The Neglect of Epistemic Considerations in Logic: The Case of Epistemic Assumptions
 867 Downloads
Abstract
The two different layers of logical theory—epistemological and ontological—are considered and explained. Special attention 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 Proofobject Demonstration Analytic, immediate inference1 Two Perspectives in Logic
Following Archbishop Whatley’s Elements of Logic from 1826 we say:
1.1 Logic may be Considered as the Science, and also as the Art, of Reasoning
When reasoning we carry out acts of passage, “inferences”, from granted premises to novel conclusions. Logic is Science because it investigates the principles that govern reasoning and Logic is Art because it provides practical rules that may be obtained from those principles. Reasoning is par excellence an epistemic matter, dependent on a judging agent. If the ultimate starting points for such a process of reasoning are items of knowledge, accordingly a chain of reasoning in the end brings us to novel knowledge.
In today’ logic, on the other hand, inferences are not primarily seen as acts, but as productionsteps in the generation of derivations among metamathematical objects known as wff’s, that is, wellformed formulae. Furthermore, by the side of this metamathematical change regarding the status of inferences, an ontological approach has largely taken over from the previous epistemological one. This ontological approach in logic began with another nineteenth century cleric, namely the Bohemian Bernard Bolzano and his Wissenschaftslehre (1837). As is by now wellknown Bolzano avails himself of certain denizens in a Platonic “Third Realm” that are known as Sätze an sich, that is, propositionsinthemselves, precisely half of which, namely the truthsinthemselves, are true. This notion of truth (initself), also considered as a Platonist initself notion, when applied to a proposition (initself), serves as the pivot for this novel rendering of logic.
The epistemic conception of traditional logic is allout Aristotelian and stems from the early sections of the Posterior Analytics. The Aristotelian conception of demonstrative science organizes a field of knowledge by using axioms that are selfevident in terms of primitive concepts and proceeds to gain novel insights by application of similarly selfevident rules of inference. Frege’s great innovation in logic can be seen as refining this traditional Aristotelian axiomatic conception by joining it to his notion of a formal language, with its concomitant notion of logical inference. Frege’s deployment of a novel form of judgement, namely proposition (“Thought”) A is true, where the content A has function/argument structure P(a), allowed him to develop a much richer view of what follows from what, in particular when drawing upon quantification theory. He did not change anything, though, with respect to epistemic demonstration (Beweis), which remains Aristotelian through and through. Thus, both the Preface to the Begriffsschrift as well §3 of Grundlagen der Arithmetik bear strong resemblance to the wellknown regress argument unto first principles, with which Aristotle opens the Posterior Analytics.
2 Two Views on Logical Language
Aristotle’s detailed account of consequence from the Prior Analytics, on the other hand, was of course superseded by Frege’s introduction of the formal ideography that comprises also quantification theory. Frege’s conception of a formal language, though, was different from our modern notion of a formal language (or perhaps better today: formal system) that distinguishes between syntax and semantics and deploys two turnstiles: one “syntactic” – that really is a metamathematical theorempredicate with respect to wff’s, and indicates the existence of a suitable formal derivation, and one semantic = that indicates “satisfaction” in a suitable model. Both turnstiles furthermore are relativized by including also assumptions in the guise of antecedentformulae to the left of the respective turnstile, thereby making matters even more complex. The second, modeltheoretic notion plays no role in Frege, and his uses of the “syntactic” turnstile is radically different from the modern one: Frege’s sign serves as a pragmatic assertion indicator, whereas the modern one is a predicate—a propositional function if you want—that is defined on wellformed formulae. This difference is symptomatic of the difference in use between Frege’s formal language, i.e. his ideography (Begriffsschrift), on the one hand, and modern formal languages that, as a rule, are construed metamathematically, on the other hand.^{2} The latter can only be talked about; they are objects of study only, but are not intended for use. For instance, in Solomon Feferman’s authoritative treatment of Gödel’s two Incompleteness Theorems one finds no “object language”; instead Feferman (1960) proceeds directly to the Gödel numbers. Since the object “language” in question is never used for saying anything—its “metamathematical expressions” are not real expressions and do not express, but instead are expressed as the referents of real expressions—there is no need to display such an object language: it is only talked about, but in contradistinction to other languages, it is not a vehicle for the expression of thoughts.^{3}
