Logic |
In General
> 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];
Johansson et al a2102 [phase space 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 paradoxes: in news pt(20)jul [St Petersburg paradox].
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. contextuality; 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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