This post will be a continuously updated commentary on Bobzien and Shogry's paper Stoic Logic and Multiple Generality published in the Philosopher's Imprint in 2020. We use the earlier 'preprint' version available online.
The main thesis of the paper is that although there is no direct evidence that the Stoics developed a full logic of multiple generality there is sufficient evidence that they were in possession of the necessary elements required for such. By this they seem to mean that the Stoics had a rigorous natural-language based (variable-free, Polish-notation style) syntax to express multiple generality but not a full deductive system. We find that the historical, logical and philosophical interest of the material presented transcends their immediate aim of arguing for, in their words, a 'somewhat more gradual development' of the logic of multiple generality from Aristotle to Frege.
On p.2 the authors state that 'certain basic natural language inferences require multiple generality'. Previously we have presented (in Ancient Natural Deduction and previous posts on this blog) evidence for such kinds of inferences in Aristotle, Euclid and Sextus, all of which cannot, prima facie, be justified according to official accounts of ancient logic. Some of these examples would have been of interest to the aims of the paper. What is required to reason (classically or intuitionistically) about multiple generality ? We contend that this is: i) a formal inductive definition of the syntax of quantified sentences involving both monadic predicates and relations and, ii) natural deduction introduction and elimination rules for the universal and existential quantifier. It is immaterial whether i) and ii) be carried out in variable-free combinatory form or not. In our Ancient Natural Deduction we produced evidence that such rules were consciously employed by Aristotle, Euclid and Galen. We then found further evidence in Sextus Empiricus for a similar use by the Stoics. The paper focuses almost entirely on i) with section 9 being dedicated to ii).
Sections 1 and 2 of the paper (pp.4 - 14) present the basics of Stoic 'Grammar' and Logic, relying heavily on Diogenes Laertius. Here are a few questions and observations:
p.4 - Were written characters also classified as part of logos ? Besides the very specialized technical use of the term logos (closed to the modern notion of 'syntax') did the Stoics also use logos in a more traditional sense, for instance, as in logoi spermatikoi ? Aristotle never used logos as pure syntax, as structured sequences of mere sounds or written characters. Is there any explicit discussion in ancient philosophy on how the meaning relation is not binary but trinary, that is, when we say "$X$ means $Y$" what we mean is "$X$ means $Y$ for $Z$" ? "Stoic contents are structured, and their structure corresponds - to some degree - to the structure of language". "In classifying the various kinds of content, the Stoics rely on grammatical properties of the linguistic items that express them". But we read at the end of p.5 that 'the grammatical properties of speech are a defeasible, and potentially mistleading, guide to the content it signifies'. Is this discussed explicitly in any surviving fragment ?
p.5 The key notions of kategorêma and axiôma (pl. axiômata) are introduced, two important kinds of lekta. Axiôma is a weird term for our modern "proposition" (truth bearers) for the root of the Greek term implies already a kind of quasi-alethic judgment regarding the proposition (i.e. it is etymologically derived from the verb aksioô, to think worthy).
Strangely the discussion of the comparison of axioma to Frege's Gedanke and kategorema to Sinne is relegated to a Hellenistic philosophy book chapter on 'Linguistics' by Barnes rather than to the first author's widely known paper on Frege and the Stoics. Kevin Mulligan told me in a conversation that Frege's Gedanken are the same as Bealer's propositions and these could arguably correspond to Stoic axiomata. However the 'mind dependency' of lekta is evidently crucial for judging such comparisons and since the footnote sends the message that this is an open question, so too is the fidelity of the correspondence between Stoic philosophical logic and early modern to contemporary propositional realism.
Simple vs. non-simple propositions: is this a grammatical distinction or a logical one ? Just because an expression is grammatically simple does not mean that it expresses is 'simple' in any obvious sense. For example, take the constant $\pi$.
