> s.a. axiom of choice; classical mechanics.
* And physics: In the classical Hamiltonian framework, the propositions about (observable of) a classical physical system are described in the Borel σ-algebra of a symplectic manifold (the phase space) and the logical connectives are the standard set operations.
@ General references: Van Heijenoort 67 [source book]; Quine 86 [philosophy of logic]; Honig 95 [non-standard]; Walicki 11, 16.
@ Application to physics: Antonsen IJTP(94) [and quantum gravity]; Kak phy/05 [history, Aristotle and Gautama]; Garola IJTP(08)qp/06 [meaning of propositions]; Marchetti & Rubele IJTP(07) [and non-commutative geometry]; Cooper a1109-talk; Clements et al a1201 [physical logic: classical rules of inference about physical events]; Takagi et al a2002 [dynamic logic of quantum field theory]; Pastorello a2004 [quantum propositions and quantized fuzzy logic]; > s.a. chaos; special relativity and spacetime models [first-order logic]; Truth.
In Mathematics, Symbolic Logic
> s.a. mathematical physics; proof theory.
* Idea: It examines the foundations of all mathematical structures.
* Propositional logic: Deals with elementary propositions, that are not further analyzed, but can be combined by means of the basic connectives ∧, \(\lor\), ¬, ⊃, ≡, to form compound propositions.
* Modus ponens: The rule of inference "If A implies B, and A is true, then B is also true" (e.g., use of lemmas).
* Modus tollens: The rule of inference "If A implies B, and B is false, then A is also false."
* Limits: Important developments that showed practical limitations of deductive logic in mathematics were Gödel's incompleteness theorem (1931), Turing, Chaitin.
@ General references: Church 56; Quine 63; Malitz 79 [II]; Gindikin 85; Carbone & Semmes BAMS(97) [without modus ponens].
@ Special emphasis: Sinaceur 06 [and algebra, Tarski's model theory and Artin-Schreier algebra of closed real fields].
Gödel's (Second Incompleteness) Theorem
> s.a. information; irreversibility.
* Idea: If a system S of formalized math is strong enough for us to do number theory with it, then some true facts cannot be proved within S, e.g., a statement implying the consistency of S.
* And physics: Not directly relevant (Landauer), since we don't perform arbitrarily long calculations/derivations.
@ References: Nagel & Newman ed-58; Smullyan 92 [II]; Boolos Mind(94) [explained in words of one syllable]; Loinger NCB(93) [comment]; Franzén 05; Goldstein 05 [I]; Barrow phy/06-conf, Myers & Madjid a1803 [and physics].
Other Topics, Puzzles and Paradoxes > s.a. differential
geometry [Synthetic Differential Geometry]; Unexpected Hanging.
* Zeno's paradox: One of a series of paradoxes developed to argue against the possibility of all motion (in support of the theories of Zeno's teacher, Parmenides); Solved using convergent infinite series and, completely, using non-standard analysis; The main point is not really infinitely many distances giving a finite sum, but the possibility to complete infinitely many acts.
@ General references: Winkler 03, 07; Smullyan 08 [puzzles and paradoxes].
@ Zeno's paradox: Sherry PhSc(88)mar; McLaughlin SA(94)nov; Ishikawa a1205.
Other Logics > s.a. category theory [categorical logic];
modified quantum theory [non-reflexive logic]; Peirce Logic.
@ Fuzzy logic: Kosko & Isaka SA(93)jul; McNeill & Freiberger 93.
@ Fuzzy logic and physics: Granik & Caulfield PE(96)qp/01 [in quantum mechanics]; Freistetter CMDA(09)-a0905 [and near-Earth-asteroids]; Dubois & Toffano a1607 [and many-valued logic].
@ Many-valued logic: Pykacz IJTP(15)-a1408 [in quantum mechanics].
Quantum Logic > s.a. foundations of quantum mechanics;
probability in quantum physics; quantum computing.
$ Def: The complete orthomodular lattice of closed subspaces of a Hilbert space.
