Kenny Anthony. Ancient Philosophy: A New History of Western Philosophy Volume 1

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(4) It is necessary that Nautilus will

not ever be seen. (from (4) and (1))

(5) It is impossible that Nautilus

will ever be seen. (necessarily not ¼

impossible that)

Though we do not know the precise form of Diodorus’ proof, it is easy

enough to gene ralize this line of argument to show that only what will

happen, can happen.

The argument is obviously akin to one that we met in discussing

Aristotle’s treatme nt of future contingents. Diodorus’ argument appears

to be Xawed by an ambiguity in the premiss that past truths are necessary .

What is a past truth? If it means a true proposition in the past tense, then

there is no guarantee that it is necessary. To see this, we have only to think

of a negative proposition in the past tense, such as ‘The Persian Empire has

not been destroyed.’ This proposition was true in the time of Socrates, but

it was not necessary: it was about to change its truth-value from true to

false. On the other hand, if a past truth is a proposition that is made true by

an event in the past, then past truths are indeed necessary; but a propos-

ition such as (4) is not a past truth and hence does not entail (5).11

Diodorus’ pupil Phi lo abandoned his master’s modal deWnitions, and

explained possibility in terms of the internal properties of a proposition

rather than in terms of its truth-values over time. We do not know how his

explanation went, but we know that on his account a piece of wood would

be capable of bein g burnt even if it was never burnt and even if it spent its

whole existence on the bed of the ocean (LS 38b).

Philo’s major contribution to logic was his deWnition of the conditional.

‘If p, then q’, he said, was false in the case in which p was true and q false, and

true in the three other pos sible cases. The truth of a conditional propos-

ition, on this view, does not depend at all on the content of the antecedent

or the consequent, but only on their truth-values. Thus, ‘If it is night, it is

day’ will be true whenever it is daytime, and on the assumption that the

atomic theory is true, ‘If there are no atoms, there are atoms’ is true. In

treating the conditional in this way Philo anticipated the truth-functional

deWnition of material implication used in modern propositional logic.

11 See A. N. Prior, Time and Modality (Oxford: Clarendon Press, 1957), 86–7; Jonathan Barnes in

CHHP 89–92.

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However, the truth-values that determine the truth or falsity of his

conditionals are changeable truth-values. This has disadvantages for the

formulation of logic, since ‘If p, then p’ is no longer a logical law: ‘If I am

sitting, I am sitting’ comes out false, as a Philonian conditional, if I rise to

my feet between the antecedent and the consequent.

Nonetheless, Philo’s deWnition seems to have been adopted by the

Stoic logicians who were the Wrst to oVer a formalization of propositional

logic. Where Aristotle used letters as variables in his logical texts, the Stoics

used numbers; this is a trivial diVerence, but more importantly, where

Aristotle’s variables stood in for terms, Stoic variables stood in for

whole sentences, or rather for elements that are capable of being whole

sentences. In ‘If the stars are shining, it is night’ neither the antecedent, ‘the

stars are shining’, nor the consequent, ‘it is night’, are complete sentences,

but each set of words is capable of standing on its own as a complete sentence.

Stoic propositional logic was embedded in an elaborate theory of

language and signiWcation. The Stoics made a distinction between sound

(phone), speech (lexis ), and saying (logos ). The roar of a beast or of the sea is a

sound, but only articulate sound counts as speech. Not all speech, how-

ever, is meaningful: humans can utter nonsense words like ‘hey nonny no’.

Only meaningful speech counts as saying anyt hing (D.L. 7. 57). The sounds

and speech of a Greek can be taken in by a non-Greek-speaking barbarian,

but the meaning is understood only by someone who knows the language

(S.E., M 8. 11–12).

The word ‘logos’, which I have translated ‘saying’, is a Greek word

of very wide meaning: in diVerent contexts it can mean ‘word’, ‘sentence’,

‘language’, ‘reason’. It is a noun connected with the common verb

‘legein’, meaning ‘to say’. The Stoics coined a new word from this verbal

root, ‘lekton’. This means literally ‘thing said’, but I will leave the

word as an untranslated technicality, since there is no exact English

equivalent.

