In this formal system, Frege developed an analysis of quantified statements and formalized the notion of a 'proof' in terms that are still accepted today.
Dale Mortensen's 1970 paper on "Job Search, the Duration of Unemployment and the Philips Curve," formalized and extended Professor Stigler's ideas.
Proponents of plural quantification claim that these theories allow plural locutions to be formalized in a way that is fundamentally different from the old set-theoretic paraphrases.
Frege formalized his system more explicitly, but his system was not precisely a set theory in the modern sense: the most that can be said is that his system is inconsistent, for basically the reason given here, and a full account of the differences between Frege's system and our "naive set theory" is beside the point (though historically certainly interesting).
Much has been written about robustness of this type (by philosophers [Wimsatt 1987; Orzack and Sober 1993; Justus 1995]) though it has never been formalized successfully.
Manders' insights lead naturally to the idea that Euclid's arguments could be formalized in a manner similar to the way Venn diagrams have been formalized in Shin 1994.
To tackle these questions, all sorts of objects—structured representations one finds in a grammar, graph-structures one might find in a taxonomic representation of causal or kin relations, schemas applied to scene or event analysis, etc. will need to be formalized and assigned probabilities.
Since there are various meanings associated with 'sortal', the logical development will depend on which conception of sortal is being formalized.
Hobbs et al. 1993 shows how an impressive range of discourse phenomena can be formalized using this idea.
If one is interested solely in how information from diagrams is situated logically in Euclid's proofs according to the Eu approach, diagrams need not formalized directly as symbols.
In other cases the rules of an institution are not explicitly formalized.
