IMPLICATVRA, in 12 volumes; vol. I

IMPLICATVRA

H. P. Grice, St. John’s Oxford

Compiled by Grice’s Playgroup, The Bodleian

a: The choice of the “a” here is because of the Latin “affirmo” versus “nego.” This gives “a” and “i” as ‘affirmatum,’ and ‘e’ and ‘o’ as ‘negatum.’ Grice knows that his problem with Strawson is the Square of Opposition (Grice 1989: )So he is well aware of the question about Barbara and Celarent. So this is the ‘universalis dedicativa.’ Vide below entries for “E” (universalis abdicative), “I” (particularis dedicativa)  and “O” (particularis abdicative). The square (figura quadrata) is generated by criss-crossing the two categories, aptly sub-divided, -- ‘quantitas’ into universalis and particularis, and qualitaas into dedicativa and the abdicative.What is “to affirm”? And what does “to affirm” have in common with “to negate”? These are deep questions that Grice tries to answer in “Negation and privation” and “Lectures on negation.” – The distinction between ‘affirmatum” and “negatum” is not clear – Etymologically, the roots are diverse. To affirm derives from ad-firmare – to make firm. To negate, rather, derives from the mere negative particle turned into a verb, “ne-gare.” It’s even more complicated in Griceain! Grice would never have given attention to this had it not be for “affirmo”and “nego” and for a little treatise he enjoyed reading, “Barbara,  Celarent”  af-firmo (better adf- ), āvi, ātum, 1, v. a. I. To present a thing in words, as fixed, firm, i. e. certain, true; to assert, maintain, aver, declare, asseverate, affirm: “dicendum est mihi, sed ita, nihil ut adfirmem, quaeram omnia,” Cic. Div. 2, 3; so id. Att. 13, 23; id. Brut. 1, 1: “jure jurando,” Liv. 29, 23: “quidam plures Deo ortos adfirmant,” Tac. G. 2; cf. id. Agr. 10: “adfirmavit non daturum se,” he protested that he would give nothing, Suet. Aug. 42.—Impers.: “atque affirmatur,” Tac. H. 2, 49.—Hence, II. To give confirmation of the truth of a thing, to strengthen, to confirm, corroborate, sanction: “adfirmare spem alicui,” Liv. 1, 1: “opinionem,” id. 32, 35: “dicta alicujus,” id. 28, 2: “aliquid auctoritate sua,” id. 26, 24: “populi Romani virtutem armis,” Tac. H. 4, 73: “secuta anceps valetudo iram Deūm adfirmavit,” id. A. 14, 22.—Hence, * affirmanter (adf- ), adv. (of the absol. P. a. affirmans), with assurance or certainty, assuredly: “praedicere aliquid,” Gell. 14, 1, 24; and: af-firmātē (adf- ), adv. (of the absol. P. a. affirmatus), with asseveration, with assurance, certainly, assuredly, positively: “quod adfirmate, quasi Deo teste promiserit, id tenendum est,” Cic. Off. 3, 29.—Sup.: “adfirmatissime scribere aliquid,” Gell. 10, 12, 9. Refs.: Grice 1989, H. P. Grice, “The implicaturums of the Square of Opposition, as dismissed by P. F. Strawson.”

∀: rendered by Grice as “all,” borrowed from Gentzen’s “All-Zeichen.” (Peano did not use it). Grice is a stickler, and uses the brackets, (∀x) –  Grice thinks that Whitehead and Russell did perfectly well with their substitutional account to ‘all,’ “even it that displeased my tutee P. F. Strawson.” Parsons, who Grice admires, suggests that one treat quantification over predicative classes substitutionally, and capture “the idea that classes are not“real” independently of the expression forthem. Grice perceives a difficulty relating to the allegedly dubious admissibility of propositions as entities. A perfectly sound, though perhaps somewhat superficial, reply to the objection as it is presented would be that in any definition of “Emissor E communicates that p” iff “Emissor E desires that p.” which Grice would be willing to countenance,  'p' operates simply as a ‘gap sign.’ ‘p’ does appear in the analysandum, and re-appears in the corresponding analysans. If Grice were to advance the not wholly plausible thesis that “to feel Byzantine” is just to have a an anti-rylean agitation which is caused by the thought that Grice is or might *be* Byzantine, it would surely be ridiculous to criticize Grice on the grounds that Grice saddles himself with an ontological commitment to feelings, or to modes of feeling. And why? Well, because, alla Parsons, if a quantifier is covertly involved at all, it will only be a universal quantifier which in such a case as this is more than adequately handled by a substitutional account of quantification. Grice’s situation vis-a-vis the ‘proposition’ is in no way different. There should be an entry for the inverted E, the first entry under the E.