The lekton plays an important part in the Stoic treatment of the distinc-

tion between signs and what they signify. Consider a sentence such as ‘Dion

is walking’, a proposition which may be true or false. Sextus Empiricus,

discussing some such sentence, tells us this:

The Stoics said that three items are linked together, the signi Wcation, the signiWer,

and the topic (tunchanon). The signiWer is a sound, such as ‘Dion’, the signiWcation is

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the matter that is portrayed (deloumenon) by it. . . . and the topic is the external object

such as Dion himself. Of these three items two, the sound and the topic, are

material, but one is intangible, the matter signi Wed, i.e. the lekton, which is what is

true or false. (S.E., M 8. 11–12)

The lekton is what is said by the sentence, namely that Dion is walking. This, as

Sextus says, is not a tangible entity like Dion himself, or the name ‘Dion’, or

the whole sentence ‘Dion is walking’. Dion, the man, is the topic of the

sentence, that is to say, what the sentence is about. Whether the sentence is

true or false depends on whether the matter it portrays12 obtains or not, i.e.

on whether Dion is or is not walking. On the basis of passages such as this,

then, we can say that a lekton is the content of a sentence in the indicative

(cf. Seneca, Ep. 117. 13).

Two qualiWcations need to be made, however, to this deWnition of lekton.

First, Diogenes Laertius tells us that the Stoics distinguished between

self-standing and defective lekta.HeoVers ‘active and passive predicates’ as a

gloss on ‘incomplete lekton’, and explains that a defective lekton is one that

has a linguistic expression that is incomplete, such as ‘is writing’, which

evokes the question ‘Who?’ A defective lekton, therefore, would be what is

said by a predicate, e.g. we may say of someone that he is writing. Such a lekton

remains defective until we make clear who we are talking about, thus

specifying a topic, e.g. Socrates (D.L. 7. 63).

Secondly, indicative sentences are not the only ones whose contents

provide examples of lekta. There are also interrogative sentences, which

come in two kinds: the questions that can be answered by ‘yes’ or

‘no’, such as ‘Is it day?’, and the questions that need more complicated

answers such as ‘Where do you live?’ Again, there are commands, like ‘Take

a bath’ and exclamation s like ‘Isn’t the Parthenon beautiful!’ (D.L. 7. 66–7).

In fact, the deWnition I oVered of lekton as the content of a sentence in the

indicative really Wts only one particular, though most important, kind of

lekton. This is what the Stoics called an axioma. Several deWnitions of axioma are

oVered. ‘An axioma is what is true or false, a complete matter capable of

assertion in and by itself.’ ‘An axioma is something which can be asserted or

denied in and by itself, such as ‘‘it is day’’ or ‘‘Dion is walking’’ ’ (D.L. 7. 65).

While an axioma is capable of being a self-standing assertion, it need not be

12 The customary translation of deloumenon as ‘revealed’ is unsatisfactory since you can only

reveal what is in fact the case. If the sentence is false, there is no matter to be revealed.

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asserted. Neither of the two quoted axiomata are asserted in ‘If Dion is

walking, then it is day’. Hence some authors translate the word as ‘asser-

table’.13 The translation is accurate, but cumbrous, and instead I shall use

‘proposition’ to render axioma, since the meaning of the Greek word, as

explained, is close to one of the standard meanings of the English word. It is

important to remember, however, that a Stoic proposition is unlike an

Aristotelian proposition in that it is not a sentence itself, but something

abstract that is said by a sentence; and that it is unlike a proposition as

discussed by modern logicians since it is something that can change its

truth-value over time.

The Stoics distinguished between simple and non-simple propositions.

Simple propositions are constantly illustrated by ‘It is day’ and ‘It is night’;

but they include three kinds of subject–predicate propositions, which diVer

depending on whether their subject is a demonstrative, a proper name, or a

pronoun functioning as a quantiWer. ‘That one is walking’ they called

adeWnite proposition, ‘Someone is walking’ an indeWnite proposition, and

‘Socrates is walking’ an intermediate proposition. Non-simple propositions

are those that are compounded out of diVeren t propositions by means of

one or more connectives (sundesmoi). Examples are ‘If it is day, it is light’ ,

‘Since it is day, it is light’, ‘Either it is day or it is night’ (D.L. 7. 71).

It is in their treatment of non-simple propositions that the Stoics

approached most nearly to the modern propositional calculus based on

truth-functional operators.14 A number of diVerences, however, need to be

marked.

In the modern calculus the negation sign is treated as a truth-functional

operator, on a par with bin ary connectives such as ‘and’, ‘or’, and ‘if ’.

