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$\neg P(x,y)\vee P(S,y)$ (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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    The symbol you used, $\rceil$ (produced using `\rceil`), is meant to be used as the right-hand element of the ceiling function $\lceil\cdot\rceil$. The symbol intended for use as a negation symbol is $\neg$ (produced using `\neg`).2012-06-05

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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.