aa r X i v : . [ m a t h . L O ] A p r On the Ja´skowski Models for IntuitionisticPropositional Logic
R.D. ArthanApril 30, 2019
Abstract
In 1936, Stanislaw Ja´skowski [1] gave a construction of an interestingsequence J , J , . . . of what he called “matrices”, which we would todaycall “finite Heyting Algebras”. He then gave a very brief sketch of a proofthat if a propositional formula holds in every J i then it is provable inintuitionistic propositional logic ( IPL ). The sketch just describes a certainnormal form for propositional formulas and gives a very terse outline ofan inductive argument showing that an unprovable formula in the normalform can be refuted in one of the J k . Unfortunately, it is far from clearhow to recover a complete proof from this sketch.In the early 1950s, Gene F. Rose [4] gave a detailed proof of Ja´skowski’sresult, still using the notion of matrix rather than Heyting algebra, basedon a normal form that is more restrictive than the one that Ja´skowskiproposed. However, Rose’s paper refers to his thesis [3] for additionaldetails, particularly concerning the normal form.This note gives a proof of Ja´skowski’s result using modern terminol-ogy and a normal form more like Ja´skowski’s. We also prove a semanticproperty of the normal form enabling us to give an alternative proof ofcompleteness of IPL for the Heyting algebra semantics. We outline adecision procedure for
IPL based on our proofs and illustrate it in actionon some simple examples.
Let H = ( H, f , t , ⊓ , ⊔ , → ) be a Heyting algebra. We will define a new Heytingalgebra Γ( H ) by adding a co-atom, i.e., a new element ∗ such that x < ∗ < t for x ∈ H \ { t } . Γ( H ) will extend H as a ( f , t , ⊓ , → )-algebra and the join in Γ( H )will agree with the join in H wherever possible. Thus, we choose some object ∗ = ∗ H that is not an element of H and let Γ( H ) = ( H ∪ {∗} , f , t , ⊓ , ⊔ , → ),where the operations ⊓ , ⊔ and → are derived from those of H as shown inthe operation tables below, in which x and y range over H \ { t } and where α : H → ( H \ { t } ) ∪ {∗} satisfies α ( x ) = x for x = t and α ( t ) = ∗ . ⊓ y ∗ t x x ⊓ y x x ∗ y ∗ ∗ t y ∗ t ⊔ y ∗ t x α ( x ⊔ y ) ∗ t ∗ ∗ ∗ tt t t t → y ∗ t x x → y t t ∗ y t tt y ∗ t B be the two-element Heyting algebra and, as usual, let us write H i forthe i -fold power of a Heyting algebra H . Then define a sequence J , J , . . . offinite Heyting algebras as follows: J = B J k +1 = Γ( J k +1 k )We take the language L of intuitionistic propositional logic, IPL , to beconstructed from a set V = { P , P , . . . } of variables, the constants ⊥ , ⊤ , andthe binary connectives ∧ , ∨ and ⇒ . We do not take negation as primitive: ¬ A is an abbreviation for A ⇒ ⊥ . The metavariables A, B, . . . , M (possibly withsubscripts) range over formulas. E and F are reserved for formulas that areeither variables or ⊥ . P, Q, . . . , Z range over variables. We assume known oneof the many ways of defining the logic of
IPL and write
IPL ⊢ A , if A is provablein IPL . IPL has an algebraic semantics in which, given a Heyting algebra H and an interpretation I : V → H , we extend I to a mapping v I : L → H byinterpreting ⊥ , ⊤ , ∧ , ∨ and ⇒ as f , t , ⊓ , ⊔ and → respectively. As usual wewrite I | = A if v I ( A ) = t , H | = A if I | = A for every interpretation I : V → H and | = A if H | = A for every Heyting algebra H . We assume known the factthat IPL is sound with respect to this semantics in the sense that, if
IPL ⊢ A ,then | = A . The converse statement, i.e., the completeness of IPL with respectto the semantics is well-known, but we do not use it: in fact we will give analternative to the usual proofs.We write A ⇔ B for ( A ⇒ B ) ∧ ( B ⇒ A ) and A [ B/X ] for the result ofsubstituting B for each occurrence of X in A . We have the following substitutionlemma: Lemma 1 (substitution)
For any formulas A , B and C and any variable X we have: (i) if IPL ⊢ C , then IPL ⊢ C [ A/X ] ; (ii) if IPL ⊢ A ⇔ B , then IPL ⊢ C [ A/X ] ⇔ C [ B/X ] ; Proof: (i) is proved by induction on a proof of C . (ii) is proved by inductionon the structure of C .We say a formula A is reduced if ⊤ does not appear in A as the operand ofany connective and ⊥ does not appear in A as the operand of any connectiveother than as the right-hand operand of ⇒ . Thus the only reduced formulacontaining ⊤ is ⊤ itself. Lemma 2
Any formula is equivalent to a reduced formula.
