47
Ancient Logic (Stanford Encyclopedia of Philosophy)First
published Wed Dec 13, 2006Logic as a discipline starts with the
transition from the more or less unreflective use of logical
methods and argument patterns to the reflection on and inquiry into
these and their elements, including the syntax and semantics of
sentences. In Greek and Roman antiquity, discussions of some
elements of logic and a focus on methods of inference can be traced
back to the late 5thcentury BCE. The Sophists, and later Plato
(early 4thc.) displayed an interest in sentence analysis, truth,
and fallacies, and Eubulides of Miletus (mid-4thc.) is on record as
the inventor of both the Liar and the Sorites paradox. But logic as
a fully systematic discipline begins with Aristotle, who
systematized much of the logical inquiry of his predecessors. His
main achievements were his theory of the logical interrelation of
affirmative and negative existential and universal statements and,
based on this theory, his syllogistic, which can be interpreted as
a system of deductive inference. Aristotle's logic is known as
term-logic, since it is concerned with the logical relations
between terms, such as human being, animal, white. It shares
elements with both set theory and predicate logic. Aristotle's
successors in his school, the Peripatos, notably Theophrastus and
Eudemus, widened the scope of deductive inference and improved some
aspects of Aristotle's logic.In the Hellenistic period, and
apparently independent of Aristotle's achievements, the logicians
Diodorus Cronus and his pupil Philo (see the entryDialectical
school) worked out the beginnings of a logic that took
propositions, rather than terms, as its basic elements. They
influenced the second major theorist of logic in antiquity, the
Stoic Chrysippus (mid-3rdc.), whose main achievement is the
development of a propositional logic, crowned by a deductive
system. Considered by many in antiquity as the greatest logician,
he was innovative in a large number of topics that are central to
contemporary formal and philosophical logic. The many close
similarities between Chrysippus' philosophical logic and that of
Gottlob Frege are especially striking. Chrysippus' Stoic successors
systematized his logic, and made some additions.The development of
logic from c. 100 BCE to c. 250 CE is mostly in the dark, but there
can be no doubt that logic was one of the topics regularly studied
and researched. At some point Peripatetics and Stoics began taking
notice of the logical systems of each other, and we witness some
conflation of both terminologies and theories. Aristotelian
syllogistic became known as categorical syllogistic and the
Peripatetic adaptation of Stoic syllogistic as hypothetical
syllogistic. In the 2ndcentury CE, Galen attempted to synthesize
the two traditions; he also professed to have introduced a third
kind of syllogism, the relational syllogism, which apparently was
meant to help formalize mathematical reasoning. The attempt of some
Middle Platonists (1stc. BCE2ndc. CE) to claim a specifically
Platonic logic failed, and in its stead, the Neo-Platonists
(3rd6thc. CE) adopted a scholasticized version of Aristotelian
logic as their own. In the monumentalif rarely creativevolumes of
the Greek commentators on Aristotle's logical works we find
elements of Stoic and later Peripatetic logic; the same holds for
the Latin logical writings of Apuleius (2ndc. CE) and Boethius
(6thc. CE), which pave the way for the thus supplemented
Aristotelian logic 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 PonensandModus
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 Other Internet Resources Related
Entries
1. Pre-Aristotelian Logic1.1 Syntax and SemanticsSome of the
Sophists classified types of sentences (logoi) according to their
force. So Protagoras (485415 BCE), who included wish, question,
answer and command (Diels Kranz (DK) 80.A1, Diogenes Laertius (D.
L.) 9.534), and Alcidamas (pupil of Gorgias, fl. 4thBCE), who
distinguished assertion (phasis), denial (apophasis), question and
address (prosagoreusis) (D. L. 9.54). Antisthenes
(mid-5thmid-4thcent.) defined a sentence as that which indicates
what a thing was or is (D. L. 6.3, DK 45) and stated that someone
who says what is speaks truly (DK49). Perhaps the earliest
surviving passage on logic is found in the Dissoi LogoiorDouble
Arguments(DK 90.4, c. 400 BCE). It is evidence for a debate over
truth and falsehood. Opposed were the views (i) that truth is
atemporalproperty of sentences, and that a sentence is true (when
it is said), if and only if things are as the sentence says they
are when it is said, and false if they aren't; and (ii) that truth
is an atemporal property of what is said, and that what is said is
true if and only if the things are the case, false if they aren't
the case. These are rudimentary formulations of two alternative
correspondence theories of truth. The same passage displays
awareness of the fact that self-referential use of the
truth-predicate can be problematican insight also documented by the
discovery of the Liar paradox by Eubulides of Miletus (mid-4thc.
BCE) shortly thereafter.Some Platonic dialogues contain passages
whose topic is indubitably logic. In theSophist, Plato analyzes
simple statements as containing a verb (rhma), which indicates
action, and a name (onoma), which indicates the agent (Soph.
261e262a). Anticipating the modern distinction of logical types, he
argues that neither a series of names nor a series of verbs can
combine into a statement (Soph. 262ad). Plato also divorces syntax
(what is a statement?) from semantics (when is it true?). Something
(e.g. Theaetetus is sitting) is a statement if it both succeeds in
specifying a subject and says something about this subject. Plato
thus determines subject and predicate as relational elements in a
statement and excludes as statements subject-predicate combinations
containing empty subject expressions. Something is a true statement
if with reference to its subject (Theaetetus) it says of what is
(e.g. sitting) that it is. Something is a false statement if with
reference to its subject it says of something other than what is
(e.g. flying) that it is. Here Plato produces a sketch of a
deflationist theory of truth (Soph. 262e263d; cf.Crat.385b). He
also distinguished negations from affirmations and took the
negation particle to have narrow scope: it negates the predicate,
not the whole sentence (Soph.257bc). There are many passages in
Plato where he struggles with explaining certain logical relations:
for example his theory that things participate in Forms corresponds
to a rudimentary theory of predication; in theSophistand elsewhere
he grapples with the class relations of exclusion, union and
co-extension; also with the difference between the is of
predication (being) and the is of identity (sameness); and
inRepublic4, 436bff., he anticipates the law of non-contradiction.
But his explications of these logical questions are cast in
metaphysical terms, and so can at most be regarded as
proto-logical.1.2 Argument Patterns and Valid
InferencePre-Aristotelian evidence for reflection on argument forms
and valid inference are harder to come by. Both Zeno of Elea (born
c. 490 BCE) and Socrates (470399) were famous for the ways in which
they refuted an opponent's view. Their methods display similarities
withreductio ad absurdum, but neither of them seems to have
theorized 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. So this is
impossible. Socratic refutation was an exchange of questions and
answers in which the opponents would be led, on the basis of their
answers, to a conclusion incompatible with their original claim.
Plato institutionalized such disputations into structured,
rule-governed verbal contests that became known as dialectical
argument. The development of a basic logical vocabulary for such
contests indicates some reflection upon the patterns of
argumentation.The 5thand early to mid-4thcenturies BCE also see
great interest in fallacies and logical paradoxes. Besides the
Liar, Eubulides is said to have been the originator of several
other logical paradoxes, including the Sorites.
