# Consistency for a set of formula (first order logic), with resolution technique.

$\neg P(x,y)\vee P(S,y)$ (1)

$\neg P(x,f(x))\vee P(S,f(x))$ (2)

$\neg P(S,f(x))\vee P(x,f(x))$ (3)

Is, this set of formulas, consistent? I think so, because I could not to obtain a contradiction. What do you think?

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Usually to speak of consistency of a set of sentences of first order logic. The above looks like they are missing some quantifiers. You should also clarify what is the first order language you are working in. –  William Jun 5 '12 at 8:55
sorry, each variable x y is to be understood universally quantified. –  Mark Jun 5 '12 at 9:14
What about $S$? Is it also a variable? –  William Jun 5 '12 at 9:16
S ground, constant. –  Mark Jun 5 '12 at 9:26
And what about $\rceil$? –  Chris Eagle Jun 5 '12 at 9:27
If $P$ is interpreted to always be true, then (no matter how $S$ and $f$ are interpreted), all three sentences are true. Thus the set has a model and hence is consistent.