Frege’s ideography, on the other hand, is an interpreted formal language, and he spent a tremendous effort on meaning explanations, for instance, in the early sections of Begriffsschrift, for the predicate logic version of the ideography from 1879, and in the opening sections §§1–32 of Grundgesetze der Arithmetik, Vol I, from 1893, especially the §§27–31. It should be noted that this Grundgesetze 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 prevalent standard of today, where theories are routinely formulated in predicate logic. In Frege’s late piece of writing, the Nachlass fragment Logische Allgemeinheit that was left uncompleted at the time of his death, we find a distinction between a Hilfssprache and a Darlegungssprache. The Editors of Frege’s “Posthumous Writings” deliberately point to Tarski and translate Hilfssprache as objectlanguage and Darlegungssprache as metalanguage. This translation, however, is not felicitous. The term Hilfssprache is the German rendering of the French langue auxiliaire, which term stands for the artificial languages that were considered in the artificial languages movement, of which Frege’s correspondents Couturat and Peano were prominent members.^{4} Examples that spring to mind are Volapük, Bolak, Esperanto, and today also Klingon, and on the scientific side Interlingua, Latine sine flexione in which Peano wrote a famous paper on differential equations. Frege’s Begriffsschrift is precisely such an artificial auxiliary language—a Hilfssprache—and the difference between it and other auxiliary languages is that it is a formal one. Nevertheless, just as Esperanto and Volapük, it was intended for expressing meaning, and accordingly one needs a “language of display” in order to set it out properly. All the languages in the Russell Tarski tower of “metalanguages” (over the first objectlanguage) are also objectlanguages, and are ultimately only spoken about.^{5} The real metalanguage is Curry’s “U language”—U for use—and it needs a vantage point outside the Russell–Tarski hierarchy in question.^{6} Frege’s Darlegungssprache matches Curry’s U language and his Hilfssprache is an auxiliary language like Volapük, Bolak, and Esperanto championed by Couturat and Peano (Interlingua, Latine sine flexione).
Of course, the two different versions of Frege’s ideography in Begriffsschrift and Grundgesetze are Hilfssprachen and must be explained, that is, dargelegt, or spelled out. The editors of the Nachlass compliment Frege for having here anticipated the precise objectlanguage/metalanguage distinction that was put firmly onto the philosophical firmament a decade later by Carnap (1934) in Logische Syntax der Sprache and by Tarski in Der Wahrheitsbegriff in den formalisierten Sprachen. However, as we saw Frege’s Hilfssprache is not an artefact void of meaning, that is, it is not an uninterpreted, “objectlanguage”: on the contrary, it is an auxiliary language in the terminology of the artificial language movement.
Up to ± 1930 every logician of note followed Frege’s lead when constructing formal calculi, marrying their formal languages to the Aristotelian conception of Science: Whitehead and Russell, Ramsey, Lesniewski, early Carnap (Aufbau and Abriss), Curry, Church, early Heyting …. ^{7}Their systems were interpreted calculi intended as epistemological tools. The mathematical study of mathematical language was naturally begun by Hilbert as part of his ideological programme of applying positivistic verificationism to mathematics. Here equations between finitistically computable terms serve as analogues of positivist observation sentences. Such formulae [s = t] are even known as “verifiable propositions” in the magisterial Hilbert and Bernays (1934, 1939).^{8}
In the Warsaw seminar of Lukasiewicz and Tarski during the second half of the 1920s, the study of formal languages and formal systems—Manyvalued Logics!—was liberated from the Göttingen finitist ideological shackles of Hilbert. From hence on ordinary mathematical means were allowed in the metamathematical study of formal systems, much in the same way that naïve set theory was used in the development of set theoretic topology and cardinal arithmetic at which Polish mathematicians then excelled. With this liberating move, yet a further radical shift of perspective occurs. The formal systems no longer serve any epistemological role per se. Instead, strictly speaking, the “wellformed formulae” lack meaning, and do not as such express. They are mathematical objects on par with other mathematical objects; in fact, formally speaking, the metamathematical expressions are elements of freely generated semigroups of strings. With this shift in the role of the “languages” of logic, epistemic matters are driven even further into the background. The logical calculi are not used for epistemological purposes anymore. One only proves theorems about them.
