Informal and formal proofs, metalogic, and the groundedness problem Mario Bacelar Valente1 Abstract: When modeling informal proofs like that of Euclid's Elements using a sound logical system, we go from proofs seen as somewhat unrigorous – even having gaps to be filled – to rigorous proofs. However, metalogic grounds the soundness of our logical system, and proofs in metalogic are not like formal proofs and look suspiciously like the informal proofs. This brings about what I am calling here the groundedness problem: how can we decide with certainty that our metalogical proofs are correct and sustain our logical system? In this paper, I will expose this problem. 1 Introduction: on the relationship between informal and formal proofs The correctness of informal proofs seems to depend on corresponding formal proofs. They are even seen as a sort of sketches that point to the underlying formal proof (Avigad 2020). There is a sense in which this view seems natural. Let us consider the proofs in planar geometry in Euclid's Elements. Avigad, Dean, and Mumma (2009) proposed a logical system called E that intends to formalize the proofs in planar geometry in Euclid's Elements. In their analysis of Euclidean proofs, they found what they called logical gaps: they mention inferential gaps due to some assumptions being implicit (in particular when Euclid assumes that the geometric configuration is nondegenerate) (Avigad, Dean, and Mumma 2009, 708); there are assumptions used with theorems without being proved (735); in what should be proofs by cases, Euclid sometimes only considers one case (738); and, sometimes, a proof is not detailed enough and lacks an argument (740). However, they consider that, in general, a line-by-line comparison of the proofs in E with that of the Elements, renders their formal proofs close to Euclid's. Because of this, they concluded that "the proofs in the Elements are more rigorous than is usually claimed" (760). This does not contradict the above view on the relationship between informal and formal proofs. It is only after we have the formal counterpart of the informal proof that we can be sure that a particular informal proof is not as sketchy as it might seem at first. For this to be the case, the formal proof must be faithful to the informal proof. According to Avigad, Dean, and Mumma, a model in E is faithful to the Euclidean proof when it reproduces line-by-line the "argumentative structure" of the proof (760). According to the creators of E, the departures of E from the Euclidean proofs are minor or can be reduced by adopting appropriate definitional extensions (734). In this way, they consider that E provides faithful models of the proofs in Euclid's Elements (731-8).2 Granted that we have a formal proof faithful to our informal Euclidean proof what does this bring to us? We can rely on all the rigor inbuilt in the formal system. In particular, Avigad, Dean, and Mumma showed that the logical system E is sound and complete (743-57). Everything that we derive in E is correct, and everything that is entailed can be derived. We have a powerful logical system that models faithfully the Euclidean proofs. We could even say that an informal proof in planar geometry in Euclid's Elements is sound in the sense that its corresponding formal proof is sound (Valente 2020b). Even if Avigad, Dean, and Mumma recognize that the less rigorous aspects of Euclidean proofs did not lead mathematicians to question the correctness of Euclidean proofs (Avigad, Dean, and Mumma 2009, 701), the existence of a sound formal system that models faithfully the Euclidean proofs would show why the mathematicians' attitude was the right one. The informal proofs in the Elements would be detailed sketches that can be modeled by formal proofs that are faithful to them. 1 Pablo de Olavide University, Seville, Spain; email: mar.bacelar@gmail.com 2 There is a problem associated with this approach. We cannot be certain that a formal proof is faithful to an informal proof. This has been called the faithfulness problem (Valente 2020b): we have no way to decide with certainty about the faithfulness of the formal proof. For the purpose of the present work, we will leave this problem lurking in the background since we have another problem to deal with. 2 Things do not seem to be as simple as this. Most of the rigor and inferential possibilities of logical systems come from general features of these; in particular their soundness and completeness. However, we do not prove that a logical system is sound and complete from within the logical system. To prove results about a logical system we need a metalogic. But here we face a problem. The metalogical proofs seem very much like informal proofs. It seems that we are applying a formal apparatus that is grounded on metatheorems that are proved informally. I call this the groundedness problem. The purpose of this work is to expose this problem (not to solve it). For that, I will consider initially an informal proof from the Elements (section 2), and then a metalogical proof of soundness (section 3). 