My understanding of a propositional function (like P(x)) is that it is a declaration that contains one or more variables, so that when values are substituted in for the variables, a proposition (a statement that is either true or false) is produced. This definition made me wonder whether or not it is valid for a propositional function to contain propositional variables as its variables. For example, can I validly define C(q,r) to be the conjunction "q ^ r" and I(q,r) to be the implication "q -> r" where q and r represent unspecified propositions since when q and r are replaced with propositions, the statements become propositions or is my provided definition/reasoning flawed?
Can variable-containing statements (propositional functions) contain propositional variables
discrete mathematicslogicpredicate-logic
Related Solutions
Represent the three conditions as you mentioned in logical form and add “and”s between to signify that all must be met simultaneously.
$(J\implies S^c)\land (S\implies K)\land (J^c\implies K^c)$
$\equiv(J^c\lor S^c)\land (S^c\lor K)\land (J\lor K^c)$
$\equiv(J^c\land S^c\land J)\lor (J^c\land S^c\land K^c)\lor (J^c\land K\land J)\lor (J^c\land K\land K^c)\lor (S^c\land S^c\land J)\lor (S^c\land S^c \land K^c)\lor (S^c\land K\land J)\lor (S^c\land K\land K^c)$
$\equiv\emptyset \lor (J^c\land S^c\land K^c) \lor \emptyset \lor \emptyset \lor (S^c\land J)\lor (S^c\land K^c)\lor (S^c\land K\land J)\lor \emptyset$
$\equiv(J^c\land S^c\land K^c) \lor (S^c\land J)\lor (S^c\land K^c)\lor (S^c\land K\land J)$
$\equiv(J^c\land S^c\land K^c) \lor \left[(S^c\land J\land K)\lor (S^c\land J\land K^c)\right]\lor \left[(S^c\land K^c\land J)\lor(S^c\land K^c\land J^c)\right]\lor (S^c\land K\land J)$
$\equiv(J^c\land S^c\land K^c)\lor (S^c\land J\land K)\lor (S^c\land J\land K^c)$
Interpreting the final line, the possible solutions are
- to invite noone,
- to invite Jasmine and neither invite Samir nor Kanti, or
- to invite Jasmine and Kanti but not Samir.
Note, the question did ask "which combinations", which is different than asking "name at least one combination." As it so happened, there are three possible solutions for these statements. Before you say a party with no guests is not much of a party, note that the way the problem was worded suggests that there are potentially other friends you are inviting which aren't going to be upset by who you do or don't invite.
The question of if this method is any faster than building a truth table is up to you. It is certainly easy to make a transcription error here. There is the chance though that we can save several steps in the case of a larger number of truth values but fewer propositions.
It certainly makes sense to have terminology to distinguish truth/falsity of propositional statements from validity/invalidity of arguments that purport to be logical reasoning. On the other hand it is very common for authors to use valid to describe a compound formula with assignments of truth and falsity to its atomic propositions when it results in the overall formula being true (resp. invalid when an assignment gives a false result).
Good authors will be careful in the use of terminology (and in the definitions), but informal usage will vary. Ultimately if you want to use the terms valid/invalid for compound propositions, you should provide consistent definitions that alert readers or listeners to your meaning.
Best Answer
In first order logic, "variables" represent individuals of a domain that you may want to consider, so not "truth values" as in propositional logic. It's not very pedagogical that two very different concepts share the same name, but that's what's happening here.
Therefore if you have variables $x,y$ in first order logic, $x\land y$ doesn't a priori make sense (it might make sense if you have a $2$-ary symbol in your language called $\land$, but then calling it $\land$ is a good way to ensure confusion).
If you want to simulate propositional variables in a first order language, you can add $0$-ary relation symbols: an interpretation of such a symbol will be a subset of $M^0=\{*\}$ (where $M$ is the universe), that is $\emptyset$ or $\{*\}$, so true or false : you can simulate propositional variables this way.