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i'm trying to understand a concept i can't understand. i don't happen to have any examples, and the explanations are quite bad. from what i understood, i'll try to explain how i tried to solve those questions:

i need to assign the noun $f(x_1,x_2)$ instead of x_1, where it is possible only. if it is not allowed, i need to explain why.

1)$ \forall x_2(R(x_1,x_2) \land \lnot \exists x_1 R(x_1,x_2)) \to \exists x_2(R(x_1,x_3) \land R(x_1,x_2))$

2)$((R(x_1,x_2)\to \forall x_2 P(x_2,x_1)) \land R(x_1,x_2))$

from what i read and understood, the assignment of the noun could only be done iff x_1 is a free variable. which means that it is not possible to do it in 1) because of $\exists x_1$. on the second x_1 doesn't seem to be bound, so it is a free variable and the assignment can be done.

however, i don't know how to actually to the assignment, and would appreciate learning how to do it correctly.

thank you very much for your help,

sorry i could not elaborate more, but i don't understand this concept and the explanations are sufficient nor good in my opinion.

BeginningMath
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1 Answers1

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You can see the post : substitution for bound variables in first-order logic.

In a nutshell, the operation of substitution in FOL replaces a free occurrence of a variable $x$ with a term $t$.

Thus, in 1) you cannot substitute the second (from left) occurrence of $x_1$ with the term $f(x_1,x_2)$, because that occurrence of $x_1$ is not free.

In addition, we have to consider also the concept of substitutability (or free for) of a term $t$ for a variable $x$ in a formula $\varphi$.

This concept is aimed at avoiding unwanted results obtained with substitution when a variable $y$ occurring in $t$ is "captured" by a quantifier $\forall y$ of $\exists y$ already present in $\varphi$.

Regarding 1), the term $f(x_1,x_2)$ is not substitutable for the variable $x_1$, because the variable $x_2$ of $f(x_1,x_2)$ will be captured by the quantifiers $∀x_2$ (LHS of $\to$) and $∃x_2$ (RHS of $\to$) respectively.

Regarding 2), the term $f(x_1,x_2)$ is not substitutable for $x_1$ in $∀x_2P(x_2,x_1)$.

See the post : definition of substitutable in first-order logic.

Conclusion : the correct answer to your question depends on the details of the syntactical specifications.

What is the formal definition of "assigning the noun $f(x_1,x_2)$ instead of $x_1$" ?

Mauro ALLEGRANZA
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