Logic |

**In General** > s.a. axiom of
choice; classical mechanics.

* __And physics__: The
logic of a physical system consists of the elementary observables of the
system.

@ __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]; > 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
[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.

$ __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).

* __Remark__: 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.

@ __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);
issue IJTP(87)#5;
Doebner & Lücke JMP(91);
Pavičić IJTP(92);
issue IJTP(92)#9,
IJTP(02)10;
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]; Rédei SHPMP(01)
[rev, philosophical]; 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.

@ __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]; > s.a. geometric
quantization; Greechie
Logic; histories quantum theory;
lattice; quantum
technology; set theory; sheaf
theory; topology [lattice of
topologies].

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