# Difference between “substitution” and “replacement”

Is there a technical difference between "substitution" and "replacement"?

For example, if I use another expression for x, am I replacing for it? Or substituting? What is I use another value of x?

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"Substitute" is the term that is usually used for this. –  user22805 Aug 11 '12 at 11:03
In general, both replacement and substitution are syntactic operations on strings. If $uvw$ and $x$ are strings, then $uxw$ is the result of replacing $v$ with $x$ in $uvw$. If $u$, $v$ and $w$ are strings, then $u[v \mapsto w]$ is the result of replacing every occurrence of $v$ in $u$ with $w$. The last operation is called substitution, and is defined in terms of replacement. The operation called instantiation is a kind of substitution which eliminates quantifiers by substituting values for free variables. –  danportin Aug 11 '12 at 11:11
I should preface this comment by saying that I don't know of an established difference. But, for instance, if someone told me to 'substitute $\cos x$ for $x$' in $\int x dx$, I would write $\int \cos x (-\sin x )dx$. But if someone told me to replace $x$ by $\cos x$, I would write $\int \cos x d(\cos x)$. So I perceive a one-step difference, in the sense that different things get written down (ignoring the same meaning). –  mixedmath Aug 11 '12 at 13:00
The rules of inference of first-order equational logic are the following (combined with the rules that equality is an equivalence relation). $$\rm A = B\ \Rightarrow\ P(A) = P(B)\qquad\qquad (Replacement)$$ $$\rm P(X) = Q(X)\ \Rightarrow\ P(A) = Q(A)\quad (Substitution)$$ –  Bill Dubuque Aug 11 '12 at 13:57