View of Valuation Semantics for Intuitionic Propositional Calculus and some of its Subcalculi

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Abstract. In this paper, we present valuation semantics for the Propositional Intuitionistic Calculus (also called Heyting Calculus) and three important subcalculi: the Implicative, the Positive and the Minimal Calculus (also known as Kolmogoroff or Johansson Calculus).

Algorithms based in our definitions yields decision methods for these calculi.

Keywords: Intuitionism, minimal calculus, valuation semantics, valuation tables.

We consider a language containing a countable set of propositional variables, a unary connective for negation (¬), three binary connectives for conjunction (&), disjunc- tion (∨) and implication (→) and parenthesis; formula and subformula are define as usual. The notions of postulate, deduction, proof, theorem, deductive consequence, etc., are used as in Kleene (1). Let us begin presenting some postulates:

P1. A→(B→A)

P2. (A→B)→((A→(B→C)→(A→C)) P3. A,(A→B)/B(Modus Ponens) P4. (A&B)→A

P5. (A&B)→B P6. A→(B→(A&B)) P7. A→(A∨B) P8. B→(A∨B)

P9. (A→C)→((B→C)→((A∨B)→C) P10. (A→B)→((A→ ¬B)→ ¬A)

P11. ¬A→(A→B)

As is well known, the Implicative Intuitionistic Calculus (I) is the logic given by P1–P3; the Positive Intuitionistic Calculus (P), by P1–P9; the Minimal Calculus (M), by P1–P10; and P1–P11 constitute a basis for the Intuitionistic (Propositional) Calculus, or Heyting’s Calculus (H). Remember, also, the deduction theorem holds for all these calculi.

Γ`CAmeans, as usual, that there is a deduction of the formulaAfrom the set of formulas Γin the calculus C. For calculi treated here (as well as other calculi using the same kind of deductive definitions), we define the concept of anF-saturated set:

Principia14(1): 125–33 (2010).

For any formula F, a set of formulas Γ is F-saturated in a calculus C, if:

it is not the case that Γ `C F; and, for every formula F<sup>0</sup>, if F<sup>0</sup> ∈/ Γ then Γ∪ {B} `CF.

It is easy to prove that ifΓisF-saturated in a calculus C, then for any formulaA, Sat1: Γ`CAiffA∈Γ.

In fact, it is required only that the following usual deductive properties hold:

i) Γ`CA, ifA∈Γ;

ii) If{B₁, . . . ,B_n} `CAand for 1≤i≤n,Γ`C B_i thenΓ`CA.

And since it holds for all these calculi that:

iii) IfΓ`CA, then for some finite subsetΓ<sup>0</sup>ofΓ,Γ<sup>0</sup>`CA, we have also for them the Lindenbaum’s style lemma:

Sat2: IfΓ0CF, there is anF-saturated set∆such thatΓ⊂∆and∆0C F.

Observe that sinceIis a subcalculus ofP, which is a subcalculus ofM, which is a subcalculus of H, using Sat1 and some almost immediate deductive properties of these calculi, we easily obtain:

Sat3: If∆is anF-saturated set — either inI,P,MorH— then i) IfB∈∆thenA→B.

ii) IfA→B∈∆thenA∈/∆orB∈∆.

iii) IfA→B∈/∆then there is aB-saturated set∆⁰(inI,P,MorH, respectively) such that∆∪ {A} ⊆∆⁰.

Sat4: If∆is anF-saturated set — either inP,MorH— then iv) (A&B)∈∆iffA∈∆andB∈∆;

v) (A∨B)∈∆iffA∈∆orB∈∆.

Sat5: If∆is anF-saturated set — either inMorH— then vi) If bothA,¬A∈∆then for allB,¬B∈∆.

Sat6: If∆is anF-saturated set inHthen vii) If¬A∈∆thenA∈/∆.

We will be using the following metalinguistic conventions:

F: for the set of formulas of this language;

→A_n: for a formula of the formA₁→(A2→. . .(An−1→A_n). . .);

→An: for→A_n, to indicate thatA_nhas not the formA→A⁰.

Semivaluations

An implicative semivaluations(I-semivaluation, or semivaluation forI) is a function fromF into{0, 1}such that, for everyA,BofF, the following conditions hold:

S0→: s(A→B) =0⇒s(B) =0;

S1→: s(A→B) =1⇒s(A) =0 ors(B) =1.

