On Negating

On Negating

Fabien Schang Moscow State University

“The Ways of Negation”, Research Stay (IERTNiL) Hyderabad, 20-22 February 2014

This research was supported by the Indo-European Research Training Network in Logic (IERTNiL) funded by the Institute of Mathematical Sciences of Chennai, the Institute for Logic, Language and Computation of the Universiteit van Amsterdam} and the Fakultät für Mathematik, Informatik und Naturwissenschaften of the Universität Hamburg

Content

1. On Negating (Problems)

2. The General Construction: QAS

3. On negating (Solutions)

4. Conclusion and Prospects

1. On Negating (Problems)

The various aspects of negation: linguistic (neg-raising, litote, presuppositional) logical (classical, non-classical) psychological (denial, refusal) philosophical (negative existentials/facts, dialectical negation)

The representations of negation: syntactic (predicate term negations vs predicate/sentence negations) semantic (operators vs operands)

Pluralism: no one-sided reduction, but a context-sensitive description

Unity in diversity: a common framework for negation an algebraic theory of negation, beyond the above applied categories

An approach by generalization (rather than elimination or representation)

“Elimination: to explain away certain syntactic types. Generalization: to give an account such that various syntactic types of negation emerge as special cases of a general construction. Representation: to represent one type of negation in terms of another type.” (Wansing, “Priest on Negation”)

An approach by generalization (rather than elimination or representation)

The general construction: Question-Answer Semantics (QAS)

A sample of philosophical problems related to negation: (a) Paradoxes of Non-Being: negative existential, negative facts (b) Dialectical negations: Heidegger’s nichtendes Nichts, Hegel’s Aufhebung (c) The Colour Exclusion Problem (d) Eastern negation(s)

QAS aims at overcoming a number of biased objections to a general theory of negation, including the following three ones:

(1) logical and linguistic negations are irreducible to each other (2) logical negation is a truth-functional operator (applied to truth-values) (3) negation has essentially to do with exclusion and incompatibility

(a) Paradox of Non-Being (I) : telling the truth about negative existentials How to tell something true about « non-being »?

How to assert anything about non-existents successfully? Russell’s solution: being as existence (negation of an existential quantifier) Problem: every sentence with negative existentials is false (cf. free logic)

The controversy: truth and modes of being No gap between “not being” (no existence) and “being nothing” (no identity)? Does truth require existence? An ontological bias about the sense of “being” “No entity without identity” (Quine) … …and no identity without existence?

“What can be spoken of and thought must be: for it is possible for it to be, but it is not possible for ‘nothing’ to be.” (Parmenides: Fragment 6)

“When you think, you think of something; when you use a name, it must be the name of something. Therefore both thought and language require objects outside themselves.” (Russell: A History of Western Philosophy)

(a) Paradox of Non-Being (I) : telling the truth about negative existentials How to tell something true about « non-being »?

How to assert anything about non-existents successfully? Russell’s solution: being as existence (negation of an existential quantifier) Problem: every sentence with negative existentials is false (cf. free logic)

The controversy: truth and the modes of being No gap between “not being” (no existence) and “being nothing” (no identity)? Does truth require existence? An ontological bias about the sense of “being” “No entity without identity” (Quine) … …and no identity without existence?

(a) Paradox of Non-Being (II) : from true negations to negative factS?

Three views of one and the same “thing”: the quality of a predication (a negative sentence: S is not P) the operation turning an affirmative into a negative sentence (not: S is P) the attitude of a speaker denying something (“no, I don’t take S to be P)

Which sort of “thing” is negation? A content of sentences: a sentential operator Problem: What makes a sentence true? Wittgenstein, Russell, Frege: true sentences express facts A content of the world: positive vs negative facts For every sentence, there is a fact corresponding to it (Russell) The result of an ontological assumption: logical atomism A content of judgments: denial Two sorts of judgment: affirmative, and negative (contra Frege)

(b) Dialectical negations (I): Heidegger’s nichtendes Nichts

Is Heidegger’s “Das Nichts nichtet” (The nothing noths) non-sensical? (Carnap:“Überwindung der Metaphysik durch logische Analyse der Sprache”)? Cf. Wittgenstein’s sinnlos (“logical propositions”) vs unsinnig (not wffs)

