Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions

Circularity in Soundness and Completeness

Richard Kaye

School of Mathematics University of Birmingham

Birmingham 15th November 2013

The strings, my lord, are false. Shakespeare, The Tragedy of Julius Caesar

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Introduction

Introduction

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Introduction

The soundness of first order logic is the statement that first order logic is semantically sound, i.e. derives truth theorems from true statements. The completeness of first order logic is the statement that first order logic is complete (for its language), i.e. any true deduction that can be expressed in the language can be written as a formal proof in the logic.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions The theorems

There are theorems (the Soundness Theorem and the Completeness Theorem1) supporting the assertions that first order logic is sound and complete. These theorems are provable within a suitable metatheory.

1I use Soundness and Completeness, capitalised, to denote the standard theorems with these names, and soundness and completeness, without capitals, to denote the desirable properties of formal systems. Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Goal

This talk looks at the status of the evidence for soundness and completeness of first order logic, presents some doubts concerning this evidence for the latter and explains why this might matter.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Soundness

The Soundness Theorem

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Soundness

The Soundness Theorem is the theorem stating

Σ ` σ implies Σ  σ. Here Σ ` σ expresses that there is a formal proof of the statement σ from assumptions in Σ, where the proof obeys some formally defined and precise rules. The conclusion Σ  σ expresses that every interpretation making Σ true (where ‘true’ usually means in terms of Tarski’s definition of Truth) also makes σ true.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Circularity

Arguably, this is circular since knowledge of semantics and whether or not Σ  σ must be founded in some system of argumentation, which, if it is the system of ` itself, quickly leads to infinite regress:

Σ ` σ implies ` ‘Σ  σ’ ` ‘Σ  σ’ implies ` ‘ ‘Σ  σ’’ ` ‘ ‘Σ  σ’’ implies ` ‘ ‘ ‘Σ  σ’’’ and so on. This is not unlike Lewis Carroll’s What the Tortoise Said to Achilles [1] and whilst not incorrect for most systems of proof ` is not usually helpful.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Discussion

This issue has had a lot of discussion in the (philosophical) literature, e.g. Dummett, Kreisel. But I want to deal with this question quickly to be able to move on to Completeness. Even without a detailed account of soundness, one can regard the Soundness Theorem as a ‘relative consistency result’.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Issues

Can we justify soundness from Soundness together with some intuitive notion of intepretation and argumentation about them? Dummett [2], for example, thought we can. He concludes that there is an idealist (or constructivist) view by which progress can be made. Kreisel’s ‘squeezing argument’ reduces the problem to understanding the differences between intuitive soundness and formal soundness using Completeness. This might be problematic. (There is an issue related to recursion and induction, however. . . )

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Is Soundness weak or strong?

In theory, the Soundness Theorem may not be of much practical use, since the existence of a formal proof of σ from Σ may be more compelling than the manipulation of this proof into semantic arguments why σ should follow from Σ. Put another way, Soundness is comparatively weak as a piece of mathematics. Nevertheless, despite the apparent limitations of Soundness when seen in this way, it remains one of the most important and useful results in mathematical logic. This is a puzzle.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Mathematical practice

Mathematicians, rightly or wrongly, are typically very comfortable with a semantic notion of Truth based on mathematical structures. They typically argue semantically with incomplete knowledge of a partially specified structure, even one that (several hundred pages later) they eventually prove cannot exist. They do not work with a formal system but might use a semi-formal system at the final write-up stage to tidy up the semantic arguments. This is also a puzzle.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Completeness

The Completeness Theorem

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Completeness

It is my contention that the converse result, the Completeness Theorem Σ  σ implies Σ ` σ, is much more problematic in its correct interpretation. It is normally used as the clinching evidence that

First order logic is complete.

However this evidence is not as clear as it might seem at first.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Formalisation

The notions of first order provability and semantic entailment (via Tarski’s definition of Truth) can be set up in one’s favourite metatheory—set theory, perhaps. The Completeness Theorem can be proved there. This gives some useful information, but does it justify the assertion that first order logic is complete?

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Applications of Completeness

If I am a mathematician working in, say, group theory, interested in properties of groups that can be expressed in the language of first order logic, I need to look for theorems (with proofs in any resonable system) or counter-examples. The Completeness Theorem states that if I cannot find a counter-example then there is a proof of the relevant theorem, in the formal first order language following the rules. I may have other knowledge showing a statement is not formally provable. Then the Completeness Theorem will furnish me with a counter-example, one I may not have been able to construct directly by other means.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions The argument for completeness

Suppose Σ 6` σ. Then the proof of Completeness constructs an interpretation of Σ in which ¬σ holds. But doesn’t this just select from the metatheory’s collection of interpretations one that fits the bill? And what if I don’t accept this collection of interpretations? What is the true ‘semantic entailment’? Moreover, the metatheory here is ZF which is based on first order logic. First order logic is providing the interpretations to show it is complete. . .

