conceptualmodalities, whereas with Diodorus’ definitions, some propositions maychange their modal value (Boeth.In Arist. De Int., sec. ed.,234–235 Meiser).

Diodorus’ definition of possibility rules out future contingents andimplies the counterintuitive thesis that only the actual ispossible. Diodorus tried to prove this claim with his famous MasterArgument, which sets out to show the incompatibility of (i)‘every past truth is necessary’, (ii) ‘theimpossible does not follow from the possible’, and (iii)‘something is possible which neither is nor will be true’(Epict.Diss. II.19). The argument has not survived, butvarious reconstructions have been suggested. Some affinity with thearguments for logical determinism in Aristotle’sDeInterpretatione 9 is likely.

On the topic of ambiguity, Diodorus held that no linguistic expressionis ambiguous. He supported this dictum by a theory of meaning based onspeaker intention. Speakers generally intend to say only one thingwhen they speak. What is said when they speak is what they intend tosay. Any discrepancy between speaker intention and listener decodinghas its cause in the obscurity of what was said, not its ambiguity(Aulus Gellius 11.12.2–3).

5. The Stoics

The founder of the Stoa, Zeno of Citium (335–263 BCE), studiedwith Diodorus. His successor Cleanthes (331–232) tried to solvethe Master Argument by denying that every past truth is necessary andwrote books—now lost—on paradoxes, dialectics, argumentmodes and predicates. Both philosophers considered knowledge of logicas a virtue and held it in high esteem, but they seem not to have beencreative logicians. By contrast, Cleanthes’ successor Chrysippus ofSoli (c. 280–207) is without doubt the second great logician inthe history of logic. It was said of him that if the gods used anylogic, it would be that of Chrysippus (D. L. 7.180), and his reputationas a brilliant logician is amply attested. Chrysippus wrote over 300books on logic, on virtually every topic logic today concerns itselfwith, including speech act theory, sentence analysis, singular andplural expressions, types of predicates, indexicals, existentialpropositions, sentential connectives, negations, disjunctions,conditionals, logical consequence, valid argument forms, theory ofdeduction, propositional logic, modal logic, tense logic, epistemiclogic, logic of suppositions, logic of imperatives, ambiguity andlogical paradoxes, in particular the Liar and the Sorites(D. L. 7.189–199). Of all these, only two badly damaged papyrihave survived, luckily supplemented by a considerable number offragments and testimonies in later texts, in particular in DiogenesLaertius (D. L.) book 7, sections 55–83, and Sextus EmpiricusOutlines of Pyrrhonism (S. E.PH) book 2 andAgainst the Mathematicians (S. E.M) book8. Chrysippus’ successors, including Diogenes of Babylon (c.240–152) and Antipater of Tarsus (2^nd cent. BCE),appear to have systematized and simplified some of his ideas, buttheir original contributions to logic seem small. Many testimonies ofStoic logic do not name any particular Stoic. Hence the followingparagraphs simply talk about ‘the Stoics’ in general; butwe can be confident that a large part of what has survived goes backto Chrysippus.

5.1 Logical Achievements Besides Propositional Logic

The subject matter of Stoic logic is the so-called sayables(lekta): they are the underlying meanings in everything wesay and think, but—like Frege’s ‘senses’—also subsist independently of us. They are distinguished from spoken and writtenlinguistic expressions: what weutter are those expressions,but what wesay are the sayables (D. L. 7.57). There arecomplete and deficient sayables. Deficient sayables, if said, make thehearer feel prompted to ask for a completion; e.g. when someone says‘writes’ we enquire ‘who?’. Complete sayables,if said, do not make the hearer ask for a completion (D. L.7.63). Theyinclude assertibles (the Stoic equivalent of propositions),imperativals, interrogatives, inquiries, exclamatives, hypotheses orsuppositions, stipulations, oaths, curses and more. The accounts ofthe different complete sayables all had the general form ‘aso-and-so sayable is one in saying which we perform an act ofsuch-and-such’. For instance: ‘an imperatival sayable isone in saying which we issue a command’, ‘an interrogativesayable is one in saying which we ask a question’, ‘adeclaratory sayable (i.e. an assertible) is one in saying which wemake an assertion’. Thus, according to the Stoics, each time wesay a complete sayable, we perform three different acts: we utter alinguistic expression; we say the sayable; and we perform aspeech-act. Chrysippus was aware of the use-mention distinction(D. L. 7.187). He seems to have held that every denoting expression isambiguous in that it denotes both its denotation and itself (Galen,On ling. soph. 4; Aulus Gellius 11.12.1). Thus theexpression ‘a wagon’ would denote both a wagon and theexpression ‘a wagon’.^[2]

Assertibles (axiômata) differ from all other completesayables in their having a truth-value: at any one time they areeither true or false. Truth is temporal and assertibles may changetheir truth-value. The Stoic principle of bivalence is hencetemporalized, too. Truth is introduced by example: the assertible‘it is day’ is truewhen it is day, and at allother times false (D. L. 7.65). This suggests some kind of deflationistview of truth, as does the fact that the Stoics identify trueassertibles with facts, but define false assertibles simply as thecontradictories of true ones (S. E.M 8.85).

