Logic and Mathematical Objects

[Noxide]

Can a statement or proposition be considered a mathematical object?

 

[Hurkyl]

That's sort of the point of studying formal logic. Also, of using second-order logic.

 

[yossell]

That's sort of the point of studying formal logic. Also, of using second-order logic.

Oh! How is this idea particularly second order logic?

 

[ZQrn]

Yes, because there are functions which take statements, entire logical formulae as their arguments.

In fact, statements have a type, which is truth value, or 'boolean' depending on context.

I can for instance define a function f : f(P) = ¬P, this function consumes a statement and produces its negation. So yes, they are objects.

Depending on context, there is no difference between a statement and a true or false value, depending on other contexts this is more nuanced, as in reduction based logic. Where the counterpart of '=' is not symmetric, so no = at all really.

I said depending on context the entire time because this is important, once you come down to this fundamental level such contexts are no longer explicit and depending on the formalism you work in, so the correct answer is:

There exists at least one mathematical formalism wherein statements can be operated upon by a higher typed operant, if you want this as your definition of 'object', then statements can be considered objects in such formalisms.

However, in a lot of formalisms, for example, first order logic, any true statement or formula can be considered the same as TRUE, and any false statement as FALSE. Where TRUE := exist x : x = x and FALSE := exist x : ¬(x = x), they are identical because they can be swapped around, so they are definitely objects if they can be identical to other objects.

I hope this argument was compelling enough to you.

 

[Hurkyl]

Oh! How is this idea particularly second order logic?

In first order logic, all of your variable and constant symbols range over "objects", whatever they may be.

In second order logic, you get a new collection of variable symbols that range over first-order propositions, functions, and the like.

In third order logic, you get a new collection of variable symbols that range over second-order propositions, functions, and the like.

And so forth.



In the end, though, set theory can be viewed as encompassing all of that; it's often more practical to simply deal with everything set-theoretically instead.

 

[ZQrn]

In first order logic, all of your variable and constant symbols range over "objects", whatever they may be.

In second order logic, you get a new collection of variable symbols that range over first-order propositions, functions, and the like.

In third order logic, you get a new collection of variable symbols that range over second-order propositions, functions, and the like.

And so forth.



In the end, though, set theory can be viewed as encompassing all of that; it's often more practical to simply deal with everything set-theoretically instead.

But a second order formula / statement is a different thing than a first order statement and you displace the problem and regress.

The what you're making is that in first order logic, predicates cannot take statements as arguments, and that's true. But implication, equality, conjunction, disconjunction et cetera are still functions in first order logic that in fact precisely take other first order formulae / statements as arguments.

-> is a function in first order logic that takes two formulae as argument to produce a new one. This is obscured by the simplicity of how it works, but in P(x) -> Q(y), -> took P(x) and Q(y) as argument and its result was ... P(x) -> Q(y)

This can be made clearer by writing: implication : implication(x,y) = x -> y, the same way we may write addition : addition(x,y) = x + y, and as we all know + is simply a function, that it's done in infix notation traditionally changes little here.

Statement are definitely objects, they are furthermore identical to the the objects TRUE or FALSE because they can always be exchanged by either.

 

[Werg22]

My understand was that propositions are simply strings of symbols, they're easily formalized into mathematical objects e.g. each symbol is identified with a natural number (symbols aren't strictly speaking mathematical objects) and then a proposition is a tuple of these numbers obeying some rules qualifying it as a WFF.

 

[ZQrn]

Formalism is basically defining objects in terms of strings of symbols, and most importantly the way we may re-arrange them.

The point where this debate hangs on is of course 'what is a mathematical object', but in most logics an object would be identical to a term, or, a thing which can stand freely. Which means that in SOL predicates are objects, but in FOL they are not.

 

[yossell]

In first order logic, all of your variable and constant symbols range over "objects", whatever they may be.

In second order logic, you get a new collection of variable symbols that range over first-order propositions, functions, and the like.

