Kleene logic and inference - Philosophy & Logic 2013

Kleene logic and inference

Grzegorz Malinowski

Department of LogicŁódź University

2013

The aims:

analysis of a distinguished three-valued 1938 Kleene logicconstruction

outlining the problem of inference, as important in view ofthe fact that no formula of Kleene logic is a K3 tautology

critical discussion of 1974 Cleve’s attempt to define atruth-preserving consequence relation within a modeltheoretic framework for inexact predicates

showing a radically different and sound ground for Kleeneinference supported by two ”idealization” operators whichconvert neutral propositions into false or true

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The aims:





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The aims:





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The aims:





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The aims:





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1. Kleene logic

Kleene three-valued logic [1938], [1952] has an epistemologicalmotivation. The third logical value was designed to markindeterminacy of some proposition at a certain stage ofscientific investigation.

is inspired by the studies of the foundations of mathe-matics, especially by the problems of algorithms, cf. alsoKleene [1952]

is based on the third category of propositions, whoselogical value of truth or falsity is not essential, undefinedor undetermined by means of accessible algorithms

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1. Kleene logic




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1. Kleene logic




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1. Kleene logic




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The third logical value was designed to mark indeterminacy ofsome proposition at a certain stage of scientific investigation.And,

its assumed status is somewhat different from the classicalvalues of truth, t and falsity, f

it is not independent since it may turn into either of thetwo classical values

the set f, u, t of three values cannot be considered aslinearly ordered

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The connectives: ¬ (negation), → (implication), ∨(disjunction), ∧ (conjunction ), and ≡ (equivalence):

α ¬αf tu ut f

→ f u tf t t tu u u tt f u t

∨ f u tf f u tu u u tt t t t

∧ f u tf f f fu f u ut f u t

≡ f u tf t u fu u u ut f u t

K3 = ({f, u, t}, ¬, →, ∨, ∧, {t}) Kleene matrix

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α ¬αf tu ut f






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α ¬αf tu ut f






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K3 defines the non-tautological logic since any valuationwhich assigns the value u to each propositional variablesends any formula into u

Kleene is aware of discarding even such law of logic as thelaw of identity p→ p and its “equivalential” version p ≡ p.

”Here “unknown” is a category into which we can regardany proposition as falling, whose value we either do notknow or choose for the moment to disregard; and it doesnot then exclude the other two possibilities ‘true’ and‘false’.” (Kleene [1952], p. 335)

We may, therefore, conclude that Kleene treated theadded logical value as apparent or as pseudo-value,distinct from the real truth-values, cf. Turquette [1963].

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2. Inexact classes

Korner [1966] gave the most accurate and compatibleinterpretation of Kleene’s views of the connectives of K3:

He introduced a non-standard notion of an inexact class asan object characterized by a non-definite classifyingprocedure determined by a partial algorithm

An inexact class is an abstract class set by a ”characteristicfunction”, which discerns between members,non-members, and neutral candidates for membership

the algebra of inexact classes leads naturally to thethree-valued logic K3

Basing on it Cleave [1974] worked out a model-theoreticframework for the logic of inexact predicates and heformulated a notion of logical consequence.

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2. Inexact classes






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2. Inexact classes






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2. Inexact classes






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2. Inexact classes






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2. Inexact classes






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Cleave’s contribution

is important in many respects, especially because of theanalysis concerning the first-order inexact structures

is faulty since Cleave adopts the idea that the Kleenevalues f, u, t are mutually comparable and linearly ordered:f ¬ u ¬ t. (-1, 0 and +1 stand for f,u and t)

The concept of a a degree of truth preserving consequenceoperation is determined by the Cleave’s variant of K3,

C3 = ( -1, 0, +1 , ¬ , → , ∨ , ∧ , ≡ , +1 ) ,

as follows: α is a C3 consequence of the set of formulasX, X |=C3 α , whenever for every valuation v of thelanguage in K3, min{v(β) : β ∈ X} ¬ v(α).

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C3 = ( -1, 0, +1 , ¬ , → , ∨ , ∧ , ≡ , +1 ) ,


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C3 = ( -1, 0, +1 , ¬ , → , ∨ , ∧ , ≡ , +1 ) ,


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C3 = ( -1, 0, +1 , ¬ , → , ∨ , ∧ , ≡ , +1 ) ,


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C3 = ( -1, 0, +1 , ¬ , → , ∨ , ∧ , ≡ , +1 ) ,


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C3 = ( -1, 0, +1 , ¬ , → , ∨ , ∧ , ≡ , +1 ) ,


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C3 = ( -1, 0, +1 , ¬ , → , ∨ , ∧ , ≡ , +1 ) ,


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3. Inference

Departing from the idea of ”neutrality” of sentences of thethird category and accepting inexactness, Korner [1966] triesto apply the classical logic to empirical discourse:

To this aim, he proposes a special evaluation deviceleading to a kind of ”modified two-valued logic” and,ultimately, to an instrument of evaluation validity ofsentences and deduction

His interesting trial ends negatively: ”The classicaltwo-valued logic as an instrument of deduction, however,presupposes that neutral propositions are treated as if theywere true, and inexact predicates as if they were exact. ...”

