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The two different layers of logical theory—epistemological and ontological—are considered and explained. Special atten- tion is given to epistemic assumptions of the kind that a judgement is granted as known, and their role in validating rules of inference, namely to aid the inferential preservation of epistemic matters from premise judgements to conclusion judgement, while ordinary Natural Deduction assumptions (that propositions are true) serve to establish the holding of consequence from antecedent propositions to succedent proposition. Keywords Assumption · Consequence · Judgement · Proof-object · Demonstration · Analytic, immediate inference 1 Two Perspectives in Logic over from the previous epistemological one. This ontologi- cal approach in logic began with another nineteenth cen- Following Archbishop Whatley’s Elements of Logic from tury cleric, namely the Bohemian Bernard Bolzano and 1826 we say: his Wissenschaftslehre (1837). As is by now well-known Bolzano avails himself of certain denizens in a Platonic 1.1 Logic may be Considered as the Science, “Third Realm” that are known as Sätze an sich, that is, and also as the Art, of Reasoning propositions-in-themselves, precisely half of which, namely the truths-in-themselves, are true. This notion of truth (-in- When reasoning we carry out acts of passage, “inferences”, itself), also considered as a Platonist in-itself notion, when from granted premises to novel conclusions. Logic is Sci- applied to a proposition (in-itself), serves as the pivot for this ence because it investigates the principles that govern rea- novel rendering of logic. soning and Logic is Art because it provides practical rules In particular, Bolzano reduces the epistemic evaluative that may be obtained from those principles. Reasoning is notions with respect to judgements and inferences, namely par excellence an epistemic matter, dependent on a judging correctness and validity, to various matters of ontology per- agent. If the ultimate starting points for such a process of taining to these propositions-in-themselves. Thus the judge- reasoning are items of knowledge, accordingly a chain of ment [A is true], in which truth is ascribed to the proposition reasoning in the end brings us to novel knowledge. (-in-itself) A that serves in the role of judgemental content, In today’ logic, on the other hand, inferences are not pri- is deemed to be right, or correct (German richtig), if the marily seen as acts, but as production-steps in the generation proposition(-in-itself) in question really is a truth. Similarly of derivations among metamathematical objects known as the inference-scheme, or figure, I: wff’s, that is, well-formed formulae. Furthermore, by the J … J 1 k side of this metamathematical change regarding the status of inferences, an ontological approach has largely taken where each judgment J is of the form proposition A is true, i i is deemed to be valid if, in modern terms, the relation of logical consequence, that is, preservation of truth “under * Göran Sundholm all variations”, holds from the antecedent propositions A , goran.sundholm@gmail.com A , ... A that serve as contents of the premise-judgements 2 k Philosophical Institute, Leiden University, P. O. Box 9515, J … J of the inference I to the proposition C that serves as 1 k 2300 RA Leiden, The Netherlands Vol.:(0123456789) 1 3 552 G. Sundholm content of the conclusion. Another way of formulating the and one semantic |= that indicates “satisfaction” in a suit- second Bolzano reduction may be found in Wittgenstein’s able model. Both turnstiles furthermore are relativized by Tractatus (5.11, 5.35.132, 5.133, 6.1201, 6.1221). The infer- including also assumptions in the guise of antecedent-for- ence J is valid if the implication A & A & ... & A ⊃ C mulae to the left of the respective turnstile, thereby making 1 2 k a logical truth, or, in the Tractarian terminology, a tautol- matters even more complex. The second, model-theoretic ogy. Both formulations of this Bolzano reduction are close notion plays no role in Frege, and his uses of the “syntactic” enough to what Bolzano actually says; his particular cavils turnstile is radically different from the modern one: Frege’s regarding the compatibility of the antecedent propositions, sign serves as a pragmatic assertion indicator, whereas the and his conjunctive, rather than the customary current dis- modern one is a predicate—a propositional function if you junctive reading of consequences with multiple consequent want—that is defined on well-formed formulae. This differ - propositions we may, at the