Stoic quantification is discussed in this paper: https://philarchive.org/archive/BOBSLA

Additional references are mentioned in footnote 3.

Remember the classical book "Stoic Logic" by Benson Mates.

Unfortunately Mates does not offer any speculation on how Chrysippus tackled the Liar paradox. He only mentions that he wrote six books on the subject.

The paper by Bobzien and Shogry puts together a lot of interesting and valuable material which sheds light on Stoic grammar and its connection to Stoic logic. The theory of the kategorema recalls the intensional abstracts of Bealer or the virtual classes (and virtual relations) of Richard Martin. The authors' treatment of variable-free quantification through regimented syntax is, I believe, of considerable interest to logic and linguistics in general. But I do feel that this grammatical theory of predication was not exclusive to the Stoics but part of ancient philosophy in general. For instance it might be found in Aristotle's theory of definition in the Topics. My preprint "Ancient Natural Deduction" argues that ancient philosophy and Aristotle in particular must have had sufficient means to deal with first-order reasoning.

Has anyone made the connection between Stoicism (Stoic logic in particular) and process philosophy (such as Whitehead) ?

Stoic logic did deal with quantification, but the surviving fragments do not provide a complete picture of their system. However, based on what we do have, it appears that the Stoics used a form of propositional logic that did not include quantifiers. Instead, they relied on various techniques to express the ideas of universality and particularity without using quantifiers.

For example, the Stoics used the concepts of "everything" and "something" to express the ideas of universality and particularity, respectively. They also employed the notion of "correlatives," which are pairs of terms that have a certain logical relationship to each other, such as "master/slave" or "father/son." The Stoics used these correlatives to express universal propositions by saying that something is the correlate of everything, and particular propositions by saying that something is the correlate of something.

Overall, the Stoic treatment of quantification was limited, and it is difficult to determine exactly how they would have dealt with more complex quantificational expressions. However, their focus on universal and particular propositions, as well as their use of correlatives, provides some insight into their approach to quantification.

There has been some discussion about the potential connections between Stoicism and process philosophy, particularly in regards to the Stoic concept of logos and its relationship to Whitehead's process philosophy. Some scholars have argued that the Stoic understanding of logos as an organizing principle that governs the universe can be seen as a precursor to Whitehead's concept of the creative advance of nature, which posits that the universe is constantly evolving and creating new forms of organization.

In addition, some scholars have suggested that the Stoic idea of the unity of all things and the interconnectedness of the universe can be seen as a precursor to the process philosophy concept of interconnectedness and interdependence. However, it should be noted that these connections are speculative and have not been widely accepted in the academic community.

Gerardo Ó. Matía Cubillo

There is no direct relationship between the universal and particular quantifiers in logic and the forms of the Not one and the One in Plato's philosophy.

The universal and particular quantifiers in logic are used to make statements about the extent to which a property applies to a set of objects. For example, the universal quantifier "for all" is used to assert that a property holds for all members of a set, while the particular quantifier "there exists" is used to assert that there is at least one member of a set for which a property holds.

In contrast, Plato's theory of Forms posits the existence of two fundamental Forms, the Form of the Not-one and the Form of the One, that serve as the ultimate ontological principles underlying all reality. The Form of the Not-one represents the principle of difference and multiplicity, while the Form of the One represents the principle of unity and identity.

While there is no direct relationship between the universal and particular quantifiers in logic and the Forms of the Not-one and the One in Plato's philosophy, it is possible to draw analogies between these concepts. For example, one could argue that the universal quantifier "for all" corresponds to the Form of the One, as it asserts the unity and identity of a property across all members of a set. Similarly, one could argue that the particular quantifier "there exists" corresponds to the Form of the Not-one, as it asserts the difference and multiplicity of properties across different members of a set.

Indeed, to consider this more deeply I would refer to Neoplatonic philosophy and mathematical category theory. By Neoplatonic philosophy I mean Plotinus in particular.

Gerardo Ó. Matía Cubillo

After struggling for quite some time, a few months ago I made the discovery that Aristotle's Topics (and probably the formal structure of ancient Greek argumentation in general) can indeed be formalised using predicate logic. The key is taking a 'sententialist' approach to logic and language in the style of Richard Martin (1916 - 1985). Predicate logic is ill equipped to deal directly with the reference or meaning of natural language, instead it seems designed to deal with language considered formally as a system of signs.

