Plato's Sophist and Type Theory (older post)

Suppose we had a type $A \rightarrow B$. Then application $a : A, f : A\rightarrow B \vdash f a : B$ can be 'internalized' as a type \[ A\rightarrow (A\rightarrow B) \rightarrow B \] But application of this type can likewise be internalised as\[ A \rightarrow (A \rightarrow B) \rightarrow (A\rightarrow (A\rightarrow B) \rightarrow B ) \rightarrow B \] and so forth. This is similar to the 'third man' argument of the Parmenides. Take $B$ to be the 'truth-value' type $\Omega$. The 'canopy' argument in fact seems to herald the idea that a type (i.e. a 'form' or 'unsaturated' propositional function) should be seen not only a 'set' but as a 'space' as in homotopy type theory. In a passage of the Sophist 237 there is a remarkable discussion on 'non-being'. You cannot talk about non-being because by doing so you already attribute it implicitly the mark of a something, an ought - both unity and being. Now this passage is in many ways an anticipation of the rule of contradiction (for $\bot$) in natural deduction as well as its logical deployment in axiomatic set theory, specially dealing with $\emptyset$ in formal proofs. But most interesting is the connection to Martin-Löf type theory: the zero type or empty type $\mathbb{O}$. This type is not inhabited by anything. But yet to use this type, to reason with it, you must assume that it is inhabited $ a : \mathbb{O}$.  Martin-Löf type theory allows us to flesh out Plato's intuition about the connection between falsehood, nothingness and absurdity. The empty set is a paradigm of the initial object of a category. There is a unique set theoretic function $f : \emptyset \rightarrow X$ for a given set $X$. Correspondingly the type $\mathbb{O}\rightarrow A$ is inhabited where $A$ is for example some non-empty inductive type. Thus we can speak meaningfully about nothing. Consider the very difficult passage Sophist 243-246 that clearly picks up on and rectifies the Parmenides. But for Plato what are 'Being', 'Unity' and the 'Whole' ?  We can consider the unit type $\mathbb{I}$ as corresponding to 'Unity' and think of it as either a set-theoretic singleton $\{u\}$ or as a contractible type where equality is interpreted as a homotopy path. In this sense it is a homogenous space much like the 'sphere' in Parmenides' poem. Plato does indeed distinguish between pure unity and a whole participating of unity. The singleton set is a paradigm of a terminal object in a category.
The 'Whole' is clearly a 'universe' type $U$ or $Type$.  It is a difficult problem to relate the 'Whole' to 'Unity'. What does it mean even for the 'Whole' to participate of 'Unity' ? That there is one supreme universe $Unity$ with only one inhabitant $Type : Unity$ ? But then we could not have accumulativity : $ a : Type$ implying that $ a : Unity$. Or, categorically, how do we interpret that all objects $A$ admit a unique morphism $A \rightarrow 1$ ? Category theoretically a ($\infty-\enspace$) groupoid is a candidate for a category 'participating in unity'. There is a tension between unital being $\mathbb{I}$ and the being-whole, the being spread out and shared by all beings $Type$. The logic in this passage is apparently 'type-free' and impredicative. 'Being' $: \Pi ( X: Type), \Omega$ and 'Unity': $\Pi (X : Type), \Omega$ seem to be able to be applied meaningfully to anything (indeed $Type : Type$). The passages on 'names' and 'reference' is quite striking. Specially when Plato conjures up 'names that refer to names' and a name referring only to itself (imagine a type only inhabited by itself). Part of Plato's argument is that no matter what in reality 'is' the fact is that there is a plurality of \names. In the later section on the five suprema genera we may ask: why is there not also a form for 'participation' itself ?What are some fundamental kinds of types ? The empty type $\mathbb{O}$, the unit type $\mathbb{I}$, the universe(s)  $\mathcal{U} $ (or $Type_i$), equality $ \Pi ( X\enspace Y: Type), Prop $, number $Nat : Type$ and the interval $\mathbf{I}$ for path spaces or cubical sets in homotopy type theory. Paths represent change, the interval represents temporality. Plato speaks of equality being or not being equal to something, just as Voevodsky speaks of equality being equivalent to equivalence.

Note on the Stoic Categories (older post)

We plan to study the formalization of Stoic logic and the Stoic categories.  I had made a close study of Bealer's  first-order intensional  logic such as presented in his book Quality and Concept. Unlike most approaches  Bealer takes propositions, unary predicates, relations, etc. to be primitive entities, units of meaning which are woven together by complex logico-combinatoric relations including the attribution of truth-value depending on states-of-affairs.  Afterwards I read Susanne Bobzien's papers on Stoic logic and specially  Did Frege plagiarize the Stoics which offered insights into both the Stoics and Frege. I then realized that the Stoic theory of lekta was an intensional version of Frege's logic and that not only did it agree with Bealer but my experience working with  dependent type theory in the Coq proof assistant suggested to me an alternative intensional type-theoretic  version of the semantics of Bealer's logic which is closely aligned to Stoic logic. There is a primitive type of 'saturared' lekta $\Lambda$  corresponding to Bealer's set of propositions $\mathcal{D}_0$,   a primitive type of truth-values $\Omega$ and a primitive type $W$ which can be thought of as indexing possible states-of-affairs or 'possible worlds'.  Our logic is essentially a logic of meaning and all operations are defined primitively on $\Lambda$ or types involving $\Lambda$ rather than on $\Omega$ (like in topos theory). The logic of truth and extensions is mediated by what I call the alethic term $\alpha : W \times \Lambda \rightarrow \Omega$ which specifies which assertibles hold in a given situation or state-of-affairs. As Bobzien writes in the Cambridge Companion (2006): The most far-reaching one is that truth and falsehood are temporal properties of assertibles. They can belong to an assertible at one time but not at another.  Note that in the Calculus of Constructions upon which Coq is based we could consider taking the type Prop as $\Lambda$. I found that  section II  of Logic and General Theory of Science (lectures from 1917/1918)  Husserl carries out an analysis of natural language which corresponds closely to the type-theoretic intensional logic I had in mind. I noted that in paragraph of section II Husserl writes: A grammatical distinction passed down from Scholasticism, and otherwise going back to the Stoics, can serve as our point of departure. This is the distinction between independent and dependent expressions.