Plato'sEuthydemuscontains a large collection of contemporary
fallacies. In attempts to solve such logical puzzles, a logical
terminology develops here, too, and the focus on the difference
between valid and invalid arguments sets the scene for the search
for a criterion of valid inference. Finally, it is possible that
the shaping of deduction and proof in Greek mathematics that begins
in the later 5thcentury BCE served as an inspiration for
Aristotle's syllogistic.2. Aristotle(For a more detailed account
see the entry onAristotle's Logicin this encyclopedia.) Aristotle
is the first great logician in the history of logic. His logic was
taught by and large without rival from the 4thto the 19thcenturies
CE. Aristotle's logical works were collected and put in a
systematic order by later Peripatetics, who entitled them
theOrganonor tool, because they considered logic not as a part but
rather an instrument of philosophy. TheOrganoncontains, in
traditional order, theCategories,De Interpretatione,Prior
Analytics,Posterior Analytics,TopicsandSophistical Refutations. In
addition,Metaphysics is a logical treatise that discusses the
principle of non-contradiction, and some further logical insights
are found scattered throughout Aristotle's other works, such as
thePoetics,Rhetoric,De Anima,Metaphysics and , and some of the
biological works. Some parts of theCategoriesandPosterior
Analyticswould today be regarded as metaphysics, epistemology or
philosophy of science rather than logic. The traditional
arrangement of works in theOrganonis neither chronological nor
Aristotle's own. The original chronology cannot be fully recovered
since Aristotle seems often to have inserted supplements into
earlier writings at a later time. However, by using logical
advances as criterion, we can conjecture that most of
theTopics,Sophistical Refutations,CategoriesandMetaphysics predate
theDe Interpretatione, which in turn precedes thePrior Analyticsand
parts of thePosterior Analytics.2.1 DialecticsTheTopicsprovide a
manual for participants in the contests of dialectical argument as
instituted in the Academy by Plato. Books 27 provide general
procedures or rules (topoi) about how to find an argument to
establish or refute a given thesis. The descriptions of these
proceduressome of which are so general that they resemble logical
lawsclearly presuppose a notion of logical form, and
Aristotle'sTopicsmay thus count as the earliest surviving logical
treatise. TheSophistical Refutationsare the first systematic
classification of fallacies, sorted by what logical flaw each type
manifests (e.g. equivocation, begging the question, affirming the
consequent,secundum quid) and how to expose them.2.2 Sub-sentential
ClassificationsAristotle distinguishes things that have sentential
unity through a combination of expressions (a horse runs) from
those that do not (horse, runs); the latter are dealt 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 this classification to be one of linguistic expressions
that can be predicated of something else; or of kinds of
predication; or of highest genera. InTopics1 Aristotle
distinguishes four relationships a predicate may have to the
subject: it may give its definition, genus, unique property, or
accidental property. These are known as predicables.2.3 Syntax and
Semantics of SentencesWhen writing theDe Interpretatione, Aristotle
had worked out the following theory of simple sentences: a
(declarative) sentence (apophantikos logos) or declaration
(apophansis) is delimited from other pieces of discourse like
prayer, command and question by its having a truth-value. The
truth-bearers that feature in Aristotle's logic are thus linguistic
items. They are spoken sentences that directly signify thoughts
(shared by all humans) and through these, indirectly, things.
Written sentences in turn signify spoken ones. (Simple) sentences
are constructed from two signifying expressions which stand in
subject-predicate relation to each other: a name and a verb
(Callias walks) or two names connected by the copula is, which
co-signifies the connection (Pleasure is good) (Int. 3). Names are
either singular terms or common nouns (An. Pr.I 27). Both can be
empty (Cat.10,Int.1). Singular terms can only take subject
position. Verbs co-signify time. A name-verb sentence can be
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 its subject. The negation particle in a
negation has wide scope (Cat. 10). Aristotle defined truth
separately for affirmations and negations: An affirmation is true
if it says of that which is that it is; a negation is true if it
says of that which is not that it is not (Met. .7 1011b25ff). These
formulations, or in any case their Greek counterparts, can be
interpreted as expressing either a correspondence or a deflationist
conception of truth. Either way, truth is a property that belongs
to a sentenceat a time. As to their quantity, sentences are
singular, universal, particular or indefinite. Thus Aristotle
obtains eight types of sentences, which are later dubbed
categorical sentences. The following are examples, paired by
quality:Singular:Callias is just.Callias is not just.
Universal:Every human is just.No human is just.
Particular:Some human is just.Some human is not just.
Indefinite:(A) human is just.(A) human is not just.
Universal and particular sentences contain a quantifier and both
universal and particular affirmatives were taken to have
existential import. (See entryThe Traditional Square of
Opposition). The logical status of the indefinites is ambiguous and
controversial (Int. 67).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 affirms of it). Aristotle assumes
thatnormallyone of these must be true, the other false. Contrary
sentences are such that they cannot both be true. The contradictory
of a universal affirmative is the corresponding particular
negative; that of the universal negative the corresponding
particular affirmative. A universal affirmative and its
corresponding universal negative are contraries. Aristotle thus has
captured the basic logical relations between monadic quantifiers
(Int. 7).Since Aristotle regards tense as part of the truth-bearer
(as opposed to merely a grammatical feature), he detects a problem
regarding future tense sentences about contingent matters: Does the
principle that of an affirmation and its negation one must be
false, the other true apply to these? What, for example, is the
truth-value now of the sentence There will be a sea-battle
tomorrow? Aristotle may have suggested that the sentence has no
truth-value now, and that bivalence thus does not holddespite the
fact that it is necessary 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 SyllogisticAristotle's non-modal
syllogistic (Prior AnalyticsA 17) is the pinnacle of his logic.
Aristotle defines a syllogism as an argument (logos) in which,
certain things having been laid down, something different from what
has been laid down follows of necessity because these things are
so. This definition appears to require (i) that a syllogism
consists of at least two premises and a conclusion, (ii) that the
conclusion follows of necessity from the premises (so that all
syllogisms arevalidarguments), and (iii) that the conclusion
differs from the premises. Aristotle's syllogistic covers only a
small part of all arguments that satisfy these conditions.Aristotle
restricts and regiments the types of categorical sentence that may
feature in a syllogism. The admissible truth-bearers are now
defined as each containing two different terms (horoi) conjoined by
the copula, of which one (the predicate term) is said of the 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. Singular
sentences seem excluded and indefinite sentences are mostly
ignored. AtAn. Pr.A 7 Aristotle mentions that by putting an
indefinite premise in place of a particular one obtains a syllogism
of the same kind.Another innovation in the syllogistic is
Aristotle's use of letters in place of terms. The letters may
originally have served simply as abbreviations for terms (e.g.An.
Post.A 13); but in the syllogistic they seem mostly to have the
function either of schematic term letters or of term variables with
universal quantifiers assumed but not stated. Where he uses
letters, Aristotle tends to express the four types of categorical
sentences in the following way (with common later abbreviations in
parentheses):Aholds of (lit., belongs to) everyB(AaB)
Aholds of noB(AeB)
Aholds of someB(AiB)
Adoes not hold of someB(AoB)
Instead of holds he also uses is predicated.All basic syllogisms
consist of three categorical sentences, in which the two premises
share exactly one term, called the middle term, and the conclusion
contains the other two terms, sometimes called the extremes. Based
on the position of the middle term, Aristotle classified all
possible premise combinations into three figures (schemata): the
first figure has the middle term (B) as subject in the first
premise and predicated in the second; the second figure has it
predicated in both premises, the third has it as subject in both
premises:IIIIII
Aholds ofBBholds ofAAholds ofB
Bholds ofCBholds ofCCholds ofB
Ais also called the major term,Cthe minor term. Each figure can
further be classified according to whether or not both premises are
universal. Aristotle went systematically through the fifty-eight
possible premise combinations and showed that fourteen have a
conclusion following of necessity from them, i.e. are syllogisms.
