Does the order of quantifiers matter in propositional calculus?

[Arian.D]

I think it matters, for example when I think of examples that I encounter in life it seems that the order of quantifiers matters and if we change the order the meaning could be interpreted differently, but does the order of quantifiers matter in propositional calculus? If yes, how could we show that it matters?

 

[DonAntonio]

I think it matters, for example when I think of examples that I encounter in life it seems that the order of quantifiers matters and if we change the order the meaning could be interpreted differently, but does the order of quantifiers matter in propositional calculus? If yes, how could we show that it matters?

Exactly because of what you said it matters: it is enough one single example that shows that we can have different

meaning in order to deduce the order of quantifiers matters, and a lot, in fact.

DonAntonio

 

[920118]

I think it matters, for example when I think of examples that I encounter in life it seems that the order of quantifiers matters and if we change the order the meaning could be interpreted differently, but does the order of quantifiers matter in propositional calculus? If yes, how could we show that it matters?

There aren't any quantifiers in propositional calculus...? If you mean predicate calculus, then yes, the order matters. e.g.,

$\exists$x$\forall$yR(x, y)
$\forall$y$\exists$xR(x, y)

The first case says that there exists something such that it stands in the relation R to everything. The second case says that everything stands in the relation R to something.
e.g., given the domain U = {1, 2, 3, ...} and R = {(x, y) | x is larger than y}, case one is false since there is no number that it is larger than every number. But case two is true, since every number has some successor. It's a matter of convention rather than something you show.

 

[mbs]

The order can be reversed in the special case of two existential or two universal quantifiers, one entirely within the scope of the other, as long as no free variable occurrences become bound and no bound variable occurrences become free in the process of reversing the order.

Quantifiers of different types can never be reversed.

 

[blue_raver22]

Yes, the order of quantifiers does matter in propositional calculus. This is because propositional calculus deals with logical statements that involve quantifiers, such as "for all" and "there exists". These quantifiers determine the scope and meaning of the logical statement.

To show that the order of quantifiers matters in propositional calculus, we can use an example. Let's consider the statement "For all x, there exists y such that x + y = 10." This statement means that for every number x, there is another number y such that their sum is 10. However, if we change the order of the quantifiers and say "There exists y, for all x such that x + y = 10," the meaning changes. Now, the statement means that there is a specific number y that, when added to any number x, will result in 10. This is a subtle but important difference in meaning that can only be captured by considering the order of quantifiers.

In addition, the rules of propositional calculus also demonstrate the importance of the order of quantifiers. For instance, the rule of universal instantiation states that if a statement is true for all values of a variable, then it is true for a specific value of that variable. This rule only holds if the quantifier "for all" comes before the quantifier "there exists". If we reverse the order, the rule would no longer be valid.

In conclusion, the order of quantifiers does matter in propositional calculus as it can significantly impact the meaning and validity of logical statements. It is important to consider the correct order of quantifiers in order to accurately represent and interpret logical statements in propositional calculus.