Proving De Morgan's Law Without Truth Tables: ~(~p∧~q) = ~p∨~q

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To prove De Morgan's Law, ~(p ∧ q) = ~p ∨ ~q, without truth tables, natural deduction can be employed. Basic rules of propositional logic, such as the rules of inference, are essential for constructing the proof. The discussion highlights confusion regarding specific rules referenced in natural deduction. Participants suggest consulting resources like Wikibooks for guidance on logical statements. The focus remains on finding a method to prove the equivalence of the expressions as required by the lecturer.
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Hi ,my lecturer ask me to prove ~(p^q) = ~pv~q i.e ~(p^q) is equivalent to ~pv~q,without using the true tables.

thanks for your help
 
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What have you tried? You should have some basic instructions for working with statements like this.
 
the usual in statement calculus. i can not get started
 
thanks,but where rules 7 and 13 are coming from what are they called?

also my version of de morgan is not proved there, probably that's why the lecturer ask me to prove that version because he knew the one already in google
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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