The Stoics, by contrast, classiWed negative propositions as simple propos-

itions. They did, however, recognize the possibility of negating a proposition

by attaching a negative sign to the entire proposition and not just to the

predicate, the procedure that is essential to the operation of the propos-

itional calculus. Thus, they preferred ‘Not: it is day’ to ‘It is not day’. They

recognized further that negation could be applied to complex as well as

simple propositions; and they realized that in such a case care needed to be

13 e.g. Suzanne Bobzien in CHHP 93 V.

14 A logical operator (i.e. a symbol that forms a new proposition out of one or more other

propositions) is truth-functional iV the truth-value of the new proposition depends only on the

truth-value of the original propositions, and not on their content.

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exercised in order to sort out genuine from spurious contradictories. ‘It is

day and it is light’ was not the contradictory of ‘It is day and it is not light’.

The contradictory must be formed by attaching the negation sign at the

beginning so that it governs the entire proposition. Thus the notion of scope

enters into the history of logic, (S.E., M. 8. 88–90).

Another diVerence between Stoic logic and modern propositional logic

comes out in the treatment of individual connectives. ‘Or’ in modern

propositional logic is treated by convention as an inclusive connective: this

is to say, ‘p or q’ comes out true if p and q are both true and not just when

only one of them is true. The Stoics seem to have been undecided between

this view and the exclusive interpretation according to which ‘p or q’ is true

if and only if one and only one of the constituent propositions is true.

Moreover, the Stoics allowed among the connectives that form complex

propositions some that are not truth-functional. Wheth er a proposition of

the form ‘Since p then q’ is true is determined not simply by the truth-

values of the constituent propositions.

With regard to the conditional connective ‘if’, there is some uncertai nty

how far the Stoics accepted Philo’s truth-functional interpretation of it,

according to which ‘If p then q’ is true in every case except when ‘p’ is true

and ‘q’ is false. Sextus Empiricus roundly attributes this view to them in the

following passage:

A sound conditional is one that does not have a true antecedent and a false

consequent. A conditional may have a true antecedent and a true consequent, e.g.

‘If it is day it is light’. It may have a false antecedent and a false consequent, e.g. ‘If

the earth Xies, the earth has wings’. It may have a true antecedent and a

false consequent, e.g. ‘If the earth exists, the earth Xies’. Or it may have a false

antecedent and a true consequent, e.g. ‘If the earth Xies, the earth exists’. Of these

they say that only the one with the true antecedent and the false consequent is

unsound, all the others are sound. (S.E., P. 2. 104–6)

The examples given here support Sextus’ assertion that the Stoics inter-

preted the conditional truth-functionally. It is characteristic of such an

interpretation that the truth of a conditional does not demand any link

between the content of the antecedent and the content of the consequent.

While ‘If the earth Xies, the earth has wings’ may be linked by the thought

that whatever Xies has wings, no such link connects ‘the earth exists’ with

‘the earth Xies’. Of course, the conditionals in which the Stoics were most

interested were ones in which such a link did exist; as in an example given

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by Sextus shortly afterwards: ‘If she has milk, she has conceived’. But the

same would be true of most of the examples in a modern textbook even

though the logic it expounds is Wrmly based on a truth-functional inter-

pretation of the basic form of conditional.

On the other hand, there are passages suggesting that at least some

Stoics took a diVerent view of the truth-conditions of condition al propos-

itions. Chrysippus is reported as saying that in ‘If p then q’ the connective

declared that q followed from p. This was glossed, by himself or by another

Stoic, in the following way:

A conditional is true when the contradictory of its consequent conXicts with its

antecedent. For instance, ‘If it is day, it is light’ is true because ‘It is not light’, the

contradictory of the consequent, conXicts with ‘It is day’. A conditional is false

when the contradictory of its consequent does not conXict with the antecedent,

such as ‘If it is day, Dion is walking’ because ‘Not: Dion is walking’ does not conXict

with ‘It is day’. (D.L. 7. 73)

Here it seems clear that ‘conXict’ must refer to some kind of incompati-

bility of content between antecedent and consequent, and not just a

diVerence of truth-value. But the exact nature of the incompatibility (is

it logical? is it discovered empirically?) remains unclear.

It is, fortunately, not necessary to resolve these uncertainties in order to

present and evaluate the Stoic theory of inference. Whereas Aristotle had

indicated each of his syllogisms by listing the conditional necessary truths

corresponding to them, the Stoics present their arguments in the form of

inference schemata, sometimes using numbers as variables and sometimes

using standard examples, and sometimes a mixture of the two as in ‘If Plato

is alive, Plato is breathing. But the Wrst, therefore the second.’ An inference,

most Stoics said, must consist of a Wrst premiss (lemma), a second premiss

(proslepsis), and a conclusion (epiphora). It was a minority view that an

inference might sometimes have only a single premiss (D.L. 7. 76).