Assertibles are simple or non-simple. A simplepredicativeassertible like ‘Dion is walking’ is generated from thepredicate ‘is walking’, which is a deficient assertiblesince it elicits the question ‘who?’, together with anominative case (Dion’s individual quality or the correlated sayable),which the assertible presents as falling under the predicate(D. L. 7.63 and 70). There is thus no interchangeability of predicateand subject terms as in Aristotle; rather, predicates—but notthe things that fall under them—are defined as deficient, andthus resemble propositional functions. It seems that whereas someStoics took the—Fregean—approach that singular terms hadcorrelated sayables, others anticipated the notion of directreference. Concerning indexicals, the Stoics took a simpledefinite assertible like ‘this one is walking’ tobe true when the person pointed at by the speaker is walking (S. E.M 100). When the thing pointed at ceases to be, so does theassertible, though the sentence used to express it remains(Alex. Aphr.An. Pr. 177–8). A simpleindefinite assertible like ‘someone is walking’is said to be true when a corresponding definite assertible is true(S. E.M 98). Aristotelian universal affirmatives(‘EveryA isB’) were to be rephrased asconditionals: ‘If something isA, it isB’ (S. E.M 9.8–11). Negations of simpleassertibles are themselves simple assertibles. The Stoic negation of‘Dion is walking’ is ‘(It is) not (the case that)Dion is walking’, and not ‘Dion is not walking’. Thelatter is analyzed in a Russellian manner as ‘Both Dion existsand not: Dion is walking’ (Alex. Aphr.An. Pr.402). There are present tense, past tense and future tenseassertibles. The—temporalized—principle of bivalenceholds for them all. The past tense assertible ‘Dionwalked’ is true when there is at least one past time at which‘Dion is walking’ was true.

5.2 Syntax and Semantics of Complex Propositions

Thus the Stoics concerned themselves with several issues we wouldplace under the heading of predicate logic; but their main achievementwas the development of a propositional logic, i.e. of a system ofdeduction in which the smallest substantial unanalyzed expressions arepropositions, or rather, assertibles.

The Stoics defined negations as assertibles that consist of a negativeparticle and an assertible controlled by this particle (S. E.M8.103). Similarly, non-simple assertibles were defined asassertibles that either consist of more than one assertible or of oneassertible taken more than once (D. L. 7.68–9) and that arecontrolled by a connective particle. Both definitions can beunderstood as being recursive and allow for assertibles ofindeterminate complexity. Three types of non-simple assertiblesfeature in Stoic syllogistic. Conjunctions are non-simple assertiblesput together by the conjunctive connective ‘both … and…’. They have two conjuncts.^[3] Disjunctions are non-simple assertibles put together by thedisjunctive connective ‘either … or … or…’. They have two or more disjuncts, all on apar. Conditionals are non-simple assertibles formed with theconnective ‘if …, …’; they consist ofantecedent and consequent (D. L. 7.71–2). What type of assertiblean assertible is, is determined by the connective or logical particlethat controls it, i.e. that has the largest scope. ‘Both notp andq’ is a conjunction, ‘Not bothp andq’ a negation. Stoic languageregimentation asks that sentences expressing assertibles always startwith the logical particle or expression characteristic for theassertible. Thus, the Stoics invented an implicit bracketing devicesimilar to that used in Łukasiewicz’ Polish notation.

Stoic negations and conjunctions are truth-functional. Stoic (or atleast Chrysippean) conditionals are true when the contradictory of theconsequent is incompatible with its antecedent (D. L. 7.73). Twoassertibles are contradictories of each other if one is the negationof the other (D. L. 7.73); that is, when one exceeds the other bya—pre-fixed—negation particle (S. E.M 8.89). Thetruth-functional Philonian conditional was expressed as a negation ofa conjunction: that is, not as ‘ifp,q’but as ‘not bothp and notq’. Stoicdisjunction is exclusive and non-truth-functional. It is true whennecessarily precisely one of its disjuncts is true. Later Stoicsintroduced a non-truth-functional inclusive disjunction (AulusGellius,N. A. 16.8.13–14).

Like Philo and Diodorus, Chrysippus distinguished four modalities andconsidered them modal values of propositions rather than modaloperators; they satisfy the same standard requirements of modal logic.Chrysippus’ definitions are (D. L. 7.75): An assertible is possiblewhen it is both capable of being true and not hindered by externalthings from being true. An assertible is impossible when it is[either] not capable of being true [or is capable of beingtrue, but hindered by external things from being true]. Anassertible is necessary when, being true, it either is not capable ofbeing false or is capable of being false, but hindered by externalthings from being false. An assertible is non-necessary when it isboth capable of being false and not hindered by external things[from being false]. Chrysippus’ modal notions differ fromDiodorus’ in that they allow for future contingents and from Philo’sin that they go beyond mere conceptual possibility.

5.3 Arguments

Arguments are—normally—compounds of assertibles. They aredefined as a system of at least two premises and a conclusion(D. L. 7.45). Syntactically, every premise but the first is introducedby ‘now’ or ‘but’, and the conclusion by‘therefore’. An argument is valid if the (Chrysippean)conditional formed with the conjunction of its premises as antecedentand its conclusion as consequent is correct (S. E.PH 2.137;D. L. 7.77). An argument is ‘sound’ (literally:‘true’), when in addition to being valid it has truepremises. The Stoics defined so-called argument modes as a sort ofschema of an argument (D. L. 7.76). The mode of an argument differs fromthe argument itself by having ordinal numbers taking the place ofassertibles. The mode of the argument

If it is day, it is light.
But it is not the case that it is light.
Therefore it is not the case that it is day.

is

If the 1^st, the 2^nd.
But not: the 2^nd.
Therefore not: the 1^st.

The modes functioned first as abbreviations of arguments that broughtout their logically relevant form; and second, it seems, asrepresentatives of the form of a class of arguments.

5.4 Stoic Syllogistic

In terms of contemporary logic, Stoic syllogistic is best understoodas a substructural backwards-working Gentzen-style natural-deductionsystem that consists of five kinds of axiomatic arguments (theindemonstrables) and four inference rules, calledthemata. Anargument is a syllogism precisely if it either is an indemonstrable orcan be reduced to one by means of thethemata(D. L. 7.78). Thus syllogisms are certain kinds of formally validarguments. The Stoics explicitly acknowledged that there are validarguments that are not syllogisms; but assumed that these could besomehow transformed into syllogisms.