Jurisprudence: McEvoy.

abderite: Grice was something of a logical atomist, so he used to refer to himself as a neo-Abderite. The reference being to Democritus and Leucippus, from Abdera. The Grecian philosophers Leucippus and Democritus, the two earliest exponents of atomism, later revived by Baron Russell as “logical” atomism, calling himself a neo-abderite. Even though Abdera, in Thrace in northern Greece, is home to three pre-Socratics  Leucippus, Democritus, and Protagoras   ‘Abderite’ and the phrase ‘School of Abdera’ are applied only to Leucippus and Democritus. We can thus distinguish between early Grecian atomism and Epicureanism, which is the later version of atomism developed by Epicurus of Athens. This usage is in one respect inapt: the corresponding Grecian “abderite” is used by some snobs in antiquity as a synonym of ‘simpleton,’ not in disparagement of any of the three philosophers of Abdera but as a regional slur, the three philosophers but not Russell included.

Aaron, r. English philosopher born in Seven Sisters, Sussex.

abdicatum: Apuleius makes an analogy that Grice (and the Grecians before) finds interesting. It is the ‘propositio dedicative’ apophatike’/’propositio abdicativa’ kataphatike distinction. The ‘abdicatum’ would be the ‘negatum.’ The ‘dedicatum’ would be the ‘affirmatum.’ Apuleius’s terms make the correlation evident and Grice preferred it to that of ‘affirmatum’ and ‘negatum,’ – “where the correlation is not that obvious.” So there is the abdicatum, the negatum, and the negation. ‘Negatum’ and ‘affirmatum’ are actually used when translating Husserl from the vernacular! For Husserl, Negation negation a noetic modification of a positing, noematic cancellation  every ‘negatum’ an Object posited as existing, reiterated negation; a ‘negatum’ not a determination produced by reflection; non-being equivalent to being validly negated. Grice’s interest in ‘not’ as a unary functor is central. Grice was ablet to tutor Strawson in philosophy in that famous term. In his “Introduction to logical theory,” Strawson alleges to show that some logical ‘laws,’ taken together, show that any truth-functional sentence or formula in which the main constant is “~ “ is the contradictory of the sentence or formula which results from omitting that sign.” Strawson goes on to say: “A standard and primary use of “not” in a sentence is to assert the contradictory of the statement which would be made by the use, in the same context, of the same sentence without “not.” Of course we must not suppose that the insertion of “not” anywhere in any sentence always has this effect. “Some bulls are not dangerous” is not the contradictory of “Some bulls are dangerous.” This is why the identification of “~” with “it is not the case that” is to be preferred to its identification with “not” simpliciter. This identification, then, involves only those minimum departures from the logic of ordinary language which must always result from the formal logician's activity of codifying rules with the help of verbal patterns : viz., (i) the adoption of a rigid rule when ordinary language permits variations and deviations from the standard use (cf. rules “ ~(p Λ ~p)” and “ ~~p ≡ p” and the discussions in 1-8, and 2-9); (ii) that stretching of the sense of ‘exemplify’ which allows, us, e.g., to regard ‘Tom is not mad’ as well as ‘Not all bulls are dangerous’ as 'exemplifications’ of  not-p.’” Strawson goes on: “So we shall call ‘~’ the negation sign, and read ‘~’as ‘not.’ One might be tempted to suppose that declaring formulae “ ~(p Λ ~p)” and “p v ~p” laws of the system was the same as saying that, as regards this system, a statement cannot be both true and false and must be either true or false. But it is not. The rules that  “ ~(p Λ ~p)” and “p v ~p” are analytic are not rules about ‘true’ and ‘false;’ they are rules about ‘~.; They say that, given that a statement has one of the two truth-values, then it is logically impossible for both that statement and the corresponding statement of the form ‘ ~p’  to be true, and for both that statement and the corresponding statement of the form ‘~p’ to be false.” A bit of palæo-Griceian history is in order. Sheffer, defines ‘not’ and negation in terms of incompatibility in ‘A set of five independent postulates for Boolean algebras, with application to logical constants,’ Trans. American Mathematical Society. Grice does refers to ‘the strokes.’ His use of the plural is interesting as a nod to Peirce’s minute logic in his ‘Boolian [sic] algebra with one constant.’ There is indeed Peirce’s stroke, or ampheck (↓), Sheffer’s stroke (|, /, ↑), and and Quine’s stroke (†, strictly Quine’s dagger). Some philosophers prefer to refer to Peirces Stroke as Peirce’s arrow, or strictly stressed double-edged sword. His editors disambiguate his ampheck, distinguishing between the dyadic functor or connective equivalent to Sheffer’s stroke and ‘nor.’ While Whitehead, Russell, and Witters love Sheffer’s stroke, Hilbert does not: ‘‘p/p’ ist dann gleichbedeutend mit ‘X̄.’ Grice explores primitiveness. It is possible, to some extent, to qualify this or that device in terms of primitiveness. As regards ‘not,’ if a communication-system did not contain a unitary negative device, there would be many things that communicators can now communicate that they would be then unable to communicate. He has two important caveats. That would be the case unless, first, the communication-system contained some very artificial-seeming connective like one or other of the strokes, and, second, communicators put themselves to a good deal of trouble, as Plato does in ‘The Sophist’ with ‘diaphoron,’ that Wiggins symbolises with ‘Δ,’ to find, more or less case by case, complicated forms of expression, not necessarily featuring a connective, but involving such expressions as ‘other than’or ‘incompatible with.’ Grice further refers to Aristotle’s ‘apophasis’ in De Int.17a25. Grice, always lured by the potentiality of a joint philosophical endeavour, treasures his collaboration with Strawson that is followed by one with Austin on Cat. and De Int. So what does Aristotle say in De Int.? Surely Aristotle could have started by referring to Plato’s Parmenides, aptly analysed by Wiggins. Since Aristotle is more of a don than a poet, he has to give ‘not’ a name: ‘ἀπόφασις ἐστιν ἀπόφανσίς τινος ἀπό τινος,’a predication of one thing away from another, i.e. negation of it. This is Grice’s reflection, in a verificationist vein, of two types of this or that negative utterance. His immediate trigger is Ryle’s contribution on a symposium on Bradley’s idea of an internal relation, where Grice appeals to Peirce’s incompatibility. ‘The proposition ‘This is red’ is imcompatible with the proposition, ‘This is not coloured.’ While he uses a souly verb or predicate for one of them, Grice will go back to the primacy of ‘potching’ at a later stage. A P potches that the obble is not fang, but feng. It is convenient to introduce this or that soul-state, ψ, sensing that …, or perceiving that … Grice works mainly with two scenarios, both involved with the first-person singular pronoun ‘I’ with which he is obsessed. Grice’s first scenario concerns a proposition that implies another proposition featuring ‘someone, viz. I,’ the first-person singular pronoun as subject, a sensory modal verb, and an object, the proposition, it is not the case that ‘the α is φ₁.’ The denotatum of the first-person pronoun perceives that a thing displays this or the visual sense-datum of a colour, and the corresponding sensory modal predicate. Via a reductive (but not reductionist) analysis, we get that, by uttering ‘It is not the case that I see that the pillar box is blue,’ the utterer U means, i. e. m-intends his addressee A to believe, U he sees that the pillar box is red. U’s source, reason, ground, knowledge, or belief, upon which he bases his uttering his utterance is U’s *indirect* mediated actual experience, belief, or knowledge, linked to a sense-datum φ₂ (red) other than φ₁ (blue). Grice’s second scenario concerns a proposition explicitly featuring the first-person singular pronoun, an introspection, involving an auditory sense-datum of a noise. Via reductive (but not reductionist) analysis, we get that, by uttering ‘It is not the case that I hear that the bell tolls in Gb,’ U means that he lacks the experience of hearing that the bell tolls simpliciter. U’s source, reason, ground, knowledge, or belief, upon which he bases his uttering his utterance is the *direct* unmediated felt absence, or absentia, or privatio or privation, or apophasis, verified by introspection, of the co-relative ψ, which Grice links to the absence of the experience, belief, or knowledge, of the sense-datum, the apophasis of the experience, which is thereby negated. In either case, Grice’s analysans do not feature ‘not.’ Grice turns back to the topic in seminars later at Oxford in connection with Strawson’s cursory treatment of ‘not’ in “Logical Theory.”‘Not’ (and ~.) is the first pair, qua unary satisfactory-value-functor (unlike this or that dyadic co-ordinate, and, or, or the dyadic sub-ordinate if) in Grice’s list of this or that vernacular counterpart attached to this or that formal device. Cf. ‘Smith has not ceased from eating iron,’ in ‘Causal theory.’ In the fourth James lecture, Grice explores a role for negation along the lines of Wilson’s Statement and Inference.’ Grice’s ‘Vacuous Names’ contains Gentzen-type syntactic inference rules for both ‘not’’s introduction (+, ~) and the elimination (-, ~) and the correlative value assignation. Note that there are correlative rules for Peirce’s arrow. Grice’s motivation is to qualify ‘not’ with a subscript scope-indicating device on ~ for a tricky case like ‘The climber of Mt. Everest on hands and knees is not to atttend the party in his honour.’ The logical form becomes qualified: ‘~₂(Marmaduke Bloggs is coming)₁’, or ‘~₂(Pegasus flies)₁.’ generic formula is ~₂p₁, which indicates that p is introduced prior to ~. In the earlier James lectures he used the square bracket device. The generic formula being ‘~[p],’ where [p] reads that p is assigned common-ground status. Cancelling the implicatura may be trickier. ‘It is not the case that I hear that the bell tolls because it is under reparation.’ ‘That is not blue; it’s an optical illusion.’ Cf. Grice on ‘It is an illusion. What is it?’ Cf. The king of France is not bald because there is no king of France. In Presupposition, the fourth Urbana lecture, Grice uses square brackets for the subscript scope indicating device. ‘Do not arrest [the intruder]!,’ the device meant to assign common-ground status. In ‘Method” Grice plays with the internalisation of a pre-theoretical concept of not within the scope of ‘ψ.’ In the Kant lectures on “Aspects,” Grice explores ‘not’ within the scope of this or that mode operator, as in the buletic utterance, ‘Do not arrest the intruder!’ Is that internal narrow scope, ‘!~p,’ or external wide scope, ‘~!p’? Grice also touches on this or that mixed-mode utterance, and in connection with the minor problem of presupposition within the scope of an operator other than the indicative-mode operator. ‘Smith has not ceased from eating iron, because Smith does not exist ‒ cf. Hamlet sees that his father is on the rampants, but the sight is not reciprocated ‒ Macbeth sees that Banquo is near him, but his vision is not reciprocated. Grice is having in mind Hare’s defense of a non-doxastic utterance. In his commentary in PGRICE, Grice expands on this metaphysical construction routine of Humeian projection with the pre-intuitive concept of  ‘not,’ specifying the different stages the intuitive concept undergoes until it becomes fully rationally recostructed, as something like a Fregeian sense. In the centerpiece lecture of the William James set, Grice explores Wilson’s Statement and inference to assign a métier to ‘not,’ and succeeds in finding one. The conversational métier of ‘not’ is explained in terms of the conversational implicaturum. By uttering ‘Smith has not been to prison yet,’ U implies that some utterer has, somewhere, sometime, expressed an opinion to the contrary. This is connected by Grice with the ability a rational creature has to possess to survive. The creature has to be able, as Sheffer notes, to deny this or that. Grices notable case is the negation of a conjunction. So it may well be that the most rational role for ‘not’ is not primary in that it is realised once less primitive operators are introduced. Is there a strict conceptual distinction, as Grice suggests, between negation and privation? If privation involves or presupposes negation, one might appeal to something like Modified Occam’s Razor (M. O. R.), do not multiply negations beyond necessity. In his choice of examples, Grice seems to be implicating negation for an empirically verifiable, observational utterance, such as U does not see that the pillar box is blue not because U does not exist, but on the basis of U’s experiencing, knowing, believing and indeed seeing that the pillar box is red. This is a negation, proper, or simpliciter (even if it involves a sense-datum phi2 incompatible with sense-datum phi1. Privation, on the other hand, would be involved in an utterance arrived via introspection, such as U does not hear that the bell is ringing on the basis of his knowing that he is aware of the absence, simpliciter, of an experience to that effect. Aristotle, or some later Aristotelian, may have made the same distinction, within apophasis between negation or negatio and privation or privatio. Or not. Of course, Grice is ultimately looking for the rationale behind the conversational implicaturum in terms of a principle of conversational helpfulness underlying his picture of conversation as rational co-operation. To use his Pological jargon in Method, in Pirotese and Griceish There is the P₁, who potches that the obble is not fang, but feng. P₁ utters p explicitly conveying that p. P₂ alternatively feels like negating that. By uttering ~p, P₂ explicitly conveys that ~p. P₁ volunteers to P₂, ~p, explicitly conveying that ~p. Not raining! Or No bull. You are safe. Surely a rational creature should be capable to deny this or that, as Grice puts it in Indicative conditionals. Interestingly, Grice does not consider, as Gazdar does, under Palmer), he other possible unitary functors (three in a standard binary assignation of values) – just negation, which reverses the satisfactory-value of the radix or neustic.  In terms of systematics, thus, it is convenient to regard Grices view on negation and privation as his outlook on the operators as this or that procedure by the utterer that endows him with this or that basic expressive, operative power. In this case, the expressive power is specifically related to his proficiency with not. The proficiency is co-related with this or that device in general, whose vernacular expression will bear a formal counterpart. Many of Grices comments addressed to this more general topic of this or that satisfactoriness-preserving operator apply to not, and thus raise the question about the explicitum or explicatum of not. A Griceian should not be confused. The fact that Grice does not explicitly mention not or negation when exploring the concept of a generic formal device does not mean that what he says about formal device may not be particularised to apply to not or negation. His big concession is that Whitehead and Russell (and Peano before them) are right about the explicitum or explicatum of not being ~, even if Grice follows Hilbert and Ackermann in dismissing Peirces arrow for pragmatic reasons. This is what Grice calls the identity thesis to oppose to Strawsons divergence thesis between not and ~. More formally, by uttering Not-p, U explicitly conveys that ~p. Any divergence is explained via the implicaturum. A not utterance is horribly uninformative, and not each of them is of philosophical interest. Grice joked with Bradley and Searles The man in the next table is not lighting the cigarette with a twenty-dollar bill, the denotatum of the Subjects being a Texas oilman in his country club. The odd implicaturum is usually to the effect that someone thought otherwise. In terms of Cook Wilson, the role of not has more to do with the expressive power of a rational creature to deny a molecular or composite utterance such as p and q Grice comments that in the case of or, the not may be addressed, conversationally, to the utterability of the disjunction. His example involves the logical form Not (p or q). It is not the case that Wilson or Heath will be prime minister. Theres always hope for Nabarro or Thorpe.  