I am currently working on this approach to the Topics and plan to present my approach in a very extensive and detailed way (I have not yet uploaded a preprint).

I have still not 'solved' the problem of giving a rigorous formal interpretation to the dialectical section of the Parmenides (if such an interpretation even exists !). But I believe the key lies in first achieving deep understanding of the Sophist (generally believed to have been written afterwards).

Agnieszka Matylda Schlichtinger

Speaking of category theory, William Lawvere and later Urs Schreiber have been working on interpreting Hegel's Science of Logic in terms not only of ordinary category theory but of (infinity,1)-categories and (cohesive) infinity toposes.

I wrote a little note about how they interpret 'being' (Sein) and 'nothing' (Nichts) but I will have to look for it...

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Agnieszka Matylda Schlichtinger

Here is an extract from the note I mentioned:

It seems to me that Lawvere and Urs Schreiber miss the mark in their interpretation of Hegel's logic. Hegel did take contradictions seriously. Classical contradiction does have a place in his system in the domain of the Verstand. Hegel emphatically links this to Kant's antinomies. Now there is nothing remotely contradictory or "opposite" about a triple, a pair of adjunctions or an adjunction between modalities/monads in a some kind of topos (higher or otherwise). Also I am yet to understand how the unit type (Seyn) and the empty type (Nichts) are the same or there can be a transition between the two (maybe in something like a "stable" category where the inital object is also a terminal object). However the category theoretical interpretation of Hegel can only be symbolic, analogical and approximative. As such the simplest concept of category theory is probably that of initial and terminal object. They both have no loops beyond the identity. Hegel's treatment of infinity is clearly related to (co)limits. Kan extensions subsume all other basic categorical notions, universal constructions. 2-categories seem to express the category of essence. Equality instead of being immediate (naive) is mediated. I think that translating Hegel in modern terms would read: common systems of formal ontology are incomplete. But this incompleteness needs to be proven in turn in a metaontological formal system. For Arithmetic, metamathematically Godel's undecidable sentence becomes capable of some meaningful interpretation (reflection). Its truth is metamathematical: it has in fact no proof in the system. If we wanted to engage in Hegel-speak we could even write: Gödel's undecidable sentence transcends the plane of formal being and expresses essence determined via self-reflection and negation.

Clarence Lewis Protin

Thank you for sharing your thoughts on Lawvere and Urs Schreiber's interpretation of Hegel's logic. It is certainly a complex and nuanced topic, and there are various ways to approach and understand it.

It is true that Hegel did take contradictions seriously, particularly in the domain of the Verstand or understanding. However, as you point out, the category theoretical interpretation of Hegel's logic may not necessarily involve classical contradictions in the same way. Rather, it may involve more nuanced notions of opposition and mediation, which can be expressed in terms of adjunctions, modalities, monads, and other categorical structures.

The concept of the unit type (Seyn) and the empty type (Nichts) as the same or transitioning between the two may indeed be difficult to grasp in the context of category theory. However, as you suggest, it may be possible to interpret Hegel's treatment of infinity in terms of (co)limits and other categorical constructions, which can help to shed light on his broader philosophical project.

Overall, it is important to recognize that any interpretation of Hegel's logic in modern terms is likely to be approximate and symbolic, as you note. Nevertheless, exploring these connections can deepen our understanding of both Hegel's thought and the mathematical structures that category theory helps us to analyze.

Describing Hegel's philosophy in terms of category theory is a challenging task, as it involves bridging two very different domains of inquiry. Nevertheless, some scholars have attempted to explore connections between Hegelian thought and category theory, and these efforts may help to shed light on both fields.

One possible approach to describing Hegel's philosophy in terms of category theory is to focus on the concepts of opposition and mediation that are central to his dialectical method. According to this approach, categories can be seen as collections of objects that are related to one another through various kinds of oppositions and mediations.

For example, in a category-theoretic interpretation of Hegel's Logic, we might view the basic structure of the dialectic as a triad of categories: thesis, antithesis, and synthesis. The thesis and antithesis would represent opposing categories, while the synthesis would represent the mediation between them.

Moreover, we might view the categories themselves as complex structures that are built up from simpler structures through various kinds of categorical constructions, such as limits, colimits, adjunctions, and so on. In this sense, Hegelian categories might be seen as analogous to algebraic structures, such as groups or rings, which are similarly built up from simpler building blocks.