There is also the enigma of the Stoic categories. An old paper by Margaret Reesor (1957) presents some interesting information about the category of 'relation'  pros ti  and specially pros ti pôs ekhôn translated as 'relative disposition'.   It seems that the older Stoics considered such  terms as 'intelligence' and 'virtue' both in themselves simply and qualified and related to something else.  For instance for Zeno courage is wisdom in things to be endured and justice is wisdom in things to be distributed.  It does not seem straightforward to present this in terms of the modern predicate calculus, for instance a binary relation $\exists y.R(x,y)$ instantiated differently as $ R_1(x):=R(x,a),  R_2(x) := R(x,b)$, etc. Rather this situation recalls polymorphism in modern type theory or its more general form involving dependent type theory with universes.  This can be understood by an example from programming languages where there is a general class for 'list'  taking a type as a parameter which then becomes a concrete object according to the specified type (we get lists of integers, lists of boolean values, lists of strings, etc.).  Thus dependent types such as $\Pi_{T: Type} Type$ would seem to capture the Stoic category of relation which includes 'virtue' and 'intelligence'. These are both  types in themselves and  types which can be further determined through specification of an external type $S$ which we obtain through application $(\Pi_{T: Type} Type) S : Type$.  Thus we can speak of 'list' simply or 'list of $X$s', 'list of $Y$s', etc. Likewise we have both 'wisdom' and 'wisdom in $X$', 'wisdom in $Y$' for types X and Y.

What of the concepts of  dependent and independent meaning ?  The type $\Lambda$ is certainly independent. The corresponding dependent types are those that can be written equivalently (i.e. via currying) in the form $X \rightarrow \Lambda$. Thus for example 'or' would have type $\Lambda\rightarrow(\Lambda \rightarrow \Lambda)$ which is equivalent to $\Lambda\times\Lambda \rightarrow \Lambda$.  But there are other types of independent and dependent meanings.  A preliminary proposal might be that independent meanings are atomic types and dependent meanings non-atomic types.

A dependent meaning can be transformed into a independent one, for instance when we talk about 'the connective or'.  This can be handled by a term $ n_X : (X\rightarrow \Lambda) \rightarrow N$ which transforms a dependent meaning type into for instance a noun term of the atomic type $N$. This is of course also Bealer's  intensional abstraction /nominalization operator.

Formalism is not clarity

Ich setze also voraus, daß man sich nicht damit begnügen will, die reine Logik in der bloßen Art unserer mathematischen Disziplinen als ein in naiv-sachlicher Geltung erwachsendes Sätze system auszubilden, sondern daß man in eins damit philosophische Klarheit in betreff dieser Sätze anstrebt (...) - Husserl I Log. Unt.

On a surface level Aristotle's Organon and Physics are formally impressive and from a contemporary mathematical point of view quite suggestive.  Yet, if we analyze things very carefully we find that at a deeper level we are in the presence of a big step backwards from Plato which also cannot really be compared to the sophistication and brilliance of the Stoics. For in Aristotle the key fundamental terms and concepts ("term", "concept", "predication", "essential predication",  "proposition",  "huparkhein", "quality", etc.) are never defined, elucidated, clarified  and perhaps not even used consistently. There is also a serious lack of grammatical and linguistic analysis . To study Aristotle it is not enough either to engage in traditional "classicist"  or commentary-based methods of exhaustive textual analysis and nitpicking or to think that somehow modern mathematical or symbolic logic in itself is sufficient tool to clarify all problems.  Rather we must deploy what is scientific and sophisticated in modern philosophy to bring to light what lies beneath the surface of the Aristotelic texts. Fortunately we do have a kind of philosophical Principia Mathematica, and this is Husserl's Logical Investigations and other subsequently published and equally important texts complementing and developing this work.

Recall Husserl's distinction between judgments of existence and judgment of essence.  Can this help us understand the universal quantifier ? Consider:

1.All ducks can swim.

2.All people in this room are under 30.

3. All prime numbers greater than 2 are odd.

 What does 1) mean, what do we mean by 1).  That swimming is part of the definition of duck, that being able to swim is a logical consequence of the definition of duck - and here we are assuming an artificial consensus - or that everything belonging to the extension of duck (for instance, we could just take a heap of things and label it "duck") happens to have the property of being able to swim (this is unlikely or at most genetic, plausible) ?  For 2) we cannot state that being under 30 somehow is a logical consequence of the concept of being in this room.  2) is definitely a Husserlian 'judgment of existence'.  3) can be given an extensional reading but it also could be given a logical reading in the sense that being odd follows from the definition of being prime and the condition of being greater than 2.  Thus 3) differs from 1) and 2) by allowing both interpretations. 3) can also mean: there is an algorithm which takes as input a prime number and a proof that this number is greater than 2 and yields as output a proof that it is odd.

But consider a model-theoretic approach.  For a model $M$, representing the current world, or current global state-of-affairs, we may well have that $M \Vdash \forall x. \phi(x) \rightarrow \psi(x)$ without it being the case that for our theory $T$ we have  that $T \vdash \forall x. \phi(x) \rightarrow \psi(x)$. But the statement $M \Vdash \forall x. \phi(x) \rightarrow \psi(x)$ must itself be proven in some metatheory $T'$ and is thus again purely logical.

The extensional interpretation of 1) can be: i) that things in the extension of "duck" have the property of being able to swim. ii) that the extension of "duck" is contained as a set in the extension of "being able to swim". 

More profound is the dependent-type theoretic interpretation $\vdash p: \Pi_{ x : D} S(x)$ which reads: there is a function $p$ which takes as input a duck and yields a proof that that particular duck can swim. Compare this to Bobzien and Shogry's interpretation of Stoic quantification:

If something is a duck then that duck can swim. 

How far we are from understanding quantifiers, concepts, extensions and predication in general !

Radical mathematical logicism is the position that logic (or pure rationality) only exists fully in mathematics (and mathematical models in science).  Natural language can only attain an approximate rationality via a mathematical pragmatics (as in computer science).

