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Philosophy & Computer Science

Gianfranco Basti

Faculty of Philosophy – Lateran University www.irafs.org

A New Mission, and New Job Opportunities for Philosophers

IRAFS website: www.irafs.org

February 7, 2017Puebla 2017 - http:/www.irafs.org 2

http://www.irafs.org/

Summary


The Next «Digital Tsunami»And the Key-Role of Philosophy: The «Philosophy & Computer Science» Study Program


The «digital tsunami» on the job market The exponential rise of computing

power, and of AI applications on allthe fields of private and public life isdestined to change deeply oursociety, economy, and culture.

One recent study by academics at Oxford University suggests that 47% of today’s jobs could be automated in the next two decades.

«Optical quantum computing isready – are you prepared for it?» (Prof. Jeremy O’Brien at Davos 2016).


D-Wave quantum computer | Optical quantum computer

http://www.irafs.org/materials/puebla/davos2016.pdf

http://www.irafs.org/materials/puebla/obrien_paper.pdf

Optical quantum computer revolution: the computational power of D-Wave in a smartphone


The emerging automation technologies: in ICT At the moment, the edge technologies in Information and Communication

Technology (ICT) concern the research & development (R&D) of advanced software for the simulation/integration of the human semantic, and intentional behavior in the artificial intelligence (AI) systems. These technologies embrace (the examples I give are limited, and for only introductory aims):

In the ICT realm. The development of largely automatized and individualized:1. Semantic large databases (“big data”) in whichever realm of ICT;2. User friendly man-machine interfaces;3. Distance learning programs and platforms for education;4. Telemedicine systems and robotized systems in surgery, nursery, diagnostics, etc.5. Administration, information, assistance (e.g., call-centers), e-government services,

interacting in human-like fashion with individuals, citizens, companies, etc.


The emerging information technologies: in ACS The automatic control systems (ACS) realm, practically in whichever component of

the private and public life of our present and future societies.1. At the public level: ever larger parts of the control of power, flight, rail, transportation and

communication networks will be managed by completely automatic systems.2. At the private level: Ever larger parts of the financial market transactions are, and will be managed by automatic

systems (actually the estimations are about 30% of the transactions); It is rapidly growing the usage of drones both for observation and transportation aims, either

in the military realm (where they are used also as unmanned weapons), or in the civilian realm;

The future massive usage of self-driving cars in our cities and streets; The usage of humanoid robots, not only as soldiers in the battlefields for military aims, but

also for civilian aims. E.g., in particularly risky rescue missions, or for the continuous 24/7 medical/home care/assistance of long-standing patients as humanoid nurses for integrating human ones, etc.


The practical issues of such a revolution The tremendous issues for our political, economical, social, educational systems,

both on a national and international dimension, are evident.

The less obvious, but probably the most important and decisive one, concerns the necessity of implementing in the human-like, autonomous behavior of ICT and ACS systems ethical principles/skills, governing their choices, given that – before all for the amazing velocity of their execution – they escape any possibility of human meta-control. These programs for the ethical behavior simulation, are already being implemented in automatic systems – at least, for very simple tasks.

Their necessity is, however, growing, as far as these machines/systems will enter massively into our lives (think only at the necessity of implementing the military ethics in robotized soldiers/drones, of implementing financial ethics in automatic traders, of implementing human ethics in self-driving cars, humanoid nurses, humanoid teachers for automatized distance learning programs, etc.).


A review of the actual perspectives and trends in ICT and AI

The economic and social impact of next automation (The Economist 2014)

Main commercial lines of the actual CS/IT/AI development (The Economist 2015)

Main trends of the actual CS/IT/AI Research&Development (Science, 2015)

Brochure of the PCS Course at Oxford (2015-16)

A survey of research questions for robust and beneficial AI (FLI Institute, 2016)

Are you ready for the next quantum optical computer? (Prof. O’Brien at Davos World Economic Forum 2016)

An example of quantum optical computer architecture (O’Brien’s group, 2015)

A urgent educational challenge (Basti, 2017)


http://www.irafs.org/materials/puebla/economist_2014.pdf

http://www.irafs.org/materials/puebla/economist_2015.pdf

http://www.irafs.org/materials/puebla/science_17_july_2015.pdf

http://www.irafs.org/materials/puebla/oxford_brochure.pdf

http://www.irafs.org/materials/puebla/research_survey.pdf

http://www.irafs.org/materials/puebla/davos2016.pdf

http://www.irafs.org/materials/puebla/obrien_paper.pdf

http://www.irafs.org/materials/puebla/an_educational_challenge.pdf

The ever more diffused awareness of the challenge

An example: www.futureoflife.org

A survey of research questions for robust and beneficial AI

Why 2016 Was Actually a Year of Hope

An educational challenge


http://www.futureoflife.org/

http://www.irafs.org/materials/puebla/research_survey.pdf

http://www.irafs.org/materials/puebla/2016_year_of_hope.pdf

http://www.irafs.org/materials/puebla/an_educational_challenge.pdf

The Computer Science & Philosophy Study ProgramA Proposal


The Philosophy & Computer Science Study Program (PCS): A Proposal

The PCS study program I proposed want to be an initial response to the tremendous educational challenge with which Christians – and more generally all good will people – are facing today in front of the new computer and information technologies.

They are destined to change ever more deeply than actually do, our society, our economy, and our culture.