His procedure was this: He assumed that the syllogisms of the first
figure are complete and not in need of proof, since they are
evident. By contrast, the syllogisms of the second and third
figures are incomplete and in need of proof. He proves them by
reducing them to syllogisms of the first figure and thereby
completing them. For this he makes use of three methods:(i)
conversion (antistroph): a categorical sentence is converted by
interchanging its terms. Aristotle recognizes and establishes three
conversion rules: fromAeBinferBeA; fromAiBinferBiA and
fromAaBinferBiA. All second and third figure syllogisms but two can
be proved by premise conversion.(ii)reductio ad impossibile(apagg):
the remaining two are proved by reduction to the impossible, where
the contradictory of an assumed conclusion together with one of the
premises is used to deduce by a first figure syllogism a conclusion
that is incompatible with the other premise. Using the semantic
relations between opposites established earlier the assumed
conclusion is thus established.(iii) exposition or setting-out
(ekthesis): this method, which Aristotle uses additionally to (i)
and (ii), involves choosing or setting out some additional term,
sayD, that falls in the non-empty intersection delimited by two
premises, sayAxBandAxC, and usingDto justify the inference from the
premises to a particular conclusion,BxC. It is debated whether D
represents a singular or a general term and whether exposition
constitutes proof.For each of the thirty-four premise combinations
that allow no conclusion Aristotle proves by counterexample that
they allow no conclusion. As overall result, he acknowledges four
first figure syllogisms (later named Barbara, Celarent, Darii,
Ferio), four second figure syllogisms (Camestres, Cesare, Festino,
Baroco) and six third figure 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 cases where the name has more than
three, indicates in order whether the first and second premises and
the conclusion were sentences of typea,e,ioro.) Aristotle
implicitly recognized that by using the conversion rules on the
conclusions we obtain eight further syllogisms (An. Pr.53a314), and
that of the premise combinations rejected as non-syllogistic, some
(five, in fact) will yield a conclusion in which the minor term is
predicated of the major (An. Pr.29a1927). Moreover, in
theTopicsAristotle accepted the rules fromAaBinferAiB and
fromAeBinferAoB. By using these on the conclusions five further
syllogisms could be proved, though Aristotle did not mention
this.Going beyond his basic syllogistic, Aristotle reduced the
3rdand 4thfirst figure syllogisms to second figure syllogisms,
thusde factoreducing all syllogisms to Barbara and Celarent; and
later on in thePrior Analyticshe invokes a type of cut-rule by
which a multi-premise syllogism can be reduced to two or more basic
syllogisms. From a modern perspective, Aristotle's system can be
represented as an argumental natural deduction systemen miniature.
It has been shown to be sound and complete if one interprets the
relations expressed by the categorical sentences set-theoretically
as a system of non-empty classes as follows:AaBis true if and only
if the classAcontains the classB.AeBis true if and only if the
classesAandBare disjoint.AiBis true if and only if the
classesAandBare not disjoint.AoBis true if and only if the
classAdoes not contain the classB. The vexing textual question what
exactly Aristotle meant by syllogisms has received several rival
interpretations, including one that they are a certain type of
conditional propositional form. Most plausibly, perhaps,
Aristotle's complete andincompletesyllogisms taken together are
understood as formally valid premise-conclusion arguments; and his
complete andcompletedsyllogisms taken together as (sound)
deductions.2.5 Modal LogicAristotle is also the originator of modal
logic. In addition to quality (as affirmation or negation) and
quantity (as singular, universal, particular, or indefinite), he
takes categorical sentences to have a mode; this consists of the
fact that the predicate is said to hold of the subject either
actually or necessarily or possibly or contingently or impossibly.
The latter four are expressed by modal operators that modify the
predicate, e.g. It is possible forAto hold of someB; Anecessarily
holds of everyB.InDe Interpretatione1213, Aristotle (i) concludes
that modal operators modify the whole predicate (or the copula, as
he puts it), not just the predicate term of a sentence. (ii) He
states the logical relations that hold between modal operators,
such as that it is not possible forAnot to hold ofB implies it is
necessary forAto hold ofB. (iii) He investigates what the
contradictories of modalized sentences are, and decides that they
are obtained by placing the negator in front of the modal operator.
(iv) He equates the expressions possible and contingent, but wavers
between a one-sided interpretation (where necessity implies
possibility) and a two-sided interpretation (where possibility
implies non-necessity).Aristotle develops his modal syllogistic
inPrior Analytics1.822. He settles on two-sided possibility
(contingency) and tests for syllogismhood all possible combinations
of premise pairs of sentences with necessity (N), contingency (C)
or no (U) modal operator: NN, CC, NU/UN, CU/UC and NC/CN.
Syllogisms with the last three types of premise combinations are
called mixed modal syllogisms. Apart from the NN category, which
mirrors unmodalized syllogisms, all categories contain dubious
cases. For instance, Aristotle accepts:Anecessarily holds of
allB.Bholds of allC.ThereforeAnecessarily holds of allC.This and
other problematic cases were already disputed in antiquity, and
more recently have sparked a host of complex formalized
reconstructions of Aristotle's modal syllogistic. As Aristotle's
theory is conceivably internally inconsistent, the formal models
that have been suggested may all be unsuccessful.3. The early
Peripatetics: Theophrastus and EudemusAristotle's pupil and
successor Theophrastus of Eresus (c. 371c. 287 BCE) wrote more
logical treatises than his teacher, with a large overlap in topics.
Eudemus of Rhodes (later 4thcent. BCE) wrote books
entitledCategories,AnalyticsandOn Speech. Of all these works only a
number of fragments and later testimonies survive, mostly in
Aristotle commentators. Theophrastus and Eudemus simplified some
aspects of Aristotle's logic, and developed others where Aristotle
left us only hints.3.1 Improvements and Modifications of
Aristotle's LogicThe two Peripatetics seem to have redefined
Aristotle's first figure, so that it includes every syllogism in
which the middle term is subject of one premise and predicate of
the other. In this way, five types of non-modal syllogisms only
intimated by Aristotle later in hisPrior Analytics(Baralipton,
Celantes, Dabitis, Fapesmo and Frisesomorum) are included, but
Aristotle's criterion that first figure syllogisms are evident is
given up (Theophrastus fr. 91, Fortenbaugh). Theophrastus and
Eudemus also improved Aristotle's modal theory. Theophrastus
replaced Aristotle's two-sided contingency by one-sided
possibility, so that possibility no longer entails non-necessity.