2 Informal proofs in Euclid's Elements Instead of a general overview of the proofs in the Elements, the approach adopted here is to look into one particular proof that I find representative as an example of Euclidean proofs. It is the proof of proposition 1 of book 1 (I.1). In the standard English version still at use, it is the following: On a given finite straight line to construct an equilateral triangle. Let AB be the given finite straight line. Thus it is required to construct an equilateral triangle on the straight line AB. With center A and distance AB let the circle BCD be described; [Post. 3] again, with center B and distance BA let the circle ACE be described; [Post. 3] and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. I] Now, since the point A is the center of the circle CDB, AC is equal to AB. [Def. I5] Again, since the point B is the center of the circle CAE, BC is equal to BA. [Def. I5] But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; [C. N. I] therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another. Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB. (Being) what it was required to do. (Heath 1956, 241-242)3 3 The model in E of this proof is as follows: Assume a and b are distinct points. Construct point c such that ab = bc and bc = ca. Proof. Let α be the circle with center a passing through b. Let β be the circle with center b passing through a. Let c be a point on the intersection of α and β. Have ab = ac [since they are radii of α]. Have ba = bc [since they are radii of β]. Hence ab = bc and bc = ca. Q.E.F. (Avigad, Dean, and Mumma 2009, 734). For an analysis of this proof see Valente(2020b). 3 The purpose of this proposition is to construct an equilateral triangle proving that it is, in fact, equilateral.4 To enable the proof one needs the triangle to be constructed according to a wellestablished prescription that will make possible the proof to be made. In the initial lines of the proof, we find the construction of the triangle. What is strictly the construction part ends when constructing the lines CA and CB. Then we have the proof proper. Let us look into it, line-by-line: 1) Now, since the point A is the center of the circle CDB, AC is equal to AB. [Def. I5] The "comment" inside the bracket gives us an indication of how one arrives at the conclusion that AC is equal to AB. Definition 15 is the definition of a circle. This definition is complemented by the definition of the center of a circle (definition 16).5 From these definitions, we know that the line segments connecting the center of a circle to the points of the circumference are equal. We call these line segments radii of the circle. Let us recall that previous to this line of the proof we have just constructed the line segment CA, connecting the points C and A, and the line segment CB, connecting the points C and B. To make the inference using definition 15, we must notice first that these line segments are radii of the circle CDB. We simply see that in the diagram. Noticing that AC and AB are not just line segments but are line segments that are radii of the circle CDB, enables us to make the inference that AC is equal to AB. This is a direct consequence of the definition of circle. The argumentation might be something like the following: since A is the center of the circle, and B and C are points of the circumference, the line segments AC and AB are radii; as such, they are of equal length. From our reasoning, we have concluded that these line segments are equal. While it is still implicit at this point of the proof, at the conclusion we return to focus on AC and AB as just line segments. 2) Again, since the point B is the center of the circle CAE, BC is equal to BA. [Def. I5] We have, mutatis mutandis, the same situation as that of the previous line. This is indicated by the term "again" (in the same way). We conclude that the line segments BC and BA are equal. 3) But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. The "but" indicates that a new premise is being "added" to the previous result. More exactly, we are recovering a previous conclusion (AC is equal to AB) and taking it as a premise to be considered together with the previous result, also taken as a premise (BC is equal to BA). The term "therefore" indicates that a new conclusion is reached. The conclusion is simply a blending of the two premises into one proposition. It can be expressed also as follows: CA is equal to AB and CB is equal to AB. 4) And things which are equal to the same thing are also equal to one another; [C. N. I] therefore CA is also equal to CB. Here, the so-called common notion 1 is part of the argument. It is used to conclude, by taking into account the conclusion of the previous line, that CA is equal to CB. In this part, it is made evident that in 3) and 4) we are considering CA, CB, and AB as line segments. That they can be radii is not relevant in this part. In this case, the "things" in the application of common notion 1 are line segments. 5) Therefore the three straight lines CA, AB, BC are equal to one another. 4 To be more exact, instead of "construct" one might say "instantiate". One is not constructing a triangle in the sense of, e.g., drawing a triangle. One is instantiating a geometrical object in the Euclidean plane (on this see, e.g., Valente 2020a). For the present work, we do not need to address this type of issue. 5 According to definition 15, a circle is a plane figure contained by one line [the circumference] such that all the straight lines falling upon it from one point among those lying within the figure are equal (Heath 1956, 153). This point is called, according to definition 16, the center of the circle (Heath 1956, 154). 4 Here, a new conclusion is inferred from previous conclusions. We have: CA is equal to AB and CB is equal to AB; and we also have: CA is also equal to CB. From these conclusions taken as premises, we conclude that the line segments CA, AB, and BC are equal. 6) Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB. A new conclusion arises from the previous one. The diagram is crucial in this part. There was no mention of triangles until this last line of the proof. In the previous line, we concluded that the line segments CA, AB, and BC, as line segments, are equal. Now, we see in the diagram that these line segments form a figure with three sides; and these three sides were proved to be equal. Taking into account the definition of a triangle (Heath 1956, 154), CA, AB, and BC are the sides of an equilateral triangle. In this way, we have proved that the adopted construction leads to an equilateral triangle. The Euclidean proofs are made adopting a regimented language; what we might consider a part of natural language in which the terms have a very specific meaning and use (Netz 1999, 89-167). For example, "therefore" and "so that" are used to introduce conclusions; "and" and "but" are used to introduce assertions to be considered together with previous assertions, in this way enabling to infer a new assertion; "for" and "since" are used to introduce an assertion that supports a previous assertion; "since" is also used to start or restart an argument (Netz 199, 115-6). Another aspect of the language adopted in the Elements is the adoption of letters to symbolize, in particular, geometrical objects that are represented in the accompanying diagram.6 Through these letters, and also by explicit mention, different mathematical notions are taken into account in the proof. Finally, the diagram itself is used in making inferences like in the case when we see line segments as radii or line segments as the sides of a geometrical object. In simple terms, the proofs in the Elements are made adopting a regimented natural language, mathematical notions (in particular from geometry and about magnitudes), and diagrams. These informal proofs are modeled in the sound formal system E. We can say that these proofs are correct in the sense that we take there to be faithful models of them in E. 3 Proofs in metalogic How do we know that the logical system E is sound? Avigad, Dean, and Mumma showed that E can be expressed in terms of an axiomatization of planar geometry made adopting first-order logic; more precisely we can have a translation between proofs in E and proofs in a fragment of this axiomatic geometry that is valid for so-called "ruler-and-compass" constructions (which are the ones modeled in E) (Avigad, Dean, and Mumma 2009, 744). And the soundness of this system, where does it come from? Ultimately, from the soundness of first-order logic. Here, we are back where we started. And the soundness of first-order logic, where does it come from? That first-order logic is sound is proved with metalogic. When proving a general feature of a logical system – in particular soundness and completeness – we have to engage with the logical system from something more general that supersedes it. We do that using our natural language. And how do we go from addressing a logical system in terms of our natural language to say that we are proving theorems (metatheorems) using a metalogic? We will see with an example that we use the natural language in the proofs in a rather specific way. We can say that we have a regimented use of language, like in the case of the use of language in the proofs in the Elements. Like Euclid that adopts letters as symbols, in particular, of geometric objects, in our use of natural language as a metalanguage we will adopt letters or even new symbols to symbolize first-order logic concepts (e.g. terms, predicates, sentences, interpretations, derivations, etc.). We will also have symbols to include the mathematics of natural numbers and set theory. In this 6 On the relationship between the diagrams and the geometrical objects they represent, see Valente(2020a). 5 way, our natural language as a metalanguage has as resources the logic system itself, arithmetic, and set theory (Yaqub 2015, 87-104). This is not that different from how language is adopted in Euclid's planar geometry. 7 In that case, we have a regimented use of language, and this language is supplemented by concepts and resources from geometry, the mathematics of magnitudes, and diagrams. In the case of logic, we have a formal language and a deductive apparatus. With metalogic, we have a similar situation. Our regimented language (supplemented by special symbolism) has associated with it also a deductive apparatus. On one side, we have the resources of the logical system itself. This does not lead to any problematic circularity (Yaqub 2015, 89; Gensler 2010, 339-40). On the other side, we have deductive resources arising from the mathematics that are taken into account in the language. The most relevant for metalogical proofs is mathematical induction (Krantz 2002, 107-20; Hunter 1971). In this work I am interested in seeing metalanguage at work in proofs about logical systems. In particular, I want to see how different these proofs are from the so-called informal proofs, like the proof of I.1. The first problem I face in the process of writing this paper is to choose the proof I will use to make my point. Taking into account the importance of soundness (e.g. to know that Euclidean proofs are being modeled using a sound logical system), it will be a proof of soundness for first-order predicate logic (PL). There are many proofs available (see, e.g., Teller 1989, 193-9; Smith 2012, 3613; Bergmann, Moor, and Nelson 2014, 244-50; Barker-Plummer, Barwise, and