Circle of philosophy

We have included in this site several essays exploring (among others) a certain 'school' or approach to philosophy which might be briefly described as one based primarily on the direct (unmediated) reflection of consciousness upon itself. Our major thesis was that the core elements for such a philosophy are found in the ancient Pali suttas as well in many places in the history of both western and eastern philosophy.  Almost everything that is of value in Husserl's 'transcendental phenomenology' is to be found in previous philosophical works and the ideal of a  'reduction' and 'epokhê' is found in its most correct and thorough form in the Pali suttas (which have a strange agreement with several elements of Aristotle's De Anima and even the first Ennead).  A complex and interesting issue is whether Hume or Kant came closest to the kind of transcendental awareness required for this approach to philosophy.

Another major approach to philosophy is skepticism as exemplified by certain Socratic dialogues and what accounts we can collect of the 'skeptical' phase of the Academy, Pyrrho and most importantly the extant works of Sextus Empiricus together with the works of Nagârjuna.  We argue for a substantial affinity between Sextus' Pyrrhonism and the Madhyamaka, just as the Yogacara (and specially some later analytical works of this school) is an important example of a consciousness-based philosophy. But paradoxically there is also an intriguing correspondence with late neoplatonism as well, with the apophaticism of Damaskios, Pseudo-Dionysus and others. We can discern in varying degrees the same kind of  equipollent amphibolous 'dialectic' in many important figures of modern philosophy such as Kant and Hegel.

Finally we have an approach to philosophy based on analytical atomism, and this 'atomism' may be (perhaps even simultaneously) physical, psychological and logical-conceptual.  Obvious examples are Leucippus, Democritus, the Vaisheshika and the various schools of pre-or-non-Mahâyâna  Abhidharma as well as the strikingly important case of Hume. There are also many examples of 'logical atomism' from ancient times to the present. In a way any doctrine of categories - specially those based on an analysis of language (cf. the Stoic lekta or the Proclean logoi) - tends towards a theory of logical atomism. One of our approaches involves a theory of categories based on finitism and computability (Turing completeness and Church's thesis) allied to a formal combinatory analysis of logic and language. Any theory of deduction or inference is at least implicitly involved with computability.

A project: history and archaeology of recursion theory and Turing completeness: search for and study ancient documents and artifacts exhibiting Turing completeness or other weaker computable properties. An important example is Panini's grammar which amounts to a term-rewriting system. A curiosity: Babbage's Analytical Engine was Turing complete.

Now a remarkable fact is that not only are these three approaches are in no wise  plainly  'contradictory' or 'incompatible' but they appear to be deeply entangled and dependent on each other (even if only in a negative sense, such as when atomism is taken as a starting point for refutation). This is patent for instance in the literature on the relationship between the Madhyamaka and Yogacara on in the text of Hume' Treatise  itself which is remarkable for containing at once elements of all three approaches.  A major issue is the relationship between these schools and ethics, the consistency with and indeed implication of a moral compassionate realism along the lines of Schopenhauer and the ancient Buddhist precepts. Notice that if we look at the best ancient representatives of all three approaches we see that the ethical spirit in fact imbues everything ! Cf. the last section of Sextus' Outlines 280: the skeptic is a philanthropist (dia to philanthrôpos einai)  who wishes to cure the dogmatist of conceit and rashness.  Now this does have a huge social and cultural implication...

It is very important to study carefully that monumental and unique work of genius which is Hume's Treatise and not only later abridgments, commentaries or 'refutations' in order to grasp all the aspects, subtleties and nuances present therein.  Our view is that the key to a correct understanding is to study the deep connection to Sextus Empiricus (properly understood) as well as the (historically paradoxical, perhaps) connection to Buddhist philosophy: certain Abhidhamma schools, the Yogacara and the Madhyamaka. Alison Gopnik has proposed a historical scenario for a direct acquaintance by Hume of the Abhidhamma literature during his stay in Paris. Our view is that in addition the Yogacara and Madhyamaka literature played a profound role. There is a certain similarity between many of Hume's arguments and those of the later Yogacara school (against external objects) and a similarity between Hume's treatment of substance, personal identity and monism in general and Nagarjuna's treatment of svabhava.  However some of these similarities may be explained by a direct influence of Sextus Empiricus (the Pyrrhonism - Madhyamaka connection has not been cleared up historically).

