The use of conventional logical connectives either in logic, in mathematics, or in both cannot determine the meanings of those connectives. This is because every model of full conventional set theory can be extended conservatively to a model of intuitionistic set plus class theory, a model in which the meanings of the connectives are decidedly intuitionistic and nonconventional. The reasoning for this conclusion is acceptable to both intuitionistic and classical mathematicians. En route, I take a detour to prove that, given strictly intuitionistic principles, classical negation cannot exist.
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Bell J.l.  Boolean-valued Models and Independence Proofs in Set Theory. Oxford Logic Guides. Oxford UK: Clarendon Press. xviii+126.
Grayson R.J.  Heyting-valued models for intuitionistic set theory in M.P. Fourman et al. (eds.) Applications of Sheaves. Proceedings Durham 1977. Lecture Notes in Mathematics. Volume 753. New York NY: Springer- Verlag. pp. 402-414.
McCarty C.  What is a logical truth? Proceedings of the XIV Congreso “Dr. AntonioMonteiro.” Institute ofMathematics. Universidad Nacional del Sur AR: Bahía Blanca.
Putnam H.  The meaning of ‘meaning.’ in K. Gunderson (ed.) Minnesota Studies in the Philosophy of Science. Volume VII. Language Mind and Knowledge.MinneapolisMN: University ofMinnesota Press. pp. 131-193.
Troelstra A.S.  Intuitionistic extensions of the reals. Nieuw Archief Voor Wiskunde. Volume XXVIII. pp. 63-113.
Troelstra A.S. & D. van Dalen  Constructivism in Mathematics. An Introduction. Volume I. Amsterdam NL: The North-Holland. xx+342+XIV.