Equivocity of negation: Used as both a name (das Nichts) and a predicate (nichten), while being uniquely viewed by Carnap as a usual logical constant

Reading Heidegger’s “pseudo-sentence” beyond Carnap’s logical syntax - Is there only one logical syntax of language? - Carnap and (the early) Wittgenstein assume some formal ontology

behind their logic: a proposition is a descriptive sentence about the world - Heidegger’s “Nichts” is related to his concept of “Sorge”; now moral

judgments are excluded by such logical positivists as Carnap

(b) Dialectical negations (II): Hegel’s Aufhebung

Aristotle’s Principle of Non-Contradiction

Two contradictories cannot hold together: 3 versions of non-contradiction (ontological , semantic, psychological)

“It is impossible that the same thing belong and not belong to the same thing at the same time and in the same respect.” (Aristotle: Metaphysics, 1005b19-20)

“The most certain of all basic principles is that contradictory propositions are not true simultaneously.” (Aristotle: Metaphysics, 1011b13-14)

“No one can believe that the same thing can (at the same time) be and not be.” (Aristotle: Metaphysics, 1005b23-24)




Hegel’s Principle of Contradiction

“Every thing is self-contradictory.”

(G.W.F. Hegel: Logik)

“Contradictio est regula veri, non contradictio, falsi.” (Hegel: Habilitationschrift’s header)




Hegel’s Principle of Contradiction Nevertheless, Hegel was said to respect Aristotle’s Principle How to do accept both “exclusive” and “inclusive” contradiction? Cf. Identity in change, preservation in difference with Hegel’s Aufhebung (logical vs dialectical negation: transformation and preservation)

(c) The Colour Exclusion Problem

About colour and their (im)compatibilities Some incompatibilities cannot, after all, be reduced to logical impossibilities (Problem of “unanalyzable statements”)

Do colours have a logical structure? Cf. psychology of perceptions (“contrary colours”)


For two colours, e.g., to be at one place in the visual field is impossible, and indeed logically impossible, for it is excluded by the logical structure of colour. (Wittgenstein: TLP, §6.3751)




Is the world full of oppositions between “objects” we can perceive? Wittgenstein: such synthetic a priori judgments are not logical

Wittgenstein’s prior assumptions: correspondence theory of truth, logical atomism


A logical language deals exclusively with what is true and false. It can not be used to make aesthetic or ethical judgments. What is beautiful or good can not be expressed, because it doesn’t concern facts. It becomes clear that a satisfactory logical description of the way things are would eradicate all questions traditional philosophy is concerned with. (Tractatus §6.53)


However, there is an essential and irreducible difference between the two propositions “A is red” + “A is blue” and “A is red” + “A is not red”. While the latter is evidently a formal, i.e. logical contradiction, the former is not as we have to rely on the “logical system of colours” to understand that both cannot be true at the same time.




Is the world full of oppositions between “objects” we can perceive? Wittgenstein: such synthetic a priori judgments are not logical

Wittgenstein’s prior assumptions: correspondence theory of truth, logical atomism How to account for mutual exclusion by means of such a logic?

(d) Eastern negation(s)s

Accepting everything? Saptabhaṅgī

In the Jain saptabhaṅgī (theory of sevenfold predication), the third basic predication “avaktavyam” is read as “asserted and denied simultaneously” A violation of Aristotle’s Non-Contradiction? A plea for true contradictions?

Rejecting everything? Catuṣkoṭi (Four-Fold Negation) In the Buddhist Madhyamaka’s school, four stances are denied in turn Does a thing or being come out itself? No. Does it come out the other? No. Does it come out both itself and the other? No. Does it come of neither. No. How to deny of these four statements at once meaningfully, consistently?

2. The General Construction: QAS

A primary reflection about the meaning of negation A problem about the meaning of logical constants: what is negation?

One, or several negations? Logical negation is currently seen as an incompatibility-forming operator Classical negation: turns the True into the False, and conversely (LNC) What is non-classical negation, accordingly? Priest: intuitionistic negation is a contrary-forming operator (rejects LEM)

An incompatibility-forming operator Slater (1995): paraconsistent negation is not a proper negation Every negation is a contradiction-forming operator What of intuitionistic (contrary-forming) negation, accordingly? A minimal criterion for negation: not a subcontrariety-forming operator

A common framework for thinking about negation: theory of opposition

A “chicken and egg” problem: truth and belief

Subjective truth: relativist theory Truth is what is taken to be the case.