There is, at first sight at least,  a huge chasm between our mathematics and the complex organic self-directed concreteness of living systems and consciousness. But this chasm can be bridged if we study mathematical theories qua theories, their diachronic and synchronic systemic articulation and organicity seen as an abstract version of consciousness and life. 

If in mathematics both formal and conceptual clarity are of great importance, in philosophy they are even more so.  While agreeing with the quote of Husserl we do not undermine the greatness of formal clarity and the huge progress in philosophy that, in the scale of things, would be achieved by a formal philosophy even if this not mean the ultimate clarity and the highest development of the philosophical project.

What is a term for Aristotle ?

One of the uniquely interesting features of the Topics is that it offers a wealth of examples illustrating the fine logical-grammatical structure of those linguistic expressions considered terms for Aristotle.  The common view that in Aristotle we have essentially a term-logic while in the Stoics we have a proposition- or sentence-logic is untenable. Not only have Bobzien and Shogry shown that the Stoics had a highly developed theory of sub-propositional elements but the evidence points to the Aristotelic term being what we now call a noun-phrase which can include embedded sentences, in particular that-clauses. Thus this opposition is merely apparent

In Plato’s Sophist there is a discussion concerning ’sentences’, ’nouns’ and ’verbs’, sentences being combinations of nouns and verbs, and all sentences must have a ’subject’. Plato gives the example ’Theaetetus, with whom I am now speaking, is flying’. This illustrates how Plato’s concept of ’noun’ corresponds to our ’noun phrase’ and very likely to the Aristotelic term or ’oros’.

1. We must be careful not to confuse Aristotle's convertability and co-extensionality with our modern notions, in particular, first-order notion. Aristotle's convertability  is more of a co-predicability. Thus the Aristotelic 'extent' of a term is not framed using individuals but other terms in general. The extent of a term B is the collection of all terms C for which B can be said of C.  Thus subordinate species and the subordinate species of theses species are all part of the 'extent' of a genus.

2.  Relative idion is a trinary predicate: $RI (x,y,z)$ read: $z$ is the relative idion of $x$ relative to $y$.

4. It will be good to have a complete glossary of all the 'terms' mentioned in the Topics and the relationships between them, either rejected or accepted (and sometimes, unfortunately both) by Aristotle. Examples of terms in book V : bear a very close resemblance to the soul , the primary element wherein the soul is naturally found,  possession of sensation,  being natural sentient, the most rarefied and lightest body, the substance to which "man" belongs as a species, that which is made of body and soul. We at once notice some grammatical issues with this translation, not to mention logical, ontological issues (the flexibility is amazing, even the meta-level predicabilia themselves can be part of the post-determiner of the noun-phrase). How much better is the original Greek. For instance the first term is : to homoiotaton psukhê(i). This appears to be an ellipsis for 'the thing ousia most similar to the soul', that is if we postulate the general structure of the Aristotelic term to be

Genus G + post-modifer

In modern logic the Peano iota operator involves individuals. But for Aristotle it is natural to conceive Peano operators for species, for instance the accepted property of animal as to ek psukhês kai somatos sugkeimenon. But this can be read: the composite consisting of soul and body. Definitions and properties of mass-nouns are not uncommon in the Topics. 

One topic is the following:

forall C.  Subexp(C,B) and Species(C,A) => not Idion(A,B)

Wait a minute ! Are we talking about the linguistic expressions or what they mean ? How dare you put the predicates for subexpression and species and idion all in the same formula !  But what if we want to deal with globs which include both the syntactical, semantic and relational aspects at once ? Fans of ordinary language and ordinary usage,  what do you mean when you say "the word 'cat' " ?   The glob ! Only linguists can invent some fancy brackets to isolate the purely phonetic or orthographic aspect, like /cat/.  Like it or not, Aristotle deals with globs - or perhaps, writing elliptically,  he does not. Maybe our Subexp is wrong, and it should be replaced with semantic inclusion ?

What is the concept of man ?  Does it differ for different people of different cultures at different times and places ? Are we after the invariant in all these concepts ? How does a concept relate to other concepts, to the mind and to the concrete things to which it is predicated ? What is the genesis of concepts and how do they relate to perception ?  What about the different modalities and intentions in which a concept takes part in ? What is predication of essence  and how can one know a concept yet definitions be defeasible ? Surely rationality, bipedalism, etc. are simpler, clearer, concepts and perhaps thus more invariant. But is the genus simpler than the species ? Does predication of essence relate to the essence qua embodied in a concrete substance or to the concept ?

Relevant logic and Aristotelic predicabilia

Relevant logic is concerned with logical concequence $\phi \vdash \psi$ but many of its aspects would seem to be transposable (or at least similar) to the conditions imposed on the predication of genus, definition and property in Aristotle's Topics. Co-extensionality (or "convertability") is a necessary but far from sufficient condition for a term $B$ to be the property (idion) or definition of $A$. For the non-redundancy, non-circularity and minimality conditions involved see our previous posts. Relevance obviously has a pragmatic dimension and can be analysed from this perspective (being careful to avoid naive materialist assumptions about the human mind). For example, for Aristotle the property $B$ must be better known than the term $A$ of which it is  a property of - this is of course rather 'pragmatic'.  But let us consider other conditions.  Repetition of the definiendum is ruled out in predication of essence (definition or genus). "What is a man ?  A man." would not be accepted just as $A\vdash A$ is not accepted in relevance logic.   Also "What is that ? It is an animal and it is an animal" would be rejected. In fact, it would seem that Aristotle tried to rule out all irrelevant information from predication of property, definition or genus.  Another example of repetition, irrelevancy and redundancy: "a rational animal that is an animal".  The idion is like a sign, a token, an indicator: "Since I heard laughing, I knew somebody was there." Thus idion and essential predication presuppose: awareness of the common knowledge between the two speakers - for the Topics, we must not forget, is essentially dialectical - and hence pragmatical with an awareness of a gradation between concepts and the undesirability of spurious and irrelevant content. The 'better known' is of course further developed by Aristotle as he distinguished between things better known to us and things better known in themselves . It is not necessarily pragmatic and social, it could follow from some measure of semantic or logical complexity or density of the concepts themselves. For Aristotle property is given for the sake of learning tou mathein kharin 131a.