"Negations and non-simple propositions are defined iteratively". This is just what we tried to argue for in Kant. This would be the first known example of a formal inductive definition of the syntax of a formal language. However operators consisting of 'connective parts' seems complicated and messy - we will return to this later. It would be nice to have simpler modern term in place of 'exclusive disjunction'. Winne-the-Pooh fans might welcome e-or.
p.6 The analysis and elimination of semantic ambiguity is an obsession that runs through Aristotle's Topics. Does 'expresses multiple contents' mean simultaneously or according to context ? What do the authors mean by 'the same content can be signified by different pieces of speech' (we take this to mean anaphora) ? The disparity between grammar and meaning involves more than just ambiguity. For instance grammar does not distinguish between two different types of predicates as in: "the house is white" and "the house is big", here "big" involves necessarily the kind of the subject and its associated standard (as in "the ant is big") whilst "white" is a simple phenomenal quality which is indifferent to what the subject is. This is what Hegel called a "judgment of reflection".
The type of ambiguity under consideration on p.6 is the syntactic ambiguity of the way logical operators occur in natural language sentences. The authors introduce general linguistic considerations involving rigid order vs. case markings which seem too general (for instance classical Latin prose is both marked but had a fairly rigid order). I find the contention about the difficulty of introducing case markings in rigid order and non-rigid order languages unconvincing. The whole discussion in pp.6-7 (and specially the Scope Principle) is difficult to follow. It would have been preferable to invoke the modern notion of Polish notation. This is an alternative syntax for formal logic which does not involve parentheses or markers and in which the implicit markings are unambiguously determined by order. It was employed extensively by Arthur Prior. For instance there is no ambiguity in \[NOAaNbNNc\] where the unary operator $N$ = not, and binary operators $A$ = and, $O$ = e-or and $a,b,c$ are statements.
What complicates things are operators such as e-or which have "parts" (for instance, in "either $a$ or $b$" "either" and "or" are part of the same connective.) The combinatorial study of the integration into Polish notation of such multi-part operators is interesting in its own right and it is nice of authors for having drawn attention to this aspect of natural language connectives. Conjecture: in "either...or" if the "or" part is put in the position required by Polish notation then the second part is spurious. Our tentative interpretation of the Scope Principle is: put all logical operators in Polish notation form and drop the second redundant part of multi-part operators.
There is an equivalent mirror version of Polish notation in which the operators are introduced from the right instead of from the left. This seems approximately closer to the syntax of many natural languages in which the verb is regimented to be in most situations occurring last. Combinatory or variable-free logics use Polish notation as does our Combinatory Intensional Logic which develops the intensional logics of Zalta and Bealer.
The interesting remarks about the kinds of negation of "Plato is walking" merit further discussion as well as their relationship to the Aristotelic syllogism.
p.7 The footnote 11 should have been part of the main text together with a more detailed and careful explanation and illustration of the Scope Principle. "oukhi": this suggests some connection to recent work in philosophical logic (bilateralism). We do not understand the sterêtikon, the alpha privativum: surely this is an operator on kategorêmata not a sentence operator ? Are indexicals also "operators" ? So we would regiment "a big mess, this is !" as "exclamation: this : is a big mess". It is nice to see the syntactic-semantic richness of the list in the footnote - it recalls the similar richness in Aristotle's Topics. What is the difference between katagoeutikon and hôrismenon (both start with a demonstrative pronoun) ? We note that on p.15 footnote 36 suggests that in the list were included different terminologies for the same notions. Aoriston did not seem to make it to the list. Here is the full list given (to which we added aoriston)
apophatikon - starts with oukhi
arnêtikon - starts with oudeis
sterêtikon - starts with alpha privativum
katêgorikon - starts with noun/name
aoriston - (someone is walking)
meson - starts with noun/name (Dio is walking)
katêgoreutikon - starts with demonstrative pronoun
hôrismenon - starts with demonstrative pronoun (this one is walking)
sunêmmenon - starts with ei
parasunêmmenon - starts with epei
sumpeplegmenon - starts with kai
apophatikê sumplokê - starts with oukhi kai
diezeugmenon - starts with êtoi or ê
aitoôdes - starts with dioti
proslêpsis - has de as second word
epiphora - has ara as second word
erôtema - starts with ara
pusma - starts with pou
homoion aksiômati - starts with hôs (Analogy - to be compared with the theory in Topics)
It is too bad that the authors did not add here at least a brief discussion or some examples to illustrate the non-obvious difference between certain elements in this intriguing list. Or maybe there are distinct pairs of conjunctions and disjunctions are in linear logic ? Later on p.16 we are, in fact, offered the examples. We have included some of these enclosed in parenthesis.
