At the time of my graduation thesis, many years ago, I used the highly appreciated volume of Chang and Lee, Symbolic Logic and Mechanical Theorem Proving. Lately I found myself picking it up again adapting it to my current research, in particular on the Herbrand Universe and Herbrand Theorem. On the web I found a ferocious review of the book by a certain C. G. Morgan. By suspending judgment on the review itself, I tried instead to verify its content. Well, at a certain point, Morgan criticizes lemma 4.1 with an example, as follows.

< "LEMMA4.1. If an interpretation over some domain D satisfies a set S of clauses, then any one of the H-interpretation I* corresponding to I also satisfies S." Given the definition by Chang and Lee, the lemma is simply false. For a counterexample, let S = { Pxx, Pxf(x)} and let I be the following interpretation D: = {1,2}; P is interpreted as true of only (1,2) and (2, 1); f is interpreted as the function mapping 1 to 2 and 2 to 1. Clearly I satisfies S. The Herbrand universe of S. is H = {a, f(a), f(f(a)), ... }. Following Chang and Lee, one I*-interpretation corresponding to I is obtained by mapping all elements of H to1; thus by their procedure, P is true of nothing in I*. Then Pxf(x) is not satisfied in I*, and hence S is not satisfiedin I*. Alternatively we could obtain I* by mapping a and f(a) to 1 and everythineg else in H to 2; again Pxf(x) would not be satisfied if we interpret P according to the procedure given by Chang and Lee. Thus the procedure outlined by the authors-must be revised . … >
My translating in symbols.
S = { \neg P(x,x), P(x,f(x))}
D = {1,2}
f(1) = 2
f(2) = 1
P(1,2)—> T
P(2,1)—> T
P(1,1)—> F
P(2,2)—>F
H = {a, f(a), f(f(a)), ... }
B_s = {P(a,a), P(a,f(a)) , P(f(a),a), P(f(a),f(a)), P(a,f(f(a))) , P(f(f(a)),a), P(f(a),f(f(a))) ….. }

Mapping of a is not specified by Morgan so we consider both cases a —> 1 and a —> 2.

a —> 1
P(a,a) = P(1,1) = F
P(a,f(a)) = P(1,2) = T
P(f(a),a) = P(2,1) = T
P(f(a),f(a)) = P(2,2) = F
P(a,f(f(a))) = P(1,1) = F
P(f(f(a)),a) = P(1,1) = F
P(f(a),f(f(a))) = P(2,1) = T
….

I*_1 = {\neg (Pa,a), P(a,f(a)) , P(f(a),a), \neg P(f(a),f(a)), \neg P(a,f(f(a))) , \neg P(f(f(a)),a), P(f(a),f(f(a))) ….. }

a —> 2
P(a,a) = P(2,2) = F
P(a,f(a)) = P(2,1) = T
P(f(a),a) = P(1,2) = T
P(f(a),f(a)) = P(1,1) = F
P(a,f(f(a))) = P(2,2) = F
P(f(f(a)),a) = P(2,2) = F
P(f(a),f(f(a))) = P(1,2) = T
…..

I*_2 = {\neg (Pa,a), P(a,f(a)) , P(f(a),a), \neg P(f(a),f(a)), \neg P(a,f(f(a))) , \neg P(f(f(a)),a), P(f(a),f(f(a))) ….. }

the only we can say is that I*_1 and I*_2 seems to be equal, but the statement of Morgan “ one I*-interpretation corresponding to I is obtained by mapping all elements of H to 1; thus by their procedure, P is true of nothing in I* ” seems unintelligible. Am I wrong ? Why “all the elemets of H to 1” ?
Is it I who don't understand or is it Morgan who made a mistake?

Thanks, Paola

ps: opinions about Chang-Lee's book, and Lemma 4.1, also very wellcome.

Morgan’s review could be found in
https://epubs.siam.org/doi/10.1137/1016071
or
https://zh.booksc.eu/book/27629332/2eb1e8

https://paolacattabriga.wordpress.com/2022/05/05/about-a-c-g-morgan-review-of-chang-lee-book-symbolic-logic-and-mechanical-theorem-proving/