All basic indemonstrables consist of a non-simple assertible asleading premiss and a simple assertible as co-assumption, and haveanother simple assertible as conclusion. They were defined by fivestandardized meta-linguistic descriptions of the forms of thearguments (S. E.M 8.224–5; D. L. 7.80–1):

• A first indemonstrable is an argument that concludes from aconditional and its antecedent the consequent <of theconditional>
• A second indemonstrable is an argument that concludes from aconditional and the contradictory of the consequent the contradictoryof the antecedent <of the conditional>.
• A third indemonstrable is an argument that concludes from thenegation of a conjunction and one of the conjuncts the contradictoryof the other conjunct.
• A fourth indemonstrable is an argument that concludes from adisjunction and one of the disjuncts the contradictory of the otherdisjunct.
• A fifth indemonstrable is an argument that concludes from adisjunction and the contradictory of one of its disjuncts the otherdisjunct.

Whether an argument is an indemonstrable can be tested by comparingit with these meta-linguistic descriptions. For instance,

If it is day, it is not the case that it is night.
But it is night.
Therefore it is not the case that it is day.

comes out as a second indemonstrable, and

If five is a number, then either five is odd or five is even.
But five is a number.
Therefore either five is odd or five is even.

as a first indemonstrable. For testing, a suitable mode of an argumentcan also be used as a stand-in. A mode is syllogistic, if acorresponding argument with the same form is a syllogism (because ofthat form). However in Stoic logic there are no five modes that can beused as inference schemata that represent the five types ofindemonstrables. For example, the following are two of the many modesof fourth indemonstrables:

Either the 1^st or the 2^nd.
But the 2^nd.
Therefore not the 1^st.

Either the 1^st or not the 2^nd.
But the 1^st.
Therefore the 2^nd.

Although both are covered by the meta-linguistic description, neithercould be singled out asthe mode of the fourthindemonstrables: If we disregard complex arguments, there arethirty-two modes corresponding to the five meta-linguisticdescriptions; the latter thus prove noticeably more economical. Thealmost universal assumption among historians of logic that the Stoicsrepresented their five (types of) indemonstrables by five modes isfalse and not supported by textual evidence.^[4]

Of the fourthemata, only the first and third are extant.They, too, were meta-linguistically formulated. The firstthema, in its basic form, was:

• When from two [assertibles] a third follows, then fromeither of them together with the contradictory of the conclusion thecontradictory of the other follows (ApuleiusInt. 209.9–14).

This is an inference rule of the kind today called antilogism. Thethirdthema, in one formulation, was:

• When from two [assertibles] a third follows, and from theone that follows [i.e. the third] together with another,external assumption, another follows, then this other follows from thefirst two and the externally co-assumed one (SimpliciusCael.237.2–4).

This is an inference rule of the kind today called cut-rule. It isused to reduce chain-syllogisms. The second and fourththemata were also cut-rules, and reconstructions of them canbe provided, since we know what arguments they together with the thirdthema were thought to reduce, and we have some of thearguments said to be reducible by the secondthema. Apossible reconstruction of the secondthema is:

• When from two assertibles a third follows, and from the third andone (or both) of the two another follows, then this other follows fromthe first two.

A possible reconstruction of the fourththema is:

• When from two assertibles a third follows, and from the third andone (or both) of the two and one (or more) external assertible(s)another follows, then this other follows from the first two and theexternal(s). (Cf. Bobzien 1996.)

A Stoic reduction shows the formal validity of an argument by applyingto it thethemata in one or more steps in such a way that allresultant arguments are indemonstrables. This can be done either withthe arguments or their modes (S. E.M8.230–8). For instance, the argument mode

If the 1^st and the 2^nd, the 3^rd.
But not the 3^rd.
Moreover, the 1^st.
Therefore not: the 2^nd.

can be reduced by the thirdthema to (the modes of) a secondand a third indemonstrable as follows:

When from two assertibles (‘If the 1^st and the2^nd, the 3^rd’ and ‘But not the3^rd’) a third follows (‘Not: both the1^st and the 2^nd’—this follows by asecond indemonstrable) and from the third and an external one(‘The 1^st’) another follows (‘Not: the2^nd’—this follows by a third indemonstrable),then this other (‘Not: the 2^nd’) also followsfrom the two assertibles and the external one.

The secondthema reduced, among others, arguments with thefollowing modes (Alex. Aphr.An. Pr. 164.27–31):

Either the 1^st or not the 1^st.
But the 1^st.
Therefore the 1^st.

If the 1st, if the 1^st, the 2^nd.
But the 1^st.
Therefore the 2^nd.

The Peripatetics chided the Stoics for allowing such uselessarguments. In agreement with contemporary logic, the Stoics insistedthat, if the arguments can be reduced, they are valid.

The fourthemata can be used repeatedly and in anycombination in a reduction. Thus propositional arguments ofindeterminate length and complexity can be reduced. Stoic syllogistichas been formalized, and it has been shown that the Stoic deductivesystem shows strong similarities with relevance logical systems likethose by Storrs McCall. Like Aristotle, the Stoics aimed at provingnon-evident formally validarguments by reducing them bymeans of accepted inference rules to evidently validarguments. Thus, although their logic is a propositionallogic, they did not intend to provide a system that allows for thededuction of all propositional-logical truths, but rather a system ofvalid propositional-logical arguments with at least two premises and aconclusion. Nonetheless, we have evidence that the Stoics expresslyrecognized many simple logical truths. For example, they accepted thefollowing logical principles: the principle of double negation,stating that a double negation (‘not: not:p’) isequivalent to the assertible that is doubly negated (i.e.p)(D. L. 7.69); the principle that any conditional formed by using thesame assertible as antecedent and as consequent (‘ifp,p’) is true (S. E.M 8.281, 466); the principlethat any two-place disjunctions formed by using contradictorydisjuncts (‘eitherp or not:p’) is true(S. E.M 8.282, 467); and the principle of contraposition,that if ‘ifp,q’ then ‘if not:q, not:p’ (D. L. 7.194, PhilodemusSign., PHerc. 1065, XI.26–XII.14).