Plato’s Protagoras: man is the measurement of everything

if so, then a pig is no more right than a taster about e.g. the quality of wine

the very meaning of “being” collapses into a mere “appearing” (being for) We need an external object to find agreement between the speakers Cf. Frege’s truth-value, as an ideal object to be grasped by thought.


Ontological truth: descriptive approach Truth is what is the case, i.e. corresponds to a fact.

Propositions are prior to judgments by expressing them Russell: an assertion is a set of beliefs that are made true by a fact a proposition stands for this set


A proposition is therefore a class of facts, psychological or linguistic, defined as standing into a certain relation (it can be either assertion or denial, according to the cases) to a certain fact. (Russell, “Truth- and meaning-functions”)



Propositions are prior to judgments by expressing them Russell: an assertion is a set of beliefs that are made true by a fact a proposition stands for this set Frege: truth is the reference of a sentence, or proposition (“Gedanke”) a proposition is the sense of an object, i.e. a way of expressing truth


A sentence proper is a proper name, and its Bedeutung, if it has one, is a truth-value: the True or the False (Frege, “On Sense and Denotation”).



Propositions are prior to judgments by expressing them Russell: an assertion is a set of beliefs that are made true by a fact a proposition stands for this set A circular relation between truth and fact Truth: property of a sentence corresponding to a fact Fact: that which makes a proposition true Another definiendum is needed for truth


Epistemic truth: normative approach What is true is what ought to be asserted/said to be the case

Judgments are prior to propositions by making them Peirce: from social knowledge to intersubjective agreement

Judgment is predication A fact, state of affairs … a predication A predication is a set-theoretical relation between two objects (subsumption).

Truth and falsity Truth: property of a sentence Truth for one/some/most of/every people: scalar quantification over beliefs Falsity: truth for no one


The question therefore is, how is true belief (or belief in the real) distinguished from false belief (or belief in fiction). Now, as we have seen (…) the ideas of truth and falsehood, in their full development, appertain exclusively to the experiential method of settling opinion. (Peirce: “The Fixation of Belief”)

A many-dimensional questioning for predication Properties come from several sorts of graded questionings (who, what, when) A sentence includes qualitative and quantitative data (cf. modal judgments)

A non-normative reduction of truth-values: “marked” values Truth-value: a property assigned to a sentence through a normative assent Truth: assent, assertion. Falsity: dissent, rejection.

From existential to scalar (generalized) quantification A sentence is assessed in accordance to what makes it “true” (accepted): the speaker (who), the time of utterance (when), the content (what) fuzzy logic is a case of generalized quantification over contents

Questioning the value of a sentence Question 1: “Is p accepted?” q1(p) Answer: “Yes”, or “No”. a1(p) Question 2: “Is rejected?” q2(p) Answer: “Yes”, or “No”. q1(p) Bivalence: a yes-answer to a1(p) entails a no-answer to a2(p), and conversely There may be reasons both for and against p (glut), or none of them (gap). A plea for Belnap’s 4-valued logic: marked values, rather than truth-values Marked value: a set of data about the value a sentential object Example: “red” is a colour, “pale red” is a grade of redness “man” is a species, “every man” is a quantification over a species Each object has a finite set of properties (definition by intension)

Properties are overlapping correlated sets (holistic view of meaning) Opposition relates things with the same string of properties

Compare with Buridan’s two negations: - negatio negans (white/black): polar negation Definite string of properties, different bits (1100/1011) - negation infinitans (white/not white): complementary negation Indefinite string of properties, different bits (110/1011)

Opposition assumes submodels: Lexical fields, categories range over a specific subset of objects What of a maximal model, i.e. one and the same string for any object? If so, then any object is properly opposed to any other one (no restriction)

How many bits are required to individuate a given object?