Both recognized that the problematic universal negative (Apossibly
holds of noB) is simply convertible (Theophrastus fr. 102A
Fortenbaugh). Moreover, they introduced the principle that in mixed
modal syllogisms the conclusion always has the same modal character
as the weaker of the premises (Theophrastus frs. 106 and 107
Fortenbaugh), where possibility is weaker than actuality, and
actuality than necessity. In this way Aristotle's modal syllogistic
is notably simplified and many unsatisfactory theses, like the one
mentioned above (that from NecessarilyAaB and BaC one can infer
NecessarilyAaC) disappear.3.2 Prosleptic SyllogismsTheophrastus
introduced the so-called prosleptic premises and syllogisms
(Theophrastus fr. 110 Fortenbaugh). A prosleptic premise is of the
form:For allX, if (X), then (X)where (X) and (X) stand for
categorical sentences in which the variableXoccurs in place of one
of the terms. For example:(1) A[holds] of all of that of all of
whichB[holds].(2) A[holds] of none of that which [holds] of
allB.Theophrastus considered such premises to contain three terms,
two of which are definite (A,B), one indefinite (that, or the bound
variableX). We can represent (1) and (2)
asX(BaXAaX)X(XaBAeX)Prosleptic syllogisms then come about as
follows: They are composed of a prosleptic premise and the
categorical premise obtained by instantiating a term (C) in the
antecedent open categorical sentence as premises, and the
categorical sentences one obtains by putting in the same term (C)
in the consequent open categorical sentence as conclusion. For
example:A[holds] of all of that of all of whichB[holds].Bholds of
allC.Therefore,Aholds 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. The number of prosleptic syllogisms was
presumably equal to that of types of prosleptic sentences: with
Theophrastus' concept of the first figure these would be sixty-four
(i.e. 32 + 16 + 16). Theophrastus held that certain prosleptic
premises were equivalent to certain categorical sentences, e.g. (1)
to Ais predicated of allB. However, for many, including (2), no
such equivalent can be found, and prosleptic syllogisms thus
increased the inferential power of Peripatetic logic.3.3
Forerunners ofModus PonensandModus TollensTheophrastus and Eudemus
considered complex premises which they called hypothetical premises
and which had one of the following two (or similar) forms:If
something isF, it isGEither something isFor it isG (with exclusive
or)They developed arguments with them which they called mixed from
a hypothetical premise and a probative premise (Theophrastus fr.
112A Fortenbaugh). These arguments were inspired by Aristotle's
syllogisms from a hypothesis (An. Pr.1.44); they were forerunners
ofmodus ponensandmodus tollensand had the following forms
(Theophrastus frs. 111 and 112 Fortenbaugh), employing the
exclusive or:If something isF, it isG.aisF.Therefore,aisG.If
something isF, it isG.ais notG.Therefore,ais notF.
Either something isFor it isG.aisF.Therefore,ais notG.Either
something isFor it isG.ais notF.Therefore,aisG.
Theophrastus also recognized that the connective particle or can
be inclusive (Theophrastus fr. 82A Fortenbaugh); and he considered
relative quantified sentences such as those containing more, fewer,
and the same (Theophrastus fr. 89 Fortenbaugh), and seems to have
discussed syllogisms built from such sentences, again following up
upon what Aristotle said about syllogisms from a hypothesis
(Theophrastus fr. 111E Fortenbaugh).3.4 Wholly Hypothetical
SyllogismsTheophrastus is further credited with the invention of a
system of the later so-called wholly hypothetical syllogisms
(Theophrastus fr. 113 Fortenbaugh). These syllogisms were
originally abbreviated term-logical arguments of the kindIf
[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's categorical syllogisms,
presumably by way of the equivalences to EveryAisB, etc. In
parallel to Aristotle's syllogistic, Theophrastus distinguished
three figures; each had sixteen modes. The first eight modes of the
first figure are obtained by going through all permutations with
notX instead of X (withXforA,B,C); the second eight modes are
obtained by using a rule of contraposition on the conclusion:(CR)
From ifX,Y infer if the contradictory ofYthen the contradictory
ofXThe sixteen modes of the second figure were obtained by using
(CR) on the 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) on the 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 could be reduced to
first figure syllogisms. If Alexander of Aphrodisias (2ndc. CE
Peripatetic) reports faithfully, any use of (CR) which transforms a
syllogism into a first figure syllogism was such a reduction. The
large number of modes and reductions can be explained by the fact
that Theophrastus did not have the logical means for substituting
negative for positive components in an argument. In later
antiquity, after some intermediate stages, and possibly under Stoic
influence, the wholly hypothetical syllogisms were interpreted as
propositional-logical arguments of the kindIfp, thenq.Ifq,
thenr.Therefore, ifp, thenr.4. Diodorus Cronus and Philo the
LogicianIn the later 4thto mid 3rdcenturies BCE, contemporary with
Theophrastus and Eudemus, a loosely connected group of
philosophers, sometimes referred to as dialecticians (see entry
Dialectical School) and possibly influenced by Eubulides, conceived
of logic as a logic of propositions. Their best known exponents
were Diodorus Cronus and his pupil Philo (sometimes called Philo of
Megara). Although no writings of theirs are preserved, there are a
number of later reports of their doctrines. They each made
ground-breaking contributions to the development of propositional
logic, in particular to the theories of conditionals and
modalities.A conditional (sunmmenon) was considered as a non-simple
proposition composed of two propositions and the connecting
particle if. Philo, who may be credited with introducing
truth-functionality into logic, provided the following criterion
for their truth: A conditional is falsewhenandonlywhenits
antecedent is true and its consequent is false, and it is true in
the three remaining truth-value combinations. The Philonian
conditional resembles material implication, except thatsince
propositions were conceived of as functions of time that can have
different truth-values at different timesit may change its
truth-value over time. For Diodorus, a conditional proposition is
true if it neither was nor is possible that its antecedent is true
and its consequent false. The temporal elements in this account
suggest that the possibility of a truth-value change in Philo's
conditionals was meant to be improved on. With his own modal
notions (see below) applied, a conditional is Diodorean-true now if
and only if it is Philonian-true at all times. Diodorus'
conditional is thus reminiscent of strict implication. Philo's and
Diodorus' conceptions of conditionals lead to variants of the
paradoxes of material and strict implicationa fact the ancients
were aware of (Sextus Empiricus [S. E.]M. 8.109117).Philo and
Diodorus each considered the four modalities possibility,
impossibility, necessity and non-necessity. These were conceived of
as modal properties or modal values of propositions, not as modal
operators. Philo defined them as follows: Possible is that which is
capable of being true by the proposition's own nature necessary is
that which is true, and which, as far as it is in itself, is not
capable of being false. Non-necessary is that which as far as it is
in itself, is capable of being false, and impossible is that which
by its own nature is not capable of being true. Diodorus'
definitions were these: Possible is that which either is or will be
[true]; impossible that which is false and will not be true;
necessary that which is true and will not be false; non-necessary
that which either is false already or will be false. Both sets of
definitions satisfy the following standard requirements of modal
logic: (i) necessity entails truth and truth entails possibility;
(ii) possibility and impossibility are contradictories, and so are
necessity and non-necessity; (iii) necessity and possibility are
interdefinable; (iv) every proposition is either necessary or
impossible or both possible and non-necessary. Philo's definitions
appear to introduce mere conceptual modalities, whereas with
Diodorus' definitions, some propositions may change their modal
value (Boeth.In Arist. De Int., sec. ed., 234235 Meiser).Diodorus'
definition of possibility rules out future contingents and implies
the counterintuitive thesis that only the actual is possible.
Diodorus tried to prove this claim with his famous Master Argument,
which sets out to show the incompatibility of (i) every past truth
is necessary, (ii) the impossible 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, but
various reconstructions have been suggested. Some affinity with the
arguments for logical determinism in Aristotle'sDe Interpretatione9
is likely.On the topic of ambiguity, Diodorus held that no
linguistic expression is ambiguous. He supported this dictum by a
theory of meaning based on speaker intention. Speakers generally
intend to say only one thing when they speak. What is said when
they speak is what they intend to say. Any discrepancy between
speaker intention and listener decoding has its cause in the
obscurity of what was said, not its ambiguity (Aulus Gellius
11.12.23).5. The StoicsThe founder of the Stoa, Zeno of Citium
(335263 BCE), studied with Diodorus. His successor Cleanthes
(331232) tried to solve the Master Argument by denying that every
past truth is necessary and wrote booksnow loston paradoxes,
dialectics, argument modes and predicates. Both philosophers
considered knowledge of logic as a virtue and held it in high
esteem, but they seem not to have been creative logicians. By
contrast, Cleanthes' successor Chrysippus of Soli (c. 280207) is
without doubt the second great logician in the history of logic. It
was said of him that if the gods used any logic, it would be that
of Chrysippus (D. L. 7.180), and his reputation as a brilliant
logician is amply testified. Chrysippus wrote over 300 books on
logic, on virtually any topic logic today concerns itself with,
including speech act theory, sentence analysis, singular and plural
expressions, types of predicates, indexicals, existential
propositions, sentential connectives, negations, disjunctions,
conditionals, logical consequence, valid argument forms, theory of
deduction, propositional logic, modal logic, tense logic, epistemic
logic, logic of suppositions, logic of imperatives, ambiguity and
logical paradoxes, in particular the Liar and the Sorites (D. L.