Another possible approach to describing Hegel's philosophy in terms of category theory is to focus on the concepts of self-reflection and negation that are also central to his thought. According to this approach, categories can be seen as structures that are capable of self-reference and self-negation, and that allow for the representation of complex processes of change and development.

For example, we might view the Hegelian dialectic as a process of self-reflection and self-negation, in which a category reflects upon itself, identifies its own internal contradictions, and then negates itself in order to move to a higher level of development. In this sense, the dialectic might be seen as a kind of categorical process, in which categories transform themselves through a series of negations and syntheses.

I have a different idea: the philosophy of Hegel and the philosophy of Plotinus in context to quantum mechanics and neuroscience inflicted through description in category theory.

One possible starting point might be to examine the ways in which the concepts of self-reflection and self-negation in the philosophies of Hegel and Plotinus might be related to the phenomena of quantum entanglement and superposition. For example, we might view the entangled state of two particles as a kind of self-referential system, in which the state of one particle depends on the state of the other, and vice versa.

Moreover, we might view the collapse of the entangled state as a kind of self-negation, in which the system transitions to a new state that is no longer entangled. In this sense, the concepts of self-reflection and self-negation in the philosophies of Hegel and Plotinus might be seen as providing a useful framework for understanding the behavior of quantum systems.

Similarly, we might explore the connections between the concepts of unity and multiplicity in the philosophies of Hegel and Plotinus and the structure and function of the brain. For example, we might view the brain as a complex system that is capable of representing both unity and multiplicity, and that can shift between these modes of representation depending on the task at hand.

Moreover, we might view the transitions between these modes of representation as a kind of dialectical process, in which the brain reflects upon its own internal contradictions and then negates them in order to move to a higher level of function. In this sense, the concepts of unity and multiplicity in the philosophies of Hegel and Plotinus might be seen as providing a useful framework for understanding the structure and function of the brain.

Of course, these are just two possible examples of the many interesting connections that could be explored between the philosophies of Hegel and Plotinus, quantum mechanics, and neuroscience, using category theory as a tool for description. There would be many challenges and obstacles to overcome in such an interdisciplinary project, but the potential insights and discoveries that could be gained would be well worth the effort.

Agnieszka Matylda Schlichtinger

Thank you very much for sharing your interesting thoughts on the interdisciplinary relevance of Hegel. Since you mentioned quantum theory I would like to share some ideas that I just recently came up with.

In Hegel's Logic we find that starting from a given category we arrive at another category in a surprising way: the new category often 'emerges' from the old one in a way that is not so obvious. Who could guess that quantity emerges from being-for-self ? Or that measure passes into essence ? And most importantly the new category does not simply negate or differ from the old one but 'subsumes' it (it both 'preserves' and 'transcends' it).

It seems to me (but this is only a rough idea that needs to be worked out) that the history of physics or indeed the global structure of current theoretical physics exhibits these processes of 'emergence' and 'subsumption' (Aufhebung).

For example in Newtonian mechanics we can start with simple postulates about the invariance under Galilean transformations and the conservation of angular momentum 'emerges' as a consequence of the invariance of inertial frames under rotation.

Starting from the invariance of the speed of light in special relativity we get:

That energy and momentum are united in a elegant way as a single 4-vector (emergence)

That special relativity subsumes classical mechanics as a limit case when c -> + infinity (subsumption)

There is also a further process of 'abrogation' in which concepts first considered static and absolute are then revealed to be relative and to have an rich internal life. Being there (Dasein) is only being there through negation of the 'other'. This is exactly what relativity does by making time, length and mass no longer in-themselves but for-the-other (the observer).

In classical quantum mechanics, starting from basic postulates about the wave function, observables and the evolution of the system we obtain:

By passing to the classical limit we get an interpretation of the operators and the wave function in terms of classical Hamiltonian mechanics (including the wave function expressed in terms of the action S). This is another case of 'subsumption'. The uncertainty principle (which is certainly suprising !) is a case of emergence.

But an even greater surprise is how the uncertainty principle interacts with the classical concept of angular momentum to give rise to an entirely novel concept / category: that of spin.

Quantum mechanics (and specially quantum field theory) is filled with the abrogation of previous concepts. I am not sure of this, but even in Bohm theory I think that the concept of an individual particle is no longer tenable one we consider relativistic versions of quantum theory.