Proof:
This follows by repeated use of the substitution lemma and the provableequivalences ⊤ ∧ A ⇔ A , ⊥ ∧ A ⇔ ⊥ etc.We define a formula to be basic if it is reduced and is either a variable orhas one of the forms P ⇒ A or A ⇒ P where P is a variable and A contains at2ost one connective. Thus a basic formula has one of the following forms . P P ⇒ Q P ⇒ Q ∧ R P ⇒ Q ∨ R P ⇒ Q ⇒ R P ⇒ ¬ Q ¬ P P ∧ Q ⇒ R P ∨ Q ⇒ R ( P ⇒ Q ) ⇒ R ¬ P ⇒ Q Note that if A is basic formula of a form other than P , ( P ⇒ Q ) ⇒ R or ¬ P ⇒ Q , then V I ( A ) = t in any Heyting algebra under the interpretation I that maps every variable to f . Our convention for the metavariables E and F allows us to write, for example, ( P ⇒ E ) ⇒ R as a metanotation for the forms( P ⇒ Q ) ⇒ R and ¬ P ⇒ R .We say a formula is a basic context if it is reduced and is a conjunction ofone or more pairwise distinct basic formulas. We say a formula is regular if it isan implication K ⇒ F where K is a basic context (and following our convention F is a variable or ⊥ ).We say A and B are equiprovable and write A ⊣⊢ B if IPL ⊢ A iff IPL ⊢ B . Lemma 3
Every formula A is equiprovable with a regular formula M ⇒ Z suchthat if H is any Heyting algebra and I is an interpretation in H with V I ( M ) = t ,then V I ( A ) ≤ V I ( Z ) . Proof:
By Lemma 2, we may assume A is reduced. If A is ⊤ , let Z beany variable and let M : ≡ Z , then A and M ⇒ Z are both provable andhence they are equiprovable. If A is ⊥ or a variable, take M and Z to bedistinct variables, then neither A nor M ⇒ Z is provable, and hence they areequiprovable. Otherwise, chose some variable Z that does not occur in A . Thenit is easy to see that A ⊣⊢ ( A ⇒ Z ) ⇒ Z (for the right-to-left direction, use thesubstitution lemma to substitute A for Z ). Our plan is to replace K : ≡ A ⇒ Z by a basic context by “unnesting” all its non-atomic subformulas. Assume K contains k non-atomic subformulas. Starting with K ≡ A ≡ B ◦ C ,enumerate the k non-atomic sub-formulas, A ≡ B ◦ C , . . . , A k ≡ B k ◦ k C k .Choose fresh variables P i , i = 1 , . . . k . Define atomic formulas, G i , H i , for i = 1 , . . . , k as follows: G i is B i if B i is atomic and is P j if B i is the j -th non-atomic subformula; H i is C i if C i is atomic and is P j if C i is the j -th non-atomicsubformula. Now define formulas L and M as follows: L : ≡ k ^ i =1 ( P i ⇔ ( G i ◦ i H i )) M : ≡ P ∧ L Recalling that B ⇔ C is just shorthand for ( B ⇒ C ) ∧ ( C ⇒ B ), and usingthe fact that A and hence K are reduced, we see that M is a basic context, so M ⇒ Z is regular.We must show that K ⇒ Z ⊣⊢ M ⇒ Z . To see this, first assume IPL ⊢ K ⇒ Z . By induction on the size of the A i , we have that IPL ⊢ L ⇒ ( P i ⇔ A i ), We elide brackets using the rules that ⇒ is right associative and that the connectives arelisted in increasing order of precedence as ⇔ , ⇒ , ∨ , ∧ , ¬ . = 1 , . . . , k . Hence, as IPL ⊢ M ⇒ L , IPL ⊢ M ⇒ ( P ⇔ A ), i.e., IPL ⊢ M ⇒ ( P ⇔ K ). As, clearly, IPL ⊢ M ⇒ P , we have IPL ⊢ M ⇒ K andthen, as IPL ⊢ K ⇒ Z by assumption, we have IPL ⊢ M ⇒ Z . Conversely,assume IPL ⊢ M ⇒ Z . Using the substitution lemma, we have also that IPL ⊢ M [ A /P , . . . , A k /P k ] ⇒ Z , but M [ A /P , . . . , A k /P k ] is K ∧ L ′ where L ′ ≡ L [ A /P , . . . A k /P K ] is a conjunction of formulas of the form A ⇔ A , hence IPL ⊢ M [ A /P , . . . , A k /P k ] ⇔ K , and as IPL ⊢ M [ A /P , . . . , A k /P k ] ⇒ Z we have that IPL ⊢ K ⇒ Z .To prove M ⇒ Z satisfies the requirement about interpretations, assume I is an interpretation such that V I ( M ) = t . Then