7.189199). Of all these, only two badly damaged papyri have
survived, luckily supplemented by a considerable number of
fragments and testimonies in later texts, in particular in Diogenes
Laertius (D. L.) book 7, sections 5583, and Sextus
EmpiricusOutlines of Pyrrhonism(S. E.PH) book 2 andAgainst the
Mathematicians(S. E.M) book 8. Chrysippus' successors, including
Diogenes of Babylon (c. 240152) and Antipater of Tarsus (2ndcent.
BCE), appear to have systematized and simplified some of his ideas,
but their original contributions to logic seem small. Many
testimonies of Stoic logic do not name any particular Stoic. Hence
the following paragraphs simply talk about the Stoics in general;
but we can be confident that a large part of what has survived goes
back to Chrysippus.5.1 Logical Achievements Besides Propositional
LogicThe subject matter of Stoic logic are the so-called sayables
(lekta): they are the underlying meanings in everything we say and
think, butlike Frege's sensessubsist also independently of us. They
are distinguished from spoken and written linguistic expressions:
what weutterare those expressions, but what wesayare the sayables
(D. L. 7.57). There are complete and deficient sayables. Deficient
sayables, if said, make the hearer feel prompted to ask for a
completion; e.g. when somone says writes we enquire who?. Complete
sayables, if said, do not make the hearer ask for a completion (D.
L.7.63). They include assertibles (the Stoic equivalent for
propositions), imperativals, interrogatives, inquiries,
exclamatives, hypotheses or suppositions, stipulations, oaths,
curses and more. The accounts of the different complete sayables
all had the general form a so-and-so sayable is one in saying which
we perform an act of such-and-such. For instance: an imperatival
sayable is one in saying which we issue a command, an interrogative
sayable is one in saying which we ask a question, a declaratory
sayable (i.e. an assertible) is one in saying which we make an
assertion. Thus, according to the Stoics, each time we say a
complete sayable, we perform three different acts: we utter a
linguistic expression; we say the sayable; and we perform a
speech-act. Chrysippus was aware of the use-mention distinction (D.
L. 7.187). He seems to have held that every denoting expression is
ambiguous in that it denotes both its denotation and itself
(Galen,On ling. soph. 4; Aulus Gellius 11.12.1). Thus the
expression a wagon would denote both a wagon and the expression a
wagon.[2]Assertibles (aximata) differ from all other complete
sayables in their having a truth-value: at any one time they are
either true or false. Truth is temporal and assertibles may change
their truth-value. The Stoic principle of bivalence is hence
temporalized, too. Truth is introduced by example: the assertible
it is day is truewhenit is day, and at all other times false (D. L.
7.65). This suggests some kind of deflationist view of truth, as
does the fact that the Stoics identify true assertibles with facts,
but define false assertibles simply as the contradictories of true
ones (S. E.M8.85).Assertibles are simple or non-simple. A
simplepredicativeassertible like Dion is walking is generated from
the predicate is walking, which is a deficient assertible since it
elicits the question who?, together with a nominative 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 predicate and subject terms as in
Aristotle; rather, predicatesbut not the things that fall under
themare defined as deficient, and thus resemble propositional
functions. It seems that whereas some Stoics took
theFregeanapproach that singular terms had correlated sayables,
others anticipated the notion of direct reference. Concerning
indexicals, the Stoics took a simpledefiniteassertible like this
one is walking to be true when the person pointed at by the speaker
is walking (S. E.M100). When the thing pointed at ceases to be, so
does the assertible, though the sentence used to express it remains
(Alex. Aphr.An. Pr.1778). A simpleindefiniteassertible like someone
is walking is said to be true when a corresponding definite
assertible is true (S. E.M98). Aristotelian universal affirmatives
(EveryAisB) were to be rephrased as conditionals: If something isA,
it isB (S. E.M9.811). Negations of simple assertibles 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. The latter is analyzed in a Russellian manner
as Both Dion exists and not: Dion is walking (Alex. Aphr.An.
Pr.402). There are present tense, past tense and future tense
assertibles. Thetemporalizedprinciple of bivalence holds for them
all. The past tense assertible Dion walked is true when there is at
least one past time at which Dion is walking was true.5.2 Syntax
and Semantics of Complex PropositionsThus the Stoics concerned
themselves with several issues we would place under the heading of
predicate logic; but their main achievement was the development of
a propositional logic, i.e. of a system of deduction in which the
smallest substantial unanalyzed expressions are propositions, or
rather, assertibles.The Stoics defined negations as assertibles
that consist of a negative particle and an assertible controlled by
this particle (S. E.M103). Similarly, non-simple assertibles were
defined as assertibles that either consist of more than one
assertible or of one assertible taken more than once (D. L. 7.689)
and that are controlled by a connective particle. Both definitions
can be understood as being recursive and allow for assertibles of
indeterminate complexity. Three types of non-simple assertibles
feature in Stoic syllogistic. Conjunctions are non-simple
assertibles put together by the conjunctive connective both and .
They have two conjuncts.[3]Disjunctions are non-simple assertibles
put together by the disjunctive connective either or or . They have
two or more disjuncts, all on a par. Conditionals are non-simple
assertibles formed with the connective if , ; they consist of
antecedent and consequent (D. L. 7.712). What type of assertible an
assertible is, is determined by the connective or logical particle
that controls it, i.e. that has the largest scope. Both notpandq is
a conjunction, Not bothpandq a negation. Stoic language
regimentation asks that sentences expressing assertibles always
start with the logical particle or expression characteristic for
the assertible. Thus, the Stoics invented an implicit bracketing
device similar to that used in ukasiewicz' Polish notation.Stoic
negations and conjunctions are truth-functional. Stoic (or at least
Chrysippean) conditionals are true when the contradictory of the
consequent is incompatible with its antecedent (D. L. 7.73). Two
assertibles are contradictories of each other if one is the
negation of the other (D. L. 7.73); that is, when one exceeds the
other by apre-fixednegation particle (S. E.M8.89). The
truth-functional Philonian conditional was expressed as a negation
of a conjunction: that is, not as ifp,q but as not bothpand notq.
Stoic disjunction is exclusive and non-truth-functional. It is true
when necessarily precisely one of its disjuncts is true. Later
Stoics introduced a non-truth-functional inclusive disjunction
(Aulus Gellius,N. A.16.8.1314).Like Philo and Diodorus, Chrysippus
distinguished four modalities and considered them as modal values
of propositions rather than modal operators; they satisfy the same
standard requirements of modal logic. Chrysippus' definitions are
(D. L. 7.75): An assertible is possible when it is both capable of
being true and not hindered by external things from being true. An
assertible is impossible when it is [either] not capable of being
true [or is capable of being true, but hindered by external things
from being true]. An assertible is necessary when, being true, it
either is not capable of being false or is capable of being false,
but hindered by external things from being false. An assertible is
non-necessary when it is both capable of being false and not
hindered by external things [from being false]. Chrysippus' modal
notions differ from Diodorus' in that they allow for future
contingents and from Philo's in that they go beyond mere conceptual
possibility.5.3 ArgumentsArguments arenormallycompounds of
assertibles. They are defined as a system of at least two premises
and a conclusion (D. L. 7.45). Syntactically, every premise but the
first is introduced by now or but, and the conclusion by therefore.