present level of generality, ence is symptomatic of the difference in use between Frege’s disregard. formal language, i.e. his ideography (Begriffsschrift), on the The epistemic conception of traditional logic is all- one hand, and modern formal languages that, as a rule, are out Aristotelian and stems from the early sections of construed meta-mathematically, on the other hand. The lat- the Posterior Analytics. The Aristotelian conception of ter can only be talked about; they are objects of study only, demonstrative science organizes a field of knowledge by but are not intended for use. For instance, in Solomon Fefer- using axioms that are self-evident in terms of primitive man’s authoritative treatment of Gödel’s two Incompleteness concepts and proceeds to gain novel insights by applica- Theorems one finds no “object language”; instead Fefer - tion of similarly self-evident rules of inference. Frege’s man (1960) proceeds directly to the Gödel numbers. Since great innovation in logic can be seen as refining this tra- the object “language” in question is never used for saying ditional Aristotelian axiomatic conception by joining it anything—its “metamathematical expressions” are not real to his notion of a formal language, with its concomitant expressions and do not express, but instead are expressed as notion of logical inference. Frege’s deployment of a novel the referents of real expressions—there is no need to display form of judgement, namely proposition (“Thought”) A is such an object language: it is only talked about, but in con- true, where the content A has function/argument struc- tradistinction to other languages, it is not a vehicle for the ture P(a), allowed him to develop a much richer view of expression of thoughts. what follows from what, in particular when drawing upon Frege’s ideography, on the other hand, is an interpreted quantification theory. He did not change anything, though, formal language, and he spent a tremendous effort on with respect to epistemic demonstration (Beweis), which meaning explanations, for instance, in the early sections remains Aristotelian through and through. Thus, both the of Begriffsschrift , for the predicate logic version of the ide- Preface to the Begriffsschrift as well §3 of Grundlagen ography from 1879, and in the opening sections §§1–32 der Arithmetik bear strong resemblance to the well-known of Grundgesetze der Arithmetik, Vol I, from 1893, espe- regress argument unto first principles, with which Aristo- cially the §§27–31. It should be noted that this Grundge- tle opens the Posterior Analytics. setze version of the Fregean ideography is not a predicate logic, but a term logic, which sometimes serves to make matters hard to understand when viewed from the preva- 2 Two Views on Logical Language lent standard of today, where theories are routinely formu- lated in predicate logic. In Frege’s late piece of writing, Aristotle’s detailed account of consequence from the Prior the Nachlass fragment Logische Allgemeinheit that was left Analytics, on the other hand, was of course superseded by uncompleted at the time of his death, we find a distinction Frege’s introduction of the formal ideography that comprises between a Hilfssprache and a Darlegungssprache. The Edi- also quantification theory. Frege’s conception of a formal tors of Frege’s “Posthumous Writings” deliberately point language, though, was different from our modern notion to Tarski and translate Hilfssprache as object-language and of a formal language (or perhaps better today: formal sys- Darlegungssprache as meta-language. This translation, tem) that distinguishes between syntax and semantics and however, is not felicitous. The term Hilfssprache is the deploys two turnstiles: one “syntactic” |– that really is a metamathematical theorem-predicate with respect to wff’s, and indicates the existence of a suitable formal derivation, Barnes (2002) convincingly argues the use of the term ideography as a translation of German Begriffsschrift. Sundholm (2002) treats in some detail of the distinction between Detailed attributions in the Wissenschaftslehre for the claims expressions and metamamthematical expressions, whereas the advan- regarding Bolzano can be found in my 2009, § 3 ‘Revolution: Bolza- tages of interpreted formal languages are argued in Sundholm (2001, no’s Annus Mirabilis’. 2003). 