During the 1920s the Grundlagenstreit came to the fore and sharp epistemological problems were raised. After Brouwer’s criticism of the unlimited use of the Law of Excluded Middle, there appear to be only two viable options with respect to logic. We may keep Platonistic impredicativity and LEM as freely used in classical analysis after the fashion of Weierstrass, or we may jettison them. We have already seen the other dichotomy of options, namely to consider formal systems based on languages with meaning, on the one hand, and based on uninterpreted formal calculi, on the other. After Gödel’s work, attempts to resuscitate Fregean logicism, for instance by Carnap, no longer seemed viable and were abandoned: retaining classical logic as well as impredicativity, while insisting on explicit meaningexplanations that render axioms and rules of inference selfevident, simply seems to be asking too much. Thus we may jettison either meaning for the full formal language, while retaining classical logic and impredicativity, which is the option chosen 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 positivism, have meaning, whereas other sentences, the “ideal” ones, strictly speaking, are not given meaningexplanations.
For the second option on the other hand we may jettison classical logic and Platonist impredicativity, but then offer meaning explanations for constructivist language after the now familiar fashion of Heyting.^{9}
The hope of Carnap and others for meaningexplanations for the full language of say, second order analysis that render evident classical logic and impredicativity appears to be forlorn. We may then follow Hilbert confining meaning only to a “real” fragment, while the “ideal sentences” of full language remain uninterpreted, or we may jettison classical logic and impredicativity, and follow Heyting’s by now wellknown way of giving constructive meaningexplanations with respect to the full language.
3 Constructive MeaningExplanations and the Two Layers of Logic
With his ConstructiveTypeTheory Per MartinLöf has given streamlined form to Heyting’s “Proof Explanation of the intuitionistic logical constants”: a proposition A is explained by laying down how its canonical proofs may be put together out of parts (and when two such canonical proofs are equal canonical proofs of the proposition A).^{10} Accordingly, for each proposition A, we have a “type” Proof(A) and define a notion of truth for propositions by means of an application of the truthmaker analysis:A is true = Proof(A) exists.^{11}
We note that propositions are given by truthconditions that are defined in terms of (canonical) proofs, and (epistemic) judgements are explained in terms of assertion conditions. Thus we get an ensuing bifurcation of notions at both the ontological level of propositions, their truth, and their proofs (that is, their truthmakers), and on the epistemic level of judgements and their demonstrations.^{13}
In the table below the epistemological and ontological two sides of logic are spelled out for a fairly large number of 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 with an epistemic assumption that a judgement is known, with as a special case an assumption that a proposition is known to be true.
Epistemic notion  Ontological (“Alethic”) notion 

Judgement (assertion)  Proposition 
Demonstration  Proof (object), truthmaker 
Truth of judgement Demonstrability  Truth of proposition Existence of proof 
Selfevident/mediated Axiomatic/derived  Direct/indirect Canonical/noncanonical 
Intuitive/discursive  Simple/composite 
Inference  Consequence 
Validity  Holding 
Assumption that a judgement is known  Assumption that a proposition is true 
Hypothetical demonstration  Dependent proofobject 
Hypothetical judgement  Implicational proposition 
Definitional (criterial) equality  Propositional identity 
(Function) Generality  Quantifier 
4 Four Different Notions of Consequence
Apart from the two changes already indicated—the metamathematical shift and the Bolzano reduction of inferential validity to logical truth (or logical consequence) in “all variations”—we then have occasion to consider another major invention of the early 1930s, namely Gentzen’s Natural Deduction derivations and his Sequent Calculi.
Within the interpreted perspective of an interpreted formal language, with respect to two propositions A and B, there are at least four relevant notions of consequence here.
 (1)
the implication proposition A⊃B, which may be true (or even logically true “in all variations”);
 (2)
the conditional [if A is true then B is true],
or, in other words,$$\begin{array}{*{20}l} {{\text{B is }}{\mathbf{true}}{\text{,}}} \hfill & {{\mathbf{on condition}}{\text{ that A is true}}} \hfill \\ {} \hfill & {{\mathbf{under hypothesis}}{\text{ that A is true}}} \hfill \\ {} \hfill & {{\text{under assumption that A is true}}} \hfill \\ \end{array}$$  (3)
the consequence [A = > B] may hold;
 (4)
the inference [A is true. Therefore: B is true] may be valid.^{14}
Fact 1 “implies” takes thatclauses, whereas “ifthen” takes complete declaratives. Ergo:implication and conditional are not the same. The conditional (2) is a hypothetical judgement in which hypothetical truth is ascribed to the proposition B. Its verificationobject is a dependent proofobject b:Proof(B) [x:Proof(A)], that is, b is a proof of B under the assumption (hypothesis, supposition) that x is a proof of A.
