Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
See: Topos, PVsNP, Nand, Type theory, CurryHowardLambek, Connectives, Set theory Predicate
Bias
Automata
Quantification
Paradoxes
Connectives and perspectives
Logic 逻辑 Wikipedia: Logic  A study of the form of argumentation, as opposed to the content. Thus logic considers general aspects of argumentation. Universal logic Readings
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Videos Darwinian axiomatic systems  complete but not consistent (bad variations go extinct)  Nicolas Lori Notes See: Logic, Workshops, Sessions, Logic conference Planar Hayting Algebras for Children Ioannis Vandoulakis Logic social process  prover and interpreters (can be teams of agents). Each agent is autonomous. Each agent has its own history, record. Agents may follow different logics of reasoning. Understanding is a semiotic morphism across semiotic spaces. Three levels of logic: philosophers, logicians, mathematicians. Mykola Nikitichenko  external/internal  pakalbėjome apie padalinimus, Hegel, duality  2012  Intentionalist theory of data. Moldavian journal Dedre Gentner  Structure Mapping Theory in cognitive psychology Analogical reasoning is creative and useful when logical reasoning does not apply. Category theory approach: Given objects A and B, choose a morphism F from A to B. Then have a functor M that takes A to M(A) and B to M(B). Then consider whether M(F) goes from M(A) to M(B)? That is, consider, whether M is a functor. Geometric proportion (division), arithmetic proportion (subtraction)  both used for derivatives. Also, harmonic proportion. Different ways of defining substitutions: by value, by name, by address? and others Three brain model:
Default mode takes over in old age.
Cutting corpus callosum  split brain. Feel normal. Triune brain:
Amygdala  Danger center, negative emotions. Helps to make meaning before signal enters sensory cortex. Creates ideas. Gets ideas from direct experience. Monitoring the situation. Is reduced upon verbalizing, naming the danger (formulating an expectation  so no fright or disgust). (So amygdala notes the inability to formulate an expectation). Example of embodied cognition. The body has an affect on the mind and interpretation. Movement and its importance. Systems mediated by dopamine (chasing the reward  suspense). Amygdala indicates if an expectation can be made. Expectations are propositions. Thus we see the source of propositions  the rational mind conceptualizes in terms of propositions  and we see where they enter, and how the emotions tell us if they are sufficient or insufficient (fright, disgust), and if they are satisfied or not satisfied (surprised, sad). NeoPythagorean thinking  generating infinite families  with pebbles. No proof is needed  just stepbystep verification  thus we have verification procedures. Non axiomatic. Verification by construction and inspection. Triangles, pyramids, but no cones. Pythagoras and Parmenides have no negative statements. For Parmenides statements are either meaningful (true) or not meaningful (false). Parmenides's semantic viewpoint: Key concepts: that which is (to eon) and thought (noema). True is expressible. Susieti žaidimų gramatiką su jausmų teorija. Įvardiję lūkestį, nepajusime baimės, nė bjauresio. O ką reiškia įvardyti žmogų? Jį galime mąstyti sąmoningai. Programs deal with components of objects, thus they are algebraic, an algebra. Programs change one variable state to another. These are mappings. Thus algebras of functions. Superposition. Naming of a variable allows diassociation of scope. Contextual understanding. The opposite is compositional semantics, which allows not to worry about the context. Integers  counting  ever introducing a new "pebble". Update  based on limitation of resources. Natural numbers  based on being able to introduce ever new item  in some dimension. Both are needed for computation. Six compositions (related to sequence, hierarchy, network?):
Study brain waves. Propositions (expectations) create the distinctions (and duality) between truth and falsehood. Amygdala related to uncertainty, related to entropy. Brain wave synchronization linked to creativity. Ambiguity makes for freedom. Alzheimer's is a decrease in synchronization. The emotional system prompts a reading of it which gives a holistic reassessment of the satsifactoriness of our logical system. And also moods can give us a simulation that tests our logical system, especially when our logical system starts to depend on our emotions and tries to game them. Our self is what obeys our logical system and the world is what is beyond it. Algorithmic algebra. On any domains. Add FloydHoare assertions, can go from programs to predicates, and so get Program specification algebra. Get a threefold mode: Quasiary predicates, quasiary functions, programs. (Semantic) consequence relation vs. inference relation. Logical systems  class of algebras considered as program models; ...?; interpretations; consequence relation; inference relation. Classical logic of nary functions. Instead of nary functions consider quasiary functions, yielding neoclassical logic. Next level abstraction is for hierarchical (nominative) data such as arrays  logic of nominative data. Quasiary logics