An argument is valid if the (Chrysippean) conditional formed with
the conjunction of its premises as antecedent and its conclusion as
consequent is correct (S. E.PH2.137; D. L. 7.77). An argument is
sound (literally: true), when in addition to being valid it has
true premises. The Stoics defined so-called argument modes as a
sort of schema of an argument (D. L. 7.76). A mode of an argument
differs from the argument itself by having ordinal numbers taking
the place of assertibles. A mode of the argumentIf 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.isIf the 1st, the 2nd.But not: the
2nd.Therefore not: the 1st.The modes functioned first as
abbreviations of arguments that brought out their logically
relevant form; and second, it seems, as representatives of the form
of a class of arguments.5.4 Stoic SyllogisticStoic syllogistic is
an argumental deductive system consisting of five types of
indemonstrables or axiomatic arguments and four inference rules,
calledthemata. An argument is a syllogism precisely if it either is
an indemonstrable or can be reduced to one by means of
thethemata(D. L. 7.78). Syllogisms are thus certain types of
formally valid arguments. The Stoics explicitly acknowledged that
there are valid arguments that are not syllogisms; but assumed that
these could be somehow transformed into syllogisms.All basic
indemonstrables consist of a non-simple assertible as leading
premiss and a simple assertible as co-assumption, and have another
simple assertible as conclusion. They were defined by five
standardized meta-linguistic descriptions of the forms of the
arguments (S. E.M8.2245; D. L. 7.801): A first indemonstrable is an
argument composed of a conditional and its antecedent as premises,
having the consequent of the conditional as conclusion. A second
indemonstrable is an argument composed of a conditional and the
contradictory of its consequent as premises, having the
contradictory of its antecedent as conclusion. A third
indemonstrable is an argument composed of a negated conjunction and
one of its conjuncts as premises, having the contradictory of the
other conjunct as conclusion. A fourth indemonstrable is an
argument composed of a disjunctive assertible and one of its
disjuncts as premises, having the contradictory of the remaining
disjunct as conclusion. A fifth indemonstrable, finally, is an
argument composed of a disjunctive assertible and the contradictory
of one of its disjuncts as premises, having the remaining disjunct
as conclusion.Whether an argument is an indemonstrable can be
tested by comparing it 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, andIf 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 argument can also be used as a stand-in. A mode
is syllogistic, if a corresponding argument with the same form is a
syllogism (because of that form). However in Stoic logic there are
no five modes that can be used as inference schemata that represent
the five types of indemonstrables. For example, the following are
two of the many modes of fourth indemonstrables:Either the 1stor
the 2nd. Either the 1stor not the 2nd.
But the 2nd. But the 1st.
Therefore not the 1st. Therefore the 2nd.
Although both are covered by the meta-linguistic description,
neither could be singled out asthemode of the fourth
indemonstrables: If we disregard complex arguments, there are
thirty-two modes corresponding to the five meta-linguistic
descriptions; the latter thus prove noticeably more economical. The
almost universal assumption among historians of logic that the
Stoics represented their five (types of) indemonstrables by five
modes is false and not supported by textual evidence.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 from either
of them together with the contradictory of the conclusion the
contradictory of the other follows (ApuleiusInt. 209.914).This is
an inference rule of the kind today called antilogism. The
thirdthema, in one formulation, was: When from two [assertibles] a
third follows, and from the one that follows [i.e. the third]
together with another, external assumption, another follows, then
this other follows from the first two and the externally co-assumed
one (SimpliciusCael. 237.24).This is an inference rule of the kind
today called cut-rule. It is used to reduce chain-syllogisms. The
second and fourththematawere also cut-rules, and reconstructions of
them can be provided, since we know what arguments they together
with the thirdthemawere thought to reduce, and we have some of the
arguments said to be reducible by the secondthema. A possible
reconstruction of the secondthemais: When from two assertibles a
third follows, and from the third and one (or both) of the two
another follows, then this other follows from the first two.A
possible reconstruction of the fourththemais: When from two
assertibles a third follows, and from the third and one (or both)
of the two and one (or more) external assertible(s) another
follows, then this other follows from the first two and the
external(s). (Cf. Bobzien 1996.)A Stoic reduction shows the formal
validity of an argument by applying to it thethematain one or more
steps in such a way that all resultant arguments are
indemonstrables. This can be done either with the arguments or
their modes (S. E.M8.2308). For instance, the argument modeIf the
1stand the 2nd, the 3rd.But not the 3rd.Moreover, the 1st.Therefore
not: the 2nd.can be reduced by the thirdthemato (the modes of) a
second and a third indemonstrable as follows:When from two
assertibles (If the 1stand the 2nd, the 3rd and But not the 3rd) a
third follows (Not: both the 1stand the 2ndthis follows by a second
indemonstrable) and from the third and an external one (The 1st)
another follows (Not: the 2ndthis follows by a third
indemonstrable), then this other (Not: the 2nd) also follows from
the two assertibles and the external one.The secondthemareduced,
among others, arguments with the following modes (Alex. Aphr.An.
Pr.164.2731):Either the 1stor not the 1st. If the 1st, if the 1st,
the 2nd.
But the 1st. But the 1st.
Therefore the 1st. Therefore the 2nd.
The Peripatetics chided the Stoics for allowing such useless
arguments, but the Stoics rightly insisted that if they can be
reduced, they are valid.The fourthematacan be used repeatedly and
in any combination in a reduction. Thus propositional arguments of
indeterminate length and complexity can be reduced. Stoic
syllogistic has been formalized, and it has been shown that the
Stoic deductive system shows strong similarities with relevance
logical systems like those by McCall. Like Aristotle, the Stoics
aimed at proving non-evident formally validargumentsby reducing
them by means of accepted inference rules to evidently
validarguments. Thus, although their logic is a propositional
logic, they did not intend to provide a system that allows for the
deduction of all propositional-logical truths, but rather a system
of valid propositional-logical arguments with at least two premises
and a conclusion. Nonetheless, we have evidence that the Stoics
expressly recognized many simple logical truths. For example, they
accepted the following logical principles: the principle of double
negation, stating that a double negation (not: not:p) is equivalent
to the assertible that is doubly negated (i.e.p) (D. L. 7.69); the
principle that any conditional formed by using the same assertible
as antecedent and as consequent (ifp,p) is true (S. E.M8.281, 466);
the principle that any two-place disjunctions formed by using
contradictory disjuncts (eitherpor not:p) is true (S. E.M8.282,
467); and the principle of contraposition, that if ifp,q then if
not:q, not:p (D. L. 7.194, PhilodemusSign., PHerc. 1065,
XI.26XII.14).5.5 Logical ParadoxesThe Stoics recognized the
importance of both the Liar and the Sorites paradoxes (CiceroAcad.
