A logic reflected into itself

Wir haben nach dem allgemeinsten Begriff der Logik gefragt. Wir haben herausgefunden, dass es einen Ausdruck geben muss, der die Wahrheit vermittelt. Wenn Wahrheitvemittelten Ausdrücke jedoch über jede Art von Ausdruck sprechen können, einschließlich wahrheitstragender Ausdrücke, wird die Logik als in sich selbst reflektiert bezeichnet.

Unser Ziel ist zunächst die Entwicklung eines formalen axiomatisch-deduktiven Systems zur Darstellung und Umsetzung philosophischer Theorien und Argumentationen. Aber ohne dass dies dem Wert und der Machbarkeit von Husserls philosophischem Projekt widerspricht.

Selbstverständlich müssen wir bereit sein, auf alle Einwände und Gegenargumente bezüglich des Werts und der Machbarkeit unseres Projekts zu antworten. Bitte beachten Sie, dass wir keinen Anspruch auf Exklusivität oder Vollständigkeit in Bezug auf ein bestimmtes System oder das gesamte Projekt erheben. 

Nur dass ein selbst unvollständiges System reich genug sein kann, um philosophisch fruchtbar zu sein und viele Unklarheiten und Verwirrungen in der aktuellen Philosophie zu klären.

Die Motivation für unser Projekt ist zweifach. Radikaler Skeptizismus in Bezug auf sprachbasierte philosophische Theorien und Argumentationen und radikale Gewissheit in Bezug auf mathematisches Wissen und moralische Gesetze (von denen wir außersprachliche Erkennbarkeit haben).

Leibniz's elegant example of abstract proof

The Kneales in their famous book The Development of Logic include an amazing document by Leibniz which shows an entirely modern axiomatic and deductive approach to  definitions and basic properties relating to (complemented) semi-lattices ($\vee$-semi-lattices) where $\vee$ is written $\oplus$. Furthermore, this structure is construed explicitly as a kind of mereology.  Leibniz uses the symbol $\infty$ for $=$.  This document can be found in the 7th volume of Gerhardt's edition of the philosophical writings of Leibniz, on p. 228.  The entire volume is in fact of the utmost interest. Gerhardt says in a footnote that Leibniz originally entitled the document 'Specimen non inelegans demonstrandi in abstracti' but then crossed out the title (Überschrift) and that the document shows signs of being reworked and revised (wiederholt überarbeitet werden) by Leibniz (?).

About some papers by Sernadas et al.

What is the 'object specification logic' of Sernadas et al., presented in various papers ? It is a the application of multi-sorted first-order temporal logic to the specification and verification of multi-agent systems but with an immediate focus on the paradigm of object oriented programming. Or rather, it is parametrization of such logics indexed by agents (and combinations of agents) which cries out for a Jacobs-type fibred category approach.  It is a multi-modal logic with operators representing 'next', 'previously', 'all time in the future (excluding the present)' , etc. It builds upon previous work in theoretical computer science and has a close connection to Joseph Goguen's category theoretic framework for software engineering and general systems theory.  The semantics are based on the combination of individual (local)  run-time histories, functions $\mathbb{N} \rightarrow P$ where $P$ is, roughly speaking, a space of  sets of possible attributes and acts.

Recall how in UDIL $D_0$ represents a space of possible propositional meanings  or just propositions in a extra-linguistic sense-oriented or semantic sense (think Fregean Gedanke or along the lines of Bolzano), quite like the Stoic lekta.  Now what Sernadas et at. did was sneak in a space (a sort) of possible actions for a given agent $c$ (or rather 'class' in OOP speak, a template for actual agents). They call this sort $\tau_c$. Since their logic is a  temporal modal logic they can have contingent equalities. That is $t_1 = t_2$ can be thought to hold at one time but not at another.  But how do they specify that an agent is performing a certain 'act' now ? With a kind of  'indexical' constant (of sort $\tau_c$ !) represented by the unpleasant notation $occ$ (short for occurs). Thus for a given agent $c$ we could write $occ = says('hello')$ to express that $c$ (or rather an object in this class) is performing the action of saying hello at the present moment, where $says$ is given the signature $string \rightarrow \tau_c$.