Logic of opposition: L,A A language L: set of objects (properties) including sentences

A: ,,,op, A,1,0 (where 1 > 0)

For every objects , in L:

ai() ai () = max(ai(),ai())

ai() ai () = max(ai(),ai())

A sentence “S is P” is true iff ai(P) = 1 ai(S) = 1, i.e. A(S) A(P) A(S) = 001001 A(P) = 000001

The sentence “S is P” is true: 001001 000001 Tautologies: only yes-answers 111…1 = T

Antilogies (“contradictions”): only no-answers 000…0 =

1. Opposition: a binary relation Op between any pairs of objects ,

Op(,): “ stands into a relation of opposition with ”

Formal ontology: meaningful objects (sentences, concepts, individuals, …)

A non-Fregean approach to meaning

Every meaningful object has a sense and a reference:

- Sense: string of m questions Symbol: Q() = q1(), …, qm()

- Reference: string of m answers Symbol: A() = a1(), …, am()

n = 2 sorts of answer: affirmation (bit: 1), denial (bit: 0)

A logical value is a bitstring of length n among a set of nm objects the logical value helps to identify an object (by a set of properties) its length depends upon how many other objects there are in a given set relative ontology/logical value, identification by differentiation

2. Opposite: an operation O inside the relation Op(,), such that:

Op(,) = Op(,op())

A general theory of negation: a difference-forming operator (“heteron”) opposition is a difference-forming operator

For every relation of opposition Op(,op()): A() A(op())

A multi-function: op is an injective function ranging upon logical values

4 sorts of opposite-forming operators op: ct, cd, sct, sb

CT(,) = CT(,ct()) ai() ai () = and ai() ai () = T

CD(,) = CD(,cd()) ai() ai () = and ai() ai () T

SCT(,) = SCT(,sct()) ai() ai () = and ai() ai () = T

SB(,) = SB(,sb()) ai() ai () = and ai() ai () = T

Subalternation: a double mixed negation, such that: sb() = cd(ct(())

Contraries are supposed to be polar (m > 2); ct = cd, otherwise Truth and falsity are mediate contraries in classical (bivalent) logic: m = 2

An abstract example of opposite values:

Let A() = 1000, where m = 4 questions and n = 2 sorts of answers. Then

cd() = 0111

ct() = {0000, 0100, 0010, 0001, 0110, 0011, 0101}

sct() =

sb() = {1100, 1010, 1001, 1100, 1011, 1101, 1111} 1 contradictory for every object, whatever its length may be 6 mediate contraries (no immediate contrariety once m > 2) 1 polar contrary ct*: ct*(1000) = 0001

ABSTRACT HEXAGON OF OPPOSITIONS (BLANCHÉ 1963)

ct()

ct()

cd(ct()) cd()

cd( ct()) = cd(cd(ct()) cd()))

ABSTRACT OPPOSITIONS (an example with mediate, non-polar contraries) 1100

1000 0100 1011 0111 0011

A sample of applied oppositions between:

Sentences

Concepts

Individuals

… ? Individuals: characterized by questions about their properties Back to the sense of being (a): interdependence of identity and membership An intensional semantics: elements are defined by all the sets they belong to (extensional definition: a set is defined by all the elements belonging to it) + An account for a number of linguistic negations: litote, neg-raising Presuppositional negation: a category-mistake-forming operator neither 1 nor 0: an “operator” without operand

A() and A(op()) don’t have the same string

QUANTIFIED SENTENCES (II)

Every S is P Every S is not-P

Not every S is not-P Not every S is P

QUANTIFIED SENTENCES (II)

1000 0001

1110 0111

q1 : true of every S? q3: false of some (but not every) S? q2: true of some (but not every) S? q4: false of every S?

QUANTIFIED SENTENCES (II) Every S is P Every S is not-P x is P x is not-P

Some S is P Some S is not-P

QUANTIFIED SENTENCES (II) 1000 0001 1100 0011

1110 0111

TERM LOGIC (I) Man is fair Man is unfair

Man is not unfair Man is not fair

TERM LOGIC (II) Man is fair Man is unfair Socrates is fair Socrates is unfair

Man is not unfair Man is not fair

TERM LOGIC (III): Englebretsen (1981) Affirmation Counter-affirmation

Counter-denial Denial

TERM LOGIC (III): Englebretsen (1981)

ct()

sb() cd()

DE DICTO ALETHIC MODALITIES (I) Necessary: p Necessary: not-p

Not necessary: not-p Not necessary: p

DE DICTO ALETHIC MODALITIES (II) Determinately: p Necessary: p Necessary: not-p

Not necessary: not-p Not necessary: p

Contingently: p

CLASSICAL BINARY SENTENCES (I)

pq (pq)

pq (pq)

CLASSICAL BINARY SENTENCES (II)