2.958, Plut.Comm.Not. 1059DE, Chrys.Log. Zet. col.IX). Chrysippus
may have tried to solve the Liar as follows: there is an
uneliminable ambiguity in the Liar sentence (I am speaking falsely,
uttered in isolation) between the assertibles (i) I falsely say I
speak falsely and (ii) Iamspeaking falsely (i.e. I am doing what
I'm saying, viz. speaking falsely), of which, at any time the Liar
sentence is uttered, precisely one is true, but it is arbitrary
which one. (i) entails (iii) Iamspeaking truly and is incompatible
with (ii) and with (iv) I truly say I speak falsely. (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 borderline sentences uttered in the context
of a Sorites series have no assertibles corresponding to them, and
that it is obscure to us where the borderline cases start, so that
it is rational for us to stop answering while still on safe ground
(i.e. before we might begin to make utterances with no assertible
corresponding to them). The latter remark suggests Chrysippus was
aware of the problem of higher order vagueness. Again, bivalence of
assertibles is preserved (cf. Bobzien 2002).6. Epicurus and the
EpicureansEpicurus (late 4thearly 3rdc. BCE) and the Epicureans are
said to have rejected logic as an unnecessary discipline (D. L.
10.31, Usener 257). This notwithstanding, several aspects of their
philosophy forced or prompted them to take a stand on some issues
in philosophical logic. (1)Language meaning and definition: The
Epicureans held that natural languages came into existence not by
stipulation of word meanings but as the result of the innate
capacities of humans for using signs and articulating sounds and of
human social interaction (D. L. 10.756); that language is learnt in
context (Lucretius 5.1028ff); and that linguistic expressions of
natural languages are clearer and more conspicuous than their
definitions; even that definitions would destroy their
conspicuousness (Usener 258, 243); and that philosophers hence
should use ordinary language rather than introduce technical
expressions (EpicurusOn Nature28). (2)Truth-bearers: the Epicureans
deny the existence of incorporeal meanings, such as Stoic sayables.
Their truth-bearers are linguistic items, more precisely,
utterances (phnai) (S. E.M8.13, 258; Usener 259, 265). Truth
consists in the correspondence of things and utterances, falsehood
in a lack of such correspondence (S. E.M8.9, Usener 244), although
the details are obscure here. (3)Excluded middle: with utterances
as truth-bearers, the Epicureans face the question what the
truth-values of future contingents are. Two views are recorded. One
is the denial of the Principle of Excluded Middle (por notp) for
future contingents (Usener 376, CiceroAcad. 2.97, CiceroFat. 37).
The other, more interesting, one leaves the Excluded Middle intact
for all utterances, but holds that, in the case of future
contingents, the component utterances p and notp are neither true
nor false (CiceroFat. 37), but, it seems, indefinite. This could be
considered as an anticipation of supervaluationism. (4)Induction:
Inductive logic was comparatively little developed in antiquity.
Aristotle discusses arguments from the particular to the universal
(epagg) in theTopicsandPosterior Analyticsbut does not provide a
theory of them. Some later Epicureans developed a theory of
inductive inference which bases the inference on empirical
observation that certain properties concur without exception
(PhilodemusDe Signis).7. Later AntiquityVery little is known about
the development of logic from c. 100 BCE to c. 250 CE. It is
unclear when Peripatetics and Stoics began taking notice of the
logical achievements of each other. Sometime during that period,
the terminological distinction between categorical syllogisms, used
for Aristotelian syllogisms, and hypothetical syllogisms, used not
only for those introduced by Theophrastus and Eudemus, but also for
the Stoic propositional-logical syllogisms, gained a foothold. In
the first century BCE, the Peripatetics Ariston of Alexandria and
Boethus of Sidon wrote about syllogistic. Ariston is said to have
introduced the so-called subaltern syllogisms (Barbari, Celaront,
Cesaro, Camestrop and Camenop) into Aristotelian syllogistic
(ApuleiusInt. 213.510), i.e. the syllogisms one gains by applying
the subalternation rules (that were acknowledged by Aristotle in
hisTopics)From Aholds of everyB infer Aholds of someBFrom Aholds of
noB infer Adoes not hold of someBto the conclusions of the relevant
syllogisms. Boethus suggested substantial modifications to
Aristotle's theories: he claimed that all categorical syllogisms
are complete, and that hypothetical syllogistic is prior to
categorical (GalenInst. Log. 7.2), although we are not told what
this priority was thought to consist in. The Stoic Posidonius (c.
135c. 51 BCE) defended the possibility of logical or mathematical
deduction against the Epicureans and discussed some syllogisms he
called conclusive by the force of an axiom, which apparently
included arguments of the type As the 1stis to the 2nd, so the
3rdis to the 4th; the ratio of the 1stto the 2ndis double;
therefore the ratio of the 3rdto the 4this double, which was
considered conclusive by the force of the axiom things which are in
general of the same ratio, are also of the same particular ratio
(GalenInst. Log.18.8). At least two Stoics in this period wrote a
work on Aristotle'sCategories. From his writings we know that
Cicero (1stc. BCE) was knowledgeable about both Peripatetic and
Stoic logic; and Epictetus' discourses (late 1stearly 2ndc. CE)
prove that he was acquainted with some of the more taxing parts of
Chrysippus' logic. In all likelihood, there existed at least a few
creative logicians in this period, but we do not know who they were
and what they created.The next logician of rank, if of lower rank,
of whom we have sufficient evidence to speak is Galen (129199 or
216 CE), whose greater fame was as a physician. He studied logic
with both Peripatetic and Stoic teachers, and recommended to avail
oneself of parts of either doctrine, as long as it could be used
for scientific demonstration. He composed commentaries on logical
works by Aristotle, Theophrastus, Eudemus and Chrysippus, as well
as treatises on various logical problems and a major work
entitledOn Demonstration. All these are lost, except for some
information in later texts, but hisIntroduction to Logichas come
down to us almost in full. InOn Demonstration, Galen developed,
among other things, a theory of compound categorical syllogisms
with four terms, which fall into four figures, but we do not know
the details. He also introduced the so-called relational
syllogisms, examples of which are Ais equal toB,Bis equal toC;
thereforeAis equal toC and Dio owns half as much as Theo; Theo owns
half as much as Philo. Therefore Dio owns a quarter of what Philo
owns. (GalenInst. Log. 1718). All relational syllogisms Galen
mentions have in common that they are not reducible in either
Aristotle's or Stoic syllogistic, but it is difficult to find
further formal characteristics that unite them. In general, in
hisIntroduction to LogicGalen merges Aristotelian Syllogistic with
a strongly Peripatetic reinterpretation of Stoic propositional
logic. This becomes apparent in particular in Galen's emphatic
denial that truth-preservation is sufficient for the validity or
syllogismhood of an argument, and his insistence that, instead,
knowledge-introduction or knowledge-extension is a necessary
condition for something to count as a syllogism.[4]The second
ancient introduction to logic that has survived is Apuleius'
(2ndcent. CE)De Interpretatione. This Latin text, too, displays
knowledge of Stoic and Peripatetic logic; it contains the first
full presentation of the square of opposition, which illustrates
the logical relations between categorical sentences by diagram. The
Platonist Alcinous (2ndcent. CE), in hisHandbook of
Platonismchapter 5, is witness to the emergence of a specifically
Platonist logic, constructed on the Platonic notions and procedures
of division, definition, analysis and hypothesis, but there is
little that would make a logician's heart beat faster. Sometime
between the 3rdand 6thcentury CE Stoic logic faded into oblivion,
to be resurrected only in the 20thcentury, in the wake of the
(re)-discovery of propositional logic.The surviving, often
voluminous, Greek commentaries on Aristotle's logical works by
Alexander of Aphrodisias (fl. c. 200 CE), Porphyry (234c. 305),
Ammonius Hermeiou (5thcentury), Philoponus (c. 500) and Simplicius
(6thcentury) and the Latin ones by Boethius (c. 480524) have their
main importance as preservers of alternative interpretations of
Aristotle's logic and as sources for lost Peripatetic and Stoic
workswith elements of Stoic logic either tacitly adopted or loudly
decried. Two of the commentators deserve special mention in their
own right: Porphyry, for writing theIsagogeorIntroduction(i.e. to
Aristotle'sCategories), in which he discusses the five notions of
genus, species, differentia, property and accident as basic notions
one needs to know to understand theCategories. For centuries,
theIsagogewas the first logic text a student would tackle, and
Porphyry's five predicables (which differ from Aristotle's four)
formed the basis for the medieval doctrine of thequinque voces. The
second is Boethius. In addition to commentaries, he wrote a number
of logical treatises, mostly simple explications of Aristotelian
logic, but also two very interesting ones: (i) HisOn Topical
Differentiaebears witness to the elaborated system of topical
arguments that logicians of later antiquity had developed from
Aristotle'sTopicsunder the influence of the needs of Roman lawyers.