Miscellaneous notes on Aristotle's Physics

Book I, 4, against Anaxagoras: proposed principles: if the whole is known then so too are the parts. Also: there is no knowledge of the infinite. The real numbers contain a subset of indefinable numbers that are individually unknowable. Therefore either they are not parts of the real numbers or the real numbers are as a whole unknowable.

Book III: no infinite body in space. In some of the proofs Aristotle's show the extreme conceptual importance and interest behind the framework of the apparently naive theory of the four elements.  The idea of opposition of a quality is intimately connected to the negative numbers, the balancing out of opposites.  Thus both dry-wetness and hot-coldness represent independent genera which can be parametrized (or scaled) by $\mathbb{R}$, where for instance wetness and coldness are negative. The concept of electric charge of modern physics is here. In the proof Aristotle basically argues that the total charge in the universe must add up to zero.  Now what is most interesting about the theory of the four elements is that we have matter endowed simultaneously with two independent kinds of charges. So to parametrize the field of fundamental qualities we need $\mathbb{R}^2$ or more suggestively $\mathbb{C}$. Thus $e^{i\theta t}$ could represent the circulation and transformation of the elements. The fact that fundamental qualities can either be scalar (positive reals) or vectors (one or higher-dimensional) (i.e. combinations of continuous genera having opposites) goes back to Aristotle.

Book VI: no finite movement in an infinite time.  The magnitude of a movement is shown by Aristotle to be closed and bounded (measured by intervals), hence compact.  Infinite time can only mean a semi-open set $[0,\infty)$.  If motion is continuous then the result follows from the fact that there is no continuous bijective function  (homeomorphism) $f: [0, \infty) \rightarrow I$ where $I$ is some closed interval (all with the topology induced by the standard topology). To us this follows immediately from the fact that $f^{-1}$ must take the compact set $I$ into a compact set. But $[0,\infty)$ is not compact.  Aristotle's argument involves dividing $I$ into subintervals $I_i$ and observing that the $f^{-1}I_i$ have to be finite and hence $f^{-1} I = f^{-1}(\bigcup_i I_i) = \bigcup f^{-1}I_0 = [0,\infty)$ would have to be finite (for finite times finite is finite) which is a contradiction.  This argument works is we observe that $f^{-1}$ is assumed to be a continuous function $g$. However Aristotle's arguments seems to rest rather on the monotonicity of the function implying that $g(I_i)$ must itself be a bounded interval $[0,a)$. Aristotle did not have our concept of 'function', he thought rather in terms of correspondences or relations, perhaps somewhat similar to adjunct pairs. Thus there is a bilateral correspondence in the Physics between time, magnitude, motion and even the extension of the moving body itself. A division of one of these quantities must correspond to a division in the others (but perhaps not the moving body itself).

Book VI: no first time for started moving. Aristotle constructs a filtration $U_1 \supset U_2...\supset U_i \supset...$ of shrinking semi-intervals all containing the starting point $t_0$. This is $dt_0$.

If at time $t$ a body $M$ is moving then it has moved before and it has been moving before. Moving times are an open set. $t$ is both a limit and there is $t' < t$ such that $M$ is moving at $t'$.

Mass nouns and count nouns

 If $P$ is predicated of $A$ then it may be the case that  for every part $B$ of $A$ $P$ is also predicated of $B$ - or it may not.   For example if some snow is white, then every portion of that snow is white.   But if a group of people are twelve then it does not follow that a part of that group, let us say five people, are also twelve.  The similarity to the distinction between finite and infinity cardinalities is striking if we force ourselves to be more rigorous in the way we dissect an object to obtain parts.  More specifically count nouns partake of finite cardinalities whilst mass nouns have the properties of the the cardinality of the continuum (more than the merely infinite countable). We associate the continuum with the Aristotelic and Platonic apeiron and with hule.  Mass nouns are generally matter.  Recall that a line has the cardinality of the continuum. If we take a standard 'part' of the line then this part (let us a say a interval) still has the cardinality of the continuum.  That is, all open sets of the topology of the real line have the same cardinality $\aleph_1$. We can also think of a counting sheaf on the category of open sets of space $X$. $\epsilon_0$, the well-ordering of the continuum, can be visualized as fractal or crystal.