The consequence (3) is a Gentzen sequent (German Sequenz). (Why, we may ask, did Gentzen drop the prefix Kon here?)
Fact 2 The judgement (1)–(3) have different meaningexplanations—their assertion conditions are not the same—and accordingly do not mean the same, are not synonymous, while (4) indicates acts of passage. The first three notions, however, are equiassertible. Given a verificationobject for one of the three, verificationobjects for the other two are readily found in a couple of trivial steps. Furthermore, all four relations are refuted by the same counterexample, namely a situation in which A is known to be true and B known to be false. This might serve to explain why the four notions have sometimes been hard to keep apart, especially from the classical point of view.^{15}
Fact 3 Bolzano deals ably with consequence, whereas his account of inference is inadequate and quite psychologistic in terms of Gewissmachungen.^{16} Frege, on the other hand, deals ably with inference, but (logical) consequence has no place in his system. Only with Gentzen’s 1936 sequential formulation of Natural Deduction, where the derivable objects are sequents, that is consequences, and where the principal introduction and elimination inferences all take place to the right of the sequentarrow, do we get a system that can cope both with inference and consequence.^{17}
Fact 4 Consequence, not logical consequence, is the primary notion. Gentzen’s system deals with arithmetic; his rules of inference that take us from premisesequent(s) to conclusionsequent are obviously valid, but they do not hold logically in all variations. They are only “arithmetically valid”.
Fact 5 A completeness theorem for an interpreted formal language would state: all truths (and in the case of Gentzen’s system: all sequents that hold) are derivable by means of these rules. For Gödelian reasons, interesting systems with theorems of the form [A is true] are not complete. ^{18}
When we now consider how one would establish that (1) to (4) obtain, we see that for (1)–(3) ordinary natural deduction derivations are involved in one way or another. In all three cases one needs a hypothetical proof b:Proof (B) [x:Proof(A)].
The implication A⊃B is established by forming the courseofvalue λ (A, B, [x]b), whereas the conditional is already established by the hypothetical, dependent proofobject in question. Finally, forming the function [x]b:Proof(A) → Proof(B) by means of “lambda” abstraction [] (Curry’s notation!) on the hypothetical proof establishes that the closed consequence (“sequent”) holds.
5 Blind Judgement and Inference
Under the Bolzano reduction, when the proofs (“verification objects”) work also in all variations, then classically one says that the inference (4) is valid. However, the Bolzano reduction validates what we may, in the excellent terminology of Brentano, call blind judgement and inference.^{19} The epistemic link to the judging reasoner has here been severed, whereas I am concerned to preserve this link.
The inference (*), certainly, is truthpreserving, in the in the light of the formalized demonstration offered and the Soundness Theorem for the Predicate Calculus: every time an NBG axiom is used in the predicate logic derivation we replace it by the proposition VNBG and then apply conjunction elimination. Hence we get a formal derivation of PNT from VNBG, whence the Soundness Theorem guarantees truthpreservation. So under the Bolzano reduction this is a valid inference, because truthpreserving under all variations, but it provides no epistemic insight at all.
6 Epistemic Assumptions
Instead, validity of inference, rather than (logical) holding of consequence, involves preservation, or transmission, of epistemic matters from premises to conclusion and it is here that epistemic assumptions that judgements are known (or granted) become helpful. In order to validate the inference I one makes the assumption that one knows the premisejudgments, 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.^{21}
The difference between the two types of assumptions is especially clear when we consider Gentzen derivations in Natural Deduction. An ordinary assumption A of Natural Deduction corresponds to an alethic, ontological assumption that proposition A is true. From such an assumption we may, for instance, obtain a conclusion that B is true, when we have already established the conditional judgement,($) B is true, on hypothesis that A is true,
Furthermore, if we wish to do so, from this we readily obtain also the outright assertion that the implication A⊃B is true by implication introduction, or, for that matter, if we so wish, but now with the aid of functional abstraction on the dependent proofobject that warrants ($), we also may conclude that the sequent [A → B] holds.
