Question

My understanding of QPL so far is that we can quantify over propositions so as to say things like, There is a proposition such that... or at least, There is anxsuch thatxis a proposition and... My initial use of this kind of sentence was in the form of, There is anaxiomsuch that...Question:would it be more proper to say, There is a proposition such that it is axiomatic, here, or can we bring inmany-sorted logicand divide axioms and theorems into at least two of the sorts of an MSQPL? It is commonplace in natural language to quantify over propositions. E.g. The witnesses contradict each other, so at least one of their statements is false, everything Smith said in his speech is nonsense, whatever the Pope says is true. A proposition can be thought of as a zero place predicate. So, if you are quantifying over propositions you are using second order logic, or at least a fragment of it. If you restrict the quantification to zero place predicates only, and you have no additional functions or operators on propositions, including identity between propositions, then you can map this into first order logic. But that is not very useful: it does not allow you to say anythingaboutpropositions, which is presumably why you wanted to quantify over them in the first place. If you want to be able to say that it is a property of some proposition that it is an axiom, i.e. (∃p)Axiom(p) then you will need some additional axioms or rules within the logic to express how to quantify into a propositional position. There are some systems that have been devised for this purpose, e.g. that of Arthur Prior. It doesn't matter much whether you prefer to treat 'axiom' as a type or as a property. Bear in mind that quantifying over propositions can lead to semantic paradoxes, especially if you include a truth predicate. Everything I say is false leads to a version of the liar paradox. a. It's possible to quantify over propositions using second-order logic or a fragment of it.

b. Quantifying over propositions without additional functions or operators can be mapped into first-order logic but might limit expressiveness.
c. To express properties like being an axiom for propositions, additional axioms or rules within the logic might be required.
d. Quantifying over propositions can potentially lead to semantic paradoxes, especially when involving truth predicates.

Answer 1

It is possible to quantify over propositions using second-order logic or a fragment of it, but additional axioms or rules might be needed to express properties like being an axiom for propositions.

Step-by-step explanation:

The question is asking about the proper way to quantify over propositions in logic and whether it would be more appropriate to use many-sorted logic to divide axioms and theorems. It is possible to quantify over propositions using second-order logic, and if you restrict the quantification to zero place predicates only, it can be mapped into first-order logic. However, if you want to express properties like being an axiom for propositions, additional axioms or rules within the logic might be required. It is important to note that quantifying over propositions can potentially lead to semantic paradoxes.