Re: compound propositions

On Mar 17, 10:25 pm, paul c <toledobythe..._at_oohay.ac> wrote: \
> Relations aren't predicates, they satisfy predicates.

I think of a predicate as essentially a boolean valued function. However I think in the formalism of the FOL a predicate is more appropriately associated with a predicate symbol or more generally any well formed formula but otherwise undefined (as also for the UoD over which quantification takes place). I think this is necessary because axiomatic set theory like ZFC uses FOL, so to avoid circular definitions FOL avoids all mention of sets which is why a predicate is not formalised as a boolean valued function (because functions by definition have domains), and the sets over which quantification takes place are not explicit in a wff of FOL.

A predicate only becomes a boolean valued function under an *interpretation*. An interpretation involves specifying sets over which quantification takes place, and values for the predicate symbols as particular boolean valued functions. Note that an interpretation must by definition be formal and precise (i.e. it is not meant to signify some informal association with real world entities).

The FOL is not concerned with particular interpretations. For example, FOL is used to determine whether one formula is a logical consequence of another (meaning that truth of the consequent follows from the antecedent under all interpretations).

So back to your statement, I would say that there is a 1:1 correspondence from relation (= set of tuples) to a boolean valued function that is true for tuples in that relation and false otherwise. This boolean valued function can be said to represent a predicate under an interpretation but I'm not sure if that's what you mean. More specifically, what do you mean by "satisfy" when you say relations satisfy predicates?

Note that this 1:1 correspondence assumes particular attribute domains are specified, so that the domain of the boolean valued function is defined.

BTW it would be nice to have some terminology for talking about the set associated with a given boolean valued function, or the boolean valued function associated with a given set. I guess one could say the "extension" of a given boolean value function. I could imagine referring to the "characteristic function" of a given set, but really the tradition is for this to map into {0,1} not {false,true}.

I wonder what is meant exactly by 'internal predicates' and 'external predicates'. I would appreciate it if someone could provide a definition.

My suspicion is that an external predicate is a contradiction because a predicate is a pure mathematical concept, and cannot be defined in terms of informal things like the entities of our real world perception. Also if 'predicate' just means boolean valued function, then what does it mean to say that the external predicate can change over time? How can a function change and not be a different function? Received on Thu Mar 18 2010 - 04:31:44 CET