Commentary on Bobzien and Shogry - Stoic Logic and Multiple Generality

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 $BeRich(b)$.

Is philosophy dead ? And why ?

Fake 'skepticism' is worse than  explicit 'dogmatism', in fact, it is the worst kind of dogmatism. Fake skepticism gives the appearance of openness, open debate, discussion, dissent, pluralism,  by frenzily moving around and nitpicking a few leaves and twigs (often employing very important sounding terminology) while leaving the root, trunk and main branches tacitly and implicitly unquestioned, unexamined, untouched - as if the implicit acceptance of this system of dogmas and the peculiar patterns of linguistic expression they entail were some kind of gate-keeping to the advancement of a philosophical career. We must never loose sight that philosophy is heavily sociologically conditioned.  Fake scepticism does bear resemblance to certain negative aspects of medieval scholasticism in so far as it might claim to be pure philosophy rather than philosophy in service of theology.  Also it is easy to claim to be 'precise' and 'analytic' when 90 percent of all questions have already been settled and agreed upon and one can engage in mere twig tweaking and leaf niggling - a mere exercise in exegesis in an unquestionably venerated authoritative historical text. In the middle ages there was very scarce direct critical engagement with Aristotle, rather there mere exegesis involving fine-grained distinctions designed to save Aristotle from himself or to reconcile him with blatantly contradictory theological premises.  

Someone may retort: this is not true, look at all the variety and dissent and difference of views in contemporary philosophy ! We live in a climate similar to that of ancient Greece, every conceivable view is represented; contemporary 'analytic philosophy' is eclectic and pluralist, gone are those dogmatic and sterile heydays !  This raises the question: what standard, what criteria should be used then to settle this debate ? How do we judge which view is correct ?  But for the present we must be content to remark:  Frege and Husserl developed extensive and finely articulated arguments for certain philosophical positions.  Why are these views and their arguments not more studied, analyzed, developed or critically engaged in by contemporary philosophy in the midst of the sea of  proponents of clearly contradictory positions ? So either Frege and Husserl's positions and arguments are so bad and weak that they simply do not merit consideration at all, or else our main contention concerning dogmatic fake skepticism is correct.

Universal duality

  By universal duality we mean a duality which permeates prima facie different regions. For instance mathematical logic, algebra,  language, ontology, topology and physics.  Despite the core ternary structure in Hegel, it is easy to argue that duality and opposition also play a similarly fundamental (if subordinate) role.  So we might speak additional of a universal structure which is a kind of 'combination' of the two sides of this universal duality - that is, a move from a more implicit or less concrete kind of mutual dependence between the two poles into a more intimate fusion. In itself, of course, all this is not new. What is remarkable is just how formal, precise and analytically this perspective can verified, developed, and argued for.  Duality is generative. That is each pole of a given concrete duality can itself split into its own duality which may reflect on its own the plane the initial duality, which in turn embodies the universal duality.  Fundamental dualities are for instance the continuous vs. the discrete, open set vs. closed set, mass nouns vs. count nouns,  imperfect vs. perfect tense, intensional vs. extensional, possibility vs. necessity,  disjunction vs. conjunction,  existential quantifier vs. universal quantifier,  colimit vs. limit,  left adjoint vs. right adjoint,  addition vs. multiplication,  introduction vs. elimination rules in natural deduction,  classical vs. intuitionistic, etc. An interesting task would be to somehow unified a great part of these dualities from a single perspective, for instance, the category theoretic one could be seen as a least a partial candidate for a universal duality, the others being special cases. In real life many of the above dualities are found to a certain extent unified and harmonized into an interconnected whole: disjunction and conjunction are both part of logic and satisfy precise interaction laws such as distributivity.

  It seems that few have delved as deeply into the symmetries and dualities of logic than Jean-Yves Girard.   For he split the members of the initial dualities between true and false, conjunction and disjunction, into a further additive vs. multiplicative pairs,  and also splitting $\rightarrow$ as well as introducing 'new particles', the pair of exponentials.  A further task would be to split the quantifiers as well and to give detailed interpretations of all these Girard pairs with special focus on Hegel's logic. One place we could start would be in the category of quantity with special focus on the mass noun vs. count noun distinction and then move through essence to finally try to attempt an interpretation of notion and its three components universality, particularity and individuality with a special focus on intensionality, induction and non-standard quantification. The duality of quantifiers  analogous to $\otimes, \oplus, \sqcup, \sqcap$ is strongly related to our previous considerations and holism and the relationship between local and global.  For instance the two extremes: global information can be contained locally or else locally the can be no trace of the non-trivial global information present.  The following are key: weakening, contraction, context dependence and context sharing.

The logic of multiple generality in antiquity

In an older preprint of ours (Ancient Natural Deduction) we tried to argue from Aristotle, Euclid and Galen that the ancients were in possession of the basic natural deduction rules for quantifiers. In Sextus' Outlines II, v, 23 (discussing a view of Democritus) we find the following theorem involving quantifiers: From the premises \[ Human(x) \leftrightarrow \forall y. (Human(y) \rightarrow Know(y,x))\tag{1}\]\[\neg \exists x. (Human(x) \& \forall y. (Human(y) \rightarrow Know(y,x)))\tag{2}\]is concluded \[ \neg \exists x. Human(x)\]

It would be interesting to find in detail how the ancients might have carried out the proof of this conclusion. Could Stoic logic have handled this ?

There also an interesting argument in II, xii, 141 (If some god has told you that this man will be rich...) which we interpret as:  \[ (\exists x. God(x)\& ToldRich(x,y)) \rightarrow BeRich(y)\tag{1}\] \[ God(a)\& ToldRich(a,b)\tag{2}\] therefore \[BeRich(b)\]

In modern natural deduction we would use $\exists$-introduction on (2) for $a$, $\forall$-introduction on (1) for variable $y$ and then $\forall$-elimination for substitution $[b/y]$ and finish with modus ponens. How would the Stoics have done this ?