5.5 Logical Paradoxes

The Stoics recognized the importance of both the Liar and the Soritesparadoxes (CiceroAcad. 2.95–8, Plut.Comm.Not. 1059D–E, Chrys.Log. Zet. col.IX).Chrysippus may have tried to solve the Liar as follows: there is anineliminable ambiguity in the Liar sentence (‘I am speakingfalsely’, uttered in isolation) between the assertibles (i)‘I falsely say I speak falsely’ and (ii) ‘Iam speaking falsely’ (i.e. I am doing what I’m saying,viz. speaking falsely), of which, at any time the Liar sentence isuttered, precisely one is true, but it is arbitrary which one. (i)entails (iii) ‘Iam speaking truly’ and isincompatible with (ii) and with (iv) ‘I truly say I speakfalsely’. (ii) entails (iv) and is incompatible with (i) and(iii). Thus bivalence is preserved (cf. Cavini 1993). Chrysippus’stand on the Sorites seems to have been that vague borderlinesentences uttered in the context of a Sorites series have noassertibles corresponding to them, and that it is obscure to us wherethe borderline cases start, so that it is rational for us to stopanswering while still on safe ground (i.e. before we might begin tomake utterances with no assertible corresponding to them). The latterremark suggests Chrysippus was aware of the problem of higher ordervagueness. Again, bivalence of assertibles is preserved (cf. Bobzien2002). The Stoics also discussed various other well-knownparadoxes. In particular, for the paradoxes of presupposition, knownin antiquity as the Horned One, they produced a Russellian-typesolution based on a hidden scope ambiguity of negation (cf. Bobzien2012)

6. Epicurus and the Epicureans

Epicurus (late 4^th–early 3^rd c. BCE) andthe Epicureans are said to have rejected logic as an unnecessarydiscipline (D. L. 10.31, Usener 257). This notwithstanding, severalaspects of their philosophy forced or prompted them to take a stand onsome issues in philosophical logic. (1)Language meaning anddefinition: The Epicureans held that natural languages came intoexistence not by stipulation of word meanings but as the result of theinnate capacities of humans for using signs and articulating soundsand of human social interaction (D. L. 10.75–6); that language islearnt in context (Lucretius 5.1028ff); and that linguisticexpressions of natural languages are clearer and more conspicuous thantheir definitions; even that definitions would destroy theirconspicuousness (Usener 258, 243); and that philosophers hence shoulduse ordinary language rather than introduce technical expressions(EpicurusOn Nature 28). (2)Truth-bearers: theEpicureans deny the existence of incorporeal meanings, such as Stoicsayables. Their truth-bearers are linguistic items, more precisely,utterances (phônai) (S. E.M 8.13, 258; Usener259, 265). Truth consists in the correspondence of things andutterances, falsehood in a lack of such correspondence(S. E.M 8.9, Usener 244), although the details are obscurehere. (3)Excluded middle: with utterances as truth-bearers,the Epicureans face the question what the truth-values of futurecontingents are. Two views are recorded. One is the denial of thePrinciple of Excluded Middle (‘p or notp’) for future contingents (Usener 376, CiceroAcad. 2.97, CiceroFat. 37). The other, moreinteresting, one leaves the Excluded Middle intact for all utterances,but holds that, in the case of future contingents, the componentutterances ‘p’ and ‘notp’are neither true nor false (CiceroFat. 37), but, it seems,indefinite. This could be regarded as an anticipation ofsupervaluationism. (4)Induction: Inductive logic wascomparatively little developed in antiquity. Aristotle discussesarguments from the particular to the universal(epagôgê) in theTopics andPosterior Analytics but does not provide a theory ofthem. Some later Epicureans developed a theory of inductive inferencewhich bases the inference on empirical observation that certainproperties concur without exception (PhilodemusDeSignis).

7. Later Antiquity

Very little is known about the development of logic from c. 100 BCE toc. 250 CE. It is unclear when Peripatetics and Stoics began takingnotice of each others’ logical achievements. At some point during thatperiod, the terminological distinction between ‘categoricalsyllogisms’, used for Aristotelian syllogisms, and‘hypothetical syllogisms’, used not only for thoseintroduced by Theophrastus and Eudemus, but also for the Stoicpropositional-logical syllogisms, gained a foothold. In the firstcentury BCE, the Peripatetics Ariston of Alexandria and Boethus ofSidon wrote about syllogistic. Ariston is said to have introduced theso-called ‘subaltern’ syllogisms (Barbari, Celaront,Cesaro, Camestrop and Camenop) into Aristotelian syllogistic (ApuleiusInt. 213.5–10), i.e. the syllogisms one gains byapplying the subalternation rules (that were acknowledged by Aristotlein hisTopics)

From ‘A holds of everyB’ infer‘A holds of someB’

From ‘A holds of noB’ infer‘A does not hold of someB’

to the conclusions of the relevant syllogisms. Boethus suggestedsubstantial modifications to Aristotle’s theories: he claimed that allcategorical syllogisms are complete, and that hypothetical syllogisticis prior to categorical (GalenInst. Log. 7.2), although weare not told what this priority was thought to consist in. The StoicPosidonius (c. 135–c. 51 BCE) defended the possibility oflogical or mathematical deduction against the Epicureans and discussedsome syllogisms he called ‘conclusive by the force of anaxiom’, which apparently included arguments of the type‘As the 1^st is to the 2^nd, so the3^rd is to the 4^th; the ratio of the1^st to the 2^nd is double; therefore the ratio ofthe 3^rd to the 4^th is double’, which wasconsidered conclusive by the force of the axiom ‘things whichare in general of the same ratio, are also of the same particularratio’ (GalenInst. Log. 18.8). At least two Stoics inthis period wrote a work on Aristotle’sCategories. From hiswritings we know that Cicero (1^st c. BCE) was knowledgeableabout both Peripatetic and Stoic logic; and Epictetus’ discourses(late 1^st–early 2^nd c. CE) prove that hewas acquainted with some of the more taxing parts of Chrysippus’logic. In all likelihood, there existed at least a few creativelogicians in this period, but we do not know who they were or whatthey created.

