Consistency vs. non-contradiction in the algebraic logic tradition

Question

In his introduction to (Skolem 1970), Wang claims that Skolem's conflations of consistency (non-contradiction) and satisfiability are

"explained by the fact that Skolem is in the [algebraic] tradition of Boole, Schroder, Korselt, and Lowenheim."

(p. 22) Wang references an observation of Bernays:

"from [the algebraic point of view] ... satisfiability is the same as consistency since no other hindrance to satisfying conditions can exist besides contradiction."

I do not understand this remark, either from a historical perspective, or from a technical one. Historically, there is plenty of evidence of algebraic logicians using "consistent" interchangeably with "satisfiable" and "non-contradictory". But I can find no evidence from the works of the algebraists referenced above that this is any kind of conflation. Indeed, "non-contradictory" and "satisfiable" seem to be used in the same semantic sense, i.e., to mean (in modern terms) that it is possible to assign truth values to atomic components so that the whole formula comes out true. If a formula contains both an atomic component A and its negation ~A then this assignment is not possible -- the formula is unsatisfiable AND contradictory.

Since the algebraists were not interested in deductive systems based on syntactic rules, I cannot understand why there would be any reason for them to understand "non-contradictory" as equivalent with "satisfiable" if "contradictory" is meant in the sense of syntactically refutable. However, this is apparently how Wang intends the remark above, because he immediately adds that as a result of this conflation, Skolem replaces the assumption of satisfiability with the assumption that a (syntactic) derivation of a contradiction exists in an implicit formal system. I suggest that the derivation is syntactic simply because from this assumption Skolem is supposed to prove that the formula is contradictory in the other semantic sense, roughly, that one of its conjuncts has no assignment of truth values that makes it true.

Does anyone have better insight into this remark?

References:

WANG, HAO

[1970] A survey of Skolem’s work in logic . In Skolem [1970], pp. 17–52.

Answer 1

Since the algebraists were not interested in deductive systems based on syntactic rules, I cannot understand why there would be any reason for them to understand "non-contradictory" as equivalent with "satisfiable" if "contradictory" is meant in the sense of syntactically refutable.

Is it true that there is some sort of confusion?

We may use Jean van Heijennort's Sourcebook for some historical references.

"Inconsistent" was used by Cantor and Dedekind in an informal sense: see Cantor (1899) [page 114] for "inconsistent multiplicities" as well as Hilbert's comment (1904) on Cantor [page 131]. See also [page 138]: "consistent existence".

In the same article [page 133] Hilbert speaks of "consistent notion*" referring to equality when defined axiomatically [reflexivity and substitution].

Thus, we may say that with Hilbert the modern sense of consistency, when referred to a "theory" (a set of axioms), is surfacing.

Zermelo (1908) [page 183-on] speaks of "consistent" and "inconsistent" sets. But see also [page 198]: "such assumption [that leads to inconsistent notions or results ] are to be excluded and that no consequences should be derived from inconsistent notions."

And see also Zermelo (1908a) [page 200]: "I have not yet even been able to prove rigorously that my axioms are consistent".

The modern clear definition is in Post (1921), and it is interesting that this definition is not present into W&R's Principia:

[page 272] Th The system of elementary propositions of Principia is consistent,

where [page 276] "the ordinary notion of consistency [of a set of postulates] involves that of contradiction [...]. Now an inconsistent system in the ordinary sense will involve the assertion of a pair of contradictory propositions ([page 272] "we have a contradiction [when] we have asserted a function and its negative")."

Skolem (1920) [page 254-on], following Löwenheim, speaks of "satisfiability (in a given domain) of a first-order proposition.

In Skolem (1922) [page 293] we have: "If the axioms are consistent there exists a domain in which the axioms hold and whose elements can all be enumerated by means of the positive finite integers.

Thus, Skolem is quite clear about the informal concept of "consistent notion" (as defined by a collection of axioms) and the "formal" notion of satisfiability.

As you said, there is still no modern sense of "syntactical consistency" (à la Post).

But what was Skolem's notion of inconsistency?

The "obvious one": see Russell's letter to Frege (1902) [page 125]: "now this view [Frege's Axiom V, i.e. the principle "that a function, too, can act as the indeterminate element"] seems doubtful to me because of the following contradiction [the well-known Russell's paradox]."

Here "a contradiction" is not defined formally (i.e. "syntactically") but is simply a statement "from which the opposite follows", and thus it is a statement than cannot be true.

In conclusion, I agree with your perplexity regarding Wang's comment: there is no "conflation", meaning some sort of confusion.

Skolem, working in the context of the "algebraic" tradition, was not interested into the "syntactical" point of view.

But, ate the same time, its notion of contradiction was the "common sense" one: a statement involving its opposite, a statement that contradicts some axiom or theorem of the theory.

See Skolem (1928) [page 508-on, and the comment by the editors on the term "Widerspruch"]:

[page 519] In the latter case the given first-order proposition contains a contradiction. In the former case, on the other end, it is consistent [widerspruchslos].

A later paper of Skolem: Some Remarks on the Foundation of Set Theory (1950), seems quite clear to me.

Skolem refers to Dedekind's "set theory" as First Set Theory [page 695]:

If we should formalize FST - which by the way we know is inconsistent [...]

[page 696] We know now that FST is inconsistent because, for example, Russell's antinomy can be deduced in it.

And [page 701]:

However a result of Gödel is known showing that this sort of reasoning [finitary reasoning as considered by Hilbert's program] is not sufficient to enable us to prove the consistency of the usual formal systems of mathematics.

Thus, the later Skolem was perfectly aware of current developments of mathematical logic and of the formalized approach to mathematical theories adopted by Hilbert's school.