This obviously means that the use of the classical logic fordrawing empirical conclusions from empirical premises isunsound.

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3. Inference





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3. Inference





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3. Inference





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3. Inference





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Grounds for new Kleene inference relation

Korner’s analysis of the structure of scientific theories andtheir relation to the experience, including:

K1.Two complementary spheres of scientific activity: thesphere of experience and the sphere of theory

K2. Deductive unification of a field of experience byclassical logic is an idealisation or a modification ofempirical discourse

K3. Idealization procedure supported by the two operatorswhich convert neutral propositions into false or true

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The relation of empirical inference:

holds between true or undefinite premises and the trueconclusion. Thus, a conclusion α may be empiricallyinferred from a set of premises X, whenever, for anyinterpretation, it is the case that if all elements of X arenot false, then α is true

may express the passing from the empirical inexact andundefinite discourse to the ”heaven” of theory but at thecost of a natural extension of the language

the latter is possible when two connectives correspondingto idealisation procedures are used. In case of (inexact)classes this means replacement by their exactcounterparts, in case of sentences turning undefinitesentences into definite, i.e. true or false

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Definition:

A relation `K3 is said to be a matrix empirical inference of K3provided that for anyX ⊆ For, α ∈ For

X `K3 α iff for every h ∈ Hom(L,A)(h(α) = t wheneverh(X) ⊆u, t.

An intuition behind this definition is that a conclusion maybe inferred empirically from a set of premises X, when forany interpretation it is the case that if all elements of Xare not false then α is true

`K3 is compatible with two possible logical idealisations ofundefinite empirical sentences

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Definition:





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Definition:





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Definition:





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Definition:





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Definition:





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The operators corresponding to these idealisations arepresented through Kleene-like tables as follows:

α Wα

f fu ft t

α Sα

f fu tt t

The letters chosen for these connectives may be read as Weakand Strong. Let us note that the empirical inference justdefined may be also interpreted as a relation which holdsbetween a set of premises X and a conclusion whenever it istrue independently from accepted idealisation(s) of empiricalsentences in X.

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α Wα

f fu ft t

α Sα

f fu tt t


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α Wα

f fu ft t

α Sα

f fu tt t


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Here are examples of theorems establishing some essentialproperties of the empirical inference of Kleene sentential logic:

(e1) If X `K3 α and X `K3 α→ β, then X `K3 β

(e2) X `K3 α→ β if and only if X ∪ α `K3 β

(e3) α `K3 S(α) and W (α) `K3 α.

The first property is an inferential modus ponens, the second aconditional deduction theorem. The inferences in (e3) statethat from a sentence its strong idealisation follows while thesentence itself is inferred from its weak idealisation.

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(e3) α `K3 S(α) and W (α) `K3 α.


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(e3) α `K3 S(α) and W (α) `K3 α.


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(e3) α `K3 S(α) and W (α) `K3 α.


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(e3) α `K3 S(α) and W (α) `K3 α.


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The diagram below shows how the Kleene values f,u,t areordered and displays the inferential relations between a formulaα and its two idealizations, S(α) and W (α):

•

• •

��

AAAAAA

f t

u•

•

•

W (α)

α

S(α)

So, any sentence α is deductively situated between its twoidealizations: W (α) `K3 α `K3 S(α).

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•

• •

��

AAAAAA

f t

u•

•

•

W (α)

α

S(α)


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•

• •

��

AAAAAA

f t

u•

•

•

W (α)

α

S(α)


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4. Final remarks

The empirical inference relation appears to be a specialcase of the semantic relation defined on q-matricesM* = (A , D* , D) ,where D*, D are disjoint subsets of Ainterpreted as rejected and accepted elements.

Kleene’s case: A = ({f, u, t}, ¬, →, ∨, ∧, ≡, S, W ),D* = {f} and D = {t}.

The Kleene q-matrix consequence is a W and Scompatible inference from non-rejected premises toaccepted conclusions.

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4. Final remarks




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4. Final remarks




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4. Final remarks




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2. References

[1] Cleave, J.P., The notion of logical consequence in the logic ofinexact predicates,Zeitschrift fur Mathematische Logik andGrundlagen der Mathematik, 20, 1974, 307-324.[2] Kleene, S.C., ‘On a notation for ordinal numbers’, The Journalof Symbolic Logic, 3(1938), pp. 150–155.[3] Kleene, S.C., Introduction to metamathematics,North-Holland,Amsterdam, 1952.[4] Korner, S., Experience and theory,Routledge and Kegan Paul,London, 1966.[5] Malinowski, G., Inferential many-valuedness [in:] Woleński, J.(ed.) Philosophical logic in Poland, Synthese Library, 228, KluwerAcademic Publishers, Dordrecht, 1994, 75-84.[6] Malinowski, G., Many-valued logics, Oxford Logic Guides, 25,Clarendon Press, Oxford, 1993.

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[7] Malinowski, G., Q-consequence operation, Reports onMathematical Logic, 24, 1990, 49-59.[8] Turquette, A. R., Modality, minimality, and many-valuedness,Acta Philosophica Fennica, XVI, 1963, 261-276.

Department of LogicUniversity of Łódź[email protected]

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Kleene logic and inference - Philosophy & Logic 2013

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