Logic| 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].
@ 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]; > s.a. chaos; special
relativity and spacetime models [first-order
logic].
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
, 
,
, 
,
, 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 05 [and algebra, Tarski's model theory
and Artin-Schreier algebra of closed real fields].
Gödel's (Second Incompleteness) Theorem > s.a. information.
* 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]; Goldstein
05 [I]; Barrow phy/06-in
[and physics].
Other Topics, Paradoxes > s.a. 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.
@ Zeno's paradox: Sherry PhSc(88)mar;
McLaughlin SA(94)nov.
Other Logics > s.a. Peirce
Logic.
@ Fuzzy: Kosko & Isaka SA(93)jul; McNeill & Freiberger;
Granik & Caulfield
PE(96)qp/01 [in
quantum mechanics]; Freistetter CMDA(09)-a0905 [and
near-Earth-asteroids].
Quantum Logic > s.a. foundations
of quantum mechanics; probability
in physics; set
theory; sheaf
theory; topology [lattice of topologies].
$ 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-in].
@ 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);
Gudder et al JMP(82);
Adler & Wirth AJP(83)may;
Mittelstaedt IJTP(83);
Kläy FP(87);
issue IJTP(87)#5;
Doebner & Lücke
JMP(91);
Pavicic IJTP(92);
issue IJTP(92)#9, IJTP(02)10;
Szabó qp/96 [against
the idea]; Svozil qp/99-in
[rev]; Coecke et al qp/00-in
[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]; Pavicic & Megill in(08)-a0812 [status].
@ Anhomomorphic logic: Sorkin JPCS(07)qp [proposal];
Ghazi-Tabatabai
& Wallden JPCS(09)-a0907 [and
probabilities]; Gudder a0910,
a0911.
@ 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.
@ 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]; Isham CP(05)
[and truth values]; Domenech & Freytes JMP(05),
et al IJTP(08)-qp/06
[contextual logic]; > s.a. Greechie
Logic; histories quantum theory; lattice; quantum
technology.
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nov
2009