1000 0001

1110 0111

JUDGMENTS (I)

S is P not: S is P

JUDGMENTS (II) | S is P | S is not P

S is not P S is P

ILLOCUTIONARY FORCES (I): VERNANT 2008 Commitment: p Assertion: p Denial: p

Non-denial: p Non-assertion: p

Consideration: p

ILLOCUTIONARY FORCES (II) Assertion Assertion: p Assertion: not-p

Denial: not-p Denial: p Supposition

REALLY-NOT NEGATION (NEG-RAISING) (I.a) It is a good day It is bad day It is not a bad day It is not a good day

REALLY-NOT NEGATION (NEG-RAISING) (II.a) It is a good day It is (really-)not a good day It is not a bad day It is not a good day

REALLY-NOT NEGATION (NEG-RAISING) (III.a) It is a good day ct(is is a good day) sb( is a good day) cd(it is a good day)

REALLY-NOT NEGATION: NEG-RAISING (I.a) x does not doubt about p

x believes that p x believes that p x does not believe that p x does not believe that p x doubts about p

REALLY-NOT NEGATION: NEG-RAISING (I.b) x does not believe that p

x believes that p x doubts that that p x does not doubt that p x does not believe that p x doubts about p

REALLY-NOT NEGATION: NEG-RAISING (I.c) x does not doubt that p x believes that p ct(x believes that p) x does not believes that p x does not believe that p x doubts about p

3.

On Negating (Solutions)

(a) Paradox of Non-Being (I) : telling the truth about negative existentials Is existence as a property of individuals, or not? (Yes: Avicenna, Leibniz, Meinong; No: Hume, Kant, Russell, Quine)

Yes: negative existentials include a no-answer about whether they exist

Not-being: something that is this and is not that a sequence of 1-0 (with 0 about existence) Identity requires at least one yes-answer about some given property

Nothingness: not something, i.e. no-thing (that is this and is not that)

a sequence of exclusive no-answers (of arbitrary length):

No identity without existence? No identity without difference! Nothing can be denied about a no-thing

(a) Paradox of Non-Being (II) : from true negations to negative facts?

Negative fact: an artifact of the correspondence theory of truth Negative fact: a corollary of logical atomism? Russell admitted negative/positive facts, not Wittgenstein Negation is not a part of the world A fact makes a sentence true (false) and its negation false (true)

Problem: what of the extremes cases: tautologies and antilogies? Tautology: a sentence that cannot be made false Antilogy: a sentence that cannot be made true What do these correspond to: “logical facts”? Two competing theories of meaning: Picture Theory vs Network Data

(b) Dialectical negations (I): Heidegger’s nichtendes Nicht

A scalar analysis of Being (Absolute) and Non-Being (Nothingness) Reines Sein: complete being without non-being

For every bit ai() of the string A(), ai() = 1 Reines Nichtsein: complete non-being without being

For every bit ai() of the string A(),ai() = 1 Dasein: incomplete being and non-being

For some bit ai() of the string A(),ai() = 1

For some bit aj() of the string A(), a () = 1

(b) Dialectical negations (I): Heidegger’s nichtendes Nicht

Is “das Nicht nichtet” analogous to “Das Regen regnet” (Carnap)? and/or

“Socrates” as “There is a x such that x socratizes” (Quine)?

Operand vs operator

“Das Nichts”: an operand A() = “nichten”: a non-involutive operation from 1 to 0 “verneinen” as denying: an involutive function from 1 to 0 and conversely

“Verneinung” as the resulting process of denial: a bit such that ai() = 0

“vernichten” as a stronger “Verneinung”: from being () to not-being ()

A non-atomistic ontology of Being and Non-Being “Das Nicht nichtet”: Non-Being participates any mere being “Das Sein seiet”: Being participates any mere being

(b) Dialectical negations (I): Heidegger’s nichtendes Nichts HEIDEGGERIAN OPPOSITIONS (?)

Reinheit = Unbestimmheit

Reines Sein Reines Nichtsein (the Absolute) (Nothingness) Nicht-reines Nichtsein Nicht-reines Sein

Dasein = Bestimmheit

(b) Dialectical negations (I): Heidegger’s nichtendes Nichts HEIDEGGERIAN OPPOSITIONS (?)