A positive semivaluation s (P-semivaluation, or semivaluation for P) is an im- plicative semivaluation such that, for everyA,BofF, the following conditions hold:

S&: s(A&B) =1⇔s(A) =1 ands(B) =1;

S∨: s(A∨B) =1⇔s(A) =1 ors(B) =1.

A minimal semivaluations(M-semivaluation, or semivaluation forM) is a posi- tive semivaluation such that, for everyA,BofF, the following condition holds:

S M¬: s(¬A) =s(A) =1⇒s(¬B) =1.

An intuionistic semivaluation s (H-semivaluation, or semivaluation for H) is a positive semivaluation such that, for everyA,BofF, the following condition holds:

SH¬:s(¬A) =1⇒s(A) =0.

Lemma 1. It is never the case that a) For everyI-semivaluations,

1.1: s(A) =1ands(B→A) =0;

1.2: s(A→B) =1,s(A→(B→C)) =1,s(A) =1ands(B) =0;

1.3: s(A) =s(A→B) =1ands(B) =0;

b) For everyP-semivaluations, 1.4: s(A& B) =1ands(A) =0;

1.5: s(A& B) =1ands(B) =0;

1.6: s(A) =1,s(B) =1ands(A& B) =0;

1.7: s(A) =1ands(A∨B) =0;

1.8: s(B) =1ands(A∨B) =0;

1.9: s(A→C) =s(B→C) =s(A∨B) =1ands(C) =0;

c) For everyM-semivaluations,

1.10: s(¬A) =s(A) =1ands(¬B) =0.

Proof. Follows immediately from the definitions ofI-,P- andM-semivaluations.

Lemma 2. Ifsis anH-semivaluation ands(¬A) =s(A) =1thens(¬B) =0.

Proof. A positive valuation which satisfiesSH¬also satisfies (vacuously)S M¬. Corollary 1. EveryH-semivaluation is anM-semivaluation.

Corollary 2. For everyM-semivaluation s it is not the case thats(A→ B) =s(A→

¬B) =s(A) =1ands(¬A) =0.

Proof. If s(A→B) =s(A→ ¬B) =s(A) =1, then s(B) =s(¬B) =1; so, byS M¬, s(¬A) =1.

Valuations

A valuationv— forI,P,MorH— is a semivaluation — forI,P,MorH— for which the conditionV→, below, holds:

V→: for every formula of the form→An, ifv(→An) =0, then there is a semivalu- ations— forI,P,Mor H, respectively — such that for everyi<n,s(A_i) =1 and

i) s(An) =0; andii)if for someB,A_n is¬B, thens(B) =1.

We may use “I-valuation” , “P-valuation” ,“M-valuation” and “H-valuation” to mean valuations forI, forP, forMand forH, respectively.

Lemma 3.

i) EveryP-valuation is anI-valuation;

ii) everyM-valuation is aP-valuation;

iii) everyH-valuation is anM-valuation.

Proof. Follows from the corresponding definitions, using, for iii), the Corollary of Lemma 2.

Lemma 4. For every valuationv(forI,P,Mor forH) and for any formula of the form

→A_n, if v(→A_n) =0, then there is a semivaluations(forI,P,MorH, respectively), such that, for every m≤n, and for every i<m,s(A_i) =1ands(A_m) =0.

Proof. By a simple induction using conditionsS1→andV→.

Corollary 1. For everyI-valuationv, and for every formula A which is an instance of postulates P1, P2,v(A) =1.

Proof. Follows from lemma1 (1.1–1.2) together with lemma 4.

Corollary 2. For everyP-valuationv, and for every formula A which is an instance of postulates P1, P2, P4–P9,v(A) =1.

Proof. Follows from lemma1 (1.1–1.9) together with lemma 4.

Corollary 3. For everyM-valuationvand for every formula A which is an instance of postulate P1, P2, P4-P10,v(A) =1.

Proof. Follows from lemma1 (1.1–1.10) together with lemma 4.

Corollary 4. For every H-valuationvand for every formula A which is an instance of postulate P1, P2, P4–P10,v(A) =1.

Proof. Follows from lemma1 (1.1–1.10), Corollary 2 of Lemma 2, and lemma 4.

Corollary 5. For everyH-valuationvand for every formula A which is an instance of postulate P11,v(A) =1.

Proof. Follows from conditionSH¬together with lemma 4.

Soundness and Completeness

In the following, we will writeÏCAto mean “for every valuationvfor the calculus C, if for every B∈Γ,v(B) =1, thenv(A) =1”.

