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Very noob to math and stackexchange: Example propositions given in Rosen's Discrete Maths textbook, such as

a. "It is raining today"; or

b. "Today is Thursday" -

Can't they be considered as propositional functions themselves:

If $x$ is a variable with all days in it, the truth value of "It is raining today" varies with $x$, and so is a propositional function of $x$ rather than a proposition, no?

Is my understanding correct and can I use it as a basis for further study?

Harish Chandra Rajpoot
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ramiah ariya
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  • What does Rosen mean by Proposition & Propositional function ? [ I am asking the Book Definition ] We can try Answering within that context. It may be that the variable must be explicit. English is inherently not very Precise. Once we "write" it in "Mathematical form" , it will be more Precise. – Prem Sep 04 '22 at 15:58
  • His definition for a proposition is: A proposition is a statement that is either true or false, but not both. The examples he gives are statements such as 1+1=2; or "Toronto is the capital of Canada". He also says that "x+1=2" is not a proposition because you cannot assign a truth value to it (the value depends on "x"). But then, it appears that 1+1=2 is always true. Another example given, of a proposition, is: "Students who have taken calculus can attend this class". I am not sure how this statement has a "true" or a "false" value. – ramiah ariya Sep 04 '22 at 16:12
  • A propositional function is a statement P(x) which is neither true or false, but the truth value takes concrete forms based on x. To me, in the above example, "Students who have taken calculus can attend this class" - given that x is a student at an university, it is true or false, depending upon x's value as it passes over all the students, right? So the statement appears to be a propositional function rather than a proposition. – ramiah ariya Sep 04 '22 at 16:15
  • The other possibility is that in the proposition "Students who have taken calculus can attend this class" the students are "objects", and are not the actual variables. Instead "Students who have taken calculus can attend this class" is always true, because it specifies a rule. It never becomes false. – ramiah ariya Sep 04 '22 at 16:42
  • In short, my question is if a proposition's truth value changes from true to false in a domain, then isn't it necessarily a propositional function? – ramiah ariya Sep 04 '22 at 16:53
  • Statements like "Noida is in India" & "India contains Noida" are Propositions, either true or not true. Statements like "X is in Y" or "X contains Y" are Propositional functions with 2 variables, which become true or not true when we use some values of X & Y. Eliminating only X (or only Y) will still give Propositional functions with 1 variable. Eg "X is in India" & "India contains Y". Eliminating both variables will give Propositions which are either true or not true. When written in "English" , it may not be clear whether we are talking about all Items or variable Items. – Prem Sep 04 '22 at 17:48
  • Here , "x+1=2" is either true (x=1) or not true(x=3) when "x" is a variable. In case we know "x=1" a constant , then "x+1=2" is true & it is a Proposition because we are not making "x" a variable. – Prem Sep 04 '22 at 17:51
  • When taking "Students who have taken calculus can attend this class" , we can write it like this "for all S in Students : Student S has taken calculus IMPLIES Student S can attend this class" & there is no variable to which we can assign a value. If it was "Student Z who has taken calculus can attend this class" , we can write it like this "P(Z) : Z has taken calculus" & "Q(Z) : Z can attend this class" , then we can not know whether these are true or not true but we know "P(Z) IMPLIES Q(Z)" where Z is a variable & we also know that whatever value of Z we take, it is always true in this case. – Prem Sep 04 '22 at 18:02
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    Great! I understand. Can you post your comments as answer, so I accept? – ramiah ariya Sep 04 '22 at 18:13
  • Nice to know that it was useful. Currently I am away from a Proper Computer , hence there will be a Delay in Posting my "Expanded" Answer. – Prem Sep 05 '22 at 04:47

1 Answers1

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The CORE Distinction is whether the Statements have some variables or not.

Propositions : No variables. Maybe true or not true. The "truth value" will not change or will not Depend on variables.

Propositional functions : must have atleast 1 variable. May have 2 or more variables. In general, Will be neither "true" nor "not true" , until all the variables are assigned some values. In general, the "truth value" will change when the variables are assigned some other values.

Statements like "Noida is in India" & "India contains Noida" are Propositions, either true or not true.
Statements like "X is in Y" or "X contains Y" are Propositional functions with 2 variables, which become true or not true when we use some values of X & Y.
Eliminating only X (or only Y) will still give Propositional functions with 1 variable.
Eg "X is in India" & "India contains Y".
Eliminating both variables will give Propositions which are either true or not true.
When written in "English" , it may not be clear whether we are talking about all Items or variable Items.

Consider "$x+1=2$" where "$x$" is a variable : When we do not know "$x$" variable value, It is neither true nor not true. When we do know "$x$" variable value, It is either true (when $x=1$) or not true (when $x=3$) otherwise.
In case we know "$x=1$" a constant , then "$x+1=2$" is true & it is a Proposition because we are not making "$x$" a variable. It is almost like "$1+1=2$" (true) or like "$3+1=2$" (not true) where we have no variables.

Consider "Students who have taken calculus can attend this class" , which we can write like this : "for all S in Students : Student S has taken calculus IMPLIES Student S can attend this class"
$\forall S \in Students : Student\ S\ has\ taken\ calculus \implies Student\ S\ can\ attend\ this\ class$
Here, there are no variables to which we can assign values.

If it was "Student Z who has taken calculus can attend this class" , we can write it like this : "P(Z) : Z has taken calculus" & "Q(Z) : Z can attend this class" :
$P(Z) : Z\ has\ taken\ calculus$
$Q(Z) : Z\ can\ attend\ this\ class$
where $Z$ is a variable.
In that case, we can not know whether $P$ & $Q$ are true or not true until we assign a value to $Z$.
We are given that "P(Z) IMPLIES Q(Z)" :
$P(Z) \implies Q(Z)$
where $Z$ is a variable.
We are told that whatever value of $Z$ we take, it is always true, in this given context.

Prem
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