The criterion for the invalidity of an inference was analogous to the one

Chrysippus oVered for the truth-value of a condition al. An inference was

valid (perantikos) if the contradictory of the conclusion conXicted with the

conjunction of the premisses; if it did not conXict, then the inference was

invalid. A typical invalid inference was ‘If it is day, it is light. But it is day.

Therefore Dion is walking’ (D.L. 7. 77). Nowadays we are accustomed to

distinguish between valid inferences and sound inferences. An inference

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may be valid but unsound if one or more of its premisses is untrue. The

Stoics made a similar distinction, but used the Greek word for ‘true’, alethes,

to correspond to ‘sound’ and ‘false’ to correspond to ‘unsound’. An

inference was unsound, they said, if either it was invalid or it contained

some falsity in its premisses (D.L. 7. 79).

Inferences came in various forms, called ‘moods’. Chrysippus listed

Wve basic forms of valid inference, or ‘indemonstrable moods’ (D.L. 7. 79).

They may be set out as follows, using cardinal numbers rather than ordinals.

(A) If 1 then 2; but 1; therefore 2.

(B) If 1 then 2; but not 2; therefore not 1.

(D) Either 1 or 2; but 1; therefore not 2.

(E) Either 1 or 2; but not 2; therefore 1.

All valid inferences, Chrysippus believed, could be reduced to these primi-

tive forms, and in his many lost works he seems to have proved many

theorems which reduced more complex and derivative moods to these

simple patterns. Thus, if we take

(F) If 1, then if 1 then 2; but 1; therefore 2,

we can show this is a valid inference schema by deriving from the two

premisses in accordance with (A) ‘If 1 then 2’, and then using (A) once more

to derive, from this conclusion and the second premiss ‘2’ (S.E., M 8. 234–6).

On the face of it, Chrysippus’ W ve primitive schemata form neither a

complete nor an irreducible basis for deductions within the propositional

calculus. There is no primitive proposition to justify the inference of ‘p’ from

‘both p and q’; this, no doubt, is because of the reluctance to consider

inferences with only a single premiss. The fourth primitive schema is valid

only if ‘or’ is given its exclusive interpretation; but if it is, then it is not needed,

since any inference that it validates will already have been validated by (C).

In late antiquity Aristotelian logic and Stoic logic were regarded as rivals,

and while the Stoics’ own writings have not survived, we have much

evidence of polemics between supporters of the two systems. With the

hindsight of millennia we can see that the systems were not in general

incompatible with each other, but were formulations of diVerent areas of

logic, each of them precursors of diVerent, but complementary, modern

developments, in the propositional and predicate calcu lus.

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Knowledge and its Limits:

Epistemology

here is a branch of philosophy nowadays called epistemology: the

inquiry into what can be known, and how we can know it. We all have

many beliefs on many topics; which, if any of them, can count as real

knowledge? What is the mark of genuine knowledge and how does it diVer

from mere belief? Is there a reliable way to acquire knowledge of the truth

to eliminate false beliefs that are mere seemings? These questions occupied

the attention of Greek thinkers from an early stage.

Presocratic Epistemology

Parmenides might well claim to be the founder of epistemology: at least he is

the Wrst philosopher to make a systematic distinction between knowledge

and belief. At the beginning of his great poem a goddess promises that he will

learn all things, both reliable truth and the untrus tworthy opinions of

mortals. The poem is in two parts: the way of truth and the way of seeming.

The way of truth sets out Parmenides’ theory of Being, which we will

consider in Chapter 6 on metaphysics. The way of seeming deals with the

world of the senses, the world of change and colour, the world of empty

names. Mortals who do not accept the way of truth, sunk in metaphysical

error, know nothing at all. Deaf, daze d, and blind, they can be called ‘two-

headed’ because of the internal inconsistencies of their beliefs (KRS 293).