Etchemendy 2011, 525-8; Yaqub 2015, 139-44). How do I choose between these proofs? One of the most important characteristics of formal proofs is that, as Avigad puts it, "a formal derivation with even a single error is simply not a derivation" (Avigad 2020, 8). There is no gradation of rigor in formal proofs. All the inferential steps must be correct. How do I know that regarding the different metalogical proofs I can choose from? Here, I am anticipating the point I am going to make in this paper. There should be no issue regarding a proof in metalogic being more or less rigorous. However, I have no formal means to attest the rigorousness of the different proofs. So, I choose the proof that I consider that enables me to make a clearer presentation (Yaqub 2015, 139-44). The statement of the soundness theorem is as follows: For every set G of PL sentences and every sentence X of PL, if X is a theorem of G, then X is a logical consequence of G. Symbolically, if G |– X, then G |= X. (Yaqub 2015, 43) This proof is made by resorting to mathematical induction. The general pattern of a proof by mathematical induction is as follows: 1. Enumeration. We assign to each object (in this case each line of the derivation) a number 1, 2, 3, ... 2. Base step. We prove that property P holds for the object numbered 1 (in this case, we prove that G |= X for the first line of the derivation). 3. Induction step. We prove that P holds for the object(s) numbered n + 1, assuming that it holds for all objects numbered 1 through n (in this case, we prove that G |= X for line n + 1of the derivation, assuming that it holds for the previous lines). (Smith 2012, 359) Since to make my point, I do not need to look into the whole proof, I will only consider the first part of the proof – the proof of the base step: Let n = 1. The first line of any derivation has no antecedents. So the sentence Z1 is either a premise, an assumption of a hypothetical rule, or an identity statement of the form s = s, which is introduced by the MDS rule Identity. If Z1 is a premise or an assumption, then S1 = {Z1}. It is obvious that, in this case, S1 |= Z1. If Z1 is of the form s = s (where s is any PL singular term), then S1 is empty. But s = s is a valid sentence, that is, it is true on all the PL interpretations that 7 That this is the case is one of the points I want to make in this work. I will make my case simply by showing part of a proof in metalogic and seeing how the language is used and how the available resources are used, just like in the case of the proof of I.1. 6 are relevant to it. By definition, all PL interpretations are models of the empty set, since an interpretation would fail to be a model of the empty set only if the empty set were to contain a sentence that is false on that interpretation; given that the empty set contains no sentences, no PL interpretation fails to be a model of it. Hence s = s is true on every model of the empty set that is relevant to it. In other words, s = s is a logical consequence of the empty set. Therefore, in all cases, S1 |= Z1. This establishes the Base Step. (Yaqub 2015, 140) A brief overview of the above extract already reveals many aspects of what a proof is in metalogic. The reference to arithmetic is already present in the choice of a proof using mathematical induction. We also have in this part of the proof arithmetic symbolism (n = 1). The reference to set theory is present, e.g., with the set S1 = {Z1} and the mention to the empty set. However, the overwhelming majority of symbols and technical terms are related to first-order predicate logic (PL). For example, we have the symbol |= that corresponds to the notion of logical consequence. In natural language, S1 |= Z1 means that Z1 is a logical consequence of S1. If we know first-order predicate logic, the meaning of the terms is clear. When this is the case, the argumentative structure of the proof becomes clear also. The crucial aspect of this proof – shared with all metalogical proofs – is the following: A proof about a formal system is a piece of meaningful discourse, expressed in the metalanguage, justifying a true statement about the system. (Hunter 1971, 11) Now, the proofs in Euclid's Elements are also examples of pieces of "meaningful discourse". There are evident differences. In the Elements, we have diagrams; here, we do not. In the Elements, the adopted regimented language is supplemented, in particular, by special symbolism regarding geometry and the mathematics of magnitudes; here, we take into account natural number theory, set theory, and first-order logic. However, this proof using metalanguage seems much closer to the proofs in the Elements than to a formal proof, like that of I.1 made using E. In fact, I do not have formal means to determine if this proof is "correct" or rigorous "enough" (or even to determine if to ask these questions makes any sense). It is the case that to understand the proof and evaluate its "correctness" I will have to look into the "discourse", like I did in the case of the proof of I.1. Here, we face what I call the groundedness problem. What would it mean to use E without knowing that it is a sound formal system? We would not know if a derivation arrives at a false conclusion, or even if this was not the case, we would not know for sure if there was not some small gap in the deductive structure of the proof. Our trust in E comes from knowing that is was proved that it is a sound system. But now we see that what is sustaining a formal system