Perhaps it's just terminology, but I understood second order logic as an extension which allows you to have variables and quantify into predicate position, such quantification typically being understood as a quantification into *all* n-tuples of the first order domain. I wouldn't say there's a particularly intimate attachment to language that quantifies over linguistic or propositional items in this idea.

http://plato.stanford.edu/entries/logic-higher-order/
http://en.wikipedia.org/wiki/Second-order_logic

 

[Hurkyl]

Statement ... are furthermore identical to the the objects TRUE or FALSE because they can always be exchanged by either.

They're certainly not identical, since the sequence of symbols contained therein (or something similar) is part of the identity of a logical statement.

While a statement can be in the same equivalence class (under implication) as TRUE or FALSE, the guarantee that all are only comes in a very restricted context -- e.g. relative to a truth valuation or a complete formal theory or somesuch.

 

[Hurkyl]

Perhaps it's just terminology, but I understood second order logic as an extension which allows you to have variables and quantify into predicate position, such quantification typically being understood as a quantification into *all* n-tuples of the first order domain. I wouldn't say there's a particularly intimate attachment to language that quantifies over linguistic or propositional items in this idea.

http://plato.stanford.edu/entries/logic-higher-order/
http://en.wikipedia.org/wiki/Second-order_logic

I can't follow what you're thinking. But I shall try explaining again.

In the wiki link, you have the statement

[tex]\forall P \forall x (x \in P \vee x \notin P)[/tex]​

The presence of "[itex]\forall P[/itex]" means that this is unambiguously a second-order statement (or higher).

The status of the string

[tex]\forall x (x \in P \vee x \notin P)[/tex]​

is a little ambiguous. It could be viewed either as a unary predicate in the variable P -- necessarily a second-order predicate, because P is a propositional variable -- one that happens to be identically true. Or, it could be viewed as a rule of deduction or a theorem schema in first-order logic. What it is definitely not, however, is a first-order predicate.

 

[ZQrn]

They're certainly not identical, since the sequence of symbols contained therein (or something similar) is part of the identity of a logical statement.

While a statement can be in the same equivalence class (under implication) as TRUE or FALSE, the guarantee that all are only comes in a very restricted context -- e.g. relative to a truth valuation or a complete formal theory or somesuch.

Nope, a statement that is true is the very same object as true because it an be exchanged for the object 'true' in all contexts without anything happening. This is why the formula 3 + 6 and the formula 13 are different formulae, but the same object. This is the idea of an object contrasting a formula.

A sequence of symbols is a formula yes, but we were talking about objects. Come to think of it, one way to define the set of objects is the type hierarchy of '=' in a formal language. A formula P is an object if the formula P = P is legal. Consequently deriving P = Q is a proof in the context of that formal system that P and Q are the same object.

This is why in first order logic predicates are not objects but in second order logic they are. In first order logic things like A = P if P is a predicate is not a legal term. But A = P(x) however is. And P(x) is a statement in first order language.

That the identity relationship is in the formula makes no difference, after all, 3 is an object, which can also be expressed by the formula 3 + 6 / 3, which is also a formula which contains 3.

 

[yossell]

I can't follow what you're thinking. But I shall try explaining again.

I disagree with none of your last post.

My thinking was that there's nothing intimately tied in the notion of second order logic to propositions or statements - and these were the terms used by the original poster.

Yes, that sentence is unambiguously a second order statement. The quantifier into predicate position is most naturally linked to a notion of quantification over *properties*, and more technically is treated as quantification over *all* (not just first order definable) subsets of the first order domain. Again, these explanations do not involve the notion of a proposition or statement.

 

[Hurkyl]

Nope, a statement that is true is the very same object as true because it an be exchanged for the object 'true' in all contexts without anything happening.

Only in contexts that care only about the equivalence class of the statement.


Examples of interesting and important contexts where such substitution would be flat-out wrong is the predicate that defines whether a list of statements constitutes a proof, or the predicate that tests whether or not the statement is a universally quantified equation, or if a predicate is geometric.

Not to mention the functions used for extracting parts of or otherwise parsing a formula!