The Philosophy & Computer Science Study Program


http://www.irafs.org/materials/puebla/pcs_project.pdf

A Paradigm Shift in Philosophy & Computer ScienceFrom the Analytic Philosophy to Formal Philosophy: Mathematical Logic versus Philosophical Logic

Premise I: Methodological solipsism and representationalism

Logic is always representational, it concerns relations among tokens, either at the symbolic or sub-symbolic level. It has always and only to do with representations, not with real things.

R. Carnap’s (1936) principle of the methodological solipsism in formal semantics extended by H. Putnam (1975) and J. Fodor (1980) to the representationalism of the functionalist cognitive science based on symbolic AI, according to the Turing paradigm in CS.

W.V.O. Quine’s (1960) opacity of reference beyond the network of equivalent statements meaning the same referentialobject in different languages.


Premise II: The issue of cognitive science: «Putnamroom», reference and the coding problem

Turing test («Turing room»: 1954) and the problem of modelling mind as a UTM, that is, as a second-order infinitistic TM Turing halting problem in relationship with Goedeltheorems

Searle’s test («Chinese room»: 1980) UTM fails in modelling intensional modalcalculations of brains as counterpart of intentional tasks of mind

Putnam test («Putnam room»: 1988) After Searle’s Chinese Room Putnam suggestedanoter room metaphor for empasizing TM limitations for representing intentional minds in formal semantics: the impossibility of solving for a TM the simplest problem of how manyobjects are in «this» room: three (a lamp, a chair, a table) or many trillions (if we considerthe molecules) and ever much more (if we consider also atoms and sub-atomic particles…).

Putnam’s problem is the main problem of CS in dealing with infinite streams of data: to reckon with the always changing hidden correlations of the data stream.


From mathematical to philosophical logic: intensional logics as interpretations of modal logics

Since the first publication of Whitehead & Russell Principia in 1912 the American philosopher Carol I. Lewis, prophetically:

1. Warned against the misuse of mathematical logic for the analysis of philosophical & religious languages as it happened thereafter through the publication of L. Wittengstein’s Tractatus Logico-Philosophicus by his mentor B. Russell (1917).

2. Recognized that the power of mathematical logic is in its formalization necessity of formalizing also the philosophical logic, so to understand clearly similarities and differences: Between science(s) and philosophy(ies) and Among the different philosophies, beyond the differences of ages, languages and

cultures.

The globalization of science during XX cent. depends on its formalization, just as the marginalization of philosophy depends on its non-formalization in the XXI cent. we have to restore the balance by formalizing philosophy.


Extensional (mathematical) vs. intensional (philosophical) logic

ML relational structures with all its intensional interpretations are what is todaydefined as philosophical logic (Burgess 2009), as far as it is distinguished from the mathematical logic, the logic based on the extensional calculus, and the extensional notions of meaning, truth, and identity.

What generally characterizes intensional logic(s) as to the extensional one(s) is that:1. neither the extensionality axiom between classes: A↔B ⇒ A=B;2. nor the existential generalization axiom: Pa ⇒ ∃x Pxof the extensional predicate calculus hold in intensional logic(s).

Consequently, also the Fegean notion of extensional truth based on the truth tables does not hold in the intensional predicate and propositional calculus.


Intensional logic and intentionality

Anyway because of the axiomatization of modal and intensional logicsstarting from Lewis’ pioneering work: There exists an intensional logical calculus, just like there exists an

extensional one, and this explains why both mathematical and philosophicallogic are today often quoted together within the realm of computer science.

This means that intensional semantics and even the intentional tasks can be simulated artificially («third person» simulation of «first person» tasks, like in human simulation of understanding, without conceptual «grasping»).

The “thought experiment” of Searle’s “Chinese Room” is becoming a reality, as it happens often in the history of science


Modal logic: from philosophy to computer science

Following (Blackburn, de Rijke & Venema, 2002) we can distinguish three eras of modal logic(ML) recent history:

1. Syntactic era (1918-1959): C.I.Lewis…: second-order modal logic and intensional semantics

2. Classic era (1959-1972): S. Kripke’s… relational semantics based on frame theory (second-order «global» semantics) and Kripke models (first-order «local» semantics) (Goranko & Otto, 2007)

3. Actual era (1972…): S. K. Thomason’s algebraic interpretation of modal logic for reducing to itTuring-like second-order semantics ML as a fundamental tool in theoretical computer sciencea. Correspondence principle: equivalence between modal formulas interpreted on models and first

order formulas in one free variable Possiblity of using ML (decidable) for individuating noveldecidable fragments of first-order logic (being first-order theories (models) not fully decidable)

b. Duality theory between ML relation semantics and algebraic semantics based on the fact thatmodels in ML are given not by substituting free variables with constants like in logistic, but by usingbinary evaluation letters in relational structures (frames/models) like in algebraic semantics.


1. The First Era: second-order modal logicAnd the related intensional semantics in philosophical logic


Main intensional logics: alethic logics

Alethic logics: they are the descriptive logics of “being/not being” in which the modal operators have the basic meaning of “necessity/possibility” in two main senses:

Logical necessity: the necessity of lawfulness, like in deductive reasoning

Ontic necessity: the necessity of causality, that, on its turn, can be of two types: Physical causality: for statements which are true (i.e., which are referring to

beings existing) only in some possible worlds. Metaphysical causality: for statements which are true of all beings in all

possible worlds, because they refer to properties or features of all beings such beings.