A sharp contrast between reality and appearance also appears in the

writing of a very diVerent philosopher, Democritus. For him, atoms and

the void are the only two realities and the qualities perceived by the senses

are mere appearances. To show that sense-appearances cannot be the truth

about things, he argues that they conXict with each other. The sick and the

healthy do not agree about the taste of things, men disagree with other

animals, and sensory properties appear diVerent even to the same individ-

ual at diVerent times (Aristotle, Metaph C 5. 1009 b7). Sense-appearances lead

only to belief, not to truth. ‘By convention sweet,’ he is quoted as saying,

‘by convention bitter; by convention hot, by convention cold; by conven-

tion colour, but in reality atoms and void’ (KRS 549). To say that a

proposition such as ‘The wind is cold’ enunciates a false belief seems not

quite the same as saying that it enunciates something that is true only by

convention; but whatever exactly Democritus meant, it is clear that he

maintained that the senses did not deliver truths about an independent

reality.

If I stand in the same wind as you and pronounce it hot, while

you pronounce it cold, Democritus would say that neither of us is speaking

the truth. The sophist Protagoras took up a quite opposite position: he

claimed that each of us is speaking the truth. (Plato, Tht. 151e). ‘Man is the

measure of all things,’ he famously said; ‘both of things that are that they

are, and of things that are not that they are not’ (KRS 551). Whatever

appears true to a particular person is true for that person. All beliefs,

therefore, are true: but they have only a relative truth. There is no such

thing as the independent, objective truth that Democritus sought, and

failed to Wnd, in sense-appearance. Democritu s objected that Protagoras’

doctrine was self-refuting. If all beliefs are true, then among true beliefs is

the belief that not every belief is true (DK 68 A114).

Protagoras might have tried to counter this objection by restricting his

claim to the case of sense-perception . The expression ‘It appears to me

that ...’,anditsequivalent in Greek, can cover either sense-impressions or

opinions, and this fact is exploited by Democritus in his refutation. Histor-

ically, however, Protagoras did not take this route of escape: his interests

extended far wider than the realm of sense-perception. Diogenes Laertius

tells us that he said that there were two opposed accounts of every matter,

and Seneca that he claimed that on every issue one could argue equally

well on either side.1 If A oVers arguments for p, and B oVers arguments for

1 D. L. 9. 51; DK 80 A20. See J. Barnes, The Presocratic Philosophers, rev. edn. (London: Routledge,

1982), ii. 243.

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not-p, and both sets of arguments are equally good, how should I decide

between them? Protagoras appears to suggest that I should not decide, but

accept both. But does not this involve accepting both sides of a contradic-

tion? On the contrary, Protagoras denied that contradiction was possible

(D.L. 9. 53). What is really accepted is not ‘p’ and ‘not-p’ but ‘ ‘‘p’’ is true for

A’ and ‘ ‘‘not-p’’ is true for B’.

For Protagoras, all truth is relative, and not just truth about obviously

subjective matters such as the feel of the wind. For this thesis, so far as we

know, he did not oVer any argument, merely the analogy between sense-

appearances and beliefs, and a personal claim to be able to match any

argument pro with an argument contra. But the thesis does give him an

escape from Democritus’ trap. He can accept ‘Some beliefs are false’ as

true—but true for Democritus. He can continue to believe that ‘No beliefs are

false’ is true—true, of course, for himself, for Protagoras. There has to be

some other way of sorting out the issue between the two of them—a way

which Plato, as we shall see, attempted to provide.

Protagoras is sometimes described as a sceptic. In one way this is an odd

description. A sceptic is someone who thinks the discovery of truth is

diYcult, perhaps impossible. For Prota goras it is all too easy: you only have

to frame a belief and, hey presto, it is true. But from the point of view of

someone like Democritus, the replacement of a universal, objective con-

cept of truth with a relative one is itself a very deep form of scepticism. The

only kind of truth really worth seeking is, for a relativist, impossible to

discover because it does not exist.

Democritus himself, however, was in no strong position to reject

scepticism. He claimed that there were two kinds of knowledge, one

through the senses and one through the intellect. Only intellectual

knowledge is legitimate knowledge; the Wve senses deliver only a bastard

version (S.E., M. 7. 130–9). There is, however, a problem: the intellectual

knowledge expressed in the atomic theory is based in part on empirical

evidence: and this comes from the cheating senses. Galen, quoting the

dictum about the conventionality of sense-properties, says, ‘Having slan-

dered appearances, [Democritus] makes the senses address the intellect

thus: ‘‘Mind, you wretch! You take your evidence from us and then you

throw us over! Our overthrow is your downfall too’’ ’ (KRS 552).

Logically, then, perhaps Democritus should have ended up not as an

atomist but as a sceptic. One of his pupils, Metrodorus of Chios, is known

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