# Substitution (logic)

**Substitution** is a fundamental concept in logic.
A **substitution** is a syntactic transformation on formal expressions.
To **apply** a substitution to an expression means to consistently replace its variable, or placeholder, symbols by other expressions.
The resulting expression is called a **substitution instance**, or short **instance**, of the original expression.

## Propositional logic

### Definition

Where *ψ* and *φ* represent formulas of propositional logic, *ψ* is a **substitution instance** of *φ* if and only if *ψ* may be obtained from *φ* by substituting formulas for symbols in *φ*, replacing each occurrence of the same symbol by an occurrence of the same formula. For example:

- (R → S) & (T → S)

is a substitution instance of:

- P & Q

and

- (A ↔ A) ↔ (A ↔ A)

is a substitution instance of:

- (A ↔ A)

In some deduction systems for propositional logic, a new expression (a proposition) may be entered on a line of a derivation if it is a substitution instance of a previous line of the derivation (Hunter 1971, p. 118). This is how new lines are introduced in some axiomatic systems. In systems that use rules of transformation, a rule may include the use of a *substitution instance* for the purpose of introducing certain variables into a derivation.

In first-order logic, every closed propositional formula that can be derived from an open propositional formula *a* by substitution is said to be a substitution instance of *a*. If *a* is a closed propositional formula we count *a* itself as its only substitution instance.

### Tautologies

A propositional formula is a tautology if it is true under every valuation (or interpretation) of its predicate symbols. If Φ is a tautology, and Θ is a substitution instance of Φ, then Θ is again a tautology. This fact implies the soundness of the deduction rule described in the previous section.

## First-order logic

In first-order logic, a **substitution** is a total mapping σ: *V* → *T* from variables to terms; many,^{[1]}^{:73}^{[2]}^{:445} but not all^{[3]}^{:250} authors additionally require σ(*x*) = *x* for all but finitely many variables *x*. The notation { *x*_{1} ↦ *t*_{1}, ..., *x*_{k} ↦ *t*_{k} }^{[note 1]}
refers to a substitution mapping each variable *x*_{i} to the corresponding term *t*_{i}, for *i*=1,...,*k*, and every other variable to itself; the *x*_{i} must be pairwise distinct. **Applying** that substitution to a term *t* is written in postfix notation as *t* { *x*_{1} ↦ *t*_{1}, ..., *x*_{k} ↦ *t*_{k} }; it means to (simultaneously) replace every occurrence of each *x*_{i} in *t* by *t*_{i}.^{[note 2]} The result *t*σ of applying a substitution σ to a term *t* is called an **instance** of that term *t*.
For example, applying the substitution { *x* ↦ *z*, *z* ↦ *h*(*a*,*y*) } to the term

*f*(*z*, *a*,*g*(*x*), *y*)yields *f*(*h*(*a*,*y*), *a*,*g*(*z*), *y*).

The **domain** *dom*(σ) of a substitution σ is commonly defined as the set of variables actually replaced, i.e. *dom*(σ) = { *x* ∈ *V* | *x*σ ≠ *x* }.
A substitution is called a **ground** substitution if it maps all variables of its domain to ground, i.e. variable-free, terms.
The substitution instance *t*σ of a ground substitution is a ground term if all of *t'*s variables are in σ's domain, i.e. if *vars*(*t*) ⊆ *dom*(σ).
A substitution σ is called a **linear** substitution if *t*σ is a linear term for some (and hence every) linear term *t* containing precisely the variables of σ's domain, i.e. with *vars*(*t*) = *dom*(σ).
A substitution σ is called a **flat** substitution if *x*σ is a variable for every variable *x*.
A substitution σ is called a **renaming** substitution if it is a permutation on the set of all variables. Like every permutation, a renaming substitution σ always has an **inverse** substitution σ^{−1}, such that *t*σσ^{−1} = *t* = *t*σ^{−1}σ for every term *t*. However, it is not possible to define an inverse for an arbitrary substitution.

For example, { *x* ↦ 2, *y* ↦ 3+4 } is a ground substitution, { *x* ↦ *x*_{1}, *y* ↦ *y*_{2}+4 } is non-ground and non-flat, but linear,
{ *x* ↦ *y*_{2}, *y* ↦ *y*_{2}+4 } is non-linear and non-flat, { *x* ↦ *y*_{2}, *y* ↦ *y*_{2} } is flat, but non-linear, { *x* ↦ *x*_{1}, *y* ↦ *y*_{2} } is both linear and flat, but not a renaming, since is maps both *y* and *y*_{2} to *y*_{2}; each of these substitutions has the set {*x*,*y*} as its domain. An example for a renaming substitution is { *x* ↦ *x*_{1}, *x*_{1} ↦ *y*, *y* ↦ *y*_{2}, *y*_{2} ↦ *x* }, it has the inverse { *x* ↦ *y*_{2}, *y*_{2} ↦ *y*, *y* ↦ *x*_{1}, *x*_{1} ↦ *x* }. The flat substitution { *x* ↦ *z*, *y* ↦ *z* } cannot have an inverse, since e.g. (*x*+*y*) { *x* ↦ *z*, *y* ↦ *z* } = *z*+*z*, and the latter term cannot be transformed back to *x*+*y*, as the information about the origin a *z* stems from is lost. The ground substitution { *x* ↦ 2 } cannot have an inverse due to a similar loss of origin information e.g. in (*x*+2) { *x* ↦ 2 } = 2+2, even if replacing constants by variables was allowed by some fictitious kind of "generalized substitutions".