1 3 The Neglect of Epistemic Considerations in Logic: The Case of Epistemic Assumptions 553 German rendering of the French langue auxiliaire, which of applying positivistic verificationism to mathematics. Here term stands for the artificial languages that were considered equations between finitistically computable terms serve as in the artificial languages movement, of which Frege’s cor - analogues of positivist observation sentences. Such formulae respondents Couturat and Peano were prominent members. [s = t] are even known as “verifiable propositions” in the Examples that spring to mind are Volapük, Bolak, Espe- magisterial Hilbert and Bernays (1934, 1939). ranto, and today also Klingon, and on the scientific side In the Warsaw seminar of Lukasiewicz and Tarski during Interlingua, Latine sine flexione in which Peano wrote a the second half of the 1920s, the study of formal languages famous paper on differential equations. Frege’s Begriffss- and formal systems—Many-valued Logics!—was liberated chrift is precisely such an artificial auxiliary language—a from the Göttingen finitist ideological shackles of Hilbert. Hilfssprache—and the difference between it and other aux- From hence on ordinary mathematical means were allowed iliary languages is that it is a formal one. Nevertheless, in the meta-mathematical study of formal systems, much in just as Esperanto and Volapük, it was intended for express- the same way that naïve set theory was used in the develop- ing meaning, and accordingly one needs a “language of ment of set theoretic topology and cardinal arithmetic at display” in order to set it out properly. All the languages which Polish mathematicians then excelled. With this liber- in the Russell -Tarski tower of “meta-languages” (over the ating move, yet a further radical shift of perspective occurs. first object-language) are also object-languages , and are The formal systems no longer serve any epistemological ultimately only spoken about. The real meta-language is role per se. Instead, strictly speaking, the “well-formed Curry’s “U language”—U for use—and it needs a vantage formulae” lack meaning, and do not as such express. They point outside the Russell–Tarski hierarchy in question. are mathematical objects on par with other mathematical Frege’s Darlegungssprache matches Curry’s U language objects; in fact, formally speaking, the meta-mathematical and his Hilfssprache is an auxiliary language like Volapük, expressions are elements of freely generated semi-groups of Bolak, and Esperanto championed by Couturat and Peano strings. With this shift in the role of the “languages” of logic, (Interlingua, Latine sine flexione). epistemic matters are driven even further into the back- Of course, the two different versions of Frege’s ideogra- ground. The logical calculi are not used for epistemological phy in Begriffsschrift and Grundgesetze are Hilfssprachen purposes anymore. One only proves theorems about them. and must be explained, that is, dargelegt, or spelled out. During the 1920s the Grundlagenstreit came to the fore The editors of the Nachlass compliment Frege for having and sharp epistemological problems were raised. After Brou- here anticipated the precise object-language/meta-language wer’s criticism of the unlimited use of the Law of Excluded distinction that was put firmly onto the philosophical firma- Middle, there appear to be only two viable options with ment a decade later by Carnap (1934) in Logische Syntax respect to logic. We may keep Platonistic impredicativity der Sprache and by Tarski in Der Wahrheitsbegriff in den and LEM as freely used in classical analysis after the fash- formalisierten Sprachen. However, as we saw Frege’s Hilf- ion of Weierstrass, or we may jettison them. We have already ssprache is not an artefact void of meaning, that is, it is not seen the other dichotomy of options, namely to consider an uninterpreted, “object-language”: on the contrary, it is formal systems based on languages with meaning, on the an auxiliary language in the terminology of the artificial one hand, and based on uninterpreted formal calculi, on the language movement. other. After Gödel’s work, attempts to resuscitate Fregean Up to ± 1930 every logician of note followed Frege’s lead logicism, for instance by Carnap, no longer seemed viable when constructing formal calculi, marrying their formal lan- and were abandoned: retaining classical logic as well as guages to the Aristotelian conception of Science: Whitehead impredicativity, while insisting on explicit meaning-explana- and Russell, Ramsey, Lesniewski, early Carnap (Aufbau and tions that render axioms and rules of inference self-evident, Abriss), Curry, Church, early Heyting …. Their systems simply seems to be asking too much. Thus we may jettison were interpreted calculi intended as epistemological tools. either meaning for the full formal language, while retaining The mathematical study of mathematical language was natu- classical logic and impredicativity, which is the option cho- rally begun by Hilbert as part of his ideological programme sen by Hilbert’s formalism. Only his “real” sentences, that is, the “verifiable” equations between finitist terms, and which serve as the analogue to the observation sentences of posi- tivism, have meaning, whereas other sentences, the “ideal” I owe my awareness of these origins of Frege’s Hilfssprache to the ones, strictly speaking, are not given meaning-explanations. scholarship of Wolfgang Künne, cf. Künne (2010), Chap. 5, §5, pp. For the second option on the other hand we may jettison 725–738. 