p.8 The authors state that Stoic logic is a "propositional sequent logic" when the whole paper presents evidence that it was far more than this and thus, strictly speaking, not a propositional sequent logic. And as for the "sequent" aspect, why is not the first author's well-known paper on "Stoic Sequent Logic and Proof Theory" (found in the bibliography) referenced here (at least it is not in the version I am using).
It is stated that Chryssipus "displayed a keen interest in sub-propositional elements" and among these elements the relationship between "says that : x" and "x is true". So we have in Stoic syntax not only elements for a quantifer logic but for an intensional logic as well ! In "say that : x", what is "x" ? A lekton, an axioma ? So sub-propositional elements of an axioma may be whole axiomata themselves. Were the Stoics explicitly aware of this ? Supposedly "x is true" is also a axioma in its own right having as sub-axioma the axioma "x".
The rest of p.8 contains a prudent caveat regarding the difficulty of reconstructing Stoic logic and the fragmentary nature of surviving materials.
p.9 Now begins section 2 of the paper in which the authors now focus on the intricacies of the katêgorêma, a special kind of lekton (at least since Cleanthes). Footnote 16 offers a first glimpse at classifications of katêgoremata: they could be, for instance, active, passive, event-predicates, indefinite or temporal.
The authors present a definition of katêgorêma found in Diogenes Laertius: an incomplete (ellipes) lekton that "can be connected" or "is connected" with an orthê ptôsis, translated as "upright case-content". As the authors observe, the above definitions strongly suggests that katêgorêma were monadic predicates which nevertheless can connect to plurality through plural case nouns or noun phrases. But there is another question. Take Caius Marcellam amat. The katêgorêma is "Marcellam amat X" connected in this instance to the nominal case proper name Caius. But what about the katêgorêma "X is loved by Caius" ? Would the Stoics have first made the transformation to the passive voice "Marcella a Caio amâtur" putting Marcella back in the orthê ptôsis and then abstracted to the new katêgorêma "X a Caio amâtur" (we are putting aside regimentation for the moment and making use of the variable X just for sake of clarity - the authors would have written "...is lived by Caius", etc. as on p.19) ? But see pp.21-22 when the enigmatic "secondary-event" predicates are discussed.
p.10 ptôsis was translated as "case-content" and this is now specified as "content that can be signified by a noun", footnote 18 further adding that this interpretation should not be considered definite. But surely it would be better to alternatively consider ptôsis as content signified by noun-phrases ?
p.11 DL 7.64 is presented as evidence that though monadic, katêgorêmata connect with pluralities. Prima facie this would be nominate case plural nouns (or noun-phrases such as 'leaping lords'). p.12 presents some more evidence for a monadic reading.
p.13 There is also a polyadic interpretation of katêgorêma, but explicit evidence is given for the definition of rhêma ("verb") in DL. 7.58. A verb must be seen as a polyadic predicate.
p.14 We do not understand why the polyadic reading should be considered a "serious step backwards". Why should this hinder the forming of katêgorêma like $Fxa \wedge p$ ? And we also not do we understand the polemic with Gaskin 1997. The argument might have been settled by observing that the concept of 0-valent verbs such as in "it is raining" are well-known in linguistics; "it is raining", Latin pluit is an example of an axiôma not connecting to nominative nouns which was nevertheless used by the Stoics (cf. the example on p.20: "to walk, since it is day", "it is day" is another example of a lekton without a connection to a nominative noun). Thus anything involving essential connectability to nominate nouns cannot be a general definition of axiôma or lekton.
p.16 footnote 39 mentions the important example in Sextus' Outlines 8.308 which we discussed in a previous post. However not in the context of quantifier rules but to illustrate the tripartite division of the content of argument-places in katêgorêmata. We do not see why the examples and discussions in pp. 15 et seq. should count as evidence for the specific functional nature (in the extensionalist mathematical sense) of kategorêma. Rather they constitute simply a description of syntax.