for each i = 1 , . . . , k , wehave V I ( P i ⇔ ( G i ◦ i H i )) = t , but this implies that V I ( P i ) = V I ( G i ◦ i H i )and hence, (by induction on the size of the A i ) that V I ( P i ) = V I ( A i ). Inparticular, V I ( P ) = V I ( A ) and since we also have V I ( P ) = t , we must have V I ( A ) = t . But by construction A ≡ A ⇒ Z , so V I ( A ⇒ Z ) = t , whichimplies V I ( A ) ≤ V I ( Z ).We now state and prove three lemmas whose purpose will become clear attheir point of use in the proof of our main theorem, Theorem 7. Lemma 4 If B is a basic formula that is not of the form P or P ⇒ Q ∨ R and P occurs in B , then IPL ⊢ P ∧ B ⇔ P ∧ C where C has fewer connectiveoccurrences than B and is either a basic formula, an atom or a basic contextcomprising a conjunction of two variables. Proof:
Routine using the fact that
IPL ⊢ P ∧ B ⇔ P ∧ B [ ⊤ /P ] (which maybe proved for arbitrary B by induction on the structure of B ). Lemma 5 If IPL ⊢ K ∧ A ∧ ( B ⇒ C ) ⇒ B , then IPL ⊢ (( K ∧ (( A ⇒ B ) ⇒ C )) ⇒ D ) ⇔ ( K ∧ C ⇒ D ) . Proof: ⇒ : easy using IPL ⊢ C ⇒ (( A ⇒ B ) ⇒ C ). ⇐ : the following gives the highlights of the natural deduction proof. K ∧ A ∧ ( B ⇒ C ) ⇒ B [Given] (1) K ∧ C ⇒ D [Assume] (2) K ∧ ( B ⇒ C ) ⇒ A ⇒ B [By (1)] (3) K ∧ (( A ⇒ B ) ⇒ C ) ⇒ A ⇒ B [By (3)] (4) K ∧ (( A ⇒ B ) ⇒ C ) ⇒ C [By (4)] (5) K ∧ (( A ⇒ B ) ⇒ C ) ⇒ D [By (5) and (2)] (6)( K ∧ C ⇒ D ) ⇒ (( K ∧ (( A ⇒ B ) ⇒ C )) ⇒ D ) [By (6), discharge (2)] (7)Here in step (4) we use IPL ⊢ (( A ⇒ B ) ⇒ C ) ⇒ ( B ⇒ C ). Lemma 6
Let B be a basic formula that is not a variable and let I be aninterpretation in a non-trivial Heyting algebra H such that V I ( B ) = t . Let α : H → ( H \ { t } ) ∪ {∗ H } be as in the definition of Γ( H ) . Define an interpretation J in Γ( H ) by J = α ◦ I . If B does not have the form ( P ⇒ E ) ⇒ R then V J ( B ) = t . (ii) If B has the form ( P ⇒ E ) ⇒ R , and if in addition V I ( P ) = V I ( E ⇒ R ) = t while V I ( E ) = t , then also V J ( B ) = t . Proof: (i) : This is easily checked for the case P ⇒ E and for the cases P ◦ Q ⇒ R and P ⇒ Q ◦ R when ◦ ∈ {∧ , ∨} . In the remaining case B ≡ P ⇒ Q ⇒ E .As B is equivalent to P ∧ Q ⇒ E , we have already covered the case when E is avariable, while if E is ⊥ , V J ( B ) = α ( p ) ⊓ α ( q ) → f , where p = I ( P ) and q = I ( Q ),but then, by inspection of the operation tables, we have α ( p ) ⊓ α ( q ) = p ⊓ q unless p = q = t , but as H is non-trivial and V I ( B ) = t , the case p = q = t cannot arise. (ii) : we have V J ( B ) = ( α ( p ) → α ( e )) → α ( r ), where p = V I ( P ), e = V I ( E )and r = V I ( R ). By assumption, p = t and e = t , so α ( p ) = ∗ and α ( e ) = e , hence α ( p ) → α ( e ) = ∗ → e = e , so that V J ( B ) = e → α ( r ) which is e → ∗ = t , if r = t , and is e → r otherwise, in which case, as we are given that V I ( E ⇒ R ) = t ,we have e → r = V I ( E ⇒ R ) = t .To state our main theorem, we define an interpretation I to be a strongrefutation of a formula of the form K ⇒ C , if V I ( K ) = t while V I ( C ) = t . Theorem 7
Let A ≡ K ⇒ F be a regular formula (so that F is either a variableor ⊥ ), let K ≡ B ∧ . . . ∧ B k display K as a disjunction of basic formulas andlet d = d ( A ) be the number of B i of the form ( P ⇒ E ) ⇒ R . Either IPL ⊢ A or A has a strong refutation in J d . Proof:
The proof is by induction on the sum s ( A ) = c ( A ) + d ( A ) + v ( A ), where c ( A ) is the number of connective occurrences in K , d ( A ) is as in the statementof the theorem and v ( A ) is the number of conjuncts of K comprising a singlevariable.Case (i) : v ( A ) = d ( A ) = 0: in this case, the interpretation in J = B thatmaps every variable to f is easily seen to be a strong refutation of A (which istherefore unprovable, by the soundness of IPL ).Case (ii) : v ( A ) >
