Do constant propositional functions exist?

[browser32 (Paul E. Mokrzecki)]

For a proposition p, it seems we can make the constant propositional function F(x) = p. But it seems we run into problems. For instance, what does it mean to say that F(x) is true for all values of x, given any domain of discourse for x? What does it mean to say that F(x) is true for some value of x? The truth value of p does not depend on the value of x. This problem is especially troublesome when p is a proposition that is sometimes true and sometimes false.

What is the solution to this problem? Is there any resource out there that talks about this issue or about constant propositional functions in general?

[Dan1 (Dan~)]

Math and truth aren’t the same things.

[Drusuz]

Rain Men who are good with numbers usually has no rationally to calculate abstarct variables, to understand sublety, just like computers who has great difficulty to preform a normal conversation with a human.

Computers can outpreform humans when it comes to liniar logic, but not abstract logic with many abstract variables, therefore we can’t calculate truth with math.

[browser32 (Paul E. Mokrzecki)]

I would appreciate a reply that directly addresses the problems I mention in my original post.

[Only_Humean]

@browser32

For a proposition p, it seems we can make the constant propositional function F(x) = p. But it seems we run into problems. For instance, what does it mean to say that F(x) is true for all values of x, given any domain of discourse for x? What does it mean to say that F(x) is true for some value of x? The truth value of p does not depend on the value of x. This problem is especially troublesome when p is a proposition that is sometimes true and sometimes false.

What is the solution to this problem? Is there any resource out there that talks about this issue or about constant propositional functions in general?

I’m not sure I follow your problem. I’m not familiar with the term “constant propositional function”. It’s late, maybe I’m not reading well.

To clarify: Take an example proposition p as “All bachelors are married”. Where F() corresponds to “is married”, then for all bachelors x, F(x) is true . “For all” shows a tautology… so I don’t follow what you mean by “the truth value of p does not depend on the value of x”. Is the complaint that “for some x:F(x)” in some way breaks the bond between p and x?

[browser32 (Paul E. Mokrzecki)]

@Only_Humean

I don’t follow what you mean by “the truth value of p does not depend on the value of x”. Is the complaint that “for some x:F(x)” in some way breaks the bond between p and x?

Let proposition p = “I am hungry.” It seems we can define propositional function F(x) such that F(x) = “I am hungry.”

Since I am sometimes hungry and sometimes not hungry, p is sometimes true and sometimes false. But then, no matter what the domain of discourse for x in F(x) is, what does it mean to say that F(x) is true for all x in the domain? And what does it mean to say that F(x) is true for some value in the domain? The truth value of F(x) = “I am hungry” does not depend on the value of x since x does not occur in “I am hungry.”

[statiktech]

I does depend on x though, doesn’t it?

If F()=hungry and x=me, F(x)=I am hungry. That can be true sometimes. If x=underpants, the proposition would probably never be true.

[phyllo]

A propositional function is formulated in terms of x.

If x has not eaten for two days then x is hungry

Or

X is human

I don’t think that ‘x is hungry’ is sufficiently defined to qualify as a function on its own.

[Dan1 (Dan~)]

Math is measurement, numbers, symbols, etc. But on its own, math doesn’t exist, even though the things math applies to will continue to exist.

Do constant propositional functions exist?
I’d say in math yes, but in the world, no.

This should be easy to understand.

[browser32 (Paul E. Mokrzecki)]

@statiktech

I does depend on x though, doesn’t it?

No, it doesn’t. Given F(x) = “I am hungry” = p, it’s true that F(x) = “I am hungry,” no matter what value x is given (and given, of course, a domain that contains that value). F(smile) = “I am hungry,” F(hello) = “I am hungry,” F(5) = “I am hungry,” and F(Obama) = “I am hungry,” as a few examples.

[phyllo]

That’s not a function.

A property of a function is that it only has one value for a given x.

F(obama)=I am hungry is true sometimes and false at other times. And BTW, who does ‘I’ refer to? Is it Obama? Is it the reader of the statement? Is it the writer of the statement?

[browser32 (Paul E. Mokrzecki)]

@phyllo

That’s not a function.

A property of a function is that it only has one value for a given x.

F(obama)=I am hungry is true sometimes and false at other times. And BTW, who does ‘I’ refer to? Is it Obama? Is it the reader of the statement? Is it the writer of the statement?

I don’t see why F(x) = “I am hungry” = p isn’t a function. It does have at most one value for each input. In fact, it has exactly one value for all inputs (whatever domain that may be defined to be), that value being “I am hungry.” By “I,” I am referring to me, browser32.

[phyllo]

The value of the function is True or False. The value of the function is not the proposition.

propositional function, in logic, a statement expressed in a form that would take on a value of true or false were it not for the appearance within it of a variable x (or of several variables), which leaves the statement undetermined as long as no definite values are specified for the variables. Denoted as a mathematical function, A(x) or A(x1, x2, · · ·, xn), the propositional function is an abstraction from propositional forms (or predicates). For example, “x is small,” “x ist klein,” and “x is not large” are all propositional forms. The substitution of some noun (or value) for x yields a specific proposition that is either true or false, but “x is small” itself has no truth-value. Abstracting from these three propositional forms yields a propositional function A(x), which has, for example, the value true when x takes the value mouse and has the value false when x is elephant.