Main intensional logics: deontic and epistemiclogics

The deontic logics: concerned with what “should be or not should be”, where the modal operators have the basic meaning of “obligation/permission” in two main senses: moral and legal obligations.

The epistemic logic: concerned with what is “science or opinion”, where the modal operators have the basic meaning of “certainty/uncertainty”.


Main axioms of Lewis’ ML syntax

For our aims, it is sufficient here to recall that formal modal calculus is an extension of classical propositional, predicate and hence relation calculus with the inclusion of some further axioms:

N: <(X→α) ⇒ (X→α)>, where X is a set of formulas (language), is the necessity operator, and α is a meta-variable of the propositional calculus, standing for whichever propositional variable p of the object-language. N is the fundamental necessitation rule supposed in any normal modal calculus


More…

D: <a→àa >, where à is the possibility operator defined as ¬¬ a. D is typical, for instance, of the deontic logics, where nobody can be obliged to what is impossible to do.

T: <a → a>. This is typical, for instance, of all the alethic logics, to express either the logic necessity (determination by law) or the ontic necessity (determination by cause).

4: <a →a>. This is typical, for instance, of all the “unification theories” in science where any “emergent law” supposes, as necessary condition, an even more fundamental law.

5: <◊a →◊a>. This is typical, for instance, of the logic of metaphysics, where it is the “nature” of the object that determines necessarily what it can or cannot do.


Main Modal Systems (Galvan 1991; Creswell & Huges 1996)

By combining in a consistent way several modal axioms, it is possible to obtain several modal systems which constitute as many syntactical structures available for different intensional interpretations.

So, given that K is the fundamental modal systems, constituted by the ordinary propositional calculus k plus the necessitation axiom N, some interesting modal systems for our aims are: KT4 (S4, in early Lewis’ notation), typical of the physical ontology;

KT45 (S5, in early Lewis’ notation), typical of the metaphysical ontology;

KD45 (Secondary S5), with application in deontic logic, but also in epistemic logic, in ontology.


Alethic vs. deontic intensional interpretations

Generally, in the alethic (either logical or ontological)interpretations of modal structures the necessity operator p is interpreted as “p is true in all possible world”, while the possibility operator àp is interpreted as “p is true in some possible world”. In any case, the so called reflexivity principle for the necessity operator holds in terms of axiom T, i.e, p → p.

This is not true in deontic contexts. In fact, “if it is obligatory that all the Italians pay taxes, does not follow that all Italians really pay taxes”, i.e.,

February 7, 2017

p p→O

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Reflexivity in deontic contexts

In fact, the obligation operator Op must be interpreted as “p is true in all ideal worlds” different from the actual one, otherwise O=, i.e., we should be in the realm of metaphysical determinism where freedom is an illusion, and ethics too. The reflexivity principle in deontic contexts, able to make obligations really effective in the actual world, must be thus interpreted in terms of an optimality operator Ot for intentional agents x, i.e,

(Op→p) ⇔ ((Ot (x,p) ∧ ca ∧ cni ) → p)


Reflexivity in epistemic context

In similar terms, in epistemic contexts, where we are in the realm of representations of the real world. The interpretations of the two modal epistemic operators B(x,p), “x believes that p”, and S(x,p), “x knows that p” are the following: B(x,p) is true iff p is true in the realm of representations believed by x. S(x,p) is true iff p is true for all the sound representations believed by x. Hence the relation between the two operators is the following:

February 7, 2017

( ) ( )( ), ,x p x p⇔ ∧S B F


Reflexivity in epistemic logic

While

because of F

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( ),x p p→B

( ),x p p→S


Finitistic, infinitistic, and finitary interpretations of the F foundation clause (second-order vs. first-order semantics)

Different interpretations of the foundation clause F depend on the different ontologies that, if formalized in the proper modal logic, are as many formal ontologies (truth conditions of predication in logic and mathematics)1. As far as we move within a second-order logic like Lewis’ modal one (Creswell & Huges 1996), we move inside

an infinitistic interpretation of F, consistent with a logicist (Plato, Descartes, Frege, Kutschera, Galvan…), conceptualist (Kant, Husserl, Stein, Cocchiarella,…), or logical atomist (Democritus, Newton, Wittengstein, Carnap, …) formal ontologies.

2. As far as we move within a first-order logic (FOL) like in all constructive approaches to logic and mathematics, we move inside a finitistic interpretation of F consistent with an anti-platonic, nominalist formal ontology (Buridan, Ockham, Bishop, Nelson,…).

3. However, FOL is consistent also with a relational (coalgebraic) modal semantics, based onto a “homomorphic duality” coalgebra-algebra in Category Theory, we move inside a finitary interpretation of F, consistent with an Aristotelian, naturalist formal ontology based on the distinction between natural kinds and logical classes (Aristotle, Aquinas, Poinsot, Peirce, Kripke, …) notion of local modal truth in Kripke models:


| ( )Algebra( *) Bounded Morphism Co-Algebra( )

n n n m horse mammmalian horse mammalian∀ >Ω Ω

∈ ← ∋

A taxonomy of formal ontologies depending on truth conditions of predication (universals issue)


Ont

olog

y Nominalism

Conceptualism

RealismLogical

NaturalAtomistic

Relational

Atomistic: without natural kinds, where mechanics is the fundamental physics, and its absolute mathematical laws with their empirical fulfillment are ultimately determining the truth conditions of the ontological propositions (Democritus, Newton, Laplace Wittengstein’s Tractatus, Carnap, …).Relational: with “natural kinds” – the “generals” of Peirce’s semiotics –, because the real relations(causes) among things ultimately determine the linguistic relations, and then the truth conditions of the ontological propositions (Aristotle, Aquinas, Poinsot, Peirce, Kripke, …).