The utterer is, at the level of the implicaturum, not now contradicting what his co-conversationalist has utterered. The utterer is certainly not denying that Wilson will be Prime Minister. It is, rather, that the utterer U wishes not to assert or state, say, what his co-conversant has asserted, but, instead, to substitute a different statement or claim which the utterer U regards as preferable under the circumstances. Grice calls this substitutive disagreement. This was a long-standing interest of Grices: an earlier manuscript reads Wilson or MacMillan will be prime minister. Let us take a closer look at the way Grice initially rephrases his two scenarios involving not as attached to an auditory and a visual sense datum. I do not hear that the bell is ringing is rationally justified by the absence or absentia of the experience of hearing it. I do not see that the pillar box is blue is rationally justified by U’s sensing that the pillar box is red. The latter depends on Kant’s concept of the synthetic a priori with which Grice tests with his childrens playmates. Can a sweater be red and green all over? No stripes allowed! Can a pillar box be blue and red all over? Cf. Ryles’ssymposium on negation with Mabbott, for the Aristotelian Society, a source for Grices reflexion. Ryle later discussing Bradleys internal relations, reflects that that the proposition, ‘This pillar box is only red’ is incompatible with ‘This pillar box is only blue.’ As bearing this or that conversational implicatura, Grices two scenarios can be re-phrased, unhelpfully, as I am unhearing a noise and That is  unred. The apparently unhelpful point bears however some importance. It shows that negation and not are not co-extensive. The variants also demonstrate that the implicaturum, qua conversational, rather than conventional, is non-detachable. Not is hardly primtive pure Anglo-Saxon. It is the rather convoluted abbreviation of ne-aught. Its ne that counts as the proper, pure, amorphous Anglo-Saxon negation, as in a member of parliament (if not a horse) uttering nay.  Grices view of conversation as rational co-operation, as displayed in this or that conversational implicaturum necessitates that the implicaturum is never attached to this or that expression. Here the favoured, but not exclusive expression, is not, since Strawson uses it. But the vernacular provides a wealth of expressive ways to be negative! Grice possibly chose negation not because, as with this or that nihilistic philosopher, such as Schopenhauer, or indeed Parmenides, he finds the concept a key one. But one may well say that this is the Schopenhauerian or the Parmenidesian in Griceian. Grice is approaching not in linguistic, empiricist, or conceptual key. He is applying the new Oxonian methodology: the reductive analysis alla Russell in terms of logical construction. Grices implies priority is with by uttering x, by which U explicitly conveys that ~p, U implicitly conveys that q. The essay thus elaborates on this implicated q. For the record, nihilism was coined by philosopher Jacobi, while the more primitive negatio and privatio is each a time-honoured item in the philosophical lexicon, with which mediaeval this or that speculative grammarian is especially obsessed. Negatio translates the ‘apophasis’ of Aristotle, and has a pretty pedigreed history. The philosophical lexicon has nĕgātĭo, f. negare, which L and S, unhelpfully, render as a denying, denial, negation, Cicero, Sull. 13, 39: negatio inficiatioque facti, id. Part. 29, 102. L and S go on to add that negatio is predicated of to the expression that denies, a negative. Grice would say that L and S should realise that its the utterer who denies. The source L and S give is ADogm. Plat. 3, p. 32, 38. As for Grices other word, there is “prīvātĭo,” f. privare, which again unhelpfully, L and S render as a taking away, privation of a thing. doloris, Cic. Fin. 1, 11, 37, and 38, or pain-free, as Grice might prefer, cf. zero-tolerance. L and S also cite: 2, 9, 28: culpæ, Gell. 2, 6, 10. The negatio-privatio distinction is attested in Grecian, indeed the distinction requires its own entry. For it is Boethius who first renders Aristotle’s ‘hexis’ into ‘habitus’ and Aristotle’s steresis’ into ‘privatio.’ So the the Grecians were never just happy with “ἀπόφασις (A)” and they had to keep multiplying negations beyond necessity. The noun is from “ἀπόφημι.” Now L and S unhepfully render the noun as as denial, negation, adding “oκατάφασις,” for which they cites from The Sophista by Plato (263e), to  give then the definition “ἀπόφασις ἐστιν ἀπόφανσίς τινος ἀπό τινος,” a predication of one thing away from another, i.e. negation of it, for which they provide the source that Grice is relying.  