Lemma 5. If C is eitherI,P,MorH, then ifÏCA andÏCA→B thenÏCB Theorem 1(Soundness). If C is eitherI,P,MorH, then ifΓ`CA thenÏCA.

Proof. Follows from corollaries of lemma 4 together with lemma 6.

Lemma 6. Letvbe the characteristic function of a set∆; then, 1) if∆is F -saturated (inI,P,MorH),vis anI-semivaluation;

2) if∆is F -saturated (inP,MorH),vis aP-semivaluation;

3) if∆is F -saturated (inMorH),vis anM-semivaluation;

4) if∆is F -saturated (inH),vis anH-semivaluation.

Proof. Use Sat3 i) and ii) to prove 1); use 1), Sat4 iv) and v) to prove 2); use 2) and Sat5 vi) to prove 3); and use 3 and Sat6 vii) to prove 4).

Lemma 7. Letvbe the characteristic function of a set∆; then, 1) if∆is F -saturated (inI,P,MorH),vis anI-valuation;

2) if∆is F -saturated (inP,MorH),vis aP-valuation;

3) if∆is F -saturated (inMorH),vis anM-valuation;

4) if∆is F -saturated (inH),vis anH-valuation.

Proof. Follows from lemma 7 using Sat3 iii).

Theorem 2(Completeness). If C is eitherI,P,MorH, then ifÏCA thenΓ`CA.

Proof. IfΓ0CAthen there is anA-saturated∆in C such thatΓ⊆∆; so the valuation for C which is the characteristic function of∆gives the value 1 to every formula of Γand 0 toA; hence,Γ2CA.

Valuation Tables

Valuation tables may be constructed to decide the calculiI,P,MandH.

Before defining this kind of construction, let us introduce some auxiliary con- cepts.

AΣ-sequence is a finite sequence of formulasF₁, . . . ,F_n such that, for 1≤i≤n, if F⁰is a subformula ofF_i, then there is ak≤isuch thatF⁰=F_k.

Avaluations table for a givenΣ-sequence is construction containing ahead line and a certain number of valued linesobtained by some given rules. The head line specify the three main vertical parts of each valued line l, which consists of:

1) σl — the justification sequence of sets of the valued linel;

2) l — the number of the valued line;

3) δ_l — the values sequence of the valued line, corresponding to the formulas of theΣ-sequence — so, to each formula of theΣ-sequence, a value is associated in that line.

AnI-,P-,M- andH-valuations tableT[F₁, . . . ,F_n]may now be defined by induc- tion on the lengthnof theΣ-sequence, as follows:

n=1 : An I-, P-, M- or H-valuations tableT[F₁]will have two valued lines, with numbers 1 and 2, each with a void set as first term of the justification sequence.

F₁ receives the value 1 in line 1 and the value 0 in line 2.

n>1: suppose that T[F₁, . . . ,F_n−1]is constructed withqlines; we will writel_­l⁰ to mean that for 1≤i<nif thei^thterm ofδl is 1 then thei^thterm ofδl⁰ is 1.

If F_n is atomic or an implication, obtain the I-, P-, M- or H-valuations table T[F1, . . . ,F_n]by extendingT[F1, . . . ,F_n−1]as follows:

• ifF_n is atomic,T[F1, . . . ,F_n]will haveqnew lines with numbersq+1 toq+q and such that: for 1≤l≤q, then^thterm ofσl is the void set and,σ_q+l=σl; the initial segment tilln−1 ofδq+l is the same as inδl; then^thvalue ofδl is 1 and ofδ_q+l is 0.

• ifF_nis(F_k→F_m)thenT[F₁, . . . ,F_n]is obtained in 4 steps:

a) first, in each linel≤qwhere thek^thvalue ofδl is 1 and them^thvalue of δl is 0, we take 0 as then^thvalue ofδl; and we take the void set as the n^thterm ofσl; these lines will be called the a-lines of then^th step;

b) second, for eachl ≤q, if there is no a-linel⁰such that l ­l⁰, we take 1 for the n^th value of δ_l; and we take the void set as the n^th term of σ_l; these lines will be called the b-lines of then^th step;

c) third, for eachl ≤ q, in the usual order, if there is a term of σl whose elements are all a-lines or lines where then^th value ofδ_l was previously obtained by this c) rule, we take 0 as then^th value ofδl; and we take the void set as then^th term ofσ_l; these lines will be called the c-lines of the n^thstep;

