## Fundamental Tautologies

Posted by allzermalmer on September 29, 2013

First I shall list all the truth tables for basic logical operators. They shall each be given their own symbol as an operator. I will give both two different symbols for them, one for symbolic notation and one in polish notation.

Φ and Ψ will be used as meta-variables, which may be replaced by propositions at any time.

Meta-Variable for proposition Φ:

Given that Φ=True then Φ=True.

Given that Φ=False then Φ=False.

Symbolic (~) and Polish (N): Not..

Given that Φ=True then NΦ=False or (~Φ=False).

Given that Φ=False then NΦ=True or (~Φ=True).

Symbolic(&) and Polish (K): Both…and…

Given that Φ=True and Ψ=True, then KΦΨ=True or (Φ&Ψ)=True.

Given that Φ=True and Ψ=False, then KΦΨ=False or (Φ&Ψ)=False.

Given that Φ=False and Ψ=True, then KΦΨ=False or (Φ&Ψ)=False.

Given that Φ=False and Ψ=False, then KΦΨ=False or (Φ&Ψ)=False.

Symbolic (↓) and Polish (X): Neither…nor…

Given that Φ=True and Ψ=True, then XΦΨ=False or (Φ↓Ψ)=False.

Given that Φ=True and Ψ=False, then XΦΨ=False or (Φ↓Ψ)=False.

Given that Φ=False and Ψ=True, then XΦΨ=False or (Φ↓Ψ)=False.

Given that Φ=False and Ψ=False, then XΦΨ=True or (Φ↓Ψ)=True.

Symbolic (<->) and Polish (E): …if and only if…

Given that Φ=True and Ψ=True, then EΦΨ=True or (Φ<->Ψ)=True.

Given that Φ=True and Ψ=False, then EΦΨ=False or (Φ<->Ψ)=False.

Given that Φ=False and Ψ=False, then EΦΨ=False or (Φ<->Ψ)=False.

Given that Φ=False and Ψ=False, then EΦΨ=True or (Φ<->Ψ)=True.

Symbolic (v) and Polish (A): Either…or…both

Given that Φ=True and Ψ=True, then AΦΨ=True or (ΦvΨ)=True.

Given that Φ=True and Ψ=False, then AΦΨ=True or (ΦvΨ)=True.

Given that Φ=False and Ψ=True, then AΦΨ=True or (ΦvΨ)=True.

Given that Φ=False and Ψ=False, then AΦΨ=False or (ΦvΨ)=False.

Symbolic (↑) and Polish (D): Not both…and…

Given that Φ=True and Ψ=True, then DΦΨor (Φ↑Ψ)=False.

Given that Φ=True and Ψ=False, then DΦΨ or (Φ↑Ψ)=True.

Given that Φ=False and Ψ=True, then DΦΨ or (Φ↑Ψ)=True.

Given that Φ=False and Ψ=False, then DΦΨ or (Φ↑Ψ)=True.

Symbolic (->) and Polish (C): If…then…

Given that Φ=True and Ψ=True, then CΦΨ or (Φ->Ψ)=True.

Given that Φ=True and Ψ=False, then CΦΨ or (Φ->Ψ)=False.

Given that Φ=False and Ψ=True, then CΦΨ or (Φ->Ψ)=True.

Given that Φ=False and Ψ=False, then CΦΨor (Φ->Ψ)=True.

Tautologies:

Symbolic (&) and Polish (K): Both…and…

~(Φ&~Φ)=NKΦNΦ

~(~Φ&Φ)=NKNΦΦ

Symbolic (↓) and Polish (X):Neither…nor…

~(~Φ↓Φ)=NXNΦΦ

~(Φ↓~Φ)=NXΦNΦ

Symbolic (<->) and Polish (E):…if and only if…

(Φ<->Φ)=EΦΦ

(~Φ<->~Φ)=ENΦNΦ

Symbolic (v) and Polish (A):Either…or…both

(Φv~Φ)=AΦNΦ

(~ΦvΦ)=ANΦΦ

Symbolic (↑) and Polish (D):Not both…and…

(~Φ↑Φ)=DNΦΦ

(Φ↑~Φ)=DΦNΦ

Symbolic (->) and Polish (C): If…then…

(Φ->Φ)=CΦΦ

(~Φ->~Φ)=CNΦNΦ

Equivalence:

The order of these equivalence follow those above: (&), (↓), (<->), (v), (->), (↑)

(K) (Φ&Ψ): (Φ&Ψ), (~Φ&~Ψ), ~(Φ&~Ψ)&~(~Φ&Ψ), ~(~Φ&~Ψ), ~(Φ&~Ψ), ~(Φ&Ψ)

(X) (Φ↓Ψ): (~Φ↓~Ψ), (Φ↓Ψ), ~((~Φ↓~Ψ)↓(Φ↓Ψ)), ~(Φ↓Ψ), ~(~Φ↓Ψ), ~(~Φ↓~Ψ)

(A) (ΦvΨ): ~(~Φv~Ψ), ~(ΦvΨ), ~(Φv~Ψ)v~(ΦvΨ), (ΦvΨ), (~ΦvΨ), (~Φv~Ψ)

(D) (Φ↑Ψ): ~(Φ↑Ψ), ~(~Φ↑~Ψ), ~(Φ↑Ψ)↑(Ψ↑~Φ), (~Φ↑~Ψ), (Φ↑~Ψ), (Φ↑Ψ)

(C) (Φ->Ψ): ~(Φ->~Ψ), ~(~Φ->Ψ), ~((Φ->Ψ)->~(Ψ->Φ)), (~Φ->Ψ), (Φ->Ψ), (Φ->~Ψ)

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