If we allow fusions into UDIL2 so that some elements of $D_{-1}$ represent pluralities then are we forced to have elements of $D_{-1}$ representing at least all finite sets ? What about the cardinality of $I$ with which $H_{-1}$ is defined ?

Rudolf Hirzel on Stoicism

Untersuchungen zu Cicero's philosophischen Schriften (vol 2, part 1)

De logica stoicorum (pp.61-78)

I managed to find online the volume (dedicated to Hermann Sauppe) containing Hirzel's De logica Stoicorum as well as the volumes (published by Solomon Hirzel) of the 1882 'Untersuchungen' on Cicero's philosophical writings. Vol 2, part 1 , alone is more than 500 pages and entirely dedicated to 'The development of Stoic philosophy' . I also got the Gabriel et al. paper. De logica Stoicorum is written in a florid Latin and is just a pedantic discussion about whether the Peripatetics or the Stoics were the first to use the term 'logic' ; it goes on to cite various ancient sources that mention or give a definition of the terms 'logic', logical' and 'dialectics' . There is no actual logical or philosophical content in this 17 page article and nothing that would have interested or influenced Frege. 

Adjectives, Adverbs and Time

As pioneered by I. Mel'cuk's lexical functions it advantageous to introduce (partial ?) function symbols into UDIL in order to adequately formalize natural language. For is not treating 'big' as a unary predicate semantically ridiculous ? And it does seem that adverbs can be seen as both modifying verbs directly and as modifying states of affairs ?  In terms of UDIL models we do this as follows. For example 'big' would be a (partial) function $b : D_1 \rightarrow D_1$.  Determination according to time would be a (partial) function : $t : D_0 \times T \rightarrow D_0$ where $T \subset D_{-1}$ (or $T \subset I$).  Partiality might be defined  using $0$.  In the language of UDIL we would likewise introduce function symbols. Thus the proposition  'John pet the big cat at 5 o'clock'  would be $at([pet(John, \iota x. b[cat(x)]_x)],5 o'clock)$. Note: we should no longer write $\iota x.\phi(x)$ but $\iota x.[\phi]_x$. How would we decompose such terms ?  $at COMB_{(0,0)}[pet(u,v)]_{uv}John\,\iota x.b[cat(x)]_x\, 5 o'clock$.  The problem is now: how does this work on the extension functions for states-of-affairs ? That is, how does for instance $H \mathcal{M}b[Cat(x)]_x$ relate to $H \mathcal{I}[Cat(x)]_x$ ? We may think that for instance unary functions are interpreted as operators $\mathcal{I}f: \mathcal{P}D^{(i)} \rightarrow \mathcal{P}D^{(i)}$. This runs into a problem for $D_0$. But on the other hand we can interpret function symbols as in first-order logic, as functions $\mathcal{I}f: D_i \rightarrow D_i$. But then how do we express in UDIL that 'a big cat is a cat' ? Is it convenient to internalize at least the logical operations so that for instance $[\phi \rightarrow \psi] =  imp([\phi],[\psi])$ ?

The above considerations are interesting for classical logic, for definitions involving the genus plus the difference. The difference seems adjectival. Man is a rational animal. 

A fundamental division of predication is between predications that can be temporally specified and those that cannot. 

Project: add function symbols to UDIL and find the UDIL analogue of  multi-sorted first-order logic. Find the UDIL version of temporal logic. Then express (and demystify) work in theoretical computer science regarding specification and verification of multi-agent systems in this new framework.  The immense advantage of this is the presence of intensional abstracts allowing us to seamlessly deal with epistemic logic and the flow of information.

What about temporal logic ? One idea is to make $\mathcal{H}$ consist of an ordered  set $H_i$ for $i \in \mathbb{Z}$ where the natural order by $i$ represents temporal succession. In this way each model represents a possible temporal evolution and we can define in the expected way the standard modal temporal operators.  Each agent is associated with such a model.  We must study the combination and interactions of agents in terms of combinations of the various agent models subject to constraints (global specifications).