The next logician of rank, if of lower rank, of whom we havesufficient evidence to speak is Galen (129–199 or 216 CE), whoachieved greater fame as a physician. He studied logic with bothPeripatetic and Stoic teachers, and recommended availing oneself ofparts of either doctrine, as long as it could be used for scientificdemonstration. He composed commentaries on logical works by Aristotle,Theophrastus, Eudemus and Chrysippus, as well as treatises on variouslogical problems and a major work entitledOnDemonstration. All these are lost, except for some information inlater texts, but hisIntroduction to Logic has come down tous almost in full. InOn Demonstration, Galen developed,among other things, a theory of compound categorical syllogisms withfour terms, which fall into four figures, but we do not know thedetails. He also introduced the so-called relational syllogisms,examples of which are ‘A is equal toB,B is equal toC; thereforeA is equal toC’ and ‘Dio owns half as much as Theo; Theo ownshalf as much as Philo. Therefore Dio owns a quarter of what Philoowns’ (GalenInst. Log, 17–18). All the relationalsyllogisms Galen mentions have in common that they are not reduciblein either Aristotle’s or the Stoic syllogistic, but it is difficult tofind further formal characteristics that unite them. In general, inhisIntroduction to Logic Galen merges AristotelianSyllogistic with a strongly Peripatetic reinterpretation of Stoicpropositional logic. This becomes apparent in particular in Galen’semphatic denial that truth-preservation is sufficient for the validityor syllogismhood of an argument, and his insistence that, instead,knowledge-introduction or knowledge-extension is a necessary conditionfor something to count as a syllogism.^[5]

The second ancient introduction to logic that has survived isApuleius’ (2^nd cent. CE)De Interpretatione. ThisLatin text, too, displays knowledge of Stoic and Peripatetic logic; itcontains the first full presentation of the square of opposition,which illustrates the logical relations between categorical sentencesby diagram. The Platonist Alcinous (2^nd cent. CE), in hisHandbook of Platonism chapter 5, is witness to the emergenceof a specifically Platonist logic, constructed on the Platonic notionsand procedures of division, definition, analysis and hypothesis, butthere is little that would make a logician’s heart beatfaster. At some time between the 3^rd and 6^th centuryCE Stoic logic faded into oblivion, to be resurrected only in the20^th century, in the wake of the (re)-discovery ofpropositional logic.

The surviving, often voluminous, Greek commentaries on Aristotle’slogical works by Alexander of Aphrodisias (fl. c. 200 CE), Porphyry(234–c. 305), Ammonius Hermeiou (5^th century),Philoponus (c. 500) and Simplicius (6^th century) and theLatin ones by Boethius (c. 480–524) are mainly important forpreserving alternative interpretations of Aristotle’s logic and assources for lost Peripatetic and Stoic works. They also allow us totrace the gradual development from a Peripatetic exegesis ofAristotle’sOrganon to a more eclectic logic that resultedfrom the absorption and inclusion of elements not just from Stoic andPlatonist theories but also from mathematics and rhetoric. Two of thecommentators in particular deserve special mention in their own right:Porphyry, for writing theIsagoge orIntroduction(i.e. to Aristotle’sCategories), in which he discusses thefive notions of genus, species, differentia, property and accident asbasic notions one needs to know to understand theCategories. For centuries, theIsagoge was the firstlogic text a student would tackle, and Porphyry’s five predicables(which differ from Aristotle’s four) formed the basis for the medievaldoctrine of thequinque voces. The second is Boethius. Inaddition to commentaries, he wrote a number of logical treatises,mostly simple explications of Aristotelian logic, but also two veryinteresting ones: (i) HisOn Topical Differentiae bearswitness to the elaborated system of topical arguments that logiciansof later antiquity had developed from Aristotle’sTopicsunder the influence of the needs of Roman lawyers. (ii) HisOnHypothetical Syllogisms systematically presents whollyhypothetical and mixed hypothetical syllogisms as they are known fromthe early Peripatetics; it may be derived from Porphyry. Boethius’insistence that the negation of ‘If it isA, it isB’ is ‘If it isA, it is notB’ suggests a suppositional understanding of theconditional, a view for which there is also some evidence in Ammonius,but that is not attested for earlier logicians. Historically, Boethiusis most important because he translated all of Aristotle’sOrganon into Latin, making these texts (except thePosterior Analytics) available to philosophers of themedieval period.

Bibliography

Greek and Latin Texts

• Alcinous,Enseignement des doctrines de Platon,J. Whittaker (ed.), Paris: Bude, 1990.
• Alexander of Aphrodisias,On Aristotle’s Prior Analytics1.Commentaria in Aristotelem Graeca, Vol. 2.1,M. Wallies (ed.), Berlin: Reimer, 1883.
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Translations of Greek and Latin Texts

