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Can anybody either verify or dispute the my proof for the following argument?

Premise 1: (E • I) v (M •U)

Premise 2: ~E

Conclusion: ~(E v ~M)

Proof:

(1) Applying DeMorgan's Second Law to the Conclusion; The Negation of a Disjunction, it is the case that ~(E v ~M) is logically equivalent to ~ E • ~~M.

(2) Applying the double negation rule to ~~M, we have M. Hence, the conclusion becomes ~ E • M.

(3) Since, by the first premise, it is the case that either both E and I must be true in conjunction or both M and U must be true in conjunction, it the case that the second premise indicates that E is false. Hence the first conjunction in the first premise is rendered false and the second conjunction in the first premise is rendered true by applying the disjunctive syllogism rule of inference to the first premise. Therefore it is the case that M and U are true.

(4) Since ~ E is given in the second premise and since it has been determined by means of simplification that if M and U is true, then M is true, it is therefore the case that the conclusion, ~ E and M, is true.

Does this seem accurate? Why or why not?

Sage Hopkins
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3 Answers3

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Correct, and you can present your argument concisely and cleanly as follows:

$(E \land I) \lor (M \land U)$.   [Premise]

$\neg E$.   [Premise]

$\neg ( E \land I )$.   [Semantics of $\land$]

$M \land U$.   [Disjunctive syllogism]

$M$.   [Conjunction elimination]

$\neg E \land M$.   [Conjunction introduction]

$\neg E \land \neg \neg M$.   [Double negation]

$\neg ( E \lor \neg M )$.   [De Morgan's]

user21820
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Although the OP's proof appears correct, it may help to use a proof checker. The drawback of using such a tool is that one is forced to use the inference rules available. The benefit is the added confidence one has that one's proof is correct.

Here is a proof using a Fitch-style proof checker.

enter image description here

Links to the proof checker and associated textbook are below.

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf

Frank Hubeny
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You proof is overly verbose, but essentially:

  • The premise $(E\land M)\lor (M\land U)$ entails $E\lor M$ (can be proven by cases)
  • $E\lor M$ and premise $\lnot E$ entails $M$ (via disjunctive syllogism)
  • $M$ and $\lnot E$ entails $\lnot E\land M$
  • $\lnot E\land M$ is equivalent to $\lnot (E\lor\lnot M)$ (via de Morgan's Laws).

That is okay.

Graham Kemp
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