The pragmatic approach of CS to formal ontology in knowledge management

“We say that an agent commits to an ontology if its observable actions are consistent with the definitions in the ontology. The idea of ontological commitments is based on the Knowledge-Level perspective (Newell, 1982). The Knowledge Level is a level of description of the knowledge of an agent that is independent of the symbol-level representation used internally by the agent. Knowledge is attributed to agents by observing their actions; an agent “knows” something if it acts as if it had the information and is acting rationally to achieve its goals.” (Gruber 2003)

“An ontology is an explicit specification of a conceptualization. The term is borrowed from philosophy, where an Ontology is a systematic account of Existence for a given group” (Gruber 1993).

Definition of standard Formal Web Ontology Languages (OWL) and tools in terms of Descriptive Logics for decidable fragments of FOL, during more than twenty years of activity of the World Wide Web Consortium for the semantic web (W3C).

Propositional ML decidable fragments are included in them.


The limit of a descriptive formal ontology The descriptive approach to formal ontology, ML included, as far as

conceptualist version of Frege’s descriptive theory of reference of the classical propositional and predicate logic, is useful for knowledge management:1. In philosophy, for comparative studies among the ontologies of different groups and

cultures using the powerful tools of formal semantics;2. In computer science, for knowledge management beyond the limits of the usual

database semantic management (Horrocks 2010a,b).

However, in both cases the descriptive approach fails when we have to deal: 1. Either with the pre-conceptual basis of human knowledge in ontology;2. Or with the change of the basic class/concept, overall when these changes are not

linkable into a tree-structure (Horrocks 2010b), i.e., with we have to deal with infinite streams of data.


2. The Second Era: Kripke Relational SemanticsKripke theory of frames and of models


A first move toward a finitary modal semantics

The fundamental correspondence theorem (Van Benthem 1976) between modal axioms and first order formulas.


Kripke relational semantics (Kripke 1963;1965) Kripke relational semantics is an evolution of Tarski formal semantics, with two

specific characters: 1) it is related to an intuitionistic logic (i.e., it considers asnon-equivalent excluded middle and contradiction principle, so to admit coherenttheories violating the first one), and hence 2) it is compatible with the necessarilyincomplete character of the formalized theories (i.e., with Gödel theoremsoutcome), and with the evolutionary character of natural laws not only in biology but also in cosmology.

In other terms, while in Tarski classical formal semantics, the truth of formulas isconcerned with the state of affairs of one only actual world, in Kripke relationalsemantics the truth of formulas depends on states of affairs of worlds differentfrom the actual one (= possible worlds).

Stipulatory character of Kripke’s possible worlds


Kripke notion of frames Kripke notion of frame main novelty in logic of the last 50

years relational structure. This is an ordered pair, <W, R>, constituted by a subset W of

possible worlds u, v, w… of the universe W, W ⊆ W, and a by a two-place relation R defined on W, i.e., by a set of ordered pairs of elements of W (R ⊆ W×W), where W×W is the Cartesian product of W.

E.g. with W = u,v,w and R = uRv, we have:


Relations defined on frames

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Seriality: (om u)(ex v)(uRv)


Transitivity: ((uRv∧vRw)→uRw; Simmetry: uRv=vRU; Riflexivity: uRu; Equivalence (KT5)

Euclidean relation (axiom 5 or E)

<(om u) (om v) (om w) (uRv et uRw ⇒ vRw)>


Derivation of KD45 by applying Euclidean and serial relations derivation of secondary transitive, symmetric, and reflexive relations among worlds accessed by u these accessed worlds form a Secondary S5 system

Ontological, metaphysical and theologicalinterpretations of the KD45 system

Of course, this procedure of a (logical) equivalence class constitution (modal S5 system) by iteration of an Euclidean and serial relation can be extended indefinitely:

As we see below:

Evidently this modal syntax is perfectly compatible with an ontological interpretation, where the inaccessible world u represents the first cause and the accessibility relations as many causal relations.

It will be a physical ontology, compatible with quantum field theory (QFT), if we interpret u as the quantum vacuum (QV) endowed with itsspontaneous symmetry breakdown (SSB) dynamic principle, by which enriching the universe(s) with always new physical systems.

It will be a metaphysical ontology, compatible with Aquinas metaphysics of participation of being, if we interpret u as a transcendentFirst Cause founding the existence of the same QV. If we move to theology, interpreting u as a Personal First Cause (God), He isinterpretable as an Intelligent Designer of the universe(s), given that KD45 is compatible also with a deontic interpretation.


KD45 as a secundary S5 (KT45) with respect to an inaccesible world u

February 7, 2017

S5(KT45) KD45


Interpretation I over W (Kripke frame semantics)


I: V x W → 0,1

Where V is a set of propositional variables p. Then I(p,u) = 0 means that p isfalse in u; whereas I(p,v) = 1 means that p is true in v.

It is to be noticed that, as in the propositional calculus k semantics all the interpretations are determined as to all the propositional variables, so in Kripke’sframe semantics all the interpretations are determined as to all the pairs V,W. I.e. as to V x W.