An epistemic assumption that a judgement [A is true] is known, or perhaps better granted, corresponds for Natural Deduction derivations to the hypothesis that we have been provided with a closed derivation of the proposition A. This is patently a different kind of assumption from the ordinary Natural Deduction assumption of the wff A.
Brouwer did not accept hypothetical proofs—I hesitate to call them proofobjects in his case. His proofs are all epistemic demonstrations: an assumption that a proposition is true amounts to an assumption that the assumed proposition is known to be true, for instance in his demonstration of the Bar Theorem.^{22}
7 Gentzen’s Two Frameworks for Natural Deduction Ans Epistemic Assumptions
Over the past decades I have had a discussion with Dag Prawitz about the status of the proofs in the BKH explanation: I have claimed that they are not demonstrations with epistemic power, but that they are mathematical witnesses, corresponding to truthmakers in currently popular theories of grounding. Prawitz, on the other hand, has held that they are epistemically binding.^{23} With my present terminology I can formulate my principal objection thus: the distinction between epistemic and alethic assumptions collapses if proofs are held to be epistemically binding. There will be no difference between assuming that proposition A is true and assuming that one knows that A is true.
In type theory the difference between the two kinds of assumption comes out in different treatments of proofobjects. An ordinary assumption has the form x:Proof(A):assume that x is a proof for A
An epistemic assumption with respect to the same proposition takes a closed proofobject as given:assume that I am given a closed proof a:Proof(A)
Against the background of these distinctions we can now explain the difference between the two Gentzen frameworks for Natural Deduction.
8 Epistemic Assumptions and Analytic Validation of Inferences
In recent work, Per MartinLöf has given an interesting dialogical twist to epistemic assumptions.^{25} Already in his first 1946 paper on performatives, etc., John Austin wrote:
If I say “S is P” when I don’t even believe it, I am lying: if I say it when I believe it but am not sure of it, I may be misleading but I am not exactly lying. ………
When I say “I know”, I give others my word: I give others my authority for saying that “S is P”.^{26}
Assertions contain implicit, firstperson knowledge claims (recall G. E. Moore and asserting that it is raining, but that one does not believe it!), so assertions grant authority.
When I first read Austin in 2009 I was led to formulate an Inference Criterion of the same kind:
When I say “Therefore” I give others my authority for asserting the conclusion, given theirs for asserting the premisses.
MartinLöf has now noted that one does not need to know that the premises are evident for the validation of an inference: what one must be prepared to undertake is to make the conclusion known or evident under the assumption that someone else grants the premises as evident.
In order to undertake that responsibility it is enough if I possess a chain of immediately evidencepreserving steps (in terms of meaningexplanations) that link premises to conclusion.^{27} Here the introduction rules of Gentzen may be seen as immediate and meaning explanatory, whereas the elimination rules are immediate, but not meaning explanatory. In Kantian terms, both the introduction and elimination rules are analytically valid, but only the introduction rules are explicitly analytic, or “identical”, whereas the analyticity of the elimination rules is implicit, and might need to be made explicit in terms of the meaning explanations offered by the introduction rules, in analogy with:

All rational animals are rational
is an explicitly analytic (identical) judgement, whereas

All humans are rational
In order to complete the comparison, we consider the question:

Why is &elimination rule valid?
We are then, in an epistemic assumption, given as evident the premisejudgement
 (i)
c:Proof (A&B)
for an application of &elimination.
Under this epistemic assumption we have to make evident the conclusion
 (ii)
p(c): Proof(A).
Since c is a proof of A&B, it executes, (evaluates, is definitionally equal) to a canonical proof of A&B that accordingly has the form
 (iii)
<a,b>: Proof(A&B) and c = < a,b>: Proof(A&B),
where we know that
 (iv)
a :Proof(A) and b:Proof(B).
But granted this, it is a meaning stipulation for the orderedpair and projectionoperators that
 (v)
p(< a,b>) = a:Proof(A)
but, since c = < a,b>: Proof(A&B), we also get p(c) = p(< a,b>) = a :Proof(A), whence we are done.
Note here these deliberations are all pursuant to the relevant meaning explanations for the notions Proof, &, < >, and p. The step from (i) to (iii) and (iv) matches the resolution step that replaces human by rational animal.