0: in this case at least one B i is a variable. If all the B i arevariables and if B i F for any i , then A has strong refutation such that I ( B i ) = t , i = 1 , . . . , k and V I ( F ) = f . Otherwise, rearranging the B i if necessary, wemay assume that K ≡ P ∧ L where P is a variable and L ≡ B ∧ . . . ∧ B k . If P ≡ F , we are done: F ∧ L ⇒ F is provable. If P F and P does not occurin L , then it is easy to see that A ⊣⊢ A ′ where A ′ : ≡ L ⇒ F . As s ( A ′ ) < s ( A ),by induction, if IPL L ⇒ F , we can find a strong refutation I of L ⇒ F , butthen, because P does not occur in L ⇒ F , by adjusting I if necessary to map P to t we obtain a strong refutation of A . If P occurs in L , let us rearrange the B i again so that K ≡ P ∧ B ∧ M where M ≡ B , . . . , B k and P occurs in B .If B does not have the form P ⇒ Q ∨ R , then, by Lemma 4, we may replace P ∧ B by an equivalent formula P ∧ C where C is either a basic formula, anatom or a basic context comprising a conjunction of two variables and containsfewer connectives then B . If C is ⊥ , A is provable and we are done. Otherwise,we may replace A by the equivalent regular formula A ′ : ≡ P ∧ C ∧ M ⇒ F P ∧ M ⇒ F , if C is ⊤ ) and we are done by induction, since s ( A ′ ) < s ( A ).If B has the form P ⇒ Q ∨ R , then IPL ⊢ P ∧ B ∧ M ⇔ K ′ ∨ K ′′ where K ′ : ≡ P ∧ Q ∧ M and K ′′ : ≡ P ∧ R ∧ M , and hence IPL ⊢ A ⇔ A ′ ∧ A ′′ where A ′ : ≡ K ′ ⇒ F and A ′′ : ≡ K ′′ ⇒ F . If A is not provable, then one of A ′ and A ′′ is not provable, in which case, as s ( A ′ ) < s ( A ) and s ( A ′′ ) < s ( A ), by inductionwe have a strong refutation in J d of either A ′ or A ′′ and this will also stronglyrefute A .Case (iii) : v ( A ) = 0 and d = d ( A ) >
0: Let X = { j , . . . , j d } be theset of i such that B i has the form ( P ⇒ E ) ⇒ R . For each i ∈ X , let K i : ≡ B ∧ . . . ∧ B i − ∧ B i +1 ∧ . . . ∧ B k and let P i , E i and R i be such that B i ≡ ( P i ⇒ E i ) ⇒ R i . We now have two subcases depending on the provabilityof the formulas C i : ≡ K i ∧ P i ∧ ( E i ⇒ R i ) ⇒ E i :Subcase (iii)(a) : for some i ∈ X , IPL ⊢ C i : By Lemma 5, A , which isequivalent to K i ∧ (( P i ⇒ E i ) ⇒ R i ) ⇒ F , is equivalent to A ′ : ≡ K i ∧ R i ⇒ F .As s ( A ′ ) < s ( A ), we are done by induction.Subcase (iii)(b) : for every i ∈ X , IPL C i : By induction, as s ( C i ) < s ( A )and d ( C i ) = d −
1, for each i ∈ X there is an interpretation I i in J d − thatstrongly refutes C i , i.e., K i ∧ P i ∧ ( E i ⇒ R i ) ⇒ E i . Now define an interpretation I in J dd − , by I ( U ) = ( I j ( U ) , . . . , I j d ( U )). Then V I ( B i ) = t for i = 1 , . . . , k (because, for i ∈ X , V I i ( P i ) = V I i ( E i ⇒ R i ) = t and B i ≡ ( P i ⇒ E i ) ⇒ R i ).But then applying Lemma 6 to I gives us an intepretation J in J d = Γ( J dd − )that strongly refutes A . Corollary 8
Let A ≡ K ⇒ F be a regular formula and let d be the number ofconjuncts of K of the form ( P ⇒ E ) ⇒ R . Then IPL ⊢ A iff J d | = A . Proof:
Immediate from the theorem given the soundness of
IPL for the Heytingalgebra semantics.
Corollary 9 IPL is complete for the Heyting algebra semantics.
Proof:
Assume | = A . We have to show that IPL ⊢ A . Consider the regularformula A ′ ≡ M ⇒ Z such that A ⊣⊢ A ′ whose existence is given by Lemma 3.If IPL A , then IPL A ′ , whence by the theorem, A ′ has a strong refutationin J k for some k , i.e., an interpretation I in J k such that V I ( M ) = t , but V I ( Z ) < t . But then Lemma 3 gives us that V I ( A ) ≤ V I ( Z ) < t , so I = A contradicting our assumption that | = A . Corollary 10 IPL has the finite model property.
Proof:
It is immediate from the theorem and soundness that a refutable regularformula has a refutation in a finite model. Argue as in the proof of Corollary 9to reduce the general case to the case of regular formulas.If H , H , . . . is a sequence of Heyting algebras, let us define J k H k tobe the subalgebra of Q k H k comprising sequences ( p , p , . . . ) such that for allsufficiently large k , the p k are either all f or all t . Our final corollary showsthat there is countably infinite Heyting algebra J , such that for any formula φ , J | = φ iff IPL ⊢ φ . 6 orollary 11 For any formula A , IPL ⊢ A iff J | = A , where J = J k J k . Proof:
The left-to-right direction is just the soundness of
IPL for Heytingalgebras. For the right-to-left direction argue as in the proof of Corollary 9 andnote that a refutation in J d gives a refutation in the subalgebra of J comprisingthe sequences ( p , p , . . . ) such that p i is constant for i > d .The statement of Theorem 7 leads to a decision procedure for IPL thatinvolves a search through all interpretations of a formula in one of the J d for acertain d . As Rose [4] observes, the size of the J k grows very rapidly with k , sothis decision procedure is impractical. However, the proof of the theorem leadsto a much better algorithm: given any formula A , we first apply the algorithmof Lemma 3 if necessary to convert A into an equiprovable regular formula andthen follow the case analysis of the proof of the theorem: if we are in Case (i) , A is unprovable and we are done; if we are in Case (ii) , the proof shows ushow to produce one or two simpler formulas whose conjunction is equivalentto A and we may proceed recursively to decide these formulas; if we are inCase (iii) , we can derive the formulas C i described in the proof and decidethem recursively; if any C i is provable, we are in Subcase (iii)(a) and we mayreplace A by an equivalent and simpler formula that we can decide recursively;if no C i is provable, we are in Subcase (iii)(b) and A is unprovable. If A isunprovable, then the proof ot the theorem yields an explicit refutation in one ofthe J k . In the appendix, we show some example calculations using this decisionprocedure. We make no claim that the decision procedure is practical on largeexamples: its time complexity involves a factor d !, where d is bounded below bythe number of implications in the input formula.Ja´skowski’s construction was used by Tarski to show the completeness ofintuitionistic propositional logic for its topological interpretation [5]. One imag-ines that Ja´skowski’s proof was well known to Polish logicians in the 1930s, butsadly the details have been lost: by the 1950s, Kleene’s student Gene F. Rosehad to reinvent a proof. The proof of Theorem 7 given here and, in particular,its use of Lemma 5 is largely due to Rose [3, 4]. Rose’s analogue of our notion ofbasic formula admits only 6 forms: P , ¬ P , P ⇒ Q , P ⇒ Q ∨ R , P ∧ Q ⇒ R and( P ⇒ Q ) ⇒ R ). To prove his analogue of our Lemma 3 involves a lengthy caseanalysis, whereas our more liberal notion of basic formula admits the simplerand more intuitive proof given here. As far as I know, the observations thatTheorem 7 leads to an alternative proof of the completeness of IPL and thatits proof leads to a syntax-driven decision procedure for
IPL are new.