Kripke frame semantics is therefore a second-order semantics (Goranko & Otto 2007)

Kripke model semantics in QML In a quantified modal logic QML (modal predicate logic), a Kripke model is an ordered

quintuple <W,R,D,Q,T>, where: W is a set of worlds, effectively a subset of the universe W, W ⊆ W; R is the binary accessibility relation; D is the domain; Q is a function assigning to each world w ∈ W a subset D(w) ⊆ D, i.e. the domain of quantification of

w; T is a valuation assigning for each world w ∈ W: an object in D to each term t, variables included; a set of ordered n-tuples of elements of D to each n-ary predicate, and the set D(w) to the existence predicate E.


Coalgebraic semantics of Kripke models

It is demonstrated that a Kripke model semantics is complete only in the framework of a coalgebraic modal logic (semantic dual equivalence between the categories of modal Boolean algebras and coalgebras in Category Theory logic).

Coalgebraic modal semantics of Kripke models, with the related notion of local truth, is the proper FOL decidable fragment of ML . Kripke frame semantics is a second-order ML semantics (total truths)

Kripke model semantics is a modal coalgebraic first-order semantics (local truths) (Goranko & Otto, 2007)

Fully computable modal logics are coalgebraic! (Venema 2007).


3. The present era: the (co-)algebraic modal logicThe coalgebraic modal logic in the framework of Category Theory Logic


Category Theory (CT) as universal language for logic and mathematics

What is CT? It is a natural recovering of Ch. S. Peirce and E. Schroeder research project of an algebraic foundation of logic and mathematics (algebra of relations, calculus of relations) suddenly interrupted by the Loewenheim-Skolem Theorem (1921) migration of set-theoreticsemantics (Boolean Algebra semantics included) to second (and higher) order logicintroduction of arbitrariness («creativity», for being more politically correct) in mathematics, because second (and higher) order predicate calulus is not complete differently from first-orderone (Cfr. Goedel demonstration (1929), as necessary premise of his famous incompletenesstheorems for first order theories (models)). These are the «strong roots of the weak thought», as I entitled one book of mine on these topics.

Anti-platonic stance (implicitely Aristotelian) of CT: even set elements are interpreted asdomain-codomain of arrows (morphisms) Possibility of discovering, both in mathematical and philosophical logics, relationships among theories

that it would be impossible to discover otherwise natural place for developing a set-theoretic modal logic in its pragmatic interpretation by Peirce and,

obviously, for giving a full foundation to QFT calculations on phase transitions (coalgebraic interpretationof quantum systems (Abramsky, Vitiello, Basti)).


Some elements of CT (Abramsky & Tzevelekos2011)

The primitives of CT language are:1. Morphisms or arrows, f,g;

2. Compositions of arrows, f g;

3. Two mappings dom (), cod () assigning to each arrow its domain and codomain.

Therefore: Any object A, B, C, characterizing a category, can be substituted by the correspondent reflexive morphism A → A

constituting a relation identity IdA.

Moreover, for each triple of objects, A,B,C, there exists a composition map written as .

A category is any structure in logic or mathematics with structure-preserving morphisms.

In this way, some fundamental mathematical and logical structures are as many categories: Set (sets and functions), Grp (groups and homomorphisms), Top (topological spaces and continuous functions), Pos(partially ordered sets and monotone functions), Vect (vector spaces defined on numerical fields and linear functions), etc.


f gA B C→ → g f

http://www.irafs.org/materials/puebla/abramsky_logic_categories.pdf

The categorical notion of functor F

Another fundamental notion in CT is the notion of functor, F, that is, a bijective mapping of objects and arrows from a category C into another D, F: C → D, so to preserve compositions and identities. In this way, between the two categories there exists a homomorphism up to isomorphism (if the bijective mapping can be reversed).

Generally, a functor F is covariant, that is, it preserves arrow directions and composition orders i.e.:

However, two categories can be equally homomorphic up to isomorphism if the functor Gconnecting them is contravariant, i.e., reversing all the arrows directions and the composition orders, i.e. G: C → Dop:


if : , then ; if , then ( ) ; if , then .A A FAf A B FA FB f g F f g Ff Fg id Fid id→ → = =

if : , then ; if , then ( ) ; but if , then .A A GAf A B GB GA f g G g f Gg Gf id Gid id→ → = =

Category duality and dual equivalence in categorical semantics

Through the notion of contravariant functor, we can introduce the notion of category duality. Namely, given a category C and an endofunctor E: C → C on a category onto itself, the contravariant application of E links a category to its opposite, i.e.: Eop: C → Cop.

In this way it is possible to demonstrate the dual equivalence between them, in symbols: CCop. In CT semantics, this means that given a statement α defined on C α is true iff the statement αop defined on Cop is also true.

In other terms, truth is invariant for such an exchange operation over the statements, that is, they are dually equivalent.

In symbols: α αop, as distinguished from the ordinary equivalence of the logical tautology: α ↔ β, defined within the very same category, i.e., on the same basis.


Duality and semantics


The dual category of Setop is more important than Set, given that a generic conditional in logic “if…then”, e.g., “for all x, if x is a horse then it is a mammalian”, is true if and only if the “mammalian set” includes the “horse set” with all its subsets.

