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Predicate logic
Basic logic in effect concerns itself with whole statements (or "propositions") that are each either True or False . … In general statements in predicate logic can contain arbitrary so-called predicates, say p[x] or r[x, y] , that are each either True or False for given x and y .

Multivalued logic
As noted by Jan Łukasiewicz and Emil Post in the early 1920s, it is possible to generalize ordinary logic to allow k values Range[0, 1, 1/(k - 1)] , say with 0 being False , and 1 being True .

But this all changed in 1931 when Gödel's Theorem showed that at least in any finitely-specified axiom system containing standard arithmetic there must inevitably be statements that cannot be proved either true or false using the rules of the axiom system.

reached—corresponding to the fact that statements must exist that cannot be proved either true or false from a given set of axioms.

For any input x one can test whether the machine will ever halt using
u[{Reverse[IntegerDigits[x, 2]], 0}]
u[list_] := v[Split[Flatten[list]]]
v[{a_, b_: {}, c_: {}, d_: {}, e_: {}, f_: {}, g___}] := Which[a == {1} || First[a] 0, True, c {}, False, EvenQ[Length[b]], u[{a, 1 - b, c, d, e, f, g}], EvenQ[Length[c]], u[{a, 1 - b, c, 1, Rest[d], e, f, g, 0}], e {} || Length[d] ≥ Length[b] + Length[a] - 2, True, EvenQ[Length[e]], u[{a, b , c, d, f, g}], True, u[{a, 1 - b, c, 1 - d, e, 1, Rest[f], g, 0}]]
This test takes at most n/3 recursive steps, even though the original machine can take of order n 2 steps to halt.

For if one looks at axiom systems that are widely used in mathematics they almost all tend to be complete enough to prove at least a fair fraction of statements either true or false.

This can be done for blocks up to length n in a 1D cellular automaton with k colors using
ReversibleQ[rule_, k_, n_] := Catch[Do[ If[Length[Union[Table[CAStep[rule, IntegerDigits[i, k, m]], {i, 0, k m - 1}]]] ≠ k m , Throw[False]], {m, n}]; True]
For k = 2 , r = 1 it turns out that it suffices to test only up to n = 4 (128 out of the 256 rules fail at n = 1 , 64 at n = 2 , 44 at n = 3 and 14 at n = 4 ); for k = 2 , r = 2 it suffices to test up to n = 15 , and for k = 3 , r = 1 , up to n = 9 .

Boole identified 1 with True and 0 with False , then noted that theorems in logic could be stated as equations in which Or is roughly Plus and And is Times —and that such equations can be manipulated by algebraic means. … The question of whether any particular statement in basic logic is true or false is always formally decidable, although in general it is NP-complete (see page 768 ).

To show that a particular problem like the halting problem is undecidable one typically argues by contradiction, setting up analogs of self-referential logic paradoxes such as "this statement is false".

[i, j, k] || If[EvenQ[n + i - j], [i, j], False] || [i + 1, j, k], {i, 0, t - 2}, {j, Max[1, n - i], n + i}, {k, 0, ktot - 1}], Table[Map[Function[ z, Outer[!