Truth, Certainty and Proof

Philosophy must avoid jargon, forbidding terminology, often with little connection to the relatively simple concepts involved. It must avoid giving the appearance of difficulty, depth or of having made more progress than is actually the case, for instance by copying the surface-appearance of technical scientific literature. 

A philosopher must be frank and honest about their positions and be capable of stating these in a plain concise way, without evasion, obfuscation or jargon.

I hold that there is such a thing as truth, that truth can be investigated and meaningful things can be said about truth. Certain knowledge is possible - attaining knowledge that is true and knowledge of the fact of this truth and certainty itself . We can obtain certain knowledge about truth itself.  I hold that there is such a thing as meaning, in particular the meaning of an expression.  I hold that truth can be predicated meaningfully, truthfully and non-redundantly. That the concept of truth and the truth predicate cannot be eliminated from discourse or knowledge. I believe that proof  and logical rules can be  instruments at arriving at the truth and help explore and expand meaning.  But truth and meaning are quite distinct from proof though proof can be sufficient to establish the truth of a proposition.  The truth and meaning of a proposition does not depend on its proof nor on any psychological or sociological factor or the particular language in which a proposition happens to be clothed. I hold that there are absolute certain moral truths which all human (or analogous rational) beings are bound to follow. I hold that there is such a thing as a priori and analytic knowledge and that this kind of knowledge can be defined. I hold that there is a true logic  which encompasses particular regional logics valid in their own domains .  At present we know only partial aspects of the one true logic (or rather intimately connected, mutually embeddable fragments, see Bealer's Quality and Concept).  I hold that a philosopher must a have a good technical knowledge of a wide range of sciences for reasons such as avoiding talking nonsense about the world and evoking inconsistent thought experiments involving arbitrary modifications of some aspect of reality. The philosopher of language must likewise have a good knowledge of linguistics. I hold that there are ideal, objective, non-physical, extra-linguistic, timeless, knowable entities such as propositions, properties, concepts, relations and ideal descriptions (templates) for individuals. I hold that there is nothing remotely 'naive' or 'default' about these positions, rather, as Plato  implies in the Theaetetus, it is physicalism, empiricism, nominalism, psychologism and relativism that can claim the authority of the poets and popular culture. I hold that mind and brain are completely distinct entities and that there is no grounds for identifying or reducing one to another or explaining one by means of another.

Algorithms and Numbers

When we investigate the concept of computability we necessarily require arithmetic. When we investigate arithmetic from a logical point of view it is inevitable that we consider computability or are lead to it. Arithmetic and computability are inseparable notions and it is likely that the Turing-Church thesis is tied to the categoricity of the theory of natural numbers N. When one defines a natural number object in a topos the universal property gives us automatically the notion of a primitive recursive function. The fundamental concept in recursion theory is that of partial recursive function, which can be embodied or implemented in a variety of abstract machine models such as Turing machines. Partial recursive functions, which are partial functions (partial because the computation need not stop) from natural numbers to natural numbers can themselves be codified by natural numbers. This allows us to have a concept of constructive functional, extending the notion of recursivity to all finite types over the natural numbers. There are several possible constructions such as HRO and HEO. This entire process can be generalised to the algebraic setting of partial combinatoric algebras. Intuitionistic arithmetic in all finite types HA, the terms of which form Gödel's system T, is an alternative to set theory, category theory and type theory for doing mathematics. It has a model in ZFC given by interpreting terms of a given finite type by means set-theoretic maps and the natural numbers. This model is extremely complex (it can represent the real numbers) and very little can be said about it in general. What is amazing is that models of HA like HRO and HEO pack this entire structure into the microcosm of N ! This gives us a vision of the mathematical universe in which, as for the ancient Pythagoreans, all things are number.
Tichý has written a monumental work on the philosophy of logic and mathematics in which computationally inspired formal semantics is given allowing a consistent rectification, interpretation and extension of Frege's Begriffschrift and Russell's Ramified Theory of Types. In this system (which we call T) the structure of an expression is isomorphic to the computational process it signifies. Thus an expression in T is like a syntactically correct program in some (typed) programming language. Its meaning is the process of execution (evaluation) of such a program, or rather, a certain abstraction of this. For instance we can give a description of how a certain function (in the programming sense) is executed for an undefined value of its argument. But it is clear that a lambda-calculus type of formalism (or functional programming in general) is the most adequate for this approach and it is not difficult to see why Russell's and Church's systems should figure prominently. We could attempt to formalise in turn such descriptions of computational processes, but this will just evade the problem again. We must stop at an extra-formal foundation in terms of definite mental-cognitive processes and acts. Tichý in his anti-formalist argumentation underestimates just how much mathematics has to be used and assumed to treat combinations of symbols in a relevant and interesting way (for instance proving soundness, completeness and consistency). Hilbertian metamathematics is not consistent with relativism.