 We plan to write a survey article discussing Bobzien and Shogry's paper Stoic Logic and Multiple Generality. These authors argue that the Stoics 'had all the elements' to develop a variable-free version of quantifier logic, even if there is no direct evidence that the Stoics actually carried out such an enterprise. One of the reason this interests us is because of our own project which involves combinatory (variable-free) intensional logics.

Hegel as a reader of Sextus

In this post we plan to explore parallels between the argumentation in Hegel's Logic and those in Sextus' Outlines.  Our first observation will concern the section on 'syllogism' in the Subjective Notion and its comparison to Sextus' analyses of proof and deduction in book II of the Outlines.  Hegel argues for the infinite regress and insufficiency of the syllogism of the form IPU and concludes that it is the individual that must mediate between the particular and the universal PIU. Sextus argues that this same syllogism is circular because if the conclusion can be reached from the premise(s), the major premise could only have been reached by taking into account the conclusion (Socrates is mortal): for to arrive at 'all men are mortal' it was necessary - by induction - to consider the case of the man Socrates and having ascertained that the man Socrates is mortal. Also Hegel and Sextus invoke in a similar way the infinite regress which results from further demanding a proof of the major premise and so on.

One of Sextus'  rather dubious arguments against genera and species is actually already found in the Aristotle's Topics serving anti-platonic purposes. This involves how the differentia demarcating different species can subsist in the genus: all at once, none at all, only potentially ? Hegel's whole theory of Notion seems to be a development of an answer to these objections (see also Sextus's arugment against common predicates whereby seemingly all predicates become inseparably individuated - this suggests a connection to Hegel's theory of individuality in the Notion).

This is a theme which merits extensive developments. In particular one is struck by the profound connection between Sextus' arguments and both Hegel and Husserl.  In particular Hegel's critique of Kant or the fact that the main components of Husserl's theory of epokhê are found in book II of the Outlines. We can consider thought and appearance without conceding the claim-to-reality. 

The 'scepticism' of Sextus is perhaps the most distorted and misunderstood 'ism' in all philosophy.  It has nothing in common with either Hume or post-modernism.  Sextus is not a theory of relativism and it is not any kind of nihilism.  Rather we must leave Western philosophy altogether to find an analogue. In a previous post we outlined a kind of formal logic version of Pyrrhonism. We do not claim that this is compatible with Sextus either, but it does share some methodological principles. Basically it is dubious if Sextus can be considered philosophy, rather we are in the presence of pure methodology and a methodic challenge which might be described as a transposition of Hilbert’s program to the analysis and critique of philosophical systems. On the most basic level the challenge is this: formalize your axioms, deductive system, definitions and proofs - and analyse consistency and completeness. But then the whole story starts again at a meta-level - yet there is one main strategy open to Sextus : produce concrete proofs in an agreed upon system $D$ of both $A$ and $\neg A$ where $A$ is, for instance, some philosophically relevant sentence.  Sextus ultimately has no other weapon besides formal logic including metalogical methodology. However we must also analyse negation, paraconsistency, etc.  Contradictions are sometimes resolved by refining concepts and predicates, by relativization and specialization (a monadic predicate, for instance, is refined to a relation, or multivalued logic is considered, and thus a cat can be black and white).

In order to suggest the unity of the Sextus-Hegel-Husserl triad, we present here a quote from Husserl's Introduction to Logic and Theory of Knowledge (tr. C. Ortiz Hill, pp.189 et seq.)

After we have made clear the meaning of epistemological skepticism as a methodological precondition of the beginnings of a theory of knowledge, the question however arises as to how theory of knowledge is still possible afterward, and how it can unfold as a scientific discipline in a series of progressive cognitions without slipping back into the prohibited psychologism. If we adopt the position-taking of the absolute epoché required, if we make use of no preestablished knowledge, if we hold each and every thing in abeyance, then we obviously do not fail to do something, but we do not retain anything either. We do not have a single bit of knowledge. And, shall we have any, be able to acquire any? The epoché is surely not itself already a method. It is at best a piece of a method. How is an actual, complete method of knowledge to be established here? We must obtain clarity about this. The possibility of a method first ensures the possibility of the discipline. The situation at first appears rather desperate. All knowledge is to be problematical. But, the epistemological knowledge we are seeking is indeed knowledge. It therefore seems that theory of knowledge is needed in order to obtain theory of knowledge, which seems to show that theory of knowledge is impossible in principle. Let us reflect. The precious core of Cartesian doubt may help us a bit further. Before we enter into it, we shall, however, be able to say the following: Theory of knowledge does not mean to be anything more than self-understanding on the part of knowledge. It is now obvious that we cannot assume a position outside of knowledge in order to throw light on dark areas of knowledge and to solve problems that it itself raises for us. Only by knowing are we able to shed light on knowledge. If all knowledge then becomes problematical for us, or, if knowledge in general becomes a problem for us, then some knowledge is already implied in this, and absolutely indubitable knowledge, namely that knowledge in general is problematical, or that knowledge in general harbors one obscurity or another and for this reason becomes a problem. It is also further an Evidenz that it can only be in knowing that the problem is solved, that the meaning of the knowledge sought unveils itself. Consequently, it is certainly unquestionable, and again completely evident, that questions concerning all knowledge also affect the knowledge in which, as already in that just realized, these reflections about knowledge itself lie. Nevertheless, this is not to say that those kinds of reflections about the meaning and possibility of knowledge are meaningless and must remain fruitless, that, say, the Evidenzen with which the reflections begin, and in which they advance, are not Evidenzen, not knowledge, something doubtful. The necessary referring back of the elucidation of knowledge to itself is manifestly something belonging to the essence of knowledge as such.

And now Hegel in the introduction to the Encyclopedia:

But with the rise of this thinking study of things, it soon becomes evident that thought will be satisfied with nothing short of showing the necessity of its facts, of demonstrating the existence of its objects, as well as their nature and qualities. Our original acquaintance with them is thus discovered to be inadequate. We can assume nothing and assert nothing dogmatically; nor can we accept the assertions and assumptions of others. And yet we must make a beginning: and a beginning, as primary and underived, makes an assumption, or rather is an assumption. It seems as if it were impossible to make a beginning at all.