The three-fold division of the kinds of content for the argument place is of extraordinary interest in light of recent research in non-standard quantifiers and the interpretation of substructural logics (Wansing, Zardini, etc.). We refer the reader to our post on "universal duality" in which the count-noun vs. mass-noun distinction is given central importance. This is reflected in a gradation of determiners. The difference between "a certain" and "some" specially as occurring in the scope of propositional attitudes. We must interpret the difference between Socrates, this man and some man. There is a clearly a progression towards epistemic vagueness at stake. The problem is that "Dio is walking" and "a human being is walking" are both classified as meson. But there was no definite article in classical Greek or Latin. So tempus volat can either mean that a particular time has passed quickly, or time itself as a genera has the property of passing quickly. For the generic use of the definite article cf. Burns' "A man is a man, for all that". Perhaps considerations such as this would have clarified the discussion in the text ?
p.17 It seems unlikely that complete lekta should be defined by being truth-bearers as this will either rule out the lekton "it will rain tomorrow" or commit oneself to metaphysical determinism. But this could be resolved by defining complete lekta as those expressions capable of receiving truth values (but this is not without its own difficulties). Footnote 42 is strange indeed, talking about the completeness or incompleteness of things signified by nouns (but what about "says : that x" and "x is true" ?). It reminds us that Stoic lekta included other things which were not predicates (saturated or unsaturated) at all.
p.19 To us the examples adduced suggest the construction of katêgorêma of the form $\lambda x. F(x,a)$. There is clearly a polyadic type of saturation at work (in fact what is related to the less-than-katêgorêma discussed on pp.24-25). The verb being in the infinite is irrelevant. Why not "...talks to Socrates" if we have "....is talking" ? Clearly there remains something to be said about tense (on p.20 infinitives are directly linked to predicates). To us the evidence suggests that the Stoics had some kind of abstraction operator which yielded a tenseless (infinite) katêgorêma as in the case of Col. XIII 17-22 where "Dio" is abstracted as if a bound variable. From $W(Dio)\& (\neg W(Dio) \rightarrow S(Dio))$ we get the katêgorêma $W(x) \& (\neg W(x) \rightarrow S(x))$ or equivalently $\lambda x.W(x) \& (\neg W(x) \rightarrow S(x))$. This agrees with formula (6) on p.20.
p.20 But what about "to walk, since it is day" ? This is quite a challenge. In fact, it includes an interesting example parallel to that of 0-valent verbs as in pluit: propositions which cannot be analysed into a predicate-argument structure, but rather constitute atomic predicate variables or predicate constants. Thus we could argue that for formalizing natural language predicate and propositional might need to coexist side-by-side.
pp.21-22 The parakatêgorêmata suggest that our previous remarks about conversion to the passive voice to generate katêgorêmata from other argument slots of a lekton is not necessary or was glossed over by the Stoics. This whole matter seems to require further elucidation (cf. footnote 54 "several of the sources seem confused").
p.24. The authors write that "for nontrivial manifestation of multiple generality dyadic predicates are necessary". Strictly speaking this is not true and should be emended to "...predicates of adicity greater than one (i.e. polyadic predicates) are necessary" (i.e. we can have multiple generality with only triadic predicates, etc. Unfortunately "less-than katêgorêmata" appear less than clearly historically attested. The Ammonius passage (H) is interesting and confirms what we remarked for p.19 on "polyadic saturation". Predicates involving less-than-full saturation in the oblique case are fundamental in Aristotle's Topics and it has long been our goal to make explicit the implicit grammatical and logical theory behind them.