Breaking the circle: we have previously discussed the complexities involved in the 'paths of ascent', 'anagogic processes'   which should accompany and are the ultimate ground of the circle of philosophy. These also have a multiplicity and form their own 'circle'. In the west we have the theory of dialectics expounded in Plato (specially the mathematically based on in the Republic which we have discussed) and Plotinus (and other Socratic, Megarian, Eleatic, Pyrrhonian and even late neoplatonic forms). Then we have the theory of eros and beauty in Plato and also in Plotinus (and the whole tradition of platonic and courtly love in Europe, for instance Dante and the Fedeli d'Amor).  And there is the theory of catharsis and detachment  and self-awareness  (watchfulness) in the Phaedo and the Enneads (and also in Sextus and the Stoics) - which probably was meant to accompany other methods.  And of course the central and disturbing role theurgy played for neoplatonism (but possibly also for earlier platonism, pythagoreanism and its relation to orphism).   In the east the methods of yoga form a circle and the situation for Buddhism is indeed likewise complex, even for early Buddhism (cf. Masefield's book on  divine revelation in Pali buddhism complemented by Dhammika's text Broken Buddha: all this suggests that the tradition represented by Theravada is the result of a certain school of Buddhism ceasing long ago to be 'operative'  and becoming merely 'speculative' and 'ritualistic'.). In Buddhism anagogic dialectics played (as we saw) a hugely important role (and we should also mention the Nalanda tradition in Tibetan buddhism).  In this complex circle there seems to be involved the crucial presence of an anagogic energy and power and illumination, a kind of transmission. The 'virtues' and habits in Bhagavad-Gita 13.8-12  have a profound anagogic significance. 

There are certainly grave errors involved in certain approaches to anagogic processes (it is at least problematic that popular meditation manuals aiming at leading one to the jhanas are  adequate or sufficient) as well as certain attitudes and practices (specially those involving a kind of collectivism or passive surrender to the authority of a group).  However we must give a central place to philosophical awareness and its liberating transcendental insight, also knowledge of the inner working of the mind through accumulation of habits and associations (specially those ingrained in earlier years or through traumatic experiences): the mind must be a clearly conscious creator, never passive and deluded before its own subconscious conditioning, delusions and projections. And look at the deep meaning of the 'five hindrances' and 'seven limbs of awakening'.  These all form a complex interwoven feedback system, like a higher-dimensional Rubik's cube or analogous puzzle.  Often to address undesirable state A we apply B but this in turn gives rise to another problem C and so forth. There is a certain similarity to the situation with the 5 types of brain wave frequencies.

Natural Term Logic

Our aim is to develop a formal logic which reflects the logical mechanisms of the grammar of natural language. In particular a formal logic which is not

tied in any way to extensionalism or standard interpretations of quantifiers and multiple generality. It is our conviction that the standard use of variables (in quantifier logic, lambda calculus and the constructivist version used in dependent type theory) while being useful and mathematically interesting, does not directly reflect the actual logical structure of natural language and its mechanisms for expressing 'quantitative' determiners and multiple generality (cf. the study of suppositio and syncategoremata in the middle ages. ). Our philosophical premise is that the conceiving, applying, checking and reasoning about Turing complete formal systems presupposes a priori a system of cognitive categories that are independent from and prior to standard quantifiers. Rather the most adequate type of formal system for logic would be in the style of Quine's paper 'Variables explained away' (this can be compared to our considerations on the philosophical significance of term-rewriting systems in our paper on Analyticity and the A Priori). The earlier work of Schönfinkel and Curry (combinatory logic) while also involving the elimination of bound variables is however structurally quite different. Quine himself also mentions the earlier work of Tarski on cylindric algebras. It is necessary to extend the combinatory approach to intensional logic and a main source of inspiration for our work is George Bealer's book 'Quality and Concept'.