Hegel thus is in dialogue with Sextus. He is also a phenomenologist in the sense of Husserl:

In other words, every man, when he thinks and considers his thoughts, will discover by the experience of his consciousness that they possess the character of universality as well as the other aspects of thought to be afterwards enumerated. We assume of course that his powers of attention and abstraction have undergone a previous training, enabling him to observe correctly the evidence of his consciousness and his conceptions.

Being entirely in different things at once

Sextus Empiricus in his Outlines, dealing with genus and its species, rips off an argument of the Parmenides. If we interpret the genus-species relation artificially in terms of ordinary geometric inclusion then obviously there will be problems.  But there is nothing contradictory about several objects $a_1,a_2,...,a_n$ all sharing the same relation with the same object $b$, i.e. $R(a_1,c), R(a_2,c),...,R(a_n,c)$. In fact they all share the same property $\lambda x. R(x,c)$. 

But what if we question set theoretic equality and replace it with path-connectedness ? For instance $x \subset y$ means that for every $z\in x$ we have that there is a continuous path $p_{zw}$ from $z$ to $w$ for some $w \in y$.  Then we can well have that $x_1,x_2...,x_n \subset y$ and at the same time $y\subset x_1,...,y \subset x_n$.

Consider the following set in the plane with the relative topology:

 Then by our definition we have that  $\{x_i\}\subset \{y\}$ and $\{y\}\subset \{x_i\}$ for $i=1,...,5$.

Also, what is a set of elements ? The elements must have a distinguishing property or relation prior to being organized into the set (this discussion needs to be greatly expanded).

Aristotle's theory of Idion in Topics V

We have speculated that in Aristotle's Topics, the more elaborate theory of definition in Books VI (and in particular the architecture of the species-genus hierarchy in Book IV) grew initially from the theory of property (idion) . Some textual evidence can be adduced in favour of this (cf. 154b). There are passages in which Aristotle relaxes all the complex structural requirements on the definiens term, even simple terms are allowed to be definitions; on the other hand the remarkable passage in 132a indicates that property must have a genus + specifications structure just like definition ! How are we to understand that "terrestrial bipedial animal" indicates the essence of man but that (132a) "tame (civilized, hêmeron ) animal"  or "animal gifted with knowledge" does not ?  It might be tempting to view book V of the Topics, dealing with property, to be both among the oldest portions of the said work and perhaps even as having constituted an independent treatise.  It is true, that definition, even difference as well as predication 'according to essence' is mentioned (and contrasted with property) in book V,  but not in detailed way, rather the passages in question could easily be seen as later additions or revisions. We propose that book V be studied on its own in a more-or-less self-contained way.

To the modern reader the whole theory is highly perplexing.  Not even the basic syntactical, grammatical and logical terminology is ever made fully explicit or defined clearly.  As as for $A$ being predicated of $B$ 'according to essence', it baffles us.  For what justification or criterion is used ? 

It is very difficult and  important to sort out what Aristotle means by 'subject',  'term', 'name' (onoma), 'phrase', 'predicate', 'verb', etc. These are obviously all quite different from our modern use. Most notable is the passage 132ba8-10 which distinguishes between 'name' and the 'logos' of the name.  For the moment we note that by 'term' we should understand any noun phrase. In particular those noun-phrases which can be seen as having an 'extension', objects of which the noun-phrase can be predicated.

In the most basic sense, property is related to co-extensionality.  A term $A$ is a property of term $B$ if both terms have the same extension, which we write $E(A) = E(B)$.  But this is not all. Originally property, idion,  seems to have been subordinated to  pragmatic, epistemic purposes. Property was a kind of  'distinguishing mark' (perhaps not unlike linga in the Indian tradition)  which served to make an object known.  Thus it is natural to suppose that the property is somehow conceptually simpler than that which it is a property of.  For example certain marks distinguishing certain species of plants. 

So even at this stage it is clear that for Aristotle co-extensionality is a necessary but not sufficient condition for being a property (cf. also 153a in Book VII: equality is not enough for there to be definition). Thus being a property is also intensional. If we take the sentence "$X$ is a property of $A$" then we cannot substitute co-extensional terms for $X$ salva veritate.  Indeed Aristotle's predicate of terms ' being better known'  (see 129b-130a) is intensional. We denote this situation by $A\prec B$ to mean that the term $B$ is better known than $A$. $\prec$ is a transitive.  We write $A\sim B$ if $A$ and $B$ are simultaneously known (this can be defined as $A\nprec B\& B\nprec A$). This can only involve the respective concepts and not the extensions.  If we denote by $A^\circ$ the contrary of the term $A$ then we have the following rule in 131a: $ A \sim A^\circ$.

One of the most interesting aspect of the theory of property (contained in the first four sections of Book V) is the series of syntactic-semantic  (intensional) conditions imposed on the property term. The theory involves implicit syntactic-grammatical notions which unfortunately are either lost or never made explicit in surviving texts. Nevertheless the theory seems be surprisingly modern.  Here are, in a modernized language, some of the conditions. For $B$ to be a property of $A$ it is required that:

i)  the term $A$ does not occur in $B$ (non-circularity) (131a)

ii)  the term $B$ is different from the term $A$ (non-identity) (135a)

iii) the term $B$ is not a conjunction $B_1 \& B_2$ in which $B_1$ or $B_2$ are already properties of $A$ (irreducibility)

iv) $B$ is not a universal predicate or does not make use of universal predicates (non-redundancy)(130b)

v) the term $B$ does not make use of terms lesser known than $A$ (epistemic priority)(129b)

vi) the term $B$ does not have repeated subterms (including via anaphora) (130b)

Perhaps iii) can be replaced with

iii)' the term $B$ does not strictly contain a term $B'$ which is a property of $A$.

As a corollary of v) we obtain that: $B$ cannot contain $A^\circ$ (131a). 

Note of these conditions would we rejected, for instance, by a modern mathematician who wishes to "characterize" (but not necessarily define) a class of objects. Even if the 'better known' relation is questionable, characterizations involving theory disproportionately more elaborate and distant than those of the the original class of objects will not always be welcomed (though this discussion needs to be refined).

