Let $\mathbb{L}$ be the 1st order language where $C = \{c_1\}$ is the set of symbols of constant, $F=\{f^1_1\}$ is the set of function symbols, and $P=\{p^1_1,p^2_1,p^3_1\}$ is the set of predicate symbols. Let $I$ be an interpretation defined by: $D_I = \mathbb{Z}$, $[f^1_1(x)]_I = x+1$, $p^1_1(d) = 1 \iff d \ge 0$, $[p^2_1]_I(d_1,d_2) = 1 \iff d_1 \ge d_2$ and $[p^3_1]_I (d_1,d_2,d_3) =1 \iff d_1 + d_2 = d_3$. Consider the following formula:
$$\phi = \lnot p^1_1 (x_1) \implies (\forall x_{10})p^2_1(x_{10},x_1)$$
a) If $\phi = \lnot p^1_1 (x_1) \implies (\forall x_{10})p^2_1(x_{10},x_1)$ true if $x(x_i) = -i$?
b) Is $\phi$ logically valid?
a) $\lnot p^1_1(x_1) = \lnot p^1_1(-1) = 1$. The right hand side must be 1 for this to be true. Then
$$ (\forall x_{10})p^2_1(x_{10},x_1) = (\forall x_{10})p^2_1(x_{10},-1)$$
If I remember right, when you have for all x_something, you treat x as a variable and not -10 in this case. So here $x_{10}$ can be any integer (I think). What is the answer then? This is true only when $x_{10} \ge -1$ ?
b) Does logically valid = tautology? Then no?