Our work on Aristotle's Topics attempts not only a formalization of this work but argues for there being present in Aristotle a variant of a natural deduction calculus for an extension of second-order logic. However it does not deal directly with variable-free logic, being focused on inference rather than logical expression.

Our system of variable-free logic is called Natural Term Logic. The basic building blocks are called primitive terms and denoted by capital roman letters $M,N,P$. Each term is assigned a unique (predication) sort which is an integer $\geq 0$. The sort $0$ should be understood as indicating that the term cannot be used a a predicate (i.e. it is unsaturated) and this will include both abstract objects (like propositions) and concrete objects. We sometimes denote the sort of a term by $M^{(s)}$. Finally we have constructors, denoted by lower-case Greek letters $\alpha, \beta, \gamma$ to which are assigned a unique signature $(s_1, s_2,..., s_n) \rightarrow s$ where $s$ is sort and the $s_i$ are sorts. The classification into sorts can be compared to the saturated and unsaturared lekta of Stoic logic and there are also important comparisons that should be made with medieval and classical grammar (our terms are slightly more general than definite grammatical categories or 'flections'). Constructors take as arguments terms of specified sorts and yield a complex term of sort $s$.

Roughly speaking, sort 0 terms correspond to proper nouns or nominalized sentences (in particular the objects of propositional attitudes), sort 1 terms are those that can be used as monadic predicates, sort 2 terms correspond to relations, etc. At this stage the same term can be used for different kinds of supposition. Thus we have the sort 1 term 'man' which also doubles as a predicate 'being a man' when conjoined with the appropriate constructor.

Basic constructors include the family $\delta$ (for simplicity we denote these basic constructor families by a single letter) which diagonalizes terms (this corresponds in particular to reflexive constructions such as 'to love oneself') and $\sigma$ which corresponds to permutations (this includes the reciprocal of relations: the reciprocal of the relation of 'somebody to know something' corresponds to 'something to be known by somebody').

But the fundamental family of constructors are $\pi$ which correspond to predication (both simple and embedded - this will be explained further ahead). We sometimes omit the constructor and use simply the concatenation of terms such as $SM$ for the nominalized sentence 'Socrates is a man'. If $L$ is of sort 2 (a relation) then predication can create a term o sort 1 $LJ$ (to love John) and then a term of 0 sort $MLJ$ 'that Mary loves John' (where we use a form of concatenation mimicking the SVO construction). As we shall see, we have a certain freedom on how we define these simple forms of predication (when combined with the basic constructors above).

But what is embedded predication ? Consider the sort 1 term Happy and the sort 1 complex term 'known by John'. Then we can form the 1 sort complex term 'known to be happy by John'. The grammatical aspect here is doubtlessly complex and varied (related to dependency and verbal complements). What is clear is that this construction cannot be reduced to simple predication involving the terms 'know' and 'happy'. Let us explain in more detail embedded predication.

Consider the following related complex terms (we use variables for convenience):\\


1 sort: 'X knows (about the) love (relation)' (knowing about the love relation)

2 sort 'X knows (about property) loving W' (somebody knowing the property of loving somebody)

2 sort 'X knows (about property) being loved by Z'

1 sort 'X knows (about property) to love Mary'

1 sort 'X knows (about property) being loved by John'

3 sort 'X knows that Z loves W'

2 sort 'X knows that Z loves Mary'

2 sort 'X knows that John loves W'

1 sort 'X knows that John loves Mary'


Only the 1 sort complex terms are examples of non-embedded predication of the terms Knowledge and Love. In all other cases the predicative aspect of the predicate term is pulled up into the complex predication term itself.

