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William Elliot wrote:
> Since your draft isn't posted here, but there for your convenience, my
> comments can not be put under your statements where they will be best
> understood. Some tables weren't clear and you didn't define theorem.
>
> For better scrutiny, post here where I can make comments instead of there
> where comments aren't possible.
Theorems
These are expressions that are tautologies or truth-functionally true,
meaning that they are evaluated as T for every assignment of truth
values to the variables. Many systems employ the rule of necessitation,
in which if P is a theorem of the logic, []P can be introduced into a
proof. This can also be used here, because according to the truth
tables, []P is true for every tautology.
This is only a preliminary list: Other theorems will be added in future
versions. Also, a bare list without interpretation or commentary isn't
very interesting. More will be added later.
General theorems
These apply to propositions in general.
1} P == ~ ~P
Double negation
2} P == P v P
Idempotent law for v
3} P == P & P
Idempotent law for &
4} (P v Q) == (Q v P)
Commutative law for v
5} (P & Q) == (Q & P)
Commutative law for &
6} ((P v Q) v R) == (P v (Q v R))
Associative law for v
7} ((P & Q) & R) == (P & (Q & R))
Associative law for &
8} ((P v (Q & R) == ((P & Q) v (P & R))
Distributive law for v
9} ((P & (Q v R)) == ((P v Q) & (P v R))
Distributive law for &
10} ~(P v Q) == ~P & ~Q
de Morgan's law for v
11} ~(P & Q) == ~P v ~Q
de Morgan's law for &
12} P => P
13} (P => Q) == (~Q => ~P)
Contrapositive rule
14} (P & (P => Q)) => Q
Modus ponens
15} ((P => Q) & (Q => R)) => (Q => R)
Transitive rule of the strict conditional
16} ((P => Q) & (Q => P)) == (P == Q)
Strict biconditional equivalence
17} (P == P)
Reflexive law of equivalence
18} (P == Q) == (~P == ~Q)
Negation law of equivalence
19} (P == Q) => (Q == P)
Symmetric law of equivalence
20} ((P == Q) & (Q == R)) => (P == R)
Transitive law of equivalence
Modal theorems
21} []P => P
22} P => <>P
23} ~[]P == <>~P
24} ~<>P == []~P
25} [][]P == []P
26} <>[]P == []P
27} []<>P == <>P
28} <><>P == <>P
29} [](P v Q) == ([]P v []Q)
30} <>(P v Q) == (<>P v <>Q)
31} [](P & Q) == ([]P & []Q)
32} <>(P & Q) == (<>P & <>Q)
Other theorems
33} !P == !~P
34} ?P == ?~P
35} ~!P == ?P
36} ~?P == !P
37} ![]P
38} !<>P
39} !?P
40} !!P
41} []P v ~[]P
42}<>P v ~<>P
43} !P v ?P
44} <>(P v ~P)
45} ~[](P & ~P)
46} (P v ~P) == ~(P & ~P)
47} !P == []P v []~P
48} ?P == <>P & <>~P
49} ?P == ( P == ~P)
50} ?P == ?(P v ~P)
51} ?P == ?(P & ~P)
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