on: Arist. Int.17a25, cf. APo. 72a14; ἀπόφασις τινός, negation, exclusion of a thing,  Pl. Cra. 426d; δύο ἀ. μίαν κατάφασιν ἀποτελοῦσι Luc. Gall.11. If he was not the first to explore philosophically negation, Grice may be regarded as a philosopher who most explored negation as occurring in a that-clause followed by a propositional complexus that contains ~, and as applied to a personal agent, in a lower branch of philosophical psychology. It is also the basis for his linguistic botany. He seems to be trying to help other philosopher not to fall in the trap of thinking that not has a special sense. The utterer means that ~p. In what ways is that to be interpreted? Grice confessed to never been impressed by Ayer. The crudities and dogmatisms seemed too pervasive. Is Grice being an empiricist and a verificationist? Let us go back to This is not red and I am not hearing a noise. Grices suggestion is that the incompatible fact offering a solution to this problem is the fact that the utterer of “Someone, viz. I, does not hear that the bell tolls” is indicating (and informing) that U merely entertains the positive (affirmative) proposition, Someone, viz. I, hears that the bell tolls, without having an attitude of certainty towards it. More generally, Grice is proposing, like Bradley and indeed Bosanquet, who Grice otherwise regards as a minor philosopher, a more basic Subjects-predicate utterance. The α is not β. The utterer states I do not know that α is β if and only if every present mental or souly process, of mine, has some characteristic incompatible with the knowledge that α is β. One may propose a doxastic weaker version, replacing the dogmatic Oxonian know with believe. Grices view of compatibility is an application of the Sheffer stroke that Grice will later use in accounts of not. ~p iff p|p or ~p ≡df p|p. But then, as Grice points out, Sheffer is hardly Griceian. If Pirotese did not contain a unitary negative device, there would be many things that a P should be able to express that the P should be unable to express unless Pirotese contained some very artificially-looking dyadic functor like one or other of the strokes, or the P put himself to a good deal of trouble to find, more or less case by case, complicated forms of expression, as Platos Parmenides does, involving such expressions as other than, or incompatible with. V. Wiggins on Platos Parmenides in a Griceian key. Such a complicate form of expression would infringe the principle of conversational helpfulness, notably in its desideratum of conversational clarity, or conversational perspicuity [sic], where the sic is Grices seeing that unsensitive Oxonians sometimes mistake perspicuity for the allegedly, cognate perspicacity (L. perspicacitas, like perspicuitas, from perspicere). Grice finds the unitary brevity of not-p attractive. Then theres the pretty Griceian idea of the pregnant proposition. Im not hearing a nose is pregnant, as Occam has it, with I am hearing a noise. A scholastic and mediæval philosopher loves to be figurative. Grices main proposal may be seen as drawing on this or that verificationist assumption by Ayer, who actually has a later essay on not falsely connecting it with falsity. Grices proposed better analysis would please Ayer, had Grice been brought on the right side of the tracks, since it can be Subjectsed to a process of verification, on the understanding that either perception through the senses (It is red) or introspection (Every present mental or souly process of mine ) is each an empirical phenomenon. But there are subtleties to be drawn. At Oxford, Grices view on negation will influence philosophers like Wiggins, and in a negative way, Cohen, who raises the Griceian topic of the occurrence of negation in embedded clauses, found by Grice to be crucial for the rational genitorial justification of not as a refutation of the composite p and q), and motivating Walker with a reply (itself countered by Cohen ‒ Can the conversationalist hypothesis be defended?). So problems are not absent, as they should not! Grice re-read Peirces definition or reductive analysis of not and enjoyed it!  Peirce discovers the logical connective Grice calls the Sheffer Stroke, as well as the related connective nor (also called Joint Denial, and quite appropriately Peirces Arrow, with other Namess in use being Quines Arrow or Quines Dagger and today usually symbolized by “/”). The relevant manuscript, numbered MS 378 in a subsequent edition and titled A Boolian [sic] Algebra with One Constant, MS 378, was actually destined for discarding and was salvaged for posterity A fragmentary text by Peirce also shows familiarity with the remarkable meta-logical characteristics that make a single function functionally complete, and this is also the case with Peirces unfinished Minute Logic: these texts are published posthumously. Peirce designates the two truth functions, nand and nor, by using the symbol “” which he called ampheck, coining this neologism from the Grecian ἀμφήκης, of equal length in both directions. Peirces editors disambiguate the use of symbols by assigning “” to the connective we call Sheffers troke while preserving the symbol “/” for nor.  