p.26 .../---F. There are some interesting problems here. For order must be taken into account in many-step saturation. This requires either strict or complicated conditions in a function-theoretic interpretative context. Our less-than-katêgorêma would have to then be of type $b^1_o \rightarrow....b^n_o \rightarrow a_n \rightarrow T$. We have to saturate imperfectly first with oblique cases and then finally saturate completely with the nominal case (this correspond to what the authors propose on p.26). But this is somewhat artificial alternative of the passive voice construction we previously suggested.
p.27 "We have yet to find examples in which both argument places are filled with the same argument". In the Topics Aristotle strictly rules out the repetition on any subterms in an expression assigning property, genus or definition to a subject. But the most probable reason why constructions such as "Dio loves Dio" are not found is that they would have been expressed through a reflexive verb, that is, the application of a reflection operation on the less-than-katêgorêma ".../---loves". Such an operation is found in Quine's "Variables explained away" (called "short and splendid" on p.31) as well as in the subsequent work on intensional logic by Bealer and Zalta and our own system of Combinatory Intensional Logic. Thus "Dio loves Dio" would be the result of the single saturation of "...loves him/herself" by "Dio". The reflection operator works parallel to the saturation by oblique cases. All this is developed in Bealer, Zalta and our own system of CIL.
p.28 Finally we start to delve into multiple generality itself. This will involve quantifier determiners somebody, something (tis) and anaphoric pronouns (cf. footnote 67). These elements are used to saturate less-than-katêgorêmata to express multiple generality. The notable example is taken from DL 7.75
If someone gave birth to something, then she is its mother.
To us it is not enough to have polyadic predicates and multiple quantifiers to obtain "problematic" multiple generality (both syntactically and deductively). For there is the clearly attested special case of Galen's relational syllogism, which involves universal quantifier elimination for sentences of the form $\forall x,y. \phi(x,y)$ where $\phi(x,y)$ is quantifier-free (generally an implication). See our Ancient Natural Deduction for more details. We could indeed read the above sentences as
\[\forall x, y. GiveBirth(x,y) \rightarrow Mother(x,y)\]
rather than the very much more complicated and tentative
\[\forall x. (\exists y. GiveBirth(x,y) )\rightarrow Mother(x,\iota z. GiveBirth(x,z)) \]
We will not go into the matter of higher-order alternatives here.
p.30 "The Stoics use iterative definitions in many parts of their logic". We have suggested in a previous post that so does Kant in his Kritik, though it is unlikely that there was any direct influence. The authors examine our favourite example from Sextus as evidence for ternary predicates. In our previous post we took the ungrammatical "told that be rich" as a binary predicate. But as the "said: that x" structure was already invoked, the ternary interpretation (which entails a Stoic intensional logic) does indeed not seem so far-fetched after all. But apparently there are reasons against this interpretation, something involving "unmethodical arguments" (the authors refer oddly enough to Alexander of Aphrodisias's commentary on the Analytics...).
p.31 The authors present the most basic kind of aoriston axioma (indefinite proposition) which saturates a katêgorêma with the determiner tis, symbolically $\tau F$. This is the analogue of $\exists x. Fx$. The authors delve into the epistemic and semantic nature of this existential quantifier: "indefinite propositions do not refer", they do not pick any particular thing, etc. I wrote "epistemic" because belief in the existential proposition in question is not the kind of belief that follows from acquaintance with a particular definite instantiation of the predicate, rather an indefinite, undefined belief which would follow from the presence of any instantiation. Thus this is a mass-noun-like $\exists$, in the terminology of Zardini, multiplicative exists $\bigoplus$. The discussion surrounding the second kind of indefinite proposition (L) seems more involved, although ultimately convincing. In a previous post we explored the idea that all quantifiers should have a domain, as in dependent type theory (cf. p.41 "virtually, all extant examples of Stoic universals are restricted univerals(...)"). The authors argue that if we restrict all universal quantification to the form $\forall x. A(x) \rightarrow B(x,..)$ then the linguistic mechanisms of the determiner tis together with anaphoric pronouns are sufficient to express any form of multiple generality. For instance $\forall x. Walks(x) \rightarrow Moves(x)$ can be expresses as "if someone walks than he moves". That is, indefinite conditionals correspond to universal quantification.