d) finally, suppose that there are still lines not yet finished and that D is the set of those lines; then iflis the j^thline in D, in the usual order, construct a new line, with number q+ j, as a copy of the part of l constructed at then−1^thstep; then take 1 as the n^th value ofδl and 0 as then^th value ofδ_q+j; take the void set as the n^th term of σl and eliminate from the previous terms ofσl all the numbers of a-lines and c-lines of then^thstep;

finally add toσq+j its n^th term ofσq+j which will be the set {l⁰≤q:l⁰ is an a-line or a c-line such thatq+ j_­ l⁰}. Calll a d1-line andq+j a d0-line of then^thstep.

If F_nis(Fk&F_m)or(Fk∨F_m), to get theP-,M- andH-valuations table T[F1, . . . , F_n], just complete every linel, by extendingδ_l in the classical way and adding the void set is the n^th term ofσl.

If F_n is ¬F_k, the M- andH-valuations table T[F₁, . . . ,F_n] is obtained by these steps:

i) if we are dealing withM-valuations then:

• if, for every m < n there is no i such that F_m = ¬F_i, then, for every line l, construct a new linel⁰ with the initial segment tilln−1 of l; then^th value of δl, will be 1 forl and 0 forl⁰; the void set is added to bothσl andσl⁰; call a lineM-typical if F_n andF_khave both the same value 1;

• else, for everyl≤q,

a) ifl is not a typical line, then then^thvalue ofδl, will be 0 if the k^thvalue is 1 and the void set is added toσl;l is called an a-line for then^th step;

b) ifl is a typicalM-line (F_m andF_i have both value 1), take 1 as the value ofF_n; add the void set as a new term of the justification sequence.

We have now to treat the lines l where thek^th term ofδl has the value 0; then we proceed exactly as in cases c) and d) whenF_nis an implication.

ii) if it is anH-valuations table, then :

a) then^th value ofδl, will be 0 if the k^thvalue is 1 and the void set is added toσl; these lines are called the a-lines for then^thstep;

b) void case inH(we maintain the letter “b” only to help comparisons) c) and d) — like c) and d) inM-valuations tables.

By methods similar to those presented in Lopari´c 1986, two lemmas can be proved:

Lemma 8. For every value line l in everyI-,P-,M- orH-valuation table T[F₁, . . . ,F_n], there is a valuation v(forI,P,MorH, respectively) which gives to F_i the value of the i^thterm ofδl , for1≤i≤n (that means,v“agree” with l in the values associated with the formulas F₁, . . . ,F_n).

Lemma 9. For every valuation v for I, P, M or H, if T[F₁, . . . ,F_n] is an I-, P-, M- or H-valuation table T[F1, . . . ,F_n], respectively, there is a line l such that for every i, 1≤i≤ n, the the i^th term ofδl is the value thatv gives to F_i (that means, l “agree”

withvin the values associated with the formulas F₁, . . . ,F_n).

Theorem 3. TheI-,P-,M- andH-valuation tables are decision methods for the calculi I,P,MandH, respectively.

The URLhttp://www.paralogics.net/tableaux/minimal_intuitionism/is a page containing a form where, when a visitor enters a formula and chooses a calculus (Mor H), a script returns a corresponding valuation table for a sequence having, as its terms, this formula and all its subformulas.

References

Kleene, S. C. 1952.Introduction to Metamathematics. Amsterdam: North Holland.

Lopari´c, A. M. A. C. 1986. A Semantical Study of Some Propositional Calculi.The Journal of Non-Classical Logic3(1): 74–95.

A^NDRÉAL^{OPARI ´C} Departamento de Filosofia Universidade de São Paulo São Paulo, SP

BRAZIL

[email protected] Resumo. Apresentamos neste trabalho semânticas de valorações para o Cálculo Proposicio- nal Intuicionista (também conhecido como Cálculo Proposicional de Heyting) e três de seus

importantes subcálculos: os cálculos proposicionais Implicativo, Positivo e Minimal (também chamado Cálculo de Kolmogoroff ou de Johansson). Provamos a correção e a completude dessas valorações com respeito aos respectivos cálculos e, em seguida, apresentamos algo- ritmos de geração das tabelas dessas valorações, algoritmos que se constituem, assim, em métodos alternativos de decisão para os esses cálculos.

Palavras-chave:Intuicionismo, cálculo minimal, semântica de valorações, tabelas de valo- rações.