We now investigate the topics involved with the relation between $B$ being the property of $A$ and the relationship between $B$ and terms $C$ in types of fundamental relation to $A$.  For instance in 131a condition i) is actually strengthened to i)'  the term $B$ does not include terms $C$ which are contained in $A$ (seemingly in the sense of species).  We denote this inclusion relation by $C\subset A$.

Thus we have the topic: $\exists C. C\subset A \&  C\notin B \,\rightarrow\,  \neg I(A,C)$.  Here $\in$ and $\notin$ represent simple predication and $I(A,B)$ the circumstance of $B$ being a property of $A$ (132a).

But what about individuals which are part of the extension of $A$, $E(A)$, how do they enter in topics of property ?  At first sight individuals would seem to have little relationship with properties of the term to whose extension they belong. For indeed one man is not distinguished from another man on account of such a property (but cf. Nietzsche's wenn du eine Tugend hast, und es deine Tugend ist, so hast du sie mit niemandem gemeinsam).   Of course just as in the topic above, the property of the species must be predicates of each individual in the extension of $A$. But does it make sense to speak of property here ? How does one extrapolate from predication of individuals to a property of their species ? Aristotle introduces the construction: '$B$ is said of all individuals $a \in E(A)$ qua $A$' (132b). How are we to interpret this ? Why would not a single individual suffice ? What about the distributivity of the 'all' ?

But first a remark on 132b, on the 'onoma' and the 'logos'.  Perhaps these refer simply to the subject and to what is proposed as the property of a subject. Thus the destructive topic would read: $\exists X. ((X\in A)\& (X\notin B) \vee (X \notin A) \& (X\in B))\rightarrow \neg I(A,B)$.  The constructive topics seems however insufficient, for this would hold for definitions.

The following passage (133b) is interesting yet quite difficult to interpret: (W.A. Pickard translation):

Next, for destructive purposes, see if the property of things that are the same in kind as the subject fails to be always the same in kind as the alleged property: for then neither will what is stated to be the property of the subject in question. Thus (e.g.) inasmuch as a man and a horse are the same in kind, and it is not always a property of a horse to stand by its own initiative, it could not be a property of a man to move by his own initiative; for to stand and to move by his own initiative are the same in kind, because they belong to each of them in so far as each is an 'animal' .

The subtext here seems to be that if $C_1$ and $C_2$ are species of a genus $G$ and $C_1$ has attribute $A_1$ then there is a canonically morphism which applied to $A_1$ yields an attribute $f_C(A_1)$ of $C_2$. There seems to be analogy involved here. But why is 'moving' to man analogous to 'standing' for horse ?

Book V is a good place to focus on for the study of opposites (treated is section/part 6). Aristotle's theory of opposite is difficult to fathom from a modern perspective.  Opposition is not classical or intuitionistic negation (Aristotle has in fact a theory of the positive and negative).  Rather it is a algebraic-like operator defined on the class of terms; and in fact there are many distinct species of opposition.  This suggests that genera have  in general  automorphisms acting on species. For the present we assume that our notation $T^\circ$ refers to contraries. We have  $T^{\circ\circ} = T$.  We interpret terms 'not having an opposite' as having $T^\circ = T$.  A striking feature is that $E(T^\circ)$ is not the complement of $E(T)$. This is substantiated by 135b in which  the contrary of  'the highest good' is 'the worst evil'.  The following is the fundamental topic for property and contraries:

\[ I(A,B) \leftrightarrow I(A^\circ, B^\circ) \]

We denote Aristotle's 'modern' notion of term-negation by $n(T)$.  The associated topics are not surprising (136a). For instance $I(A, n(B)) \rightarrow \neg I(A,B)$ and $I(A,B) \rightarrow \neg I(A,n(B))$. 

Part 8 of book V deals with degrees of terms as well as with what might be called probabilistic logic. The aspect of degree of a term seems very much to act like an operator on terms. Thus we can denote 'more','most', 'less' and 'least' $T$ by $T^+, T^\triangle, T^-, T^\bigtriangledown$.  If a term does not admit degree then we assume that $T^+ = T$, etc.  The related topics can be condensed into

\[ I(A,B) \leftrightarrow I(A^+, B^+) \leftrightarrow I(A^-, B^-) \leftrightarrow I(A^\triangle, B^\triangle) \leftrightarrow I(A^\bigtriangledown, B^\bigtriangledown)\]

It does not seem that degree has directly comparative aspect. Rather the degree is taken as relative to some implicit absolute standard. But for probability degree the situation is otherwise.  Let us write $\mathbb{P}I(A,B) > \mathbb{P}I(A'.B')$ for 'it is more likely for $B$ to be the property of $A$ then $B'$ to be the property of $A'$.  Then Aristotle's rather strange supposition is that

\[ \mathbb{P}I(A,B) > \mathbb{P}I(A'.B')\,\&\, \neg I(A,B) \rightarrow \neg I(A',B') \]

In 138a we see variants on the topics corresponding to the cases in which $A = A'$ and $B= B'$. Then follows topics relating to equal likelihood which can be reduced to

 \[ \mathbb{P}I(A,B) = \mathbb{P}I(A'.B') \rightarrow ( I(A,B) \leftrightarrow I(A',B')) \]

and cases in which $A= A'$, etc.

These topics are strange because Aristotle does not state clearly the intrinsic relationship between $A$ and $A'$ or $B$ and $B'$. The following difficult passage seems relevant:

The rule based on things that are in a like relation' differs from the rule based on attributes that belong in a like manner,' because the former point is secured by analogy, not from reflection on the belonging of any attribute, while the latter is judged by a comparison based on the fact that an attribute belongs.

We have deliberately refrained from discussing important modal and temporal aspects of the 'logic' of book V of the Topics.  Book V ends with the curious rejection of property given by superlatives - surely not the same superlative as $T^\triangle$ but rather something like a modifier of a noun, something of the form 'the Xest Y'. Aristotle basically says that such a definite description will not denote uniquely for different states of affairs.