Therefore, the semantics of a given statement is set theoretically defined on the power set (X) of a given set X. Categorically, indeed, the power set functor is a covariant endofunctorSet → Set, mapping each set X to its power set (X) and sending each function f : X → Y to the map sending U ⊆ X to its image f (U) ⊆ Y. Namely:

X (X), (f : X → Y (f ) := S f (x) | x ∈ S

Vice versa, the contravariant set functor op: Setop ® Set sends each function f : X → Y to the map which sends V ⊆ Y to its inverse image f -1(V) ⊆ X, but, of course, preserving all the objects. Therefore:

op (X) := (X); op (f : X → Y) (Y ) ® (X) := T x ∈ X | f (x) ∈ T

Several useful dual constructions in CT

Moreover, in CT other useful categorical dual constructions can be significantly formalized that we cannot define here.

For instance, “left” and “right adjoints” of functions and operators;

“universality” and “couniversality”;

“products” and “coproducts”,

“limits” and “colimits” interpreted, respectively, as “final” and “initial” objects of two categories related by a third category of “indexing functors”, so to grant the mapping, via a “diagonal functor”, of all the objects and morphisms of one category into the other.


A comparison with set-theoretic logic and mathematics Practically, all the objects and the operations that are usefully formalized in set

theory, and then in calculus and logic – included the “exponentiation” operation for forming function spaces, and the consequent “evaluation function” over function domains –, can be usefully formalized also in CT, with a significant difference, however.

Instead of considering objects and operations for what they “are” as it is in set theory, in CT we are considering them for what they “do” (Abramsky & Tzevelekos, 2011, p. 53).

That is, there exists a natural affinity of CT with applied computational semantics and with quantum physics and then with quantum computations. In both cases, indeed, we are dealing with observables behaviors being the underlying internal states of a communication agent and/or of a quantum system absolutely hidden.


Coalgebraic semantics of infinite streams As we know classical computational and philosophical logics fail in dealing with infinite

streams as far as their semantics is strictly related with the “Putnam room” paradox (sudden and unpredictable changes of data inner correlations inadequacy of the system representation space at coping with them, as far as we use standard statistical updating tools). A possible solution comes from the coalgebraic semantics of Boolean Algebras:1. Stone representation theorem for Boolean algebras (BA) (1936): A BA is isomorphic with a partial

ordered set (a set ordering, ≤, satisfying the reflexive, transitive, and antisymmetric relations) of clopen sets (real number intervals) defined on a Stone topological space. Moreover, BA and Stone are dual categories; and Stone category share the same topologies of the category of topological spaces dually associated with the C*-Alg category in QM and QFT operator algebra formalism.

2. Usage of non-wellfounded (NWF) set theory (Aczel 1988), where set self-inclusion and then unbounded chains of set inclusions are allowed ( no set total ordering), for defining on them modal coalgebras in order to model infinite streams. Final Coalgebra Theorem (1989): sets represented as oriented graph, where edges are inclusion relations, nodes are subsets and the root is the superset, share the same ultimate root powerful notion of coalgebraic coinduction as dual to algebraic induction, both as method of set definition and proof.


Continuing…3. Dual equivalence of the respective categories, by a contravariant vectorial mapping *

(“Vietoris construction”) from the category of coalgebras of NWF-sets defined on Stone spaces, Scoalg, onto the category of Boolean Algebras, BAlg. That is, SCoalg BAlg* /^operators of BA algebraically lower bounded and coalgebraically upper bounded notion of finitary computation coalgebraic semantics of Boolean logics (and then of propositional and predicate calculus) (Abramsky 1988; Venema 2007). Notion of bounded morphism for the dual equivalence in Kripke model semantics:

4. Definition of “Universal Coalgebra” as dual to “Universal Algebra” and interpreted as general theory of systems interpreted as labelled state transition systems (LTS) (Rutten 2000) construction of the “infinite state black-box machine” for modelling infinite streams through the dual equivalence between a final coalgebra and an initial algebra.

Applicability of this computational logic to a new class of quantum computer based on thermal QFT (Basti, Capolupo & Vitiello 2016).


| ( )Algebra( *) Bounded Morphism Co-Algebra( )

n n n m horse mammmalian horse mammalian∀ >Ω Ω

∈ ← ∋

http://www.irafs.org/materials/puebla/arxiv_1701.00527v1.pdf

A Paradigm Shift in Natural SciencesA Paradigm Shift in Fundamental Physics and the Emergence of a Naturalistic Ontology

Hilbert space and Dirac Delta Function in the QM interpretation of quantum systems

As we know, for the Heisenberg uncertainty principle, the canonical variables of Newtonian mechanics position, x, and momentum, p, do not commute between themselves for quantum particles in QM necessarily statistical and not geometrical representation of quantum states Schroedinger wave function.

However the outstanding Hilbert’s discovery that the canonical variables commute also in QM each with the Fourier transform of the other notion of canonical commutation relations (CCR).

This commutability opened the way to a geometrical representation also of a QM system in terms of a Hilbert space to which it is possible to apply the powerful algebraic tool of the vectorial calculus, and hence of the matrix calculus of the classical statistical mechanics to the wave function of QM.


Continuing…

However, quantum calculations are so faced with the problem of undesired infinite magnitudes in two ways: 1. On one side, Hilbert vectorial space is necessarily infinite-dimensional Stone-Von Neumann

theorem (1931) a finite number of unitarily equivalent CCR’s is necessary and sufficient for representing a QM system choice of a finite orthonormal basis – i.e. a finite number of degrees of freedom of the system – of the Hilbert space, sufficient for representing adequately a given quantum system, depends on the human observer epistemological problem of the intrinsic observer-dependent character of QM calculations;

2. On the other side, in QM matrix calculus the infinity appears with the problem of the so-called Dirac Delta Function: if I want to reduce the statistical variance on one of the two correlated magnitudes, I have to suppose necessarily infinite the other one, and vice versa usage of the so-called renormalization groups for solving the problem (‘tHooft).


