Usage of (x, 1) (1) in natural deduction

Question

When neither one of premises and conclusion includes a number like "1", like the following, I could at least proceed to some extent (although I don't know how to connect Q(y) of the first premise and Q(x, y) of the second premise...)

Premises: 
∀x∀y(P(x,y) ↔ Q(y)) 
∀x∀y(Q(x,y) ↔ R(x,y)) 

Conclusion: 
∀x∀y(P(x,y) ↔ R(x,y)) 

1    ∀x∀y (P(x, y) ↔ Q(y))                   Premiss
2    ∀x∀y (Q(x, y) ↔ R(y))                   Premiss
3    ∀y (P(x, y) ↔ Q(y))                     1 (∀E)
4    (P(x, y) ↔ Q(y))                        3 (∀E)
5    ∀y (Q(x, y) ↔ R(y))                     2 (∀E)
6    (Q(x, y) ↔ R(y))                        5 (∀E)
7    (P(x, y) → Q(y))                        4 (↔E)
..... don't know how to keep going... 

, but when formulas involve some concrete numbers like "1" shown below, I don't even know where to start from. What's the purpose of using a concrete number instead of general expression like x or y? And how do I work on this problem?

Premises: 
∀x∀y(P(x,y)→Q(x,y)) 
∀x(Q(x,1) V R(1))

Conclusion: 
∀xƎy(Q(x,y) V R(y))

Answer 1

We can use terms of the language, i.e. variables, constant and "complex" terms built up from "simpler" terms using function symbols.

Thus, we can use $1$ if it is part of the language.

In the example above, $1$ is used as a constant and it has no special meaning.

Thus, the subformula $R(1)$ simply means that the "object" referred to by $1$ has "property" $R$.

From premise : $∀x(Q(x,1) \lor R(1))$ we derive (using $(\forall \text E)$) :

$Q(a,1) \lor R(1)$

and then we have to use $(\exists \text I)$ to get :

$Ǝy(Q(a,y) \lor R(y))$.

Answer 2

For the first derivation I assume there is a typo in the first or second premise because the predicate $Q$ is used in the first premise with only one variable, but it has two variables in the second premise. Assuming the predicate $Q$ has two variables, this would be a proof:

For the second derivation, view $1$ as a name for a particular value of the variable y. For the proof checker that I am using I have to use letters and so I used $a$ as the name instead of $1$. Here is a proof.

Notice for this result I only used the second premise.

The proof checker and a text explaining the rules are in the links below.

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Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/