9 Axiom and Lemma from an Epistemic Point of View
Finally, what does this mean for axioms in the traditional sense? Such axioms were selfevident judgements, and known as such. The work of Pasch and Hilbert in geometry initiated a change that led to a hypotheticaldeductive conception, which replaced the epistemic notion of inference from selfevident axioms with the modeltheoretic notion of logical consequence “under all variations” or “in all models”. Natural Deduction added one more feature here to the dethroning of axioms: they now become ordinary assumptions among other ordinary assumptions, but as such they are privileged, because they need never be discharged, and may be discounted, when standing in antecedent position in consequences. Nevertheless, contrary to axioms in the oldfashioned sense, they are not known, nor are they asserted whenever they occur. An axiom in the old sense was not an assumption: it was asserted, whereas now that epistemic status is gone, and instead axioms are unasserted assumptions among other assumptions, with the privilege of not carrying the onus of discharge on them.
In conclusion then let me just note that epistemic assumptions are well known in mathematical practice when one draws upon a lemma, the demonstration of which is left out until the main demonstration has been completed. Nevertheless, within the main demonstration, the lemma does not work as an additional assumption, but avails itself of assertoric force, even though proper grounding by means of a demonstration is as yet absent. A very clear case here is the socalled Zorn’s Lemma, whose epistemic status is highly debatable from the point of view of constructivism, but classically is granted axiomatic status.
Footnotes
 1.
Detailed attributions in the Wissenschaftslehre for the claims regarding Bolzano can be found in my 2009, § 3 ‘Revolution: Bolzano’s Annus Mirabilis’.
 2.
Barnes (2002) convincingly argues the use of the term ideography as a translation of German Begriffsschrift.
 3.
 4.
I owe my awareness of these origins of Frege’s Hilfssprache to the scholarship of Wolfgang Künne, cf. Künne (2010), Chap. 5, §5, pp. 725–738.
 5.
 6.
Curry (1963), Chap 2, §§1 and 2, is the locus classicus for the U language.
 7.
Sundholm (2001).
 8.
((1934, §6) section c, third part: Verifizierbare Formeln.
 9.
The various options regarding retention of classical reasoning and meaning explanations are spelled out in some details in my 1998a.
 10.
MartinLöf (1984).
 11.
A fairly comprehensive introduction to MartinLöf’s CTT can be found in my (1977). See also the paper by Ansten Klev in the present issue of TOPOI. That Heyting’s explanation of truth as existence of a proof (object) is a kind of truthmaker analysis was first suggested in my (1994a).
 12.
As is well known, Tarski’s definition of truth does not on its own yield the Law of Excluded Middle for the notion of truth thus defined. Classical reasoning in the metatheory is required for that. In my (2004) I carry out the pendant reasoning and show that, when classical metatheory is allowed, it is very easy to validity LEM, also under the Heyting semantics.
 13.
In my (1997), (2000), and (2012) the demonstration versus proof distinction is given more substance.
 14.
My (1998) and (2012) explain the interrelations of notions (1)–(4) in considerable detail.
 15.
The afterword to my (2012) gives more details concerning the kinds of function—EulerFrege functions, Dedekind mappings, and coursesofvalue—that serve as verification witnesses for, respectively, conditionals, closed consequences (“sequents”), and implicational propositions.
 16.
Volume III of the Wissenschaftslehre contains Bolzano’s account of Gewissmachungen.
 17.
In (2006), at p. 632, and (2009), at p. 298, the links between Frege and Gentzen are explored further.
 18.
I explore these Gödel phenomena in (2004a, §8).
 19.
 20.
Mendelson (1964, Chap. 4) contains a rich exposition of NBG.
 21.
MartinLöf (1984), for instance at p. 41, avails himself of epistemic 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).
 22.
Brouwer’s Demonstration of the Bar Theorem, with its particular use of an epistemic assumption, is discussed in detail by Sundholm and Van Atten (2008).
 23.
For an early instalment in this debate, see my 2000, with a reply by Prawitz in the same issue of Theoria.
 24.
My (2006) is devoted to spelling out the differences, with respect to an interpreted calculus, between Gentzen’s 1932 and 1936 ways of setting out his derivations.
 25.
