Metatheory - University of Cambridge

Substitution 2
In this chapter, we shall put our new familiarity with induction to good work,
and prove a cluster of results that connect with the idea of substituting one
sentence in for another.
Before going further, we should state the idea of ‘substitution’ a bit more
precisely. Where A, C and S are sentences, A[S/C] is the result of searching
through A and finding and replacing every occurrence of C with S. As an
example, when A is ‘(Q ∧ (¬P ∨ (Q ↔ R)))’, when C is ‘Q’ and when S is
‘¬(D → B)’, then A[S/C] is ‘(¬(D → B) ∧ (¬P ∨ (¬(D → B) ↔ R)))’.
All of the results in this chapter employ this surprisingly powerful idea.
2.1 Two crucial lemmas about substitution
Given your familiarity with TFL, here is a line of thought which should strike
you as plausible. Suppose you have a valid argument in TFL. Now, run through
that argument, uniformly substituting every instance of some atomic sentence
that occurs within that argument, with some other sentence. The resulting
argument is still valid.
It is worth realising that this line of thought is not generally plausible in
natural languages. For example, here is a valid argument in English:
• Isaac is an eft; so Isaac is a newt.
Its validity, plausibly, connects with the meanings of its component words.
But recall that the semantics of TFL is purely extensional, and knows nothing
about meanings. Its only dimension of freedom is over how we assign truth
and falsity to components of sentences. And uniform substitution – of the sort
mentioned in the previous paragraph – cannot, in a relevant sense, increase our
options for assigning truth values to such components.
But to convince ourselves that this line of thought is correct, we of course
require a rigorous proof. Here it is.
Lemma 2.1. For any atomic sentence C and any sentences A 1 , . . . , A n+1 , S:
if A 1 , . . . , A n ⊨ A n+1 , then A 1 [S/C], . . . , A n [S/C] ⊨ A n+1 [S/C].
Proof. Suppose A 1 , . . . , A n ⊨ A n+1 . Suppose also that v is a valuation which
makes all of A 1 [S/C], . . . , A n [S/C] true. Let w be a valuation which differs from
v (if at all) only by assigning a value to C as follows: w makes C true iff v
makes S true. I now make a claim:
Claim: for any sentence A: w makes A true iff v makes A[S/C] true.
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