It is possible (using formal semantics) to translate natural language sentences into lambda expressions. So, is it possible to translate those lambda expressions into some logic, e.g. into first-order logic or into modal logic?

I am aware of the Curry-Howard correspondence, but I have not found actual translation, it is more conceptual one. I am aware of translation of lambda expressions into Answer Set Programming http://costantini.di.univaq.it/pubbls/aspopc10-Costantini.pdf and I have heard about translations into combinatory logic.

But are there translations into first order or modal logic?

Such translation would be big step into natural language processing.

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    $\begingroup$ What properties do you want the translation to have? Otherwise I can translate every lambda expression into the logical formula "True". $\endgroup$ – D.W. Oct 23 '17 at 14:07

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