In our framework, however, every is the following (using qeq conditions, as in the LinGO ERG): So these composes via op spec to yield every dog: A slight complication is that the determiner is also syntactically selected by the N via the SPR slot (following Pollard and Sag (1994)).
from the standpoint of the compositional semantics, the determiner is the semantic head, and it is only its SPEC hole which is involved: the N must be treated as having an empty SPR hole.