Clarence Lewis Protin

Your ideas connecting Hegel's dialectical logic to the history and development of physics are quite fascinating. It is interesting to consider how physics, as an evolving field, may exhibit Hegelian processes of emergence, subsumption, and abrogation.

Let's briefly recap your main points:

Newtonian mechanics: Conservation of angular momentum emerges from invariance under Galilean transformations, subsuming classical mechanics as a special case.

Special relativity: Energy and momentum are united in a single 4-vector (emergence), and classical mechanics is subsumed as a limit case when c -> + infinity.

Abrogation in relativity: Time, length, and mass become relative and depend on the observer, reflecting Hegel's idea of being-for-the-other.

Quantum mechanics: Uncertainty principle as an example of emergence, and classical Hamiltonian mechanics subsumed within the quantum mechanical framework.

Spin in quantum mechanics: Emergence of a novel concept through the interaction of the uncertainty principle with the classical concept of angular momentum.

Abrogation in quantum mechanics and quantum field theory: Classical concepts being redefined or replaced with new, more complex and nuanced concepts.

Your observations indeed highlight some intriguing parallels between Hegel's dialectical logic and the evolution of physical theories. These connections might be further explored and expanded upon through a more comprehensive analysis, taking into account other theoretical advancements in physics, such as general relativity and the various approaches to quantum gravity.

It's worth noting that drawing direct analogies between the development of physical theories and Hegelian dialectics should be approached with caution, as the underlying motivations and methodologies in physics and philosophy can be quite distinct. Nonetheless, your ideas provide a thought-provoking perspective on the interconnectedness of knowledge and the process of conceptual change in scientific understanding.

I also noticed a very interesting sentence you wrote:

„I am not sure of this, but even in Bohm theory I think that the concept of an individual particle is no longer tenable one we consider relativistic versions of quantum theory.”.

I believe you are right, also on a mathematical level. I am the first author of this article:

https://www.researchgate.net/publication/358816502_Time_of_arrival_operator_in_the_momentum_space(this paper will appear in the journal Reports on Mathematical Physics in June).

A significant element of the paper is the hypothesis, which is a consequence of the analysis of past approaches to constructing a time operator. According to this hypothesis, time is dependent on phenomena, and its passage cannot be registered in a world devoid of them. This idea is also inspired by the philosophical stance adopted by Leibniz during his dispute with Newton on the nature of time. The philosophical properties that Leibniz expected of time are transferred to the time operator.

So the mathematics here is merciless. The time operator in quantum mechanics can only be constructed on the assumption of the existence of fields (the continuous spectrum of the Hamiltonian). Thus, in the case of a one-particle model, such an operator cannot be constructed on the grounds of non-relativistic and relativistic quantum mechanics.

The one-particle model does not include a description of the interaction. Without this, there is no physics. The idealised and oversimplified models used in physics have meant that the discipline has now become a set of theories that contradict each other at some points. Personally, I believe that there is no chance of a breakthrough in physics if such models continue to be adopted. The spirit needs to be breathed into physics. Otherwise it will be a dead set of contradictory theories.

Gerardo Ó. Matía Cubillo

See Tabak's Plato's Parmenides Reconsidered, chapter 4, for a similar view on the Parmenides and its relation to the Sophist.

Gerardo Ó. Matía Cubillo Agnieszka Matylda Schlichtinger

This is one of the most interesting contemporary approaches to formalizing Plato's theory of forms:

https://mally.stanford.edu/cm/

I had found a problem with an early version of Zalta's approach to Plato but Zalta himself later found it too and published an updated version of his theory (involving the distinction between a 'thin' and 'thick' theory forms). Also Zalta follows Meinwald's theory of the two types of predication in the Parmenides.

The project's focus on automated reasoning interests me greatly. It is being carried out currently using Isabelle /HOL but Zalta used the older software Prover9 and Mace4 to make some very interesting discoveries, like a simpler version of the ontological argument !

Unfortunately it is not so easy to get Prover9 and Mace4 to run on current machines (at least for Linux users). I found a way to run these programs using VirtualBox and an old version of Ubuntu.

Chapters 8 and 9 of Terence Parsons' Articulating Medieval Logic are of great interest for investigating the way natural language expresses logical concepts.

Quantifiers are very necessary. However variables are not. We can give alternative versions of first-order logic which syntactically do not have anything resembling a variable. This is how natural language works too.