Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them.
"self-interpreting" propositions that mean what they do irrespective of what speakers or thinkers ever do with the signs or judgements that express them.
that explaining how it is possible for propositions to be so tightly related to their truth-makers without being committed to an abstruse ontology of self-interpreting propositions.
One aspect of information that Shannon's definition explicitly does not cover is the actual content of the messages interpreted as propositions.
We thus find the usual candidate truth-bearers linked in a tight circle: interpreted sentences, the propositions they express, the belief speakers might hold towards them, and the acts of assertion they might perform with them are all connected by providing something meaningful.
See King [2009] for discussion of in what sense we interpret the propositional relation of the proposition in this way.
We have seen versions of it which take beliefs, propositions, or interpreted sentences to be the primary bearers of truth.
For Bolzano, the use of genuine sentential forms is just a convenient linguistic procedure yielding results which can be interpreted in terms of propositions.
In other words, AC interpreted à la propositions-as-types is tautologous[6], yet construed set-theoretically it is anything but, since so construed its affirmation yields classical logic.
The idea is to add, to an ordinary classical propositional language, a new propositional operator stit, interpreting stitiφ, where i stands for an agent and φ for a proposition, as i sees to it that φ.