• Ackrill, J. L., (trans. & comm.), 1961,Aristotle’sCategories and De Interpretatione, Oxford: Clarendon Press.
• Annas, J. and J. Barnes, (trans.), 2000,SextusEmpiricus.Outlines of Scepticism, 2nd edition, New York:Cambridge University Press.
• Barnes, J., (trans. & comm.), 1975,Aristotle, PosteriorAnalytics, Oxford: Clarendon Press. 2nd Ed. 1996.
• –––, (trans. and comm.), 1987,Early GreekPhilosophy, London: Penguin Books. (Zeno)
• –––, (trans.), 2003,Porphyry’s Introduction,Oxford: Oxford University Press. (Porphyry:Isagoge).
• Barnes, J., and S. Bobzien, K. Ierodiakonou, (trans.), 1991,Alexander of Aphrodisias on Aristotle’s Prior Analytics1.1–7, London: Duckworth.
• Blank, D., (trans.), 1998,Ammonius On Aristotle’s On Interpretation 9 (withN. Kretzmann, trans.),Boethius On Aristotle’s On Interpretation 91.1–7, London: Duckworth.
• Brittain, C. (trans.), 2006, Cicero:On AcademicScepticism (= Academica) Indianapolis: Hackett. (Stoics,Epicureans)
• Bury R. G., (trans.), 1933–1949,Sextus Empiricus,4 vols., Loeb Classical Library. Cambridge, Massachusetts: HarvardUniversity Press; London: William Heinemann Ltd., Loeb ClassicalLibrary, vols 1 and 2.
• De Lacy, Ph. H. and E. A. De Lacy, (trans.), 1978,Philodemus.On Methods of Inference, 2nd Ed.,Naples: Bibliopolis. (Epicureans)
• Dillon, J. M., 1993,Alcinous. The Handbook of Platonism,Oxford: Oxford University Press.
• Dorion, L.-A., (trans & comm.), 1995,Aristote: Lesrefutations sophistiques, Paris: J. Vrin.
• Hicks, R.D., (trans.), 1925,Diogenes Laertius,Livesof Eminent Philosophers, 2 volumes, Cambridge, Massachusetts:Harvard University Press; London: William Heinemann Ltd., LoebClassical Library. (Protagoras Alcidamas, Antisthenes, Eubulides,Stoics)
• Huby, P.M., (trans.), 1992, in W. W. Fortenbaugh (ed.),Theophrastus of Eresus: Sources for his Life, Writings, Thoughtand Influence, texts & tr., Leiden: Brill, 114–275.
• Hülser, K. (trans.), 1987–8,Die Fragmente zurDialektik der Stoiker, 4 volumes, Stuttgart-Bad Cannstatt:Frommann-Holzboog. (Stoics; Chrysippus)
• Kieffer, J. S. (trans), 1964,Galen’s Institutio logica,Baltimore: Johns Hopkins University Press.
• Lee, D. (trans. & comm.), 1955, 1974,Plato.TheRepublic, New York: Penguin Books.
• Londey, D. and C. Johanson, (trans.), 1988,The Logic ofApuleius, Leiden: Brill.
• McCabe, M.M., (trans. & comm.), 2005,Plato,Euthydemus, Cambridge: Cambridge University Press.
• Mueller I., with J. Gould, (trans.), 1999,Alexander ofAphrodisias on Aristotle’s Prior AnalyticsI.8–13.andI,14–22, 2 volumes, London: Duckworth.
• Oldfather, W. A., (trans.), 1925–8,Epictetus,The Discourses, The Manual and Fragments, 2 vols, Cambridge,Massachusetts: Harvard University Press; London: William HeinemannLtd., Loeb Classical Library. (Stoics)
• Ophuisen, J. M. van, (trans.), 2001,Alexander of Aphrodisiason Aristotle’s Topics 1, London: Duckworth.
• Pickard-Cambridge, W. A. (trans.), 1984, Aristotle,TopicsandSophistical Refutations, inThe Complete Worksof Aristotle, The Revised Oxford Translation, vol. 1, J. Barnes(ed.), Princeton: Princeton University Press.
• Ross, W. D. (trans.), Aristotle,Metaphysics, inTheComplete Works of Aristotle, The Revised Oxford Translation,vol. 2, J. Barnes (ed.), Princeton: Princeton University Press,1984.
• Sharples, R. W., 1991,Cicero: On Fate & Boethius: TheConsolations of Philosophy IV.5–7, V, Warminster: OxbowBooks. (Stoics, Epicureans)
• Smith, A., (trans.), 2014,Boethius on Aristotle’s OnInterpretation 1–3 1.1–7, London: Bloomsbury.
• –––, (trans.), 2014,Boethius on Aristotle’s OnInterpretation 4–6 1.1–7, London: Bloomsbury.
• Smith, R., (trans. & comm.), 1989,Aristotle’s PriorAnalytics, Indianapolis: Hackett.
• –––, (trans. & comm.), 1997,Aristotle,Topics I, VIII, and Selections, Oxford: Clarendon Press.
• Striker, G., (trans. & comm.), 2009,Aristotle, PriorAnalytics: Book I, Oxford: Oxford University Press.
• Stump, E., (trans.), 1978,Boethius’s ‘De topicisdifferentiis’, Ithaca/London: Cornell UniversityPress.
• Waterfield, R., (trans.), 2000,The First Philosophers: ThePresocratics and The Sophists, Oxford: Oxford University Press(Dissoi Logoi and Sophists).
• Weidemann, H., (trans. & comm.), 1994,Aristoteles, DeInterpretatione, Berlin: Akademie Verlag.
• White N. P., (trans.), 1993,Plato: Sophist,Indianapolis: Hackett.
• Whittaker, J. (trans.), 1990,Alcinous.Enseignementdes doctrines de Platon, Paris: Bude.

Secondary Literature

General

• Anderson, A. R. and N. D. Belnap Jr., 1975,Entailment: TheLogic of Relevance and Necessity, vol. I, Princeton: PrincetonUniversity Press.
• Barnes, J., 2007,Truth, etc., Oxford: Oxford UniversityPress.
• Barnes, J., et al., 1999, “Logic”, in Keimpe Algra, etal. (eds.),The Cambridge History of Hellenistic Philosophy,Cambridge: Cambridge University Press, 77–176.
• Kneale, M. and W. Kneale, 1962,The Development of Logic,Oxford: Clarendon Press.