Propositional functions permit the treatment in symbolic logic of propositions the truth of which is contingent upon variable components.

britannica.com/EBchecked/top … l-function

[Only_Humean]

@browser32

I don’t see why F(x) = “I am hungry” = p isn’t a function. It does have at most one value for each input. In fact, it has exactly one value for all inputs (whatever domain that may be defined to be), that value being “I am hungry.” By “I,” I am referring to me, browser32.

That’s not a function. You’ve just called a proposition “F(x)” instead of “p”, and not made a propositional function F(x) from p.

[browser32 (Paul E. Mokrzecki)]

@phyllo

The value of the function is True or False. The value of the function is not the proposition.

False, the value of the function for any input is the proposition “I am hungry.” The domain of F(x), for my purposes, can be any set of values. The range of F(x) is the sole value “I am hungry.” And “I am hungry” is a proposition, not a truth value.

@Only_Humean

That’s not a function. You’ve just called a proposition “F(x)” instead of “p”, and not made a propositional function F(x) from p.

Also false, it seems I can make propositional function F(x) such that F(x) = p. It’s like the constant function f(x) = 4 from algebra, but instead of a numerical constant we have a propositional constant. Once again, the domain of F(x), for my purposes, can be any set of values. The range of F(x) is the sole value “I am hungry.” The propositional function F(x) is the mapping of each element in the domain to the proposition “I am hungry.”

Unfortunately, it appears I already know more about this elementary subject than everybody else here.

[phyllo]

A function is different from a propositional function.

You are getting answers from me which are similar to the ones you got on the other sites where you posted the same question. We all told you that a propositional function is equal to the truth value of the proposition. Maybe if everyone is telling you something and you disagree, then you should at least consider that you are wrong.

Anyway, if you don’t want to know, then I won’t bother.

[browser32 (Paul E. Mokrzecki)]

@phyllo

A function is different from a propositional function.

Of course it is. A propositional function is a type of function.

@phyllo

We all told you that a propositional function is equal to the truth value of the proposition.

I’m not talking about “that” definition of propositional function. This should be clear from the first sentence in my original post,

@browser32

For a proposition p, it seems we can make the constant propositional function F(x) = p.

The widely used “Discrete Mathematics and Its Applications, Sixth Edition” by Kenneth H. Rosen, the book by which I am going, says on page 31 the value of a propositional function P at x is the statement P(x),

The statement P(x) is also said to be the value of the propositional function P at x. Once a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value.

It’s made clear that the value of a propositional function is a statement, not a truth value. That’s the type of propositional function I am talking about.

@phyllo

Anyway, if you don’t want to know, then I won’t bother.

I do want to know. Just keep in mind that when I refer to a propositional function, the range of that propositional function is a set of propositions.

[phyllo]

Then he gives examples(DM7):

Let P(x) denote the statement “x > 3.” What are the truth values of P(4) and P(2)?
Solution: We obtain the statement P(4) by setting x = 4 in the statement “x > 3.” Hence,
P(4), which is the statement “4 > 3,” is true. However, P(2), which is the statement “2 > 3,”
is false

Let A(x) denote the statement “Computer x is under attack by an intruder.” Suppose that of the
computers on campus, only CS2 and MATH1 are currently under attack by intruders. What are
truth values of A(CS1), A(CS2), and A(MATH1)?
Solution: We obtain the statement A(CS1) by setting x = CS1 in the statement “Computer x
is under attack by an intruder.” Because CS1 is not on the list of computers currently under
attack, we conclude that A(CS1) is false. Similarly, because CS2 and MATH1 are on the list of
computers under attack, we know that A(CS2) and A(MATH1) are true.

Notice that he asks about the truth value of P().
The value of x is mapped to True or False through the proposition.

What does the proposition ‘2 > 3’ mean on its own? Nothing. But it’s either logically true or logically false. That’s the useful result.

[browser32 (Paul E. Mokrzecki)]

Just because Rosen asks for truth values in those examples, doesn’t mean anything he has previously said is wrong.

@phyllo

What does the proposition ‘2 > 3’ mean on its own? Nothing. But it’s either logically true or logically false. That’s the useful result.

I disagree. Propositions, including the proposition “2 > 3,” have much meaning on their own, well beyond their simple truth value. We learned the meaning of “2 > 3” in middle school, if not earlier. People utter propositions for a wide array of reasons to communicate ideas of all sorts. Rosen’s analysis is more than just analyzing truth values. But all this is besides the point, anyways.

I would appreciate it if you would appreciate the other points Rosen has brought up, the points I have mentioned, that are relevant to this discussion.

[phyllo]

I think that Rosen used unfortunate wording. This sentence sums up what I believe he is trying to communicate:

Once a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value.

Propositions, including the proposition “2 > 3,” have much meaning on their own, well beyond their simple truth value. We learned the meaning of “2 > 3” in middle school, if not earlier.

‘2 > 3’ is just 3 symbols which evaluate to true or false for me. If it was said ‘2+3’, then I could replace it with the symbol ‘5’ but since it’s a comparison, then my equivalent symbol must be true/false. I can safely replace the statement with .F. and proceed to manipulate other symbols. Don’t know what else to say about it.

As for the discussion, I think some form of identity is the easiest way to produce a constant propositional function. For example:
3=3 ( which is the reduced form of 0*x+3=3)
or
The universe contains all that is.

But you already have those suggestions so I am not adding anything new.

Constant propositional functions are not particularly interesting. The point of the concept was to produce a function with the variable x in it.