# Thread: Question on duality theorem[Mathematical logic]

1. ## Question on duality theorem[Mathematical logic]

Hi everybody.
in VanDallen's "Logic and structure", he defines a "duality mapping" d from PROP to PROP as follows(note that by !phi i mean negation of phi):

d(phi)=phi for atomic phi.
d(phi ^ psi)=d(phi) or d(psi)
d(phi or psi)=d(phi) ^ d(psi)
d(!phi)= !(d(phi))
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then he proves the following theorem(duality theorem):[note by p == q, i mean p and q are equivalent]
phi == psi iff d(phi) == d(psi) [in fluent English, if two propositions are equivalence, then so are their duals.]

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Now my question:

We know that (p ^ (!p or q)) == q. Then why their duals are not so? i.e., why the following does not hold? : (p or (!p ^ q)) == q.

thanks.

2. ## Re: Question on duality theorem[Mathematical logic]

(p ^ (!p or q)) implies q but is not equivalent to q.

3. ## Re: Question on duality theorem[Mathematical logic]

Hehe you are right... I was sleepy when created the example.
Thanks.