Hopf bialgebra and its self-duality in QM


Generally the algebraic tool for performing calculations on a lattice of quantum numbers is the Hopf Algebra, that is a bialgebra composed by an algebra H x H H (e.g., for calculating the energy of a single particle in a lattice of quantum numbers: right half in diagram) and a coalgebra H H x H (e.g. for calculating the total energy for two particles (coproducts are sums): left half in diagram), perfectly isomorphic, linked by a linear mapping K (= functor) on a vector space. Because both products (algebra) and coproducts (coalgebra) pairs commute within themselves they are covariant Hopf bialgebra is self-dual.

Hopf bialgebra commutative diagram

The dynamic choice by the system of itsrepresentation space in QFT On the contrary, in thermal QFT, given that, each quantum system corresponds to

one of the indefinitely many “spontaneous symmetry breakdowns” (SSB) of the QV, splitting locally the QV into a thermodynamic pair system-thermal bath, the proper mathematical formalism for quantum calculations in QFT are based on q –deformed Hopf co-algebras that are contravariant and hence dual as to the corresponding q –deformed Hopf algebras because the q-deformation parameter that is a thermal parameter broke the symmetry of a Hopf bialgebra.

Coproducts of q –deformed Hopf co-algebras indeed do not commute between themselves, since one term represents a system state, the other one a “mirroring” thermal bath state, and then cannot be represented on the same basis.

q indeed is a thermal parameter, linked to the Bogoliubov transform, i.e., linked to the so-called “operator of particle annihilation-creation in the QV” (Blasone et al., 2011, ch. 5).


The Doubling of the Degrees of Freedom (DDF) and the QV-foliation in QFT of dissipative systems In the corresponding Hilbert space is then doubled because the non-commutativity of

coproducts implies that at each state of the system corresponds the mirroring state of the thermal bath i.e., the Hamiltonian character of the system is recovered by inserting systematically the thermal bath in the Hilbert space.

I.e., limiting ourselves to the bosonic case, so that working on the hyperbolic function basis e+q, e-q , we obtain the commuting operators acting on this doubled Hilbert space given by the application of the Bogoliubov transform (Basti, Capolupo & Vitiello, 2016)

They give a concrete realization of the vectorial mapping of the q-deformed Hopfcoalgebra: A ® A x A

Because each of the system represents a QV local degeneracy at the ground state, it is very robust principle of the QV-foliation, each “sheet” be labelled by a q-value with the corresponding foliation of the doubled Hilbert space = robust dynamic mechanism of memory and construction used by nature.



Minimum of free energy as an evaluationfunction (Boolean operator) of the doubled qubit

In fact, on the basis of the QFT duality principle, it is the dynamic system (coalgebra) that chooses how many terms there are, and then maps this choice on the algebra it is the dynamic (not the observer) that chooses the orthonormal basis of the Hilbert space composed by “doubled terms” i.e., the principle of the doubling of the degrees of freedom between algebra and coalgebra.

The contravariance between algebra and coalgebra with the reversal of all the arrowsand the compositions has therefore a (thermo-)dynamic control: the energy balance = minimum of the free energy when the two subsystem are prefectly matchingbetween each other.

On this basis it is possible to design a revolutionary architecture of quantum computer based on QFT where the maximum of entropy (minimization of free-energy) plays the role of a first-order evaluation function for the local semantics, implemented in the dual coalgebra of the corresponding Boolean Algebra (i.e., notion of a semantic q-bit in QFT computing vs. the synctactic qubit of QM computing: Basti, Capolupo & Vitiello, 2016)



The epistemological and anthropologicalrelevance of QFT

This relevance is immediately evident when we realize that in the notion of the QFT dualitybetween the brain states and the environment states is the theoretical and observationalsolution of the issue of the neurophisiological basis of intentionality (Freeman & Vitiello, Basti).

I.e., the matching in real time between a complex brain state (composed of sensory, motorand emotional neurons all correlated among themseves) and an environment complex state.

Solution of the Putnam room paradox of the intentional mind.

To sum up, Freeman and his group used several advanced brain imaging techniques such as multi-electrode EEG, electro-corticograms (ECoG), and magneto-encephalogram (MEG) for studying what neurophysiologist generally consider as the background activity of the brain, often filtering it as “noise” with respect to the synaptic activity of neurons they are exclusively interested in.


Continuing… By studying these data with computational tools of signal analysis to which

physicists, differently from neurophysiologists, are acquainted, they discovered the massive presence of patterns of AM/FM phase-locked oscillations.

They are intermittently present in resting and/or awake subjects, as well as in the same subject actively engaged in cognitive tasks requiring interaction with the environment. In this way, we can describe them as features of the background activity of brains, modulated in amplitude and/or in frequency by the “active engagement” of a brain with its surround.

These “wave packets” extend over coherence domains covering much of the hemisphere in rabbits and cats, and regions of linear size of about 19 cm in human cortex, with near zero phase-dispersion. Synchronized oscillations of large-scale neuron arrays in the β and γ ranges are observed by MEG imaging in the resting state and in the motor-task related states of the human brain.