(ii) HisOn Hypothetical Syllogismssystematically presents wholly
hypothetical and mixed hypothetical syllogisms as they are known
from the 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 the
conditional, a view for which there is also some evidence in
Ammonius, but that is not attested for earlier logicians.
Historically, Boethius is most important because he translated all
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http://plato.stanford.edu/entries/logic-ancient/ 2/2/2014
Medieval Theories of ConsequenceFirst published Mon Jun 11,
2012Latin medieval theories of consequence are systematic analyses
by Latin medieval authors[1]of the logical relations between
sentences[2], in particular the notions of entailment and valid
inference. When does a sentence B follow from a sentence A? (For
example, from Some human is an animal one may infer Some animal is
a human.) What are the grounds for the relation of
entailment/consequence? Are there different kinds of consequences?
These and other questions were extensively debated by these
authors.Theories of consequence explicitly acquired an autonomous
status only in the 14thcentury, when treatises specifically on the
concept of consequence began to appear; but some earlier
investigations also deserve the general title of theories of
consequence, in view of their scope, sophistication and
systematicity. Taken as a whole, medieval theories of consequence
represent the first sustained attempt at adopting a
sentential/propositional perspective[3]since the Stoics in Greek
antiquity, and unlike Stoic logic, which had little historical
influence provide the historical background for subsequent
developments leading to the birth of modern logic in the
19thcentury. Indeed, it will be argued that the medieval concept
ofconsequentia(in its different versions) is the main precursor of
the modern concept of logical consequence. 1. Preliminary
Considerations 1.1 A genealogy of modern conceptions of consequence
1.2 What are medieval theories of consequence theoriesof? 2. Early
Theories of Consequence 2.1 Predecessors 2.2 Abelard 2.3
13thCentury 3. 14thCentury Theories of Consequence 3.1 The
emergence of treatises on consequence in the 14thcentury 3.2 Burley
and Ockham 3.3 Buridan and the Parisian tradition 3.4 The British
School 4. Conclusion Bibliography Primary Literature Secondary
Literature Academic Tools Other Internet Resources Related
Entries
1. Preliminary Considerations1.1 A genealogy of modern
conceptions of consequenceIn his much-discussed 1936 paper On the
concept of logical consequence, Tarski presents two criteria of
material adequacy for formal accounts of logical consequence, which
jointly capture the common notion of logical consequence (or so he
claims). They are formulated as the following condition:If in the
sentences of the classKand in the sentenceXwe replace the constant
terms which are not general-logical terms correspondingly by
arbitrary other constant terms (where we replace equiform constants
everywhere by equiform constants) and in this way we obtain a new
class of sentencesK and a new sentenceX, then the sentenceX must be
true if only all sentences of the classK are true. (Tarski 2002,
2.3)In more mundane terms, the two core aspects that Tarski
attributes to the so-called common notion of logical consequence
can be formulated as:(TP)necessary truth-preservation: it is
impossible for the antecedent to be true while the consequent is
not true;(ST)substitution of terms: the relation of consequence is
preserved under any (suitable) substitution of the non-logical
terms of the sentences in question; this is now often referred to
as theformality criterion.Different accounts of logical consequence
can be (and have been) formulated on the basis of (TP) and/or (ST):
they can be viewed as both necessary but independent components of
the notion of logical consequence, as Tarski seems to suggest in
the passage above; they can also be viewed as closely related, in
particular if (TP) can bereducedto (ST) (i.e. satisfaction of (ST)
would entail satisfaction of (TP) and vice-versa) a view that
Etchemendy (1990) attributes to Tarski; or one may hold that the
actual core of the notion of (logical) consequence is (TP), and
that (ST) simply specifies a particular subclass of valid
consequences, often referred to asformal consequences(Read
1994).Tarski correctly identified these two features as key
components of the notion of logical consequence as entertained by
philosophers and mathematicians of his time (and also today). But
the question arises: why these two features and not others? In
particular, through what (historical) processes have they come to
constitute the conceptual core of the notion of logical
consequence? These questions are even more pressing in view of the
fact that both features have recently been questioned as to whether
they truly capture the conceptual core of logical consequence see
e.g. Etchemendy (1990) for the centrality of formality and (ST);
Fields (2008) for the centrality of necessary truth-preservation in
view of the semantic paradoxes.To make further progress in these
debates, an important element is arguably the historical
development of the notion of (logical) consequence over the
centuries, so that we may come to understand where the so-called
pre-theoretical notion of logical consequence comes from. Engaging
in what could be described as a project of conceptual genealogy may
allow for a better grasp of the reasons why this notion (now widely
endorsed) established itself as such in the first place. If these
are compelling reasons, then they may count as arguments in favor
of the centrality of formality and necessary truth-preservation;
but if they rest on disputable, contentious assumptions, then the
analysis may provide elements for a critical evaluation of each of
these two components as truly constitutive of the concept of
logical consequence.From this point of view, the historical
developments in the Latin Middle Ages, in particular from the
12thto the 14thcentury, occupy a prominent position. As will be
argued, it is in this period that concepts and ideas inherited from
Greek Antiquity (Aristotle in particular, but also the ancient
commentators) were shaped and consolidated into conceptions of
consequence that bear a remarkable resemblance to the Tarskian
condition of material adequacy presented above. Thus, an analysis
of these historical developments is likely to contribute
significantly to our understanding of the notion(s) of logical
consequence as currently entertained.Naturally, as with any
historical analysis, an investigation of these developments has
intrinsic historical value in and of itself, independently of its
possible contribution to modern debates. Indeed, medieval theories
of consequence are a genuine medieval contribution: while medieval
authors are clearly taking ancient Greek sources and ideas as their
starting point, the emergence of theories of consequence as such is
a Latin medieval innovation. But as it turns out, following the
thread provided by the two key notions (TP) and (ST) as formulated
above provides a suitable vantage point to investigate the
development of the notion of consequence in the Latin Middle Ages.
In other words, historical and conceptual analysis can easily be
combined in this case.1.2 What are medieval theories of consequence
theoriesof?At first sight, it is not immediately clear what the
object of analysis of medieval theories of consequence is (Boh
1982). Is it the semantics of conditional sentences? Is it the
validity of inferences and arguments? Is it the relation of
consequence, construed as an abstract entity? In fact, at times it
seems that medieval authors are conflating these different notions,
perhaps betraying some conceptual confusi