In MS 378, A Boolian Algebra with One Constant, by Peirce, tagged “to be discarded” at the Department of Philosophy at Harvard, Peirce reduces the number of logical operators to one constant. Peirce states that his notation uses the minimum number of different signs and shows for the first time the possibility of writing both universal and particular propositions with but one copula. Peirce’s notation is later termed Sheffers stroke, and is also well-known as the nand operation, in Peirce’s terms the operation by which two propositions written in a pair are considered to be both denied. In the same manuscript, Peirce also discovers what is the expressive completeness of ‘nor,’ indeed today rightly recognized as the Peirce arrow. Like Sheffer, of Cornell, independently does later (only to be dismissed by Hilbert and Ackermann), Peirce understands that these two connectives can be used to reduce all mathematically definable connectives (also called primitives and constants) of propositional logic. This means that all definable connectives of propositional logic can be defined by using only Sheffers stroke or nor as the single connective. No other connective (or associated function) that takes one or two variables as inputs has this property. Standard, two-valued propositional logic has no unary functions that have the remarkable property of functional completeness. At first blush, availability of this option ensures that economy of resources can be obtained—at least in terms of how many functions or connectives are to be included as undefined. Unfortunately, as Grice, following Hilbert and Ackermann realise, there is a trade-off between this philosophical semantic gain in economy of symbolic resources and the pragmatically unwieldy length and rather counterintuitive, to use Grices phrase, appearance of the formulas that use only the one connective.  It is characteristic of his logical genius, however, and emblematic of his rather under-appreciated, surely not by Grice, contributions to the development of semiotics that Peirce grasps the significance of functional completeness and figure out what truth functions — up to arity 2 — are functionally complete for two-valued propositional logic, never mind helping the philosopher to provide a reductive analysis of negation that Grice is looking for. Strictly, this is the property of weak functional completeness, given that we disregard whether constants or zero-ary functions like 1 or 0 can be defined. Peirce subscribes to a semeiotic view, popular in the Old World with Ogden and Welby, and later Grice, according to which the fundamental nature and proper tasks of the formal study of communication are defined by the rules set down for the construction and manipulation of symbolic resources. A proliferation of symbols for the various connectives that are admitted into the signature of a logical system suffers from a serious defect on this view. The symbolic grammar fails to match or represent the logical fact of interdefinability of the connectives, and reductive analysis of all to one. Peirce is willing sometimes to accept constructing a formal signature for two-valued propositional logic by using the two-members set of connectives, which is minimally functionally complete. This means that these two connectives — or, if we are to stick to an approach that emphasizes the notational character of logical analysis, these two symbols —are adequate expressively. Every mathematically definable connective of the logic can be defined by using only these two. And the set is minimally functionally complete in that neither of these connectives can be defined by the other (so, as we say, they are both independent relative to each other.) The symbol   can be viewed as representing a constant truth function (either unary or binary) that returns the truth value 0 for any input or inputs. Or it can be regarded as a constant, which means that it is a zero=ary (zero-input) function, a degenerate function, which refers to the truth value 0. Although not using, as Grice does, Peanos terminology, Peirce takes the second option. This set has cardinality 2 (it has exactly 2 members) but it is not the best we can do. Peirces discovery of what we have called the Sheffer functions or strokes (anachronistically and unfairly to Peirce, as Grice notes, but bowing to convention) shows that we can have a set of cardinality 1 (a one-member set or a so-called singleton) that is minimally functionally complete with respect to the definable connectives of two-valued propositional logic. Thus, either one of the following sets can do. The sets are functionally complete and, because they have only one member each, we say that the connectives themselves have the property of functional completeness. / is the symbol of Sheffers stroke or nand and /is the symbol of the Peirce Arrow or nor. Grice stipulates as such, even though he does not introduce his grammar formally. It is important to show ow these functions can define other functions. Algebraically approached, this is a matter of functional composition In case one wonders why the satisfaction with defining the connectives of the set that comprises the symbols for negation, inclusive disjunction, and conjunction, there is an explanation. There is an easy, although informal, way to show that this set is functionally complete. It is not minimally functionally complete because nor and nand are inter-definable. But it is functionally complete. Thus, showing that one can define these functions suffices for achieving functional completeness. Definability should be thought as logical equivalence. One connective can be defined by means of others if and only if the formulae in the definition (what is defined and what is doing the defining) are logically equivalent. Presuppose the truth-tabular definitions of the connectives.  