Dynamic neural fields of intentional behavior

Puebla 2017

(Left). Schematic representation of human cortex (top) and limbic system (down). (Right: left). Evidence of the intentional behavior of olfactory bulb: the same olfactory stimulus induces a modulation in amplitude (top) when the cat is hungry, and no modulation when it is full; (Right: right). Dynamic attractors (closed curves: coherent states) in the overall unstable brain field dynamics related with intentional pattern recognition. Their occurrency is of the order of ≈10-1sec.!

http://www.stoqatpul.org - [email protected] 65

Conclusion: a naturalistic formal ontologyAnd its theological consequences


Naturalistic formal ontology and the “transcendental relativity” principle

Thermal QFT is consistent with a naturalistic formal ontology, based on the principle of transcendental relativity of everything as a post-modern re-proposition of Aquinas ontology

According to this principle every being emerges from a web of relations (Deely, 2001, p. 72).

Historically, this is an Aristotelian ontology in which the natural forms of things emerge from the material substrate of the próte dynamis (“primary dynamism”) of the so-called “first matter”.

Aquinas extended in the Middle Age this ontology to a metaphysics of the creation ex nihilo sui (form) et subiecti (matter)” of everything. That is, a metaphysics in which, differently from Aristotle, the same dynamic material substrate (próte dynamis) of everything, in potency to all the forms and including everything (no mechanical vacuum there exists in Aristotle’s physics), is the main “codomain” of the causality of the “participation of being” from the Primary Cause (see slide 40 for the structure of the correspondent formal ontology)


Creation as Participation of Being That is, the First Cause is the “Pure Act” (as opposed to the “pure potency” of

matter), or the “Subsistent Being”, in which no composition potency-act, or essence-existence holds.

Therefore it is the ultimate common root of any causal tree terminating into the existence of some subsistent thing, composed of form and matter, and then of essence and existence.

This metaphysical Arché of everything is theologically consistent with God of the three Biblical Religions (Judaic, Christian, and Islamic), whose first term of His creative action in Gen.1:2 is the thou wabou. Namely, “the primordial abyss with wavy waters”, from which everything is created during the further six days of the biblical creation tale, by successive separations, hierarchically ordered into a tree of bifurcations.


Reconciliation of Creation and Evolution This has today an evident remainder into the QV-foliation of thermal QFT

cosmology, but meta-physically extended to the same causal foundation “outside time” (the “in principle” of Gen.1:1) of what any fundamental physics has necessarily to suppose: the QV existence itself.

If this naturalistic formal ontology of metaphysics, On the one side, is able to confute any confusion between creatio ex nihilo and SSB’s

of the QV like in (Hawking & Mlodinow, 2010; Krauss, 2012), On the other side, it is able to reconcile definitively “creation and evolution”, even at the

cosmological and not only biological level.

It frees, indeed, the Christian and the Islamic theologies (Judaic tradition was never affected by it) from any dependence on Neo-Platonism – the physical ontologies of the “grand design”, and of the “strong anthropic principle” included. The cosmological finalism is supposing theology – that is, the supposition by faith of a personal nature of the Primary Cause –, and then it cannot found it. In this the criticism of Hawking is right, even though it does not apply to Aquinas’ metaphysics.


The QFT vision of the Universe (and its compatibility with a theology of the First Cause: “Why the QV does exist?” (Gn. 1,1)


The universe of things (circles) emerging causally(arrows) from the QV (dots), all dependingfrom the participation of being by God (big tridimensional arrows).

Each thing (cat) isemerging from a stratified web of mirroring causalarrows, on which the stability of its form-in-matter depends(shape), alldepending ultimatelyby God (big arrows)

The functorial dual equivalence between the ontic (coalgebraic) and logic (algebraic) component of the notion of ontological local truth

At the same time, a coalgebraic interpretation of modality in CT logic (Abramsky, 2005; Venema, 2007; Goranko & Otto, 2007; Kupke, Kurz, & Venema, 2004), is able to justify formally which is the syntactic root of the “finalistic confusion” in ontology and theology that Aquinas emphasizes in several passages of his works (see slide 40).

That is, both local truths in modal ontology, and deontic truths in modal ethics, share the same “arrow reversal” as to the logical conditional, typical of the notion of functorial dual equivalence.

In fact, the deontic principle of “what ought to be, in order that the goal be” (quod debeat esse utbonum sit), and the ontic (causal) foundation of logical truth in ontology, so that “intellect is measured by things”, therefore, neither “it is measured by the divine intellect” (Neo-Platonism), nor “ it is measuring things”, like in the conventionalism of the Sophistic and Nominalist ontologies, can be properly formalized by a semantic principle of functorial dual equivalence in Aquinas metaphysics and ethics, but in two distinct intensional interpretations (Basti, 2015).


The compositional contravariant character of the ontological truth

According to Aquinas De Veritate (q. 1 a. 2), indeed, the ontological theory of local truths for human intellects is based on the dual homomorphism between the “ontic” direction: Primary Cause (Divine Intellect, in theology) → things → intellect, and the “logic” direction: intellect → things → Primary Cause.

And that for Aquinas it is a local truth principle in ontology is evident from the fact that for him the ontological truths are as many as intellects and things are.

Even though all these indefinitely many local truths “participate” in the unique Absolute Truth immanent in the Divine Intellect.

As we see, only by CT logic has become today possible formalizing this Aquinas doctrine as a further proof of the incredible powers of formal philosophy.


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