Two substitutions are considered **equal** if they map each variable to structurally equal result terms, formally: σ = τ if *x*σ = *x*τ for each variable *x* ∈ *V*.
The **composition** of two substitutions σ = { *x*_{1} ↦ *t*_{1}, ..., *x*_{k} ↦ *t*_{k} } and τ = { *y*_{1} ↦ *u*_{1}, ..., *y*_{l} ↦ u_{l} } is obtained by removing from the substitution { *x*_{1} ↦ *t*_{1}τ, ..., *x*_{k} ↦ *t*_{k}τ, *y*_{1} ↦ *u*_{1}, ..., *y*_{l} ↦ *u*_{l} } those pairs *y*_{i} ↦ *u*_{i} for which *y*_{i} ∈ { *x*_{1}, ..., *x*_{k} }.
The composition of σ and τ is denoted by στ. Composition is an associative operation, and is compatible with substitution application, i.e. (ρσ)τ = ρ(στ), and (*t*σ)τ = *t*(στ), respectively, for every substitutions ρ, σ, τ, and every term *t*.
The **identity substitution**, which maps every variable to itself, is the neutral element of substitution composition. A substitution σ is called **idempotent** if σσ = σ, and hence *t*σσ = *t*σ for every term *t*. The substitution { *x*_{1} ↦ *t*_{1}, ..., *x*_{k} ↦ *t*_{k} } is idempotent if and only if none of the variables *x*_{i} occurs in any *t*_{i}. Substitution composition is not commutative, that is, στ may be different from τσ, even if σ and τ are idempotent.^{[1]}^{:73-74}^{[2]}^{:445-446}

For example, { *x* ↦ 2, *y* ↦ 3+4 } is equal to { *y* ↦ 3+4, *x* ↦ 2 }, but different from { *x* ↦ 2, *y* ↦ 7 }. The substitution { *x* ↦ *y*+*y* } is idempotent, e.g. ((*x*+*y*) {*x*↦*y*+*y*}) {*x*↦*y*+*y*} = ((*y*+*y*)+*y*) {*x*↦*y*+*y*} = (*y*+*y*)+*y*, while the substitution { *x* ↦ *x*+*y* } is non-idempotent, e.g. ((*x*+*y*) {*x*↦*x*+*y*}) {*x*↦*x*+*y*} = ((*x*+*y*)+*y*) {*x*↦*x*+*y*} = ((*x*+*y*)+*y*)+*y*. An example for non-commuting substitutions is { *x* ↦ *y* } { *y* ↦ *z* } = { *x* ↦ *z*, *y* ↦ *z* }, but { *y* ↦ *z*} { *x* ↦ *y*} = { *x* ↦ *y*, *y* ↦ *z* }.

## See also

- Substitution property in Equality (mathematics)#Some basic logical properties of equality
- First-order logic#Rules of inference
- Universal instantiation
- Lambda calculus#Substitution
- Truth-value semantics
- Unification (computer science)
- Metavariable
- Mutatis mutandis
- Rule of replacement
- Substitution (algebra) — about applying substitutions to polynoms and other algebraic expressions

## Notes

- ↑ some authors use [
*t*_{1}/*x*_{1}, ...,*t*_{k}/*x*_{k}] to denote that substitution, e.g. M. Wirsing (1990). Jan van Leeuwen, ed.*Algebraic Specification*. Handbook of Theoretical Computer Science.**B**. Elsevier. pp. 675–788. , here: p. 682; - ↑ From a term algebra point of view, the set
*T*of terms is the free term algebra over the set*V*of variables, hence for each substitution mapping σ:*V*→*T*there is a unique homomorphism σ:*T*→*T*that agrees with σ on*V*⊆*T*; the above-defined application of σ to a term*t*is then viewed as applying the function σ to the argument*t*.

## References

- Crabbé, M. (2004).
*On the Notion of Substitution*. Logic Journal of the IGPL, 12, 111--124. - Curry, H. B. (1952)
*On the definition of substitution, replacement and allied notions in an abstract formal system*. Revue philosophique de Louvain 50, 251–269. - Hunter, G. (1971).
*Metalogic: An Introduction to the Metatheory of Standard First Order Logic*. University of California Press. ISBN 0-520-01822-2 - Kleene, S. C. (1967).
*Mathematical Logic*. Reprinted 2002, Dover. ISBN 0-486-42533-9

- 1 2 David A. Duffy (1991).
*Principles of Automated Theorem Proving*. Wiley. - 1 2 Franz Baader, Wayne Snyder (2001). Alan Robinson and Andrei Voronkov, ed.
*Unification Theory*(PDF). Elsevier. pp. 439–526. - ↑ N. Dershowitz; J.-P. Jouannaud (1990). "Rewrite Systems". In Jan van Leeuwen.
*Formal Models and Semantics*. Handbook of Theoretical Computer Science.**B**. Elsevier. pp. 243&mdash, 320.

## External links

- Substitution in
*nLab*