Mathematica Question about mathematical logic

[Werg22]

Is an assertion that could be true but isn't always considered to be false or just lacking information?

 

[PowerIso]

Do you have an example?

I am thinking you mean something like if x is a member of Z(integer) then x> 5.

The assertion is true for all x greater than 6, but not for x equal to or less then 5. So what do you think, is the statement true or false?

 

[Werg22]

It's not such an assertion, it's in terms of arguments.

 

[PowerIso]

I guess what I'm trying to say is that if in an argument, you present point a. If point a is only true within certain situations then while discussing point a make sure the decision is within those restrictions, then you can say it's a true statement. If point a is moved beyond those guidelines, and point a is being discussed when it is not true, then it is a false statement.

 

[Hurkyl]

It depends on how you want to formulate things. IMHO, the cleanest formulation is that a relation is a truth value if and only if it has no free variables. You wouldn't talk about the truth of the expression x > 5, but instead of the subset of Z that it defines. Or equivalently, you think of x > 5 as a truth-value-valued function on Z.

However, if you're thinking in terms of semantics, then each interpretation of your language includes a variable assignment: it selects an actual integer to represent the symbol x. So, x > 5 would hold in some interpretations, but it would not hold in other interpretations.

If you're thinking about truth valuations, then because x > 5 is independent of the axioms of the integers, some truth valuations will assign "true" to this statement, and other truth valuations will assign "false" to this statement.

Incidentally, the notions of having a truth valuation on your language and having an interpretation of your language are equivalent. (In a certain technical sense)

 

[HallsofIvy]

An "assertion that could be true but isn't always" is NOT an "assertion", it is an "open sentence" that contains some variable. It will be true for some values of that variable but not others. Add a "quantifier" will change it into a true or false proposition.

 

[Trail_Builder]

im not that experienced with math, but since i think one of the founding axioms of logic is a statement can ONLY be true or false, i guess the answer to your topic is "lacking information".

 

[HallsofIvy]

It's not math, it's logic (which is a foundation of mathematics). A "proposition" is a statement that is either a true of false (it is not necessary to know which). An "open sentence" is something like "2x+ 1= 0" or "x²>= 0". The first is true only when x= -1/2 so we add quantifiers like "for SOME x, 2x+ 1= 0" or "for all x, x²>= 0". A non-mathematical example might be something like "That man has a well-paying job" where "that man" is not specified. We could make that a "proposition" by specifying the person we are talking about or by saying something like "There exist men who have well-paying jobs" (a true statement- or so I'm told) or "All men have well-paying jobs" (definitely false).

The OP may be referring to a situation where there is not a "variable" but where we simply do not know whether the statement is true of false. That doesn't matter- as long as it must be one or the other, we don't need to know which to know it is a "proposition".

 

[LukeD]

What about statements that are independent of the theory in which you are working? E.g. Axiom of Choice in Z.F. Set Theory

Obviously, the statement is neither true nor false when considering Set Theory without the axiom, but it is also already quantified (unless you want to add another quantifier stating that their exists a theory in which the axiom of choice is true)