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,