* Example: The law of the excluded middle does not hold (a particle's spin can be both up and down, and a proposition and its negation can both be false).
* Remark: It seems to be based on Heyting rather than Boolean algebra [@ Markopoulou NPPS(00)ht/99].
@ I: in Gibbins 88; Hughes SA(81).
@ Books: Hooker ed-73; Enz & Mehra ed-74; Beltrametti & Cassinelli 81; Rota ed-81; Wallace Garden 84; Cohen 89; Svozil 98; Dalla Chiara et al 04.
@ Conceptual: Rédei SHPMP(01) [rev, philosophical]; Maudlin a1802 [Putnam's views and the difficulty trying to solve physical problems with logic]; Nurgalieva & Del Río a1804 [inadequacy of modal logic]; Del Santo SHPMP-a1910 [Popper's campaign against quantum logic].
@ General references: Birkhoff & von Neumann AM(36); Finkelstein TNYAS(63), IJTP(87); Lahti IJTP(80); Gudder et al JMP(82); Adler & Wirth AJP(83)may; Mittelstaedt IJTP(83); Kläy FP(87); Cohen & Svetlichny IJTP(87); Doebner & Lücke JMP(91); Pavičić IJTP(92); issue IJTP(92)#9; Szabó qp/96 [against the idea]; Svozil qp/99-conf [rev]; Calude et al FP(99)-a1402 [embedding quantum logics into classical logics]; Coecke et al in(00)qp [operational, overview]; Dalla Chiara & Giuntini qp/01; Coecke et al m.LO/01-in [and dynamics]; Garola qp/05 [pragmatic interpretation]; Lehmann JLC(08)qp/07 [based on "and then" connective]; Pavičić & Megill in(08)-a0812 [status]; Vol IJTP(13)-a1205; Aerts & Sozzo ACN(14)-a1401 [modeling quantum conceptual combinations]; Griffiths FP(14) [and conceptual difficulties of quantum mechanics]; Kramer a1406 [classical logic as the completion of the quantum logic]; Duncan & Panagaden EPTCS(14)-a1407-proc [Quantum Physics and Logic workshop]; Fritz a1607 [undecidability].
@ Anhomomorphic logic: Sorkin JPCS(07)qp [proposal]; Ghazi-Tabatabai & Wallden JPCS(09)-a0907 [and probabilities]; Gudder JPA(10)-a0910, JMP(10)-a0911; Niestegge AMP(12)-a0912; Gudder JPA(10)-a1002 [reality filters]; Sorkin a1003 [and "tetralemma"].
@ And spacetime, quantum gravity: Finkelstein & Hallidy IJTP(91) [and quantum topology]; Mugur-Schächter FP(92); > s.a. causality; measure theory [quantum measure]; non-commutative geometry; spacetime models.
@ Quantum representation of numbers, words: Benioff Alg(02)qp/01, PRA(01)qp, JPA(02)qp/01.
@ Truth values: Isham CP(05); Karakostas Ax(14)-a1504; Bolotin a1810 [emergence of bivalence in the classical limit].
@ Related topics: Aerts et al IJTP(93) [for macroscopic entities], FP(00)qp ['or']; Zapatrin HPA(94) [without negation]; Coecke SL(02)m.LO/00 [intuitionistic]; Mittelstaedt IJTP(04) [and decoherence]; Battilotti & Zizzi qp/04-in [2 qubits, separable vs entangled]; Domenech & Freytes JMP(05), et al IJTP(08)-qp/06 [contextual logic]; Döring a1004-conf [topos quantum logic]; Hermens EPTCS(14)-a1408 [LQM extended to a classical logic CLQM]; Ellerman a1604 [quantum logic of direct-sum decompositions]; Bolotin a1802 [and contextuality]; Bolotin a1807 [principle of the excluded middle]; > s.a. geometric quantization; Greechie Logic; histories quantum theory; lattice; quantum technology; set theory; sheaf theory; topology [lattice of topologies].
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