5 classical logic and Platonist impredicativity, but then offer Russell’s Introduction to Wittgenstein (1922) and Tarski (1936). Curry (1963), Chap 2, §§1 and 2, is the locus classicus for the U language. 7 8 Sundholm (2001). ((1934, §6) section c, third part: Verifizierbare Formeln. 1 3 554 G. Sundholm meaning explanations for constructivist language after the terms of the Heyting proof-explanation does not force us to now familiar fashion of Heyting. abandon classical logic. This, however, yields no epistemic benefits, and so I prefer to use the Brouwer–Weyl construc- tive notion of existence with respect to types α. When α is Classical logic a type (general concept), /\ exists Accept Reject Language with content is a judgement and its assertion condition is given by a rule of instantiation Yes Logicism Intuitionism a is an No Formalism ? exists We note that propositions are given by truth-conditions The hope of Carnap and others for meaning-explanations that are defined in terms of (canonical) proofs, and (epis- for the full language of say, second order analysis that ren- temic) judgements are explained in terms of assertion condi- der evident classical logic and impredicativity appears to tions. Thus we get an ensuing bifurcation of notions at both be forlorn. We may then follow Hilbert confining meaning the ontological level of propositions, their truth, and their only to a “real” fragment, while the “ideal sentences” of full proofs (that is, their truthmakers), and on the epistemic level language remain uninterpreted, or we may jettison classical of judgements and their demonstrations. logic and impredicativity, and follow Heyting’s by now well- In the table below the epistemological and ontological known way of giving constructive meaning-explanations two sides of logic are spelled out for a fairly large number of with respect to the full language. notions, and in other writings I have dealt with most of the lines. In the sequel of the present paper I intend to deal with the line contrasting an assumption that a proposition is true 3 Constructive Meaning‑Explanations with an epistemic assumption that a judgement is known, and the Two Layers of Logic with as a special case an assumption that a proposition is known to be true. With his Constructive Type Theory Per Martin-Löf has given streamlined form to Heyting’s “Proof Explanation Epistemic notion Ontological (“Alethic”) notion of the intuitionistic logical constants”: a proposition A is Judgement (assertion) Proposition explained by laying down how its canonical proofs may Demonstration Proof (-object), truthmaker be put together out of parts (and when two such canoni- Truth of judgement Truth of proposition cal proofs are equal canonical proofs of the proposition Demonstrability Existence of proof A). Accordingly, for each proposition A, we have a “type” Self-evident/mediated Direct/indirect Proof(A) and define a notion of truth for propositions by Axiomatic/derived Canonical/non-canonical means of an application of the truthmaker analysis:A is Intuitive/discursive Simple/composite true = Proof(A) exists. Inference Consequence Here the relevant notion of existence cannot be, on pain Validity Holding of an infinite regress, that of the existential quantifier. Classi- Assumption that a judgement is Assumption that a proposition is cally, we may choose it to be Platonist set-theoretic existence known true and drawing upon classical reasoning one readily checks that Hypothetical demonstration Dependent proof-object the semantics verifies the Law of Excluded Middle. Thus, Hypothetical judgement Implicational proposition if we are prepared to reason Platonistically when justifying Definitional (criterial) equality Propositional identity the rules of inference and axioms, casting the semantics in (Function) Generality Quantifier The various options regarding retention of classical reasoning and As is well known, Tarski’s definition of truth does not on its own meaning explanations are spelled out in some details