The topics involving analogy deserve special study; also those involving a term being identically related to other terms (137a). But what is analogy ? An isomorphism between the structure of two different structures.  We could compare this with the Indian system of  Nyâya: the Topics invokes strongly a 'case-based' logic according to the interpretation of Ganieri. We have $Fb$ and $Gb$ as well as $Fa$ and we want to argue that $Ga$.  This is based on analogy, or relative similarity of $a$ and $b$,

Proclus' Elements of Theology as Holology and Aetiology

 We adopt the framework of $\infty$-groupoids (which are the higher counterpart of $Set$). But first consider an ordinary groupoid $G$.  The most basic kind of 'unity'  (for a structured collection) is defined in terms of objects: for any two objects $a,b$ there exists a (iso)morphism $f: a \rightarrow b$.  Let us call this property $\mathbb{U}^1$. Then given a groupoid $G$ either $\mathbb{U}^1(G)$ or $G$ admits a disjoint decomposition $\bigsqcup_{i\in I} G_i$ in which $\mathbb{U}^1(G_i)$ for $i \in I$.

But we can go further.  Given $A$ such that $\mathbb{U}^1$ we can require that for objects $a,b$ and morphisms $f,g: a \rightarrow b$ there exists a 2-(iso)morphism $F: f \Rightarrow g$. We call this property $\mathbb{U}^2$. And analogously we define $\mathbb{U}^i$ for $i\geq 3$. We define  $\mathbb{U}^\infty(X) \equiv \forall n\geq 1. \mathbb{U}^n(X)$.  

Given $G$ we can decompose it as $\bigsqcup_{i\in I} G_i$ where for each $i$ either $\mathbb{U}^\infty(G_i)$ or $\mathbb{U}^n(G_i)$ for some $n$. This is one possible interpretation of Proclus' proposition 6 : every manifold is composed either of unified groups or henads.

But what about aetiology, i.e. causality ? It seems that we should view causality in Proclus as essentially  a paradigm of relation (order) between substances, qualities, processes, events. For topological spaces $Y$ and $X$ a causal relationship would be just a continuous map $h: Y \rightarrow X$. For instance if $Y$ is a fiber bundle with fiber $F$ and base space $X$ then it does make some sense to speak of the projection to $X$ expressing that $Y$ is the 'cause' of $X$. Recall that neoplatonic causality is closer to that of ancient philosophy (in particular Aristotle) rather than our modern notion. 

Thus if we view causality as a higher functor between $\infty$- groupoids, $F: G_1 \rightarrow G_2$ then Proclus' proposition 7 could be interpreted as saying something about the $\mathbb{U}^i(G_1)$ and $\mathbb{U}^i(G_2)$.  However, there is a first glance a problem. If we restrict ourselves to topological spaces then given a continuous map $f: X \rightarrow Y$ we have that $Y$ may have a greater level of connectivity than $X$. The image of a connected set is connected; but the image of a non-connected set can also be connected. Thus the 'cause' $X$ does not necessarily have a higher level of connectivity relative to $Y$. 

Consider the two situations. In in the first we have a simpler object, a free object, $F$ and an epi $f : F \rightarrow G$ which can be seen as introducing relations on $F$ (typically given by the kernel of $f$). In this case $F$ is like a material cause (for example a quotient topological space).  In the second situation we have an embedding $j : G \rightarrow H$.  In some sense $H$ has more information than $G$ and allows certainly potentialities in $G$ to unfold. For example, deformations, "unfoldings" in the technical sense. The varying of the base ring in scheme theory, etc. In this case we could see $H$ as a kind of universal "form". Also in this situation $H$ can manifest (rather hylomorphically) as a "completion" (see our previous post on Hegel's Logic) or cover (for instance an algebraic closure, a normal closure or the Galois group of the algebraic closure of $\mathbb{Q}$.

Proclus' elements is all about ontological hierarchies and in such a way that we can say that : a Lie group is more 'perfect' than a mere manifold, an algebraic group than a mere variety. This is because there is an effective unity and relationship between the parts and the whole.

A very interesting illustration of Proclus' causal scheme, in particular for the 'monad' which integrates, unifies and determines each 'level' is given by the concept of classifying space.  The idea is that a more limited, internal variation and aspect of an object $C$ determines a larger, more exterior, more substantial spectrum of objects $O$ : $C$ determines the $O$.  For instance, for topological groups $G$, homotopy classes of continuous maps $X \rightarrow BG$ (where $BG$ is the classifying space) determine isomorphism classes of principal $G$-bundles over $X$. This is also called a 'representation',  a plurality is mirrored and represented within the central classifying object.

A Model for $\lambda$IL

We postulate two notions of saturation. A semantic one (which corresponds to the 'sort' used in    $\lambda$IL and the $i$ in $D_i$ in CIL models) and a merely syntactic one which corresponds to the number of free variables. This is justified by the situation of BL. Thus we have a two-dimensional saturation scheme and we shall consider models having a disjoint decomposition now into $\bigcup_{i,j} M^i_j$ for $j\in \mathbb{N}_0$ where $i$ is to be interpreted as the number of free variables, the syntactic saturation level. We will have two basic operations: application and $\lambda$-abstraction:

 \[App : M^i_j \times M^k_l \rightarrow M^{i+k}_{j-1}\]

\[ \Lambda_x: M^i_j \rightarrow M^{i-1}_{j+1} \text{ or $M^i_{j-1}$ if $x \notin V(t)$} \] 

If free variables are numbered then each $M^i_j$ has a disjoint decomposition indexed by $P_n$, the set of subsets of cardinality $i$ of $\mathbb{N}$. We thus have a map $V: M^i_j \rightarrow P_i$. Thus we can express the second operation as \[ \Lambda_n: M^i_j \rightarrow M^{i-1}_{j+1} \cup M^i_{j+1} \] where $\Lambda_n(t) \in M^{i-1}_{j+1}$ iff $n \in V(t)$. With these two operations we can express all the Bealer operations (in particular as restricted to the $M^0_j$). We introduce relations on these operations to express $\alpha\beta\eta$-equivalence. $M_{-1}^j$ is empty for $j \geq 2$. $M_{-1}^1$ can be interpreted as single variables.