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The linear-algebraic lambda-calculus (arXiv:quant-ph/0612199) extends the lambda-calculus with the possibility of making arbitrary linear combinations of lambda-calculus terms a.t+b.u. In this paper we provide a System F -like type system for the linear-algebraic lambda-calculus, which keeps track of \'the amount of a type\' that is present in a term. We show that this scalar type system enjoys both the subject-reduction property and the strong-normalisation property, which constitute our main technical results. The latter yields a significant simplification of the linear-algebraic lambda-calculus itself, by removing the need for some restrictions in its reduction rules - and thus leaving it more intuitive. More importantly we show that our type system can readily be modified into a probabilistic type system, which guarantees that terms define correct probabilistic functions. Thus we are able to specialize the linear-algebraic lambda-calculus into a higher-order, probabilistic lambda-calculus. Finally we discuss the more long-term aims of this reseach in terms of establishing connections with linear logic, and building up towards a quantum physical logic through the Curry-Howard isomorphism. Thus we begin to investigate the logic induced by the scalar type system, and prove a no-cloning theorem expressed solely in terms of the possible proof methods in this logic.