in my 1998a. yield the Law of Excluded Middle for the notion of truth thus defined. Martin-Löf (1984). Classical reasoning in the meta-theory is required for that. In my (2004) I carry out the pendant reasoning and show that, when classi- A fairly comprehensive introduction to Martin-Löf’s CTT can be cal meta-theory is allowed, it is very easy to validity LEM, also under found in my (1977). See also the paper by Ansten Klev in the present the Heyting semantics. issue of TOPOI. That Heyting’s explanation of truth as existence of a proof (-object) is a kind of truth-maker analysis was first suggested in In my (1997), (2000), and (2012) the demonstration versus proof my (1994a). distinction is given more substance. 1 3 The Neglect of Epistemic Considerations in Logic: The Case of Epistemic Assumptions 555 where b is a dependent proof of B, under the assumption 4 Four Different Notions of Consequence that x is a proof(A), and have a special application function ap(y,x), whereas application in the case of f is primitive: Apart from the two changes already indicated—the meta- mathematical shift and the Bolzano reduction of inferential when a ∶ Proof(A), then f(a) ∶ Proof(B). validity to logical truth (or logical consequence) in “all vari- Fact 2 The judgement (1)–(3) have different meaning- ations”—we then have occasion to consider another major explanations—their assertion conditions are not the same— invention of the early 1930s, namely Gentzen’s Natural and accordingly do not mean the same, are not synonymous, Deduction derivations and his Sequent Calculi. while (4) indicates acts of passage. The first three notions, Within the interpreted perspective of an interpreted for- however, are equi-assertible. Given a verification-object mal language, with respect to two propositions A and B, for one of the three, verification-objects for the other two there are at least four relevant notions of consequence here. are readily found in a couple of trivial steps. Furthermore, all four relations are refuted by the same counter-example, (1) the implication proposition A⊃B, which may be true namely a situation in which A is known to be true and B (or even logically true “in all variations”); known to be false. This might serve to explain why the four (2) the conditional [if A is true then B is true], notions have sometimes been hard to keep apart, especially or, in other words, from the classical point of view. B is , that A is true Fact 3 Bolzano deals ably with consequence, whereas his that A is true account of inference is inadequate and quite psychologistic under assumption that A is true in terms of Gewissmachungen. Frege, on the other hand, deals ably with inference, but (logical) consequence has (3) the consequence [A = > B] may hold; no place in his system. Only with Gentzen’s 1936 sequen- (4) the inference [A is true. Therefore: B is true] may be tial formulation of Natural Deduction, where the derivable valid. objects are sequents, that is consequences, and where the principal introduction and elimination inferences all take Fact 1 “implies” takes that-clauses, whereas “if-then” place to the right of the sequent-arrow, do we get a system takes complete declaratives. Ergo:implication and condi- that can cope both with inference and consequence. tional are not the same. The conditional (2) is a hypotheti- Fact 4 Consequence, not logical consequence, is the cal judgement in which hypothetical truth is ascribed to the primary notion. Gentzen’s system deals with arithmetic; proposition B. Its verification-object is a dependent proof- his rules of inference that take us from premise-sequent(s) object b:Proof(B) [x:Proof(A)], that is, b is a proof of B to conclusion-sequent are obviously valid, but they do not under the assumption (hypothesis, supposition) that x is a hold logically in all variations. They are only “arithmeti- proof of A. cally valid”. The consequence (3) is a Gentzen sequent (German Fact 5 A completeness theorem for an interpreted formal Sequenz). (Why, we may ask, did Gentzen drop the prefix language would state: all truths (and in the case of Gentzen’s Kon here?) system: all sequents that hold) are derivable by means of The judgement these rules. For Gödelian reasons, interesting systems with theorems of the form [A is true] are not complete. [A => B] When we now consider how one would establish that is a generalization of [A is true] and demands for its veri- (1) to (4) obtain, we see that for (1)–(3) ordinary natural fication a mapping (higher-level function) f: Proof(A) → deduction derivations are involved in one way or another. In Proof(B). Since implication and conditional are