What is a Set ?

A set is a tree. To specify a set we can list all the elements that belong to it and then all the elements that belong to each one of these elements and so forth (rigorously this is to be understood in terms of ordinals and assuming the axiom of choice). In Cohen's forcing technique there is the notion of a $P$-name, where $P$ is some set endowed with a preorder. A $P$-name is simply a set viewed as a tree but with each node labelled with an element of $P$. An ordinary set can be seen as a $P$-name with its tree's nodes tagged with the top element 1 of $P$. Given a $P$-name and a subset $G\subseteq P$ we can obtain a normal set by going through the tree and prunning off nodes whose labels do not belong to $G$ and then forgetting the labels and considering the resulting set - this is the interpretation or evaluation map. A particular case is in which $P$ is the set of all Borelian sets of the real interval $I$. A $P$ -name in then a probabilistically described set.

Recap on Vertical vs. Horizontal Proofs

Horizontal mathematics is like a maze. It is flat and expansive. Its concepts have limited complexity and abstraction or universality. Proofs are obtained by joining together clever tricks and ad hoc techniques. Clear insight and intuition can be quite lost. As a body of knowledge it is fragmented and specialized. This is particularly so when the classical negation rule (or indeed any negation rule) is used. 

Vertical mathematics is architectural and structured. It soars upwards. Its concepts are built upon each other as a winding staircase of ever higher abstraction and universality.  Yet this universality is natural and intuitive and has a powerful unifying aspect. Proofs follow almost immediately from the gradual unpacking of definitions, much like a computation. Their is an inherent necessity in the best proof. Negation rules are rarely used. Clear intuition and insight into the concepts and their mutual relation and unfolding is obtainable. Concepts reach such high levels of abstraction that they can mirror each other and subsume each other.

Hegel's attack on mathematical knowledge in his Preface to the Phenomenology of Spirit is very much weakened if instead of horizontal mathematics we consider vertical mathematics, for instance category theory.

Model theory and Philosophy

The problem with model theory is: in what system are there results of model theory carried out ? Is model theory nothing but an internal reflection-into-self of ZF(C) itself ? If so, it is interesting, but hardly philosophically conclusive or as weighty as sometimes suggested. Also then all results of model theory can then all be represented in a countable model.  There is a certain analogy between the Downwards Löwenheim-Skolem theorem and the topological property of $\sigma$-compactness.

Theory and metatheory

ZF(C) is also arguably a mirror,  a microcosm which can reflect and represent second and higher-order logic.  In what system is proven the categoricity of certain second-order theories (naturals, reals) ? 

How arbitrary and full of presuppositions it is to consider first-order models are representing the world ! For this simply involves doing a Henkin model move involving an arbitrary restriction on power sets and relations. No, we should be prepared to consider full second-order models and not necessarily as reflected into set theory. Then we can know what we are talking about when we define and talk about $\mathbb{N}$ and $\mathbb{R}$.  There is a huge difference from the conclusion that we cannot deduce everything about these objects (true) from the claim that we cannot know anything for certain about mathematical objects at all (false).

If model theory be considered to set-theoretic, its category theoretic version needs to be considered. Just as in model theory we are lead naturally to consider classes of models, so too in category theory it is very natural to consider categories of categories (on specific kinds). Thus a 'category' is both the generalization of a model and a generalization of a class of models.