In lectures at SND, Paris 2015, and at Marseille 2016, at the meeting that provides the source for the present issue of TOPOI.
 26.
Austen (1946, p. 171).
 27.
I suggested this treatment of inferential validity in an invited lecture at LOGICA 1996, and published it the next year in the LOGICA Yearbook; it is now readily available in my (2012), p. 950. It is also dealt with in (2004a), pp. 454–455.
 28.
Leibniz considered patently analytic judgements under the nomer “identical” and preferred Latin resolution to Greek analysis used by Kant. The Kantian Jäsche Logik §37 Tautologische Sätze is helpful here.
Notes
Acknowledgements
I have written about these topics since 1996, and spoken since 2013 at workshops in Groningen (2013), Paris (2014), Petropolis (2014), Heijnice (2014), Hamburg (2015), Marseille (2016), and Prague (2016). I am indebted to the organizers for generous invitations and to participants for welcome comments and objections. Since there is a lot of material already in print, I have not endeavoured to make the present text selfcontained, but have referred to fuller presentations of mine that are readily available on line. I am indebted to Ansten Klev, Per MartinLöf, and Dag Prawitz for longterm discussion 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 colleague Arthur Schipper read a penultimate draft and offered help with proof reading.
Compliance with Ethical Standards
Conflict of interest
The author declared that he has no conflict of interest.
References
 Austen J (1946) Other minds. Proc Aristot Soc 148:148–187Google Scholar
 Barnes J (2002) ‘What is a Befgriffsschrift? Dialectica 56:65–80CrossRefGoogle Scholar
 Berka K, Kreiser L, LogikTexte, 3e Auflage. Akademie Verlag, BerlinGoogle Scholar
 Bolzano B (1837) Wissenschaftslehre, von Seidel J, SulzhbachGoogle Scholar
 Brentano F (1889) Vom Ursprung sittlichher Erkenntnis (Philosophische Bibliothek 55). Felix Meiner, Hamburg, 1955Google Scholar
 Brentano F (1930) Wahrheit und Evidenz (Philosophische Bibliothek 201) Felix Meiner, Hamburg, 1974Google Scholar
 Carnap R (1928) Der logische Aufbau der Welt. Weltkreis, BerlinGoogle Scholar
 Carnap R (1929) Abriß der Logistik. Springer, WienCrossRefGoogle Scholar
 Carnap R (1934) Logische Syntax der Sprache. Springer, WienCrossRefGoogle Scholar
 Curry HB 1976 (1963) Foundations of mathematical logic. Dover Publications, New YorkGoogle Scholar
 Feferman S (1960) Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae 49:35–92CrossRefGoogle Scholar
 Frege G (1879) Begriffsschrift. Louis Nebert, HalleGoogle Scholar
 Frege G (1884) Die Grundlagen der Arithmetik. W. Koebner, BreslauGoogle Scholar
 Frege G (1893, 1903) Grundgesetze der Arithmetik, Band I, Band II, H. Pohle, JenaGoogle Scholar
 Frege G (1979) English abbreviated translation by Peter Long and Roger White of the first edition Frege 1983. In Posthumous writings. Basil Blackwell, Oxford.Google Scholar
 Frege G (1983) Nachgelassense Schriften. In: Hermes H, Kambartel F, Kaulbach F (eds), Felix Heiner, HamburgGoogle Scholar
 Hilbert D, Bernays P (1934, 1939), Die Grundlagen der Mathematik. Springer, BerlinGoogle Scholar
 Jäsche B (1800) Imaanuel Kants Logik, 3rd edn. Felix Meiner, Leipzig, 1904Google Scholar
 Künne W (2010) Die Philosophische Logik Gotlob Freges. Klostermann, Frankfurt a. M.Google Scholar
 MartinLöf P (1984) Intuitionistic type theory. Bibliopolis, NapoliGoogle Scholar
 Mendelson E (1964) Introduction to mathematical logic. Van Nostrand, New YorkGoogle Scholar
 Sundholm G (1994a) Existence, proof and truthmaking: a perspective on the intuitionistic conception of truth. TOPOI 13:117–126CrossRefGoogle Scholar
 Sundholm G (1994b) Ontologic versus epistemologic. In: Prawitz D, Westerståhl D (eds) Logic and philosophy of science in Uppsala. Kluwer, Dordrecht, pp 373–384CrossRefGoogle Scholar