different, this is not the proof-object demanded for the truth of an implication: these have the canonical form λ (A, B, [x]b), or if your prefer the logical formulation, rather than the set- theoretical one: The afterword to my (2012) gives more details concerning the kinds of function—Euler-Frege functions, Dedekind mappings, and ⊃ I(A, B, [x]b), courses-of-value—that serve as verification witnesses for, respec- tively, conditionals, closed consequences (“sequents”), and implica- tional propositions. Volume III of the Wissenschaftslehre contains Bolzano’s account of Gewissmachungen. In (2006), at p. 632, and (2009), at p. 298, the links between Frege and Gentzen are explored further. My (1998) and (2012) explain the inter-relations of notions (1)–(4) in considerable detail. I explore these Gödel phenomena in (2004a, §8). 1 3 556 G. Sundholm all three cases one needs a hypothetical proof b:Proof (B) 6 Epistemic Assumptions [x:Proof(A)]. The implication A⊃B is established by forming the Instead, validity of inference, rather than (logical) holding course-of-value λ (A, B, [x]b), whereas the conditional of consequence, involves preservation, or transmission, of is already established by the hypothetical, dependent epistemic matters from premises to conclusion and it is here proof-object in question. Finally, forming the function [x] that epistemic assumptions that judgements are known (or b:Proof(A) → Proof(B) by means of “lambda” abstraction granted) become helpful. In order to validate the inference [] (Curry’s notation!) on the hypothetical proof establishes I one makes the assumption that one knows the premise- that the closed consequence (“sequent”) holds. judgments, or that they are being given as evident, and under this epistemic assumption one has to make clear that also the conclusion can be made evident. 5 Blind Judgement and Inference The difference between the two types of assumptions is especially clear when we consider Gentzen derivations in Under the Bolzano reduction, when the proofs (“verification Natural Deduction. An ordinary assumption A of Natural objects”) work also in all variations, then classically one Deduction corresponds to an alethic, ontological assump- says that the inference (4) is valid. However, the Bolzano tion that proposition A is true. From such an assumption we reduction validates what we may, in the excellent terminol- may, for instance, obtain a conclusion that B is true, when ogy of Brentano, call blind judgement and inference. The we have already established the conditional judgement,($) epistemic link to the judging reasoner has here been severed, B is true, on hypothesis that A is true, whereas I am concerned to preserve this link. Furthermore, if we wish to do so, from this we readily Consequence preserves truth from antecedent proposi- obtain also the outright assertion that the implication A⊃B tions to consequent proposition, and logical consequence is true by implication introduction, or, for that matter, if we does so “under all variations”. The demonstration of the so wish, but now with the aid of functional abstraction on Prime Number Theorem (PNT) by De la Vallée-Poussin the dependent proof-object that warrants ($), we also may and Hadamard in 1896 certainly could be formalized within conclude that the sequent [A → B] holds. NBG, the set theory of Von Neumann, Bernays and Gödel. An epistemic assumption that a judgement [A is true] is Since this theory is finitely axiomatized, we may conjoin its known, or perhaps better granted, corresponds for Natural axioms into one proposition VNBG and then consider the Deduction derivations to the hypothesis that we have been inference provided with a closed derivation of the proposition A. This is patently a different kind of assumption from the ordinary VNBG is true ( ) Natural Deduction assumption of the wff A. PNT is true Brouwer did not accept hypothetical proofs—I hesitate to The inference (*), certainly, is truth-preserving, in the call them proof-objects in his case. His proofs are all epis- in the light of the formalized demonstration offered and the temic demonstrations: an assumption that a proposition is Soundness Theorem for the Predicate Calculus: every time true amounts to an assumption that the assumed proposition an NBG axiom is used in the predicate logic derivation we is known to be true, for instance in his demonstration of the replace it by the proposition VNBG and then apply conjunc- Bar Theorem. tion elimination. Hence we get a formal derivation of PNT from VNBG, whence the Soundness Theorem guarantees truth-preservation. So under the Bolzano reduction this is a valid inference, because truth-preserving under all varia- tions, but it provides no epistemic insight at all. Martin-Löf (1984), for instance at p. 41, avails himself of epis- temic assumptions”“Assuming that we know the premisses …” (my emphasis). He does not, however, then formulate the explicit notion, which, or so it appears, was introduced in my (1997, p. 210). Brentano (1889, Anm. 27, pp. 64–72) and Brentano (1930), Brouwer’s Demonstration of the Bar Theorem, with its particular where, in particular, the fragments in part IV are important. use of an epistemic assumption, is discussed in detail by Sundholm Mendelson (1964, Chap. 4) contains a rich exposition of NBG. and Van Atten (2008). 