substitutions in propositional logic

Uniform Substitution

One area of mathematics where substitution plays a prominent role is mathematical logic. In this entry, we are mainly interested in propositional logicPlanetmathPlanetmath. Recall that a substitution is a function s:Σ1*P(Σ2*) preserving the empty wordPlanetmathPlanetmathPlanetmath and concatenationMathworldPlanetmath. In a propositional logic, s has the following additional characteristics:

s is also called a uniform substitution because for any propositional variable p that occurs in A, s replaces each and every occurrence of p in A by s(p). In practice, we write


to mean change all occurrences of p in A by B, and leave all other variablesMathworldPlanetmath fixed. This includes the case when p does not occur in A, in which case A[B/p]=A. A[B/p] corresponds to a substitution that sends p to B and fixes all variables in A not equal to p. This is called an individual substitution. More generally,


means: change all occurrences of pi in A to Bi, for each i=1,,n, and leave all other variables fixed. This is called a simultaneous substitution, and corresponds to a substitution that sends pi to Bi and fixes all other variables in A. Simultaneous substitutions are not the same as iterated individual substitutions:


For example, if A=pq, then A[q/p,p/q]=qppp=A[q/p][p/q].

Recursive Definition of Substitution

Substitutions can also be defined inducitvely, starting from propositional variables. For the sake of simplicity, we only define uniform substitutions on one variable.

Definition. Suppose A and B are wff’s, and p a propositional variable. Then

  1. 1.

    A is a propositional variable. If A is p, then A[B/p]:=B. Otherwise, A[B/p]:=A.

  2. 2.


  3. 3.

    If A is CD, then A[B/p]:=C[B/p]D[B/p].

  4. 4.

    If A is CD, then A[B/p]:=C[B/p]D[B/p].

  5. 5.

    If A is CD, then A[B/p]:=C[B/p]D[B/p].

Since ¬A is A, we see that (¬A)[B/p] is ¬(A[B/p]). In additionPlanetmathPlanetmath, if the languagePlanetmathPlanetmath of the logic contains a modal connective, say , we have

  1. 6.

    If A is C, then A[B/p]:=C[B/p].

Sets Closed under Uniform Substitution

A set Λ of wff’s is said to be closed underPlanetmathPlanetmath uniform substitution if for any AΛ, s(A)Λ for any (uniform) substitution s. We also say the set is closed under US (for uniform substitution), or obeys the rule of US. The smallest set containing a wff A that is closed under US is called a schema based on A, and is denoted by 𝐀, the bold face version of A. An element of 𝐀 is called a substitution instance, or simply an instance of A. For example, if A is p(qp), where p and q are propositional variables, then


is a substitution instance of A, where p is replaced by DB and q by (DB)C.

It is easy to see that a set is closed under US iff it is closed under individual substitution (IS). Obviously, one direction is clear, as IS is just special case of US. Conversely, suppose AΛ, which is closed under IS. Let p1,,pn be all the propositional variables in A, and X1,,Xn are arbitrary wff’s. Let q1,,qn be propositional variables, none of which occur in any of A,X1,,Xn. Then


There are in general two ways to specify a given axiom system for a propositional logic:

  • list wff’s A1,A2, as axioms, and R1, as inference rules, including US, or

  • list schemas 𝐀𝟏,𝐀𝟐, as axiom schemasMathworldPlanetmath, and R1, as inference rules, excluding US

Both specifications are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, in that they produce the same set of theoremsMathworldPlanetmath.

Non-Uniform Substitution

It is also possible to consider substitutions that only replace some, but not all, occurrences of a propositional variable in a formulaMathworldPlanetmathPlanetmath A, or replace a variable at different locations in A by different formulas. For example, if A is (pq)(qp), then

  • (Bq)(Bp) is obtained by replacing the first occurrences of p and q by B;

  • (Bq)(qC) is obtained by replacing the first and second occurrences of p by B and C respectively.

Replacements such as these are called non-uniform substitutions. Technically, these are no longer substitutions, for they are no longer functions on Σ*, as individual variables may be mapped to different things depending on their location in the formula. In the first example above, p is mapped to both B and p, depending on whether it is the first or second occurrence in A.

To present a non-uniform substitution, let p¯ be the list all the propositional variables p1,,pn in A in order. Note that since a propositional variable pi may occur multiple times in A, p may occur multiple times in p¯. Suppose each pi is replaced by Bi. Let B¯ be the list B1,,Bn. Then we denote


by this non-uniform substitution. In the two examples above, A[(B,q,B,p)/(p,q,q,p)] is the first wff, while A[(B,q,q,C)/(p,q,q,p)] is the second.

Nevertheless, non-uniform substitution is useful in one respect: preservation of theoremhood. This fact is the famous substitution theorem, which says, if p1,,pn are all the propositional variables (not necessarily distinct) in a wff A that are listed in order of appearance in A, then replacing each variable by deductively equivalent formulas produces equivalent result. In short, if BiCi for i=1,,n, then


A set closed under non-uniform substitution (NUS) is defined in the same way as that of uniform substitution. It is easy to see that the smallest set closed under NUS containing the formula A is the schema 𝐀[𝐪¯/𝐩¯], where q¯ is a list of pairwise distinct propositional variables. For example, the smallest set closed under NUS containing (pq)(qp) is (𝐩𝐪)(𝐫𝐬). It is not hard to see that if the NUS closure of a formula is used as an axiom schema for a logic, with modus ponensMathworldPlanetmath as a rule of inference, then the logic is inconsistent.

First Order Logic

In a first order logic, substitutions are more complicated. Given a wff A, A[B/p] does not necessarily mean replacing all occurrences of p by B. Here, again, a substitution is no longer a substitution in the sense above. In fact, replacements of symbols, like non-uniform substitutions, are conditionalMathworldPlanetmathPlanetmath on the locations of the symbols in the wff A. These conditions are collectively known as the substitutability of a term B for the variable p, and is discussed in more detail here (

Title substitutions in propositional logic
Canonical name SubstitutionsInPropositionalLogic
Date of creation 2013-03-22 19:32:41
Last modified on 2013-03-22 19:32:41
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 22
Author CWoo (3771)
Entry type Definition
Classification msc 03B99
Synonym instance
Synonym US
Related topic Substitutability
Defines simultaneous substitution
Defines uniform substitution
Defines schema
Defines substitution instance