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Is first order logic strong enough?

Thus, rather than the usual concern that first order logic is too strong (and some other logic, possibly a constructive logic based on intuitionistic logic should be preferred), I am concerned with whether first order logic is strong enough for practical mathematics, and whether there are not in fact other new logical principles that do not follow from first order logic that should be accepted.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Example from set theory

There is a model of set theory containing an abelian group A with Ext1(A, Z) = 0 and A is not free [4]. This is not in itself enough to allow all such A to be allowable structures in our universe, even though this is a question that has often been raised as part of general mathematical research. Many contributions from set theory are similarly complicated and potentially confusing to the non-expert. Most therefore duck such issues, in this case by saying that ‘the Whitehead problem is independent of the axioms of mathematics2.’ What are the allowable interpretations? How can mathematical logic allow others to contribute?

2By which is meant ZFC, of course. Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions The strength of Completeness

One (mathematical) problem that complicates the issue but must be given careful examination, and is that the interpretations provided by Completeness are not given constructively but require (in general) a weak form of the Axiom of Choice (more precisely, the Boolean Prime Ideal Theorem, BPIT, or a statement equivalent to this) to do its work.

In special cases, weaker axioms (such as WKL0) may suffice, but these still have a nonconstructive nature.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions The circularity

The main issue is the observation that the metatheory (some flavour of set theory or arithmetic, including the BPIT) begs the question of what interpretations or structures should be available, by presenting them all in advance. Thus the proof of the Completeness Theorem simply fits some logic to the (known) class of interpretations or structures for it, and checks that the rules of that logic are adequate for this class of interpretations.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Example

Suppose a mathematician rejected uncountable structures and insisted that all countable structures be recursive. He has a reasonable and self-consistent notion of semantic entailment, Σ Rec σ. He would reject ZFC as a metatheory. The logic based on Σ Rec σ is not unreasonable. For example it is consistent, i.e. Σ 6 Rec ⊥ for many Σ, since there are recursive models. All rules for first order logic are sound, since Σ ` σ implies Σ Rec σ. This mathematician would not accept that first order logic is complete, since G¨odel’sSecond Incompleteness Theorem PA 6` Con(PA) for PA in first order logic can be given adequate syntactic arguments, but PA Rec Con(PA) since no model of PA + ¬Con(PA) is recursive by Tennenbaum’s result [5].

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Propositional logic

It is well known that the Completeness Theorem for propositional logic in finitely many propositional letters,

Σ 6` ⊥ implies there is a valuation making each τ ∈ Σ true

is provable directly by constructive means.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Is propositional logic complete?

There are issues of nonconstructivity of the Completeness Theorem for propositional logic with infinitely propositional letters, however. Our mathematician who accepts only recursive objects will believe that propositional logic is not complete. One needs only that there are recursive trees of infinite 0, 1-branching trees with no recursive infinite path. It is then easy to set up a consistent recursive set of propositional formulas Σ with no recursive valuation. More generally, the Completeness Theorem for propositional logic is equivalent to WKL0 over RCA0.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Reactions

I do not hold that recursively presented interpretations are the only interesting ones, and certainly I would not myself restrict the mathematical universe in such a way. But the idea that the class of interpretations is not that that is provided by ZFC is attractive. It might conceivably be the case that some flavour of mathematics with a different class of interpretations for our logical languages is more appropriate for scientific work in understanding our physical universe, and in particular some stronger logical principles should be adopted.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Towards an analysis

Towards an analysis of completeness

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Starting assumptions

We look at statements in first order languages. We allow that truth for these logical statements is defined in a way very similar to Tarski’s definition. We take a potential collection of sets, models and interpretations for these statements. These may come from a Platonistic account of mathematics, and are thus ‘existing abstract objects’; they may come from a physics and are thus ‘existing physical objects’; or they might be virtual or ideal objects posited by a (formal or informal) theory. The universe of interpretations and our arguments about them is the metatheory. This, together with the definition of Truth, gives rise to a notion of semantic entailment , called true semantic entailment. The problem is to identify the nature of .