References [1] S. Ja´skowski. Recherches sur le syst`eme de la logique intuitionistique. In
Actes du Congr`es International de Philosophie Scientifique 6 , pages 58–61. Paris, 1936. http://gallica.bnf.fr/ark:/12148/bpt6k383699 (Alsoavailable in an English translation in [2, pp. 259–263]).72] Storrs McCall, editor.
Polish Logic 1920–1939 . Oxford University Press,1967.[3] Gene F. Rose.
Ja´skowski’s Truth-Tables and Realizability . PhD thesis, Uni-versity of Wisconsin, 1952.[4] Gene F. Rose. Propositional calculus and realizability.
Trans. Am. Math.Soc. , 75:1–19, 1953.[5] Alfred Tarski. Der Aussagenkalk¨ul und die Topologie.
Fundam. Math. ,31:103–134, 1938. Available in an English translation in [ ? , pp. 421–454]. Appendix: examples of the decision procedure
Throughout the examples “Case” and “Subcase” refer to the proof of Theorem 7.We use the following tabular format for the regular formulas B ∧ . . . ∧ B k ⇒ F that occur as the goals we are trying to decide: B , . . . , B k F Example 1: A : ≡ ( P ∨ Q ) ∧ ¬ Q ⇒ P Noting that A already has the form B ⇒ Q , we can skip the first step inthe algorithm of Lemma 3 and simply “unnest” B . Listing the subformulas of( P ∨ Q ) ∧ ¬ Q as shown by the subscripts, our initial goal is: P , P ⇔ P ∧ P , P ⇔ P ∨ Q, P ⇔ ¬ QP We are in Case (ii) and we replace the occurrence of P in P ⇔ P ∧ P by ⊤ and simplify giving; P , P , P , P ⇔ P ∨ Q, P ⇔ ¬ QP We are again in Case (ii) , but now P appears in a subformula of the form P ⇒ P ∨ Q and replacing P by ⊤ in that formula gives us two subgoals: P , P , P , P, P ⇔ ¬ QP P , P , P , Q, P ⇔ ¬ QP Both subgoals are in Case (ii) . In the first, the succedent of the goal appearsin the antecedent while in the second, replacing first P and then Q by ⊤ in P ⇔ ¬ Q and simplifying gives the antecedent ⊥ . So both subgoals and hencealso our original formula are provable. 8 xample 2: Peirce’s law: A : ≡ (( P ⇒ Q ) ⇒ P ) ⇒ P A is already regular, so we take it as our initial goal:( P ⇒ Q ) ⇒ PP We are in Case (iii) and our next step is to decide the goal:
P, Q ⇒ PQ This is in Case (ii) and replacing P by ⊤ in Q ⇒ P and simplifying leads to PQ This is again in Case (ii) and is refuted by the interpretation { P t , Q f } .Following Lemma 6, this lifts to the refutation { P
7→ ∗ , Q f } of Peirce’s lawin J = B ∪ {∗} . Example 3: prelinearity: A : ≡ ( P ⇒ Q ) ∨ ( Q ⇒ P ) Following the first part of Lemma 3, we replace A by the equiprovable formula( A ⇒ Z ) ⇒ Z and list its subformulas as indicated by the subscripts in