Definition:Substitution (Mathematical Logic)

Mapping
Let $$S$$ be a set.

Let $$f: S^t \to S$$ be a mapping.

Let $$\left\{{g_1: S^k \to S, g_2: S^k \to S, \ldots, g_t: S^k \to S}\right\}$$ be a set of mappings.

Let the mapping $$h: S^k \to S$$ be defined as:
 * $$h \left({s_1, s_2, \ldots, s_k}\right) = f \left({g_1 \left({s_1, s_2, \ldots, s_k}\right), g_2 \left({s_1, s_2, \ldots, s_k}\right), \ldots, g_t \left({s_1, s_2, \ldots, s_k}\right)}\right)$$.

Then $$h$$ is said to be obtained from $$f, g_1, g_2, \ldots, g_k$$ by substitution.

The definition can be generalized in the following ways:
 * It can apply to mappings which operate on variously different sets.
 * Each of $$g_1, g_2, \ldots, g_t$$ may have different arities. If $$g$$ is a mapping of $$m$$ variables where $$m > k$$, we can always consider it a mapping of $$k$$ variables in which the additional variables play no part. So if $$g_i$$ is a mapping of $$k_i$$ variables, we can take $$k = \max \left\{{k_i: i = 1, 2, \ldots, t}\right\}$$ and then each $$g_i$$ is then a mapping of $$k$$ variables.

Partial Function
Let $$g: \N^t \to \N$$ be a partial function.

Let $$\left\{{g_1: \N^k \to \N, g_2: \N^k \to \N, \ldots, g_t: \N^k \to \N}\right\}$$ be a set of partial functions.

Let the partial function $$h: \N^k \to \N$$ be defined as:
 * $$h \left({n_1, n_2, \ldots, n_k}\right) \approx f \left({g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)}\right)$$

where $$\approx$$ is as defined in Partial Function Equality.

Then $$h$$ is said to be obtained from $$f, g_1, g_2, \ldots, g_k$$ by substitution.

Note that $$h \left({n_1, n_2, \ldots, n_k}\right)$$ is defined only when:
 * All of $$g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)$$ are defined;
 * $$f \left({g_1 \left({n_1, n_2, \ldots, n_k}\right), g_2 \left({n_1, n_2, \ldots, n_k}\right), \ldots, g_t \left({n_1, n_2, \ldots, n_k}\right)}\right)$$ is defined.