We will not go into the details of the argumentation pp.34 - 42 for the simple nature of the indefinite conditionals. Rather we examine the situation from the point of view of a variant of combinatory logic. From a combinatory point of view it does not make much sense to speak of restricting logic to monadic predicates as will be clear by what follows. In combinatory logic we view katêgorêma as objects (we use the notation $\lambda x. F(x)$ or $\lambda x,y. G(x)$) on which certain operators (unary or binary) act. A basic kind of operation is diagonalization or reflection $D(\lambda x,y. F(x,y)) = \lambda x. F(x,x)$. Logical connectives, as our authors show, have incarnations as parts of katêgorêma not just saturated lekta. We can define a general operation like $A (\lambda x. F(x), \lambda y. G(y)) = \lambda x,y. F(x) \wedge G(y)$ which is now a less-than-katêgorêma, a binary predicate, and then proceed to apply $D$ to obtain a unary predicate. We can define similarly $I$ for the conditional and other operations for e-or and negation. Quantification can be seen as reducing the adicity, saturating a unary predicate to a proposition $Q(\lambda x. F(x)) = \forall x. F(x)$ and a binary predicate to a unary predicate $Q(\lambda x,y. F(x,y)) = \lambda y. \forall x. F(x,y)$. We can of course express our Stoic universal quantification as a binary operation defined in terms of these operations $S = QDI$ but we can also make it a primitive. Saturation itself is an operation working outwards first $\Delta(\lambda x.F(x), a) = F(a)$, $\Delta(\lambda x,y. G(x,y), a) = \lambda y. G(a,y)$. There is no way to write $S$ in the form $O...$ in which $O$ is a logical connective, thus if simplicity is defined in terms of syntactic decomposability into logical connectives, then propositions or predicates in the image of $S$ cannot be non-simple.
Now lets look a the $\tau-\epsilon$ construction, tis + anaphora. This suggests an alternative combinatory calculus. A first attempt would be to read tis an existential operation: $Tis(\lambda x. F(x)) = \exists x. F(x))$. This, as we have seen, agrees with our authors for the Stoic analogue of existential quantification. But how can we express $S$ in terms of the operations of $T$, $D$, $I$ and negation $N$ ? This would have to be something like $N Tis N D I (\lambda x. F(x), \lambda y. G(y))$ which we would read as
not: there is something such that: not : if F hold of it then G holds of it
which we understand as
if F holds of something then G holds of it
These considerations suggest that there is something quite interesting and not yet understood in the underlying natural language quantification mechanisms and a great merit of the present paper is to have brought this to light. We will reserve out commentary on sections 6-8 until we have investigated these matters more thoroughly. We now move on to section 9 concerned with inference.
p.63 The authors claim that the Stoics had a tropos involving instantiation of Stoic universal quantification followed by modus ponens and an example is given from Augustine and Cicero. Let us call this trope $S\forall$-elimination: from $\forall x. A(x) \rightarrow B(x)$ and $A(a)$ deduce $B(a)$. But the example from Sextus Outlines 8.308 ("if some god told you that you will be rich...") that we discussed in a previous post and which is also discussed by the authors, seems equally interesting, specifically because it illustrates that in Stoic universal quantification the antecedent of the condition can in turn be a quantified proposition and for providing evidence for a Stoic rule of introduction of the existential quantifier. We read the inference as follows:\[ \forall y. (\exists x. God(x)\& ToldRich(x,y)) \rightarrow BeRich(y)\tag{1}\] \[ God(a)\& ToldRich(a,b)\tag{2}\] therefore \[BeRich(b)\]
The Stoics probably proved this by first deducing $\exists x. G(x) \& ToldRich(x,b)$ (3) from (2) by $\exists$-introduction and then applying $S\forall$-elimination to (3) and (1) to obtain $BeRich(b)$.
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