1 3 The Neglect of Epistemic Considerations in Logic: The Case of Epistemic Assumptions 557 7 Gentzen’s Two Frameworks for Natural 8 Epistemic Assumptions and Analytic Deduction Ans Epistemic Assumptions Validation of Inferences Over the past decades I have had a discussion with Dag In recent work, Per Martin-Löf has given an interesting dia- Prawitz about the status of the proofs in the BKH explana- logical twist to epistemic assumptions. Already in his first tion: I have claimed that they are not demonstrations with 1946 paper on performatives, etc., John Austin wrote: epistemic power, but that they are mathematical witnesses, If I say “S is P” when I don’t even believe it, I am corresponding to truthmakers in currently popular theories lying: if I say it when I believe it but am not sure of it, of grounding. Prawitz, on the other hand, has held that they I may be misleading but I am not exactly lying. ……… are epistemically binding. With my present terminology When I say “I know”, I give others my word: I give I can formulate my principal objection thus: the distinc- others my authority for saying that “S is P”. tion between epistemic and alethic assumptions collapses if proofs are held to be epistemically binding. There will be Assertions contain implicit, first-person knowledge no difference between assuming that proposition A is true claims (recall G. E. Moore and asserting that it is raining, but and assuming that one knows that A is true. that one does not believe it!), so assertions grant authority. In type theory the difference between the two kinds When I first read Austin in 2009 I was led to formulate an of assumption comes out in different treatments of Inference Criterion of the same kind: proof-objects. An ordinary assumption has the form When I say “Therefore” I give others my authority for x:Proof(A):assume that x is a proof for A asserting the conclusion, given theirs for asserting the An epistemic assumption with respect to the same propo- premisses. sition takes a closed proof-object as given:assume that I am given a closed proof a:Proof(A) Martin-Löf has now noted that one does not need to know Against the background of these distinctions we can now that the premises are evident for the validation of an infer- explain the difference between the two Gentzen frameworks ence: what one must be prepared to undertake is to make for Natural Deduction. the conclusion known or evident under the assumption that The 1932 format from the dissertation is the usual one someone else grants the premises as evident. with assumption formulae as top nodes in derivations In order to undertake that responsibility it is enough if I possess a chain of immediately evidence-preserving steps (in A … A 1 k terms of meaning-explanations) that link premises to conclu- Π∶ .. sion. Here the introduction rules of Gentzen may be seen as immediate and meaning explanatory, whereas the elimi- nation rules are immediate, but not meaning explanatory. In 1936 format, on the other hand, is an axiomatic calculus for Kantian terms, both the introduction and elimination rules deriving consequences of the form, where the assumption are analytically valid, but only the introduction rules are formulae are listed explicitly analytic, or “identical”, whereas the analyticity of A … A → C 1 k the elimination rules is implicit, and might need to be made 1936 derivations are best seen as demonstrations of judg- explicit in terms of the meaning explanations offered by the ments of the form: introduction rules, in analogy with: A … A → C 1 k All rational animals are rational Derivations in the 1932 format, on the other hand, are to my mind best seen, not as epistemic demonstrations, but as is an explicitly analytic (identical) judgement, whereas dependent proof-objects Π of the form Π∶ C x A … x ∶ A , 1 1 k k that is, Π is a proof of C under the assumptions that x … x 1 k are proofs of A … A , respectively. 