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions First order logic

By ‘first order logic’ I mean its syntactical part, and system of proof, `FO. The the statement that first order logic is sound is

Σ `FO σ implies Σ  σ and the converse is the statement that first order logic is complete. The Completeness Theorem (formulated in ZFC) is the statement

Σ ZFC σ implies Σ `FO σ

of ZFC, where I have placed a subscript on  to indicate that this notion of semantic entailment is the one internal to ZFC and may not correspond to true semantic entailment. Weaker theories (ACA0, WKL0, PA, . . . ) also prove versions of Completeness, sometimes restricted in some way.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Scott sets

A Scott set is a boolean algebra X ⊆ N closed under relative recursion and the Completeness Theorem. An old theme in Logic (Scott, Friedman, Wilmers, Kreisel,. . . ) is to understand Scott sets as the context for study of the Completeness Theorem. (Or equivalently, but more fashionably, to use the theory WKL0; equivalent, since the ω-models of WKL0 are precisely (N, X ) for Scott sets X .) E.g.: what is the constructive content of the Completeness Theorem as proved from WKL0? (The ‘unwinding’ programme of Kriesel.)

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions

Proof of Completeness in WKL0

Fact: in the proof (in WKL0, say, or by a direct tree argument) of the Completeness Theorem for a recursive theory, only one recursive tree needs to be traversed to find an infinite path in it. (This can be arranged to get a completion of the theory, or a full description of a model.) By additional coding, the proof of Completeness for all recursive first order theories simultaneously can given using a single infinite path.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions A Scott set X as metatheory

Could our collection of allowable interpretations be a Scott set X ? Scott sets contain nonrecursive sets. (This is because of recursive trees with no recursive paths, or alternatively by completions of PA.) A recursive binary tree that has no recursive path necessarily has many paths. How does the Scott set X choose one such path? Could we take out ‘true set of interpretations’ to be those in X ?

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Choice in a Scott set

Fact: the intersection of all Scott sets is the set of recursive (computable) sets. Fact: There is no minimal Scott set. Thus, in some sense, a Scott set X chooses a path in a binary tree in an essentially arbitrary way. If X is the ‘true set of interpretations’ it might be ‘asymmetric’ as seen from the point of view of recursion theory. I don’t have a theorem in this regard, but the least symmetric Scott set might be the set of arithmetical sets.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions

Metamathematics of WKL0

There is an argument that RCA0 (or PRA) is a reasonable initial metatheory. Then WKL0 can assumed by a Hilbert-style argument:

Fact: PRA proves the equiconsistency of PRA, RCA0 and WKL0.

However, the ‘ideal’ objects that WKL0 introduces are not canonical and this does not provide us with and concrete ‘true set of interpretations’ other than the recursive ones. Note too that the Completeness Theorem would only hold for consistent recursive theories.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Arithmetical interpretations

The smallest ‘symmetric’ Scott set which might provide a ‘true set of interpretations’ is (probably) the collection of arithmetical sets. The Completeness Theorem would hold for consistent arithmetical theories in this case.

The logic of Arith may be of interest, especially with respect to (ω) theories such as Th(N) of degree 0 .

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Conclusions

Conclusions

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions First order logic

First order logic is a remarkable achievement. The Completeness and Soundness Theorems, in particular, show it to be robust and natural and deserving of our attention. The Completeness Theorem does not show it to be strong enough for any particular application, however, except where the metatheory can be taken to be a particular flavour of set theory or arithmetic which proves the theorem. Thus some of the claims for first order logic have to be treated with considerable care, to say the least.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Standard mathematical practice

As a foundational device, first order logic (along with other logics) provides a framework for discussing our foundational issues in mathematics, and it is invaluable. But we should not allow the Completeness and Soundness Theorems for first order logic to mislead us into thinking that first order logic necessarily captures usual mathematical practice.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Benefits, and historical perspective

Whatever structures were originally envisaged in the mathematical universe, for example whether one might have accepted the existence of nonstandard models of arithmetic, techniques from the proof of the Completeness Theorem and more generally the use of set theory as a standard all-encompassing theory for mathematics force us to extend what might have been a rather limited mathematical view, to examination of structures built by nonconstructive means. First order logic, and set theory (which was developed alongside) became the practical logic of modern mathematics through these successes and their strength. First order logic is complete in the context of set theory, but there are other contexts, and other areas for investigation.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions

Lewis Carroll. What the tortoise said to Achilles. Mind, 104(416):691–693, 1995. Michael A. E. Dummett. The justification of deduction. In Truth and Other Enigmas, pages 290–318. Duckworth, 1978. British Academy Lecture, 1973. Kurt G¨odel. Uber¨ die Vollst¨andigkeit der Axiome des logischen Funktionenkalk¨us. Monatshefte f¨urMathematik und Physik, 37:349–360, 1930. Saharon Shelah.

Richard Kaye Circularity Introduction The Soundness Theorem The Completeness Theorem Towards an analysis of completeness Conclusions Infinite abelian groups, Whitehead problem and some constructions. Israel J. Math., 18:243–256, 1974. Stanley Tennenbaum. Non-archimedian models for arithmetic. Notices of the American Math. Soc., 270:270, 1959.

Richard Kaye Circularity