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Suppose we have the predicate I(x,y), which means that x has influenced y. And we have C, which is the set of all composers.

If I translate the following sentence:

All composers are influenced by Bach.

to first-order logic I believe I will get:

∀x(C(x) → I(Bach,x))

Say now I need to translate the following sentence and I can re-use all my predicates etc.:

All russian composers are influenced by Bach.

What would be a good way of writing this senctence to first-order logic? Should I define an entire new set R, being the set of all Russian composers? That doesn't seem efficient, but I can't figure out what else to do.

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  • $\begingroup$ Yes; "for all $x$, if $x$ is a Composer and Russian, then Bach Influenced $x$". $\endgroup$ – Mauro ALLEGRANZA Sep 16 '17 at 19:45
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A more analytic, thus "better" way than simply defining a set of Russian composers ad hoc is to break this set down into a set of Russians $R$ and a set of composers $C$, and then picking those individuals that satisfy both predicates; thus:

$\forall x (R(x) \land C(x) \to I(Bach, x)) $

"All individuals who are Russian and composers have been influenced by Bach."

In set-theoretic terms, this is taking the intersection between two sets of individuals.

This may look clumsy at first, but is the most common way to handle adjective + noun combinations in predicate logic.
(Note that this procedure does not work without exceptions - an "alleged murderer" is not meaningfully someone who is $Murderer(x) \land Alleged(x)$. But it's good enough for most cases you'll have to deal with.)

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