If and when if -clauses can restrict quantifiers

The interpretation of if -clauses in the scope of ordinary quantifiers has provoked semanticists into extraordinary measures, such as abandoning compositionality or claiming that if has no meaning. We argue that if -clauses have a normal conditional meaning, even in the scope of ordinary quantifiers, and that the trick is to have the right semantics for conditionals.
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Conditionals and Propositions in Semantics.Daniel Rothschild - 2015 - Journal of Philosophical Logic 44 (6):781-791.
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