The Beginnings

• Bailey, D.T.J., 2008, “Excavating Dissoi Logoi4”,Oxford Studies in Ancient Philosophy, 35:249–264.
• Frede, M., 1992, “Plato’sSophist on falsestatements”, inThe Cambridge companion to Plato,R. Kraut (ed.), Cambridge: Cambridge University Press,397–424.
• Kapp, E., 1942,Greek Foundations of Traditional Logic,New York: Columbia University Press.
• Mueller, I., 1974, “Greek Mathematics and Greek Logic”,in J. Corcoran (ed.),Ancient Logic and its ModernInterpretation, Dordrecht: Kluwer Academic Publishers,35–70.
• Netz R., 1999,The Shaping of Deduction in Greek Mathematics:a study in cognitive history, Cambridge: Cambridge UniversityPress.
• Robinson, R., 1953,Plato’s Earlier Dialectic, 2nd edition,Ithaca, N.Y.: Cornell University Press.
• Salmon, W. C., 2001,Zeno’s Paradoxes, 2nd edition,Indianapolis: Hackett Publishing Co. Inc.

Aristotle

• Barnes, J. , 1981, “Proof and the Syllogism”, inE. Berti (ed.),Aristotle on Science: the ‘Posterior Analytics’,Padua: Antenore, 17–59.
• Corcoran, J., 1974, “Aristotle’s Natural DeductionSystem”, in Corcoran, J. (ed.)Ancient Logic and its ModernInterpretation, Dordrecht: Kluwer Academic Publishers,85–131.
• Evans, J.D.G., 1975, “The Codification of False Refutationsin Aristotle’s De Sophistici Elenchis”,Proceedings of theCambridge Philological Society, 201: 45–52.
• Frede, D., 1985, “The sea-battle reconsidered. A defence ofthe traditional interpretation”,Oxford Studies in AncientPhilosophy, 3: 31–87.
• Frede, M., 1987, “The Title, Unity, and Authenticity of theAristotelianCategories”, in M. Frede,Essays in AncientPhilosophy, Minneapolis: University of Minnesota Press,11–28.
• Kretzmann, N., 1974, “Aristotle on Spoken Sounds Significantby Convention”, in J. Corcoran (ed.),Ancient Logic and itsModern Interpretation, Dordrecht: Kluwer Academic Publishers,3–21.
• Lear, J., 1980,Aristotle and Logical Theory, Cambridge:Cambridge University Press.
• Łukasiewicz, J., 1957,Aristotle’s Syllogistic from theStandpoint of Modern Formal Logic, 2nd edition, Oxford: ClarendonPress.
• Malink, M., 2013,Aristotle’s Modal Syllogistic, Cambridge, MA:Harvard University Press.
• Owen, G. E. L., (ed.) 1968,Aristotle on Dialectic: TheTopics (Proceedings of the Third Symposium Aristotelicum),Cambridge: Cambridge University Press.
• Owen, G.E.L., 1965, “Inherence”,Phronesis,10: 97–105.
• Patterson, R., 1995,Aristotle’s Modal Logic: Essence andEntailment in the Organon, Cambridge: Cambridge UniversityPress.
• Patzig, Günther, 1969,Aristotle’s Theory of theSyllogism, J. Barnes (trans.), Dordrecht: D. Reidel.
• Primavesi, O., 1996,Die aristotelische Topik, Munich:C. H. Beck.
• Smiley, T., 1974, “What Is a Syllogism?”,Journal ofPhilosophical Logic, 1: 136–154.
• Smith, R., 1983, “What is Aristotelian Ecthesis?”,History and Philosophy of Logic, 24: 224–32.
• –––, 1994, “Logic”, inThe CambridgeCompanion to Aristotle, J. Barnes (ed.), Cambridge: CambridgeUniversity Press, 27–65.
• –––, “Aristotle’s Logic”,The Stanford Encyclopedia of Philosophy (Fall 2004 Edition),Edward N. Zalta (ed.), URL =<https://plato.stanford.edu/archives/fall2004/entries/aristotle-logic/>.
• Steinkrüger, P., 2015, “Aristotle’s assertoricsyllogistic and modern relevance logic”,Synthese, 192:1413–1444.
• Striker, G., 1979, “Aristoteles über Syllogismen‘Aufgrund einer Hypothese’”,Hermes, 107:33–50.
• –––, 1994, “Modal vs. AssertoricSyllogismsc”,Ancient Philosophy, 14: 39–51.
• Whitaker, C. W. A., 1996,Aristotle’s De Interpretatione:Contradiction and Dialectic, Oxford: Clarendon Press.

Theophrastus and Eudemus

• Barnes, J., 1985, “Theophrastus and HypotheticalSyllogistic”, in J. Wiesner (ed.),Aristoteles: Werk undWirkung I, Berlin, 557–76.
• Bobzien, S., 2000, “Wholly hypothetical syllogisms”,Phronesis, 45: 87–137.
• –––, 2012, “How to give someone Horns –Paradoxes of Presupposition in Antiquity”,Logical Analysis and History of Philosophy, 15: 159–184.
• Bochenski, I.M., 1947,La Logique de Théophraste,Fribourg: Librairie de l’Université; reprinted 1987.
• Lejewski, Czesław, 1976, “On proslepticpremisses”,Notre Dame Journal of Formal Logic, 17:1–18.
• –––, 1961, “On proslepticsyllogisms”,Notre Dame Journal of Formal Logic, 2:158–176.

Diodorus Cronus and Philo the Logician

• Bobzien, S., 1993, “Chrysippus’ modal logic and its relationto Philo and Diodorus”, inDialektiker und Stoiker,K. Döring and Th. Ebert (eds.), Stuttgart: Franz Steiner.
• Denyer, N.C., 1981, “Time and Modality in DiodorusCronus”,Theoria, 47: 31–53.
• Prior, A.N., 1955, “Diodorean Modalities”,ThePhilosophical Quarterly, 5: 205–213.
• –––, 1967,Past, Present, and Future, Oxford:Clarendon Press, chapters II.1–2 and III.1.
• Sedley, D., 1977, “Diodorus Cronus and HellenisticPhilosophy”,Proceedings of the Cambridge PhilologicalSociety, 203 (NS 23): 74–120.

