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];
> 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 [*L*_{QM}
extended to a classical logic *CL*_{QM}];
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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