Concepts are the products (outcomes) developed by games. Topics/Comments vs. Subjects/Predicates. Concepts are subjects/topics, what you ask questions about, and comments/predicates/propositions are the answers. How are questions related to perspectives? Then you can ask questions about the questions (syntax) or ask questions directly (semantics). What is the difference between subject/predicate and topic/comment distinctions? Perhaps there are two different types of "about". A predicate is not a selfstanding statement, but requires a subject. Whereas a comment is a selfstanding statement without the topic. All concepts are tentative, conventional, contingent, secondary  we're actually interested in something else  so what we should focus on. Kevin Scharp  Replacing truth. Truth is best understood as an inconsistent concept. Planar Hayting Algebras for Children Concepts as exemplars, prototypes and theories. Extensive definition, intensive definition. Cognitive sandwich  criticized. MANY 1  A Transversal Imaginative Journey across the Realm of Mathematics JeanYves Beziau Michel Serfati. La revolution symbolique. Jean Louis Maxime van Heijenoort. From Trotsky to Goedel. From Frege to Goedel. Frege's square of opposition. Turnstile: Content stroke and judgment stroke. Dasarathy sensory fusion. What is the minimum number of sensors to recognize any object? Jerzy Tomasik. Carnap: Semantic (truth) vs. Syntax (proof). Better is: model theory (truth) vs. proof theory (proof). Post's completeness theorem. Model theory makes the sevensome meaningful. It distinguishes between True and NotFalse, NotTrue and False, etc. Wittgenstein Notes 19301932, his most clear writing. Logical square: Exist good, exist bad imply there exists, thus not empty. Thus not a true implication  negation acknowledges the nonemptiness of a system. Lindenbaum Maximalization Theorem. Bourbaki: three fundamental structures: Order (sequence?), Topology (network), Algebra (hierarchy)  based on the structure of the real numbers. And then cross structures. Colimits identify elements  union, disjoint union, equivalence Adjoint problem for naming, etc: sets and free groups If we restructured a sequence with a sequence, then that second structure would behave like a tree or a network. Likewise for the other restructurings. Relate the axioms of set theory with logic. David Corfield: Towards a Philosophy of Real Mathematics. Adjoints? x < Delta(epsilon), then f(x) < epsilon. https://golem.ph.utexas.edu/category/2009/07/thomas_nolls_talks_at_chicago.html topos and music and presheaves Test (question) and state (probability distribution). Probability. Tensor product  f while g (network?). Composition  f after g (sequential) Circuits built from tensors and compositions. Strict normal monoidal category. Tensor product related to pullback? Cupstate cupeffect  a wire can have slack and the past and future can overlap (like in the present) Trace (AB) = Trace (BA). Bell state and adjoint  combined give a trace. Transpose implicit in the wire yank. Circuits  symmetric categories. Diagrams  monoidal categories. Classical system  single wire. Quantum system  double wire. Complex number times its conjugate gives a real. Relate Clifford algebras and the slack in string diagrams. Basel Hiley  slack, collaborator of Bohm. The Undivided Universe. John Baez  quaternionic string diagrams. Two readings  organization? and implementation. B Coecke. Categories for the practising physicist. Observation disentangles what is entangled. We live in a countable disentangled world embedded in an entangled uncountable world. Probability describes the disconnect between the primary and secondary structure. This could be the basis for noncountability. Paradox of Sleeping Beauty Seven paradoxes in probability Whether  empty set (or not); What  range of function; How  domain of function; Why  infinity. LawvereTierney topology Emotion and Meaning in Music  Leonard Meyer. Page 31  expectation activation, inhibition. Huron. Sweet anticipation. Double response. System 1 and 2. Musicians play with our tendency to overreact. Guerino Mazzola, Moreno Andreatta, denotators. PKnets. Alexandre Popoff, Carlos Agon, Moreno Andreatta, Andree Ehresmann (2016a) Adjointness  enforcing rules vs. creativity. Series A (subjective) and Series B (objective) time  example of music using both. Linguistics uses both  see aspect. Tuščioje sistemoje nėra 1 ir 0, tiesos ir netiesos. Perturbation (as in real valued logic) can be modeled by the possibility that a system is not empty. Piecewise linear functions. Semantic truth (statement truth calculated from truth tables) matches syntactic truth (statements generated from algorithms). The Form and Function of Duality in Modern Mathematics Ralf Krömer, David Corfield Emotions: naming allows not worrying. Dominique  logic is a sketch  presentation for category with limits. Verifying (NP) and solving (P) problems  compare with using variables to create problems (by adding information) and to solve problems (by removing information). There is one master function which computes all of the computable functions. So study only that one master function. Relate Ehrhart polynomials to qanalogue of simplex. https://en.wikipedia.org/wiki/Ehrhart_polynomial Arithmetic hierarchy is based on negation. There exists a y = It is Not that For all y It is Not
So consciousness, awareness, perspectives, conditionality are all based on negation and universality. And similarly:
And so we have a dual set of chains in terms of existence and universality but they are off by a half level so we have to think in terms of both chains, which is why we need the sevensome. Math variables create problems  in math, both sides are equal (commutative equality)  we have state equality. Computer variables solve problems  the two sides are not equal (noncommutative)  because we are setting the values. Nondeterministic methods are needed for the P vs. NP proof. My symmetric functions of the eigenvalues of a matrix can be considered such a nondeterministic method. Consider the various P vs. NP problems, what they say about a generic matrix and its eigenvalues. And consider the problem of determining whether two matrices have the same eigenvalues. And recall my combinatorial interpretation of K1K and whether a monoid can be factored. K1K  the inverse matrix  is the complement world. Aštuonerybė sieja algebra ir analizę žinojimo rūmuose, pirm sistemos. Reikėtų bene apversti brėžinį. Mat, algebra remiasi tuščia aibe  vienu lapu sprendimams  o analizė remiasi begaline seka lapų. Ar ta begalinė seka tiek pirmyn, tiek atgal? Ar jinai dvilypė ar ne? Aplinkos aplinka yra sistema. Delineating yra sistemoje ir pirmsistemos yra jo aplinka. Tad sistema ir pirmsistemos turi po 12 narių ir vienas kitą papildo. Turing paper of 1936 on the diagonal argument. Definition of logic duality: known and unknown. That is the basis of evoltion. Define music as purely syntactic. Four levels. Žodis "katė" išverčia pasaulį ir protą.
Pasaulye  prote
Palyginti su požiūrių grandine. Paskutiniame lygmenyje iškyla bendro tikslo (joint intentionality) galimybė, nes yra bendra sąvoka. Gali būti teiginys apie kates. Dviprasmiškumas  ar taip mąsto vienas protas ar visi protai. What is a statement of noncontradiction? Logical duality is very complete, universal, contrived, artificial, expensive, consequential. Shift from internally structured thing (whether/what) to external relationships (why/how). Syntax is conscious hierarchy, semantics is unconscious network, sequence relates them. But they all can model each other. Yet how? Variables are pronouns. Outcome of game: single trial learning. Inside of game: multiple trial learning. And it is a matter of perspective. Abstraction is based on duality of variable and value. Double bookkeeping. Distance: Fourfold relation  how related ar they? Completely (a thing in itself) or separated (as a symbol). Abstraction is separation of symbol from thing and then working with the symbol (in a game). Symbol is object of joint intentionality. Ehrhart polynomial  triangulated polytopes. Ludewigs conductor Memel music school. Conceptual Engineering: A Systematic Unified Framework Manuel Gustavo Notes
GOD, CONTRADICTION AND PARACONSISTENCY Sept 16  4pm CET  LARAWebiner Is the concept of God contradictory? But what a contradiction really is? Is it a basis or limitation of our thinking? Is the principle of noncontradiction a key to the understanding of reality? If it is, how can we use it properly? Can logical systems that relativize this principle, paraconsistent logics, as promoted by Newton da Costa, help us to clear the way and have a better understanding of God? These are some of the questions that will be addressed in this session, which will be based on some recent works by the speakers:  N.A.da Costa and JYBeziau "Is God Paraconsistent?" http://www.jyblogic.org/GOD  Jc Beall, "The Contradictory Christ" https://global.oup.com/academic/product/thecontradictorychrist9780198852360  Paul Weingartner, "Theodicy  From a Logical Point of View" https://www.peterlang.com/view/title/75192
The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein. (A.Mickūnas siūlė) https://www.amazon.com/OriginLogicSymbolicMathematicsContinental/dp/0253356717 Sevensome
Paraconsistency