Grice enjoyed that. Meanwhile, at Corpus, Grice is involved in serious philosophical studies under the tutelage of Hardie. While his philosophical socialising is limited, having been born on the wrong side of the tracks, first at Corpus, and then at Merton, and ending at St. Johns, Grice fails to attend the seminal meetings at All Souls held on Thursday evenings by the play group of the seven (Austin, Ayer, Berlin, Hampshire, MacDermott, MacNabb, and Woozley). Three of them will join Grice in the new play group after the war: Austin, Hampshire, and Woozley. But at St. Johns Grice tutors Strawson, and learns all about the linguistic botany methodology on his return from the navy. Indeed, his being appointed Strawson as his tutee starts a life-long friendship and collaboration. There are separate entries for the connectives: conjunction, disjunction, and conditional. Abdicatum -- double negation. 1 The principle, also called the law of double negation, that every proposition is logically equivalent to its double negation. Thus, the proposition that Roger is a rabbit is equivalent to the proposition that Roger is not not a rabbit. The law holds in classical logic but not for certain non-classical concepts of negation. In intuitionist logic, for example, a proposition implies, but need not be implied by, its double negation. 2 The rule of inference, also called the rule of double negation, that permits one to infer the double negation of A from A, and vice versa.  Refs.: Allusions to negation are scattered, notably in Essay 4 in WoW, but also in “Method in philosophical psychology,” and “Prejudices and predilections” (repr. in “Conception”), and under semantics and syntax. While one can draw a skull communicating that there is danger; one can then cross out the skull indicating that there is no danger. So the emissor communicates that there is no danger. Or rather, the emissor communicates that it is not the case that there is danger. Since this involves a ‘that’-clause, it is not unreasonable to speak of a ‘propositio,’ and such would be ‘abdicativa.’ In his earliest reflections on the topic, Grice draws on sub-perceptual illustrations rendered more or less as involving two items of ‘propositio dedicativa’ and their negation and privation: ‘The bell tolls in Gb” and “The pillar box is red.” For the latter, “The pillar box is not blue” can be uttered as a conclusion (“If the pillar box is red, it is not the case that the pillar box is blue.”). For the former case, “The bell tolls in Ab” may do. “If the bell tolls in Gb, it is not the case that the bell tolls in Ab.” For Grice, the métier of a propositio abidcativa has to do with the abdicatum of a conjunctum. For a more primitive rationale, Grice does not see the complete justification. That means that Grice sees that there are OPTIONS TO introducing a ‘propositio abdicativa’. These options are of two kinds. One is the ‘stroke.’ If you draw a skull, a stroke, and a skull, you communicate that it is not the case there is danger. The other involves “other than” or “incompatible.” Again, drawsing a skull and writing INCOMPATIBLE and drawing another skull and you communicate that it is not the case that there is danger. Refs.: There are specific essays of different dates, in s. V, in two separate folders, in BANC.

abductum: an implicaturum is abductum, i. e., it is not something that it is inductum or deductum. It is indeed a demonstratum, an argumentum, but qua abductum. Grice favours the form ‘implicaturum’ rather than ‘implicaturum’ in that the implicaturum is strictly what follows a ‘that’-clause. Ditto for ‘abductum.’ As opposed to in-duction and de-duction, abduction refers to canons of reasoning for the discovery, as opposed to the justification, of scientific hypotheses or theories. Reichenbach distinguished the context of justification and the context of discovery, arguing that philosophy legitimately is concerned only with the former, which concerns verification and confirmation, whereas the latter is a matter for psychology. Thus he and other logical positivists claimed there are inductive logics of justification but not logics for discovery. Both hypotheticodeductive and Bayesian or other probabilistic inductive logics of justification have been proposed. Close examination of actual scientific practice increasingly reveals justificatory arguments and procedures that call into question the adequacy of such logics. N. R. Hanson distinguishes the reasons for accepting a specific hypothesis from the reasons for suggesting that the correct hypothesis will be of a particular kind. For the latter he attempted to develop logics of retroductive or abductive reasoning that stressed analogical reasoning, but did not succeed in convincing many that these logics were different in kind from logics of justification. Today few regard the search for rigorous formal logics of discovery as promising. Rather, the search has turned to looking for “logics” in some weaker sense. Heuristic procedures, strategies for discovery, and the like are explored. Others have focused on investigating rationality in the growth of scientific knowledge, say, by exploring conditions under which research traditions or programs are progressive or degenerating. Some have explored recourse to techniques from cognitive science or artificial intelligence. Claims of success generally are controversial.