1 k In lectures at SND, Paris 2015, and at Marseille 2016, at the meet- ing that provides the source for the present issue of TOPOI. 23 26 For an early instalment in this debate, see my 2000, with a reply Austen (1946, p. 171). by Prawitz in the same issue of Theoria. I suggested this treatment of inferential validity in an invited lec- My (2006) is devoted to spelling out the differences, with respect ture at LOGICA 1996, and published it the next year in the LOGICA to an interpreted calculus, between Gentzen’s 1932 and 1936 ways of Yearbook; it is now readily available in my (2012), p. 950. It is also setting out his derivations. dealt with in (2004a), pp. 454–455. 1 3 558 G. Sundholm All humans are rational models”. Natural Deduction added one more feature here to the dethroning of axioms: they now become ordinary is also an analytic judgement, but only implicitly so, and assumptions among other ordinary assumptions, but as such one resolution-step, replacing the term human by its defini- they are privileged, because they need never be discharged, tion rational animal, is needed to bring this judgement to and may be discounted, when standing in antecedent posi- explicitly analytic form. tion in consequences. Nevertheless, contrary to axioms in In order to complete the comparison, we consider the the old-fashioned sense, they are not known, nor are they question: asserted whenever they occur. An axiom in the old sense was not an assumption: it was asserted, whereas now that Why is &-elimination rule valid? epistemic status is gone, and instead axioms are unasserted assumptions among other assumptions, with the privilege of We are then, in an epistemic assumption, given as evident not carrying the onus of discharge on them. the premise-judgement In conclusion then let me just note that epistemic assump- tions are well known in mathematical practice when one (i) c:Proof (A&B) draws upon a lemma, the demonstration of which is left out for an application of &-elimination. until the main demonstration has been completed. Never- Under this epistemic assumption we have to make theless, within the main demonstration, the lemma does not evident the conclusion work as an additional assumption, but avails itself of asser- (ii) p(c): Proof(A). toric force, even though proper grounding by means of a Since c is a proof of A&B, it executes, (evaluates, is demonstration is as yet absent. A very clear case here is the definitionally equal) to a canonical proof of A&B that so-called Zorn’s Lemma, whose epistemic status is highly accordingly has the form debatable from the point of view of constructivism, but clas- (iii) <a,b>: Proof(A&B) and c = < a,b>: Proof(A&B), sically is granted axiomatic status. where we know that Acknowledgements I have written about these topics since 1996, and (iv) a :Proof(A) and b:Proof(B). spoken since 2013 at workshops in Groningen (2013), Paris (2014), But granted this, it is a meaning stipulation for the Petropolis (2014), Heijnice (2014), Hamburg (2015), Marseille (2016), ordered-pair- and projection-operators that and Prague (2016). I am indebted to the organizers for generous invita- (v) p(< a,b>) = a:Proof(A) tions and to participants for welcome comments and objections. Since there is a lot of material already in print, I have not endeavoured to but, since c = < a,b>: Proof(A&B), we also get make the present text self-contained, but have referred to fuller pres- p(c) = p(< a,b>) = a :Proof(A), whence we are done. entations of mine that are readily available on line. I am indebted to Ansten Klev, Per Martin-Löf, and Dag Prawitz for long-term discus- Note here these deliberations are all pursuant to the rel- sion of these issues. By now they are probably responsible for some things said in this paper, but they cannot be held to be so. My Leiden evant meaning explanations for the notions Proof, &, < >, colleague Arthur Schipper read a penultimate draft and offered help and p. The step from (i) to (iii) and (iv) matches the resolu- with proof reading. tion- step that replaces human by rational animal. Compliance with Ethical Standards 9 Axiom and Lemma from an Epistemic Conflict of interest The author declared that he has no conflict of in- Point of View terest. Open Access This article is distributed under the terms of the Crea- Finally, what does this mean for axioms in the traditional tive Commons Attribution 4.0 International License (http://creat iveco sense? Such axioms were self-evident judgements, and mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- known as such. 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Published: Jun 4, 2018
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