We have two versions of the analogues of binary connectives for terms. For 2 sort terms $L$ (love) and $R$ (respect) we have $L\& R$ the relation of somebody loving and respecting somebody. But we can also combine terms into a juxtaposed conjunction relation $L\otimes R$ resulting in a degenerate complex term of sort 4. Negation applies to terms of all sorts. Thus we have 'not being the case that Mary loves John', 'not loving Mary', 'not loving', etc. The juxtaposed conjunction together with $\delta$ allow us to define the product of relations, for example 'someone being the father of someone who is the father of someone'. This can be given an alternative (perhaps preferable) presentation using the natural term logic versions of the Peano operator and its indefinite version (selector). For instance for a 1 sort $M$ we form 0-terms 'the M' and 'a M'. But there are also 'functional' versions for 2 sort term yielding 1 sort terms: for instance 'the father of somebody' and 'a father of somebody'. We can also venture into the quicksand of extensions of terms as well as mereology (which was already developed in the middle ages in the analysis of the syncategoremata totus and pars).

Finally let us come to quantifiers. Though this is not strictly necessary for a formal study of our system we make the philosophcial assumptions that all quantification is bounded in some form, even if only implicitly.

Let us consider 1 sort terms only. Then 'all M is N' will be expressed by a constructor $\kappa MN$ with obvious signature $(1, 1) \rightarrow 0$ and analogously for 'some'. We can wonder if (grammatically oriented) we could also conceive of $\kappa$ as being a constructor of signature $1 \rightarrow 1$ so that $(\kappa M)N$ could be read in the same way. Or what about Bobzien's proposal for a Stoic implicational reading ?

Of great importance is how $\delta$ combines with quantifier operators by entangling together different subterms thus allowing us to express multiple generality in a way formally similar to the mechanisms of natural language and in particular the use of anaphoric pronouns. Examples can be found for instance in Boethius' discussion of conditional propositions in \emph{De topicis differentiis} (1176B) in which what in modern notation would be written $\forall x S(x) \rightarrow R(x)$ is expressed "If it is spherical, it is revolvable.", the second occurrence of the pronoun 'it' being anaphoric. In Natural Term Logic this could correspond to first applying $\delta$ to the term corresponding to $\lambda x y S(x) \rightarrow R(y)$ (the relation of one thing being spherical implying that the other revolvable) and then applying an unbounded quantification operator.

We cannot discuss here 'judgement' vs. proposition nor the interesting possibilities of formalizing directly much of the medieval theory of obligations (much richer than the proposition vs. judgment distinction) as well as a even closer interpretation of the Topics and medieval developments inspired by this work..

Consider the sentence 'every man has a father and has somebody who gave that man a name'. Suppose 'John is a man' is accepted. Then we must be able to infer that 'John has a father and there is somebody who gave John a name' : can we devise a syntactical algorithm for tracking the 'free variable' 'that man' ? And things get even more complicated when we are in the presence of embedded predication (verbal complements, etc.).

And what about 1 sort terms built from $\kappa$ such as 'having every brother liking Winnie-the-Pooh' ? We will see further ahead how this is done.

And surely we need a determiner-constructor corresponding for instance to adjectival constructions and other determiners in noun sentences. How is the complex term 'rational animal' formed (this kind of juxtaposition of terms can be given, it seems, a conjunctive reading) ? It would be interesting to study the subtle problem of semantic differences in the order of longer strings of determiners as well as apposition. The Topics permit us to formulate a much more fine-grained theory of noun sentence determiners.

Our Natural Term Logic is typed but only in a very broad sense. It turns out that the above approach has some very interesting applications to standard type theories such as the simply typed lambda calculus and Girard's system F.

In this paper we shall see that we can give a variable-free equivalent presentation of the simply typed lambda calculus and furthermore that this approach can be internalized in system F.