Proposition and Judgment

Should we not consider propositions to be judgments, that is to say, claims with truth-values dependent on states-of-affairs ? Then the claim that a proposition is a claim is true. The original meaning of $\vdash P$ is the claim (or judgment) that $P$.  Let the predicate $C$ signify 'being a claim'. Thus $C(\vdash P)$ or rather $\vdash C(\vdash P)$ and hence $C(\vdash C(\vdash P))$.

Truth and judgment are primitive concepts.

There are judgments, for instance the one above.

The judgement that a judgment is a judgment is true.

Hence: there are true judgments and knowledge is possible.

After I wrote this I discovered that Bolzano had already written in the Theory of Science, Theory of Fundamentals §40:

During the time that a person is such a complete sceptic, he cannot form any judgement, no matter what its content. For a judgement is nothing but a proposition which the subject, with more or less confidence, takes to be true. If anybody forms a judgement, he gives us to understand that there is at least one proposition which he takes to be true (with more or less confidence) at the moment when he judges; he believes (with more or less confidence) that there is at least one truth; hence he is not to be called a complete sceptic. Consequently, in order to heal a sceptic, all we have to do is to get him to make a judgement whose truth is so irresistible to him that he tries in vain to doubt it when, a moment later, we call his attention to the fact that in this judgement he has recognised at least one truth.

Husserl, in the Logical Investigations,  distinguishes between judgment and proposition. Propositions correspond to states of affairs.  If we want to deal with propositional attitudes, then there must be a predicate for claiming, for the judgmental aspect discussed above.  Rather than A believes that P or A knows that P we should investigate the more basic A claims that P.  Frege's symbol $\vdash P$ involves an unspoken hidden subject, the subject claiming that P. Of course we can distinguish between explicit manifest claims and implicit claims such as bound-to-hold-that, forced-to-hold-that, etc. Otherwise how can we interpret logical rules of the form: if it is claimed that X then it follows that it is claimed that Y. Thus $\vdash$ should mean something like 'claimable', 'assertible' - the Stoic lekton !

Identity involves synthesis of the manifold via relations, the using of relations in a manifold for identification.  But one may want to distinguish again these elements fused together by identification: fine-grainedness. The act of synthesis presupposes a perception of difference. The elements fused together might themselves be fusions. Thus every category is a quotient category of quotient categories.

There is more than one absolute certainty. Likewise what objection is there that there should be more than one path or proof for arriving at the knowledge that there is absolute certainty ? I.e. both the Leibniz-Bolzano path and the Husserlian path are valid. 

There is much confusion regarding the nature of formal logic and the terms 'formal' and 'formalization'. To us a 'formal logic' is above all a language. A language different from English and other natural languages, but a different language is no other sense.  Formal logic aims at being more syntactically and semantically explicit, unambiguous and detailed than ordinary language. But it is still a language. Our concept of 'formal logic'  might be more properly called a Leibniz-inspired universal formal philosophical language. Here 'formal' thus pertains to its structure rather than its fundamental linguistic nature.

Truth is a relationship between thought and reality and as such both thinkable and a part of reality itself.

Transcendental Idealism and Metaphysics Husserl's Critique of Heidegger

 https://link.springer.com/book/10.1007/978-3-031-39586-4

At stake in Husserl's critique of Heidegger's philosophy in Being and Time is the refusal to transcendentalize the irrational aspects and nature of our human existence.

Beyond Subjective and Objective

What defines  the 'subjective' and the 'objective' ? Is it the nature of the domain itself or the mode, attitude and intention regarding such domains ? We might say that we normally have an objective attitude towards that which we usually call 'objective' and a subjective attitude (i.e. 'first-person' ) towards what we usually call 'subjective'.   Philosophically we should investigate the 'objective' domain's dependence on the 'subjective' domain. But most importantly we should investigate (by a shift of perspective and attitude) the 'subjective' domain in a thoroughly objective manner, rather than subjectively as  usual.  This autoreification, autotranscendence (which is in fact both theoretical and practical) is what was possibly at the core of Husserl's phenomenology and at the same time totally at odds with subsequent phenomenology, existentialism, structuralism and postmodernism.