The present paper intends to report two results. It is shown that the formula P(x)=∀y∀z[¬Gx, y→¬M(z)] provides the logic underlying gauge symmetry, where M denotes the predicate of being massive. For the logic of spontaneous symmetry breaking, by Higgs mechanism, we have P(x)=∀y∀z[Gx, y→M(z)]. Notice that the above two formulas are not logically equivalent. The results are obtained by integrating four components, namely, gauge symmetry and Higgs mechanism in quantum field theory, and Gödel's incompleteness theorem and Tarski's indefinability theorem in mathematical logic. Gödel numbering is the key for arithmetic modeling applied in this paper.