are a series of metatheorems that seem, too much, to be proved by what we called informal proofs. This is the ground that is sustaining the formal system. And it looks, discomfortingly, that we are skating on thin ice. And if this is not the case, we will not show it by relying on formal languages. It will be through our thinking, at least in part, with natural language that we will conclude that our proof is "correct". Ultimately, we decide on the correctness of the metalogical proof by relying on our reasoning expressed in natural language. So, let us begin to look into the proof and see where it leads us to: 1) Let n = 1. The first line of any derivation has no antecedents We start at the first line of the derivation; as such, it has no antecedents. From or knowledge of PL, we know that the sentence of line 1 – Z1 – can only be one of three possibilities: a premise, an assumption of a hypothetical rule, or an identity statement: 2) So the sentence Z1 is either a premise, an assumption of a hypothetical rule, or an identity statement of the form s = s, which is introduced by the MDS rule Identity. From this, we move into a proof by cases, since there are two cases that need a different treatment: a) Z1 is a premise or an assumption; b) Z1 is an identity statement. We start with a): 7 3) If Z1 is a premise or an assumption, then S1 = {Z1}. It is obvious that, in this case, S1 |= Z1. Sn is the set of all the premises and "active" assumptions that are members of G and appear at line n or before. In the case of line 1, there are no previous lines. In this case, we only have Z1. In this way, S1 = {Z1}. From the definition of logical consequence, it is evident that S1 |= Z1. We have finalized our proof of the base step for case a). We are going now to consider case b). Here, we see that the proof is a "meaningful discourse" at its most. This part is self-explanatory. It consists of a detailed argumentation where the aspects of PL that are necessary to take into account are made explicitly in it: If Z1 is of the form s = s (where s is any PL singular term), then S1 is empty. But s = s is a valid sentence, that is, it is true on all the PL interpretations that are relevant to it. By definition, all PL interpretations are models of the empty set, since an interpretation would fail to be a model of the empty set only if the empty set were to contain a sentence that is false on that interpretation; given that the empty set contains no sentences, no PL interpretation fails to be a model of it. Hence s = s is true on every model of the empty set that is relevant to it. In other words, s = s is a logical consequence of the empty set. With this "meaningful discourse" we finalized our proof of the base step for case b). Having proved that S1 |= Z1 for both cases we can then conclude this part of the proof: Therefore, in all cases, S1 |= Z1. This establishes the Base Step. How do I find this proof convincing? I do not have any formal means to show it; I only rely on my reasoning expressed in natural language to address the text and conclude this. Is this proof more rigorous than Euclid's proofs? That is very unlikely. In both cases, the argumentation is made with a regimented natural language. I would have to argue, e.g., that relying on diagrams is less rigorous than to rely on number theory and set theory. But if we are to trust Avigad, Dean, and Mumma's formal system E, diagrams do not bring any lack of rigor into the proofs. I might try to make the case that, nevertheless, number theory and set theory bring more rigor into the reasoning. But I would be playing with the idea of degrees of rigorousness due to the mathematics that we put in the mix. All this is very far from the rigor of formal proofs. This seems a deadlock. This is why I speak of the groundedness problem. Metalogic simply does not seem to be formal enough for the high task it is commissioned for. 4 Conclusion Informal proofs like the ones in Euclid's Elements might seem to be untrustworthy; they might not be rigorous enough – there is always lurking the suspicion of gaps in the reasoning. We can make amends. We adopt proofs made using a formal system that are faithful to the Euclidean proofs. Disregarding the faithfulness problem, we might say that our informal proofs are correct because of this. But where does the soundness of our formal system comes from? From a proof made using metalogic. The proof in metalogic is made using a regimented language (like the informal proofs we started with). It is a "meaningful discourse" that, it is fair to say, is convincing. But, is it more convincing than a Euclidean proof? It seems very difficult to make a definitive statement regarding this question. Is a metalogical proof sound, or rigorous, or rigorous enough? We have no formal means to decide this. As it is, there is a reasonable doubt regarding the grounding of logic by metalogic. This, I have called the groundedness problem. 8 References Avigad, J. (2020). Reliability of mathematical inference. Synthese, https://doi.org/10.1007/s11229-019-02524-y Avigad, J., Dean, E., and Mumma, J. (2009). A formal system for Euclid's Elements. The Review of Symbolic Logic 2, 700-768. Barker-Plummer, D., Barwise, J., and Etchemendy, J. (2011). Language, proof and logic. Stanford: CSLI Publications. Bergmann, M., Moor, J., and Nelson, J. (2014). The logic book. New York: McGraw-Hill. Gensler, H. (2010). Introduction to logic. New York: Routledge. Heath, T. H. (1956). 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