 Sundholm G (1997) Implicit epistemic aspects of constructive logic. J Logic Lang Inform 6:191–212CrossRefGoogle Scholar
 Sundholm G (1998a) Intuitionism and logical tolerance. In: Wolenski J, Köhler E, Alfred Tarski and the Vienna Circle (Vienna Circle Institute Yearbook), vol 6. Kluwer, Dordrecht, pp 135–149Google Scholar
 Sundholm G (1998b) Inference, consequence, implication. Philos Mathematica 6:178–194CrossRefGoogle Scholar
 Sundholm G (2000) Proofs as acts versus proofs as objects: some questions for Dag Prawitz. Theoria 64:187–216 (for 1998, published in 2000): 2–3 (special issue devoted to the works of Dag Prawitz, with his replies)CrossRefGoogle Scholar
 Sundholm G (2001) A plea for logical atavism. In Logica Yearbook 2000. Filosofia Publishers, Czech Academy of Science, Prague, pp 151–162Google Scholar
 Sundholm G (2002) What is an expression? In Logica Yearbook 2001. Filosofia Publishers, Czech Academy of Science, Prague, pp 181–194Google Scholar
 Sundholm G (2003) Tarski and Lesniewski on Languages with meaning versus languages without use: a 60th birthday provocation for Jan Wolenski. In: Hintikka J, Czarnecki T, KijaniaPlacek K, Placek T, Rojszczak A (eds) Philosophy and logic. In search of the polish tradition. Kluwer, Dordrecht, pp 109–128CrossRefGoogle Scholar
 Sundholm G (2004a) Antirealism and the roles of truth. In: Niniluoto M, Sintonen J, Wolenski (eds) Handbook of epistemology. Kluwer, Dordrecht, pp 437–466CrossRefGoogle Scholar
 Sundholm G (2004b) The proofexplanation is logically neutral. Revue Internationale de Philosophie 58(4):401–410Google Scholar
 Sundholm G (2006) Semantic values of natural deduction derivations. Synthese 148(3):623–638CrossRefGoogle Scholar
 Sundholm G (2009) A century of judgment and inference: 1837–1936. In: Haaparanta L (ed) The development of modern logic, Oxford University Press, pp 262–317Google Scholar
 Sundholm G (2012) “Inference versus consequence” revisited: inference, consequence, conditional, implication, Synthese, 187:943–956. Orig. pub. In Logica Yearbook 1997, Filosofia Publishers, Czech Academy of Science, Prague, 1998, pp. 26–35.CrossRefGoogle Scholar
 Sundholm G (2013) Demonstrations versus Proofs, being an afterword to Constructions, Proofs and The Meaning of the Logical Constants. In: van der Schaar M (ed) Judgement and the epistemic foundation of logic. Springer, Dordrecht, pp 15–22CrossRefGoogle Scholar
 Sundholm G, van Atten M (2008) The proper explanation of intuitionistic logic: on Brouwer’s demonstration ofthe Bar Theorem. In: van Atten M, Boldini P, Heintzmann G, Bourdeau M (eds) One hundred years of intuitionism (1907–2007). Birkhäuser, Basel, pp. 60–77 (joint work with Mark van Atten)CrossRefGoogle Scholar
 Tarski A (1935) Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica I (1936). Polish Philosophical Society, Lemberg. Offprints in monograph form dated 1935. Reprinted in Berka and Kreiser [1983, pp 443–546] and translated into English as ‘The Concept of Truth in Formalized Languages’, in Tarski [1956, pp. 152–278]Google Scholar
 Tarski A, translated by Woodger JH (1956) Logic, Semantics, Metamathematics. Clarendon Press, Oxford (Papers from 1923 to 1938)Google Scholar
 Van Atten M, Sundholm G (2017) LEJ Brouwer’s ‘unrealiability of the logical principles’: a new translation with an introduction. Hist Philos Logic 38(1):24–47CrossRefGoogle Scholar
 Whately R (1826) Elements of Logic. Comprising the Substance of the Article in the Encyclopaedia Metropolitana, with Additions, & c. J. Mawman, LondonGoogle Scholar
 Wittgenstein L (1922) Tractatus logicophilosophicus. Routledge and Kegan Paul, LondonGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.