The Stoics

• Atherton, C., 1993,The Stoics on Ambiguity, Cambridge:Cambridge University Press.
• Bobzien, S., 1996, “Stoic Syllogistic”,OxfordStudies in Ancient Philosophy, 14: 133–92.
• –––, 1997, “Stoic Hypotheses and HypotheticalArgument”,Phronesis, 42: 299–312.
• –––, 1999, “Stoic Logic”, in K. Algra,J. Barnes, J. Mansfeld, & M. Schofield (eds.),The CambridgeHistory of Hellenistic Philosophy, Cambridge: CambridgeUniversity Press, 92–157.
• –––, 2002, “Chrysippus and the EpistemicTheory of Vagueness”Proceedings of the AristotelianSociety, 102: 217–238.
• –––, 2011, “The Combinatorics of StoicConjunction”,Oxford Studies in Ancient Philosophy, 40:157–188.
• Bronowski, A., 2019,The Stoics on Lekta, Oxford: OxfordUniversity Press.
• Brunschwig, J., 1994, “Remarks on the Stoic theory of theproper noun”, in hisPapers in Hellenistic PhilosophyCambridge: Cambridge University Press, 39–56.
• –––, 1994, “Remarks on the classification ofsimple propositions in Hellenistic logics” , in hisPapersin Hellenistic Philosophy, Cambridge: Cambridge University Press,57–71.
• Cavini, W., 1993, “Chrysippus on Speaking Truly and theLiar”, inDialektiker und Stoiker, K. Döring andTh. Ebert (eds), Stuttgart: Franz Steiner.
• Crivelli, P., 1994, “Indefinite propositions andanaphora in Stoic logic”Phronesis, 39:187–206.
• Ebert, Th., 1993, “Dialecticians and Stoics on ClassifyingPropositions” in K. Döring and Th. Ebert (eds.),Dialektiker und Stoiker. Zur Logik der Stoiker und ihrerVorläufer, Stuttgart: Steiner, 111–127.
• Frede, M., 1974,Die stoische Logik, Göttingen:Vandenhoek & Ruprecht.
• –––, 1975, “Stoic vs. AristotelianSyllogistic”,Archiv für Geschichte der Philosophie,56(1): 1–32.
• –––, 1994 “The Stoic notion of alekton”, inCompanion to ancient thought 3:Language, Stephen Everson (ed.), Cambridge: Cambridge UniversityPress, 109–128.
• Gaskin, R., 1997, “The Stoics on Cases, Predicates and theUnity of the Proposition,” inAristotle and After,R. Sorabji (ed.), London: Institute of Classical Studies,91–108.
• Lloyd, A. C., 1978, “Definite propositions and the conceptof reference”, in J. Brunschwig (ed.),Les Stoïciens etleur logique, Paris: Vrin, 285–295.
• Long, A. A., 1971, “Language and Thought in Stoicism”,in A. A. Long (ed.),Problems in Stoicism, London: Duckworth,75–113.
• Mates, B., 1961,Stoic Logic, Berkeley-LosAngeles: University of California Press.
• McCall, S., 1966, “Connexive Implication”,TheJournal of Symbolic Logic, 31: 415–433.
• Schenkeveld, D.M., 1984, “Stoic and Peripatetic Kinds ofSpeech Act and the Distinction of Grammatical Moods”Mnemosyne, 37: 291–351.

Epicurus

• Atherton, C., 2005, “Lucretius on what language isnot”, in D. Frede and Brad Inwood (eds.),Language andLearning, Cambridge: Cambridge University Press.
• Barnes, J., 1988, “Epicurean Signs”,OxfordStudies in Ancient Philosophy (Supplementary Volume),135–44.
• Manetti, G., 2002, “Philodemus’ ‘Designis’: An important ancient semioticdebate”,Semiotica, 138: 279–297.

Later Antiquity

• Barnes, J., 1993, “A Third Sort of Syllogism: Galen and theLogic of Relations” inModern Thinkers and AncientThinkers, R. W. Sharples (ed.), Boulder, CO: Westview Press.
• –––, 1997,Logic and the Imperial Stoa,Leiden: Brill.
• Bobzien, S., 2002, “The development ofmodus ponensin antiquity: From Aristotle to the 2nd century AD”,Phronesis, 47(4): 359–394.
• –––, 2002, “Propositional logic inAmmonius” in H. Linneweber-Lammerskitten / G. Mohr(eds.),Interpretation und Argument, Würzburg:Königshausen & Neumann, 103–119.
• –––, 2004, “Hypothetical Syllogistic inGalen—Propositional logic off the rails?”Rhizai:Journal for Ancient Philosophy and Science, 2: 57–102.
• Ebbesen, S., 1990, “Porphyry’s legacy to logic”, in R.Sorabji,Aristotle Transformed—The Ancient Commentators andtheir Influence, London: Duckworth, 141–171.
• –––, 1990, “Boethius as an AristotelianCommentator” in R. SorabjiAristotle Transformed—TheAncient Commentators and their Influence, London: Duckworth,373–91.
• Lee, T. S., 1984,Die griechische Tradition deraristotelischen Syllogistik in derSpätantike (Hypomnemata 79), Göttingen:Vandenhoeck & Ruprecht.
• Martin, C. J., 1991, “The Logic of Negation inBoethius”,Phronesis, 36: 277–304.
• Sullivan, W. M., 1967,Apuleian Logic. The Nature, Sources andInfluences of Apuleius’ Peri Hermeneias, Amsterdam:North-Holland Publishing Co.
• Stump, E., 1989, “Dialectic and Boethius’sDe topicisdifferentiis”, in E. Stump,Dialectic and Its Place inthe Development of Medieval Logic, Ithaca, NY: Cornell UniversityPress, 31–56.

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