abuse of logical notation?
Apr. 21st, 2007 05:15 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
guslPatrick Suppes has discussed different usages of the word "model", in fields from mathematical logic, to physics, engineering, psychology and econometrics.
I often have the urge to do just that. Suppes himself made many structuralist formalizations this way. The basic form of these is:
Set-theoretic structure M is a model of theory T iff ___ .
e.g.: M = < P,T,... > is a model of Classical Particle Mechanics iff
* P is a set of particles, T is a set of time points, ...
* each particle only occupies one location at any given time.
* etc
But does this style of formalism capture all that is being said when scientists talk about models? When physicists say "Quantum Mechanics and Relativity haven't been unified", how can that be expressed this way?
Can this Tarskian models approach deal with the little hacks and tricks and less-than-kosher sleights-of-hand that scientists do? My inclination is to say yes, but this is an empirical question.
-------------------------------
Anyway, here's my abuse of logical notation:
I like to express the fact that Gaussians are characterized by the first two moments this way:
Gaussian |= E(X^3) = f ( E(X^2) , E(X) )
i.e. if two Gaussians have the same 1st moment and 2nd moment, then they have the same 3rd moment. (I realize this is weaker than the above statement)
'Gaussian' is the set of all Gaussian distributions (i.e. the set of all Gaussian "models"). A sentence is true in this set of models if it is true in every model.
But I don't even know what logic/language I'm using here. Whatever it is, it needs to have an equality, and to allow me to use the function f there. I also need to be able to get the first 3 moments out of each model.
It gets worse if I express the full statement, because now I need to be able to interpret quantified variables in terms of my models:
forall n ( exists f_n ( Gaussian |= ( E(X^n) = f_n ( E(X^2) , E(X) ) ) )
i.e. for all moments n, there exists a function that determines the nth moment from the first two moments.
But is it useful to think of probability distributions as models, besides the notational shorthand? i.e. can we come up with a syntactic way of answering questions about probability distributions, namely a set of rules with "|-"s that is sound (and hopefully complete) with respect to "|="... and also more efficient than alternative ways of reasoning?
I claim that the concept of model in the sense of Tarski may be used without distortion and as a fundamental concept in all of the disciplines from which the above quotations are drawn. In this sense I would assert that the meaning of the concept of model is the same in mathematics and the empirical sciences. The difference to be found in these disciplines is to be found in their use of the concept.
I often have the urge to do just that. Suppes himself made many structuralist formalizations this way. The basic form of these is:
Set-theoretic structure M is a model of theory T iff ___ .
e.g.: M = < P,T,... > is a model of Classical Particle Mechanics iff
* P is a set of particles, T is a set of time points, ...
* each particle only occupies one location at any given time.
* etc
But does this style of formalism capture all that is being said when scientists talk about models? When physicists say "Quantum Mechanics and Relativity haven't been unified", how can that be expressed this way?
Can this Tarskian models approach deal with the little hacks and tricks and less-than-kosher sleights-of-hand that scientists do? My inclination is to say yes, but this is an empirical question.
-------------------------------
Anyway, here's my abuse of logical notation:
I like to express the fact that Gaussians are characterized by the first two moments this way:
Gaussian |= E(X^3) = f ( E(X^2) , E(X) )
i.e. if two Gaussians have the same 1st moment and 2nd moment, then they have the same 3rd moment. (I realize this is weaker than the above statement)
'Gaussian' is the set of all Gaussian distributions (i.e. the set of all Gaussian "models"). A sentence is true in this set of models if it is true in every model.
But I don't even know what logic/language I'm using here. Whatever it is, it needs to have an equality, and to allow me to use the function f there. I also need to be able to get the first 3 moments out of each model.
It gets worse if I express the full statement, because now I need to be able to interpret quantified variables in terms of my models:
forall n ( exists f_n ( Gaussian |= ( E(X^n) = f_n ( E(X^2) , E(X) ) ) )
i.e. for all moments n, there exists a function that determines the nth moment from the first two moments.
But is it useful to think of probability distributions as models, besides the notational shorthand? i.e. can we come up with a syntactic way of answering questions about probability distributions, namely a set of rules with "|-"s that is sound (and hopefully complete) with respect to "|="... and also more efficient than alternative ways of reasoning?