T

T

cd() cd(T)


Problems with pure being and nothingness: - a degenerate diagram

a collapse between edges: (T ) = T, and ct(T) = cd(T) =

- mere being should be a subaltern of pure being: = sb(T)

nothingness should be a subaltern of mere being: = sb()

A plea for mereology? Beyond part-whole relations

Pure being is not a union of affirmed properties: T = (1 2 … n)

Likewise for nothingness, accordingly: = cd(1 2 … n) Absolute differs from “every-thing”, just as Nothingness from “no-thing”

Another diagram for T and

T and collapse in the center of the diagram An echo of Hegel’s view of Selbigkeit as non-informativity?


“Das reine Sein und das reine Nichts ist also dasselbe, besteht zu Recht.” (Hegel : Wissenschaft der Logik)


Problems with pure being and nothingness: - a degenerate diagram

a collapse between edges: (T ) = T, and ct(T) = cd(T) =

- mere being should be a subaltern of pure being: = sb(T)

nothingness should be a subaltern of mere being: = sb()

A plea for mereology? Beyond part-whole relations

Pure being is not a union of affirmed properties: T = (1 2 … n)

Nothingness is not a union of denied properties: = cd(1 2 … n)

An enriched diagram including T and

T and collapse in it An echo of Hegel’s view of Selbigkeit as non-informativity?


An alternative characterization: primary bits A cumulative ontology, from a model with one all-encompassing property

Aufhebung: dialectical negation as negative participation a transformation-and-conservation-forming operator

A conservative extension of things, from m = 1 to an indefinite number m Transformation: adding negativity to being (e.g. from 1 to 10) Conservation: preserving the former values/bits of the string

Assumption: a dynamic model from pure being (Absolute) to mere being Dialectical negation: N(1) = 10 Cf. Plato’s Sophist: definition by dichotomy

A dual negation: “Vernichtung” as a destruction-operator towards m = 0?


A logical algebraic system of colours Colours can be depicted as structured objects with opposite relations A scalar quantification upon wavelengths/activation of cone types in retina

Two polar cases of colour A(White) = 111 = T

A(Black) = 000 = Blackness: no activation of cones types Whiteness: activation of all the cone types

Quantitative oppositions Colours are structures similar to any other modes of predication Compare e.g. with quantification on time (never-sometimes-always)

(c) The Colour Exclusion Problem LOGIC OF COLOURS (Jaspers 2009)

Magenta Red Blue

Yellow Cyan

Green

(c) The Colour Exclusion Problem LOGIC OF COLOURS (Jaspers 2009)

101 100 001

110 011

010

(d) Eastern negation

Two opposite attitudes towards conventional truth (“saṃvṛti-satya”): Jain’s perspectivism: everything can be assigned to subject from some perspective (“anekāntavāda”: non-one-sidedness) Madhyamaka’s skepticism: nothing can be truly predicated of a subject

Two ways of negating: Locutionary negation, or sentential negation: “paryudāsapratiṣedha” Illocutionary negation, or denial (no-answer) : “prasajyapratiṣedha”

Two opposite attitudes towards nonsense Nothing can be truly predicated of the soul (self-produced entity) Sense is questionability: yes- and no-answers about properties No sense = no relevant question (property) to individuate a thing An echo to Hegel’s stance: pure being and nothingness are the “same”

4. Conclusion and Prospects

Conclusion

A formal ontology of objects, with an ensuing logic (as calculus) A question-answer game without specific agents (humans, the Nature, etc) Every object is a meaningful information including sense and reference Objects are strings of being (so) and non-being (so)

Negation: a difference-forming operator op Expression of a disagreement (in at least one bit of the opposed strings) To negate, that is the question (contributes to the sense of the object).

Denial: a no-answer ai() = 0 To deny, that is the answer (contributes to the reference of the object).

A general theory of negation A Boolean framework of Boolean bitstrings, including a variety of negations Negation is a central operation to think about identity and difference

Prospects

A formal tool for philosophy A more comprehensive logic of opposition, enlightening the meaning of: “identity”, “sameness”, “equality”, “value”, and the like A conceptual framework to think about art, politics, economics, etc.

A developed reflection about the limits of sense About the possibility of “true”, inclusive contradictions

An abstract logic of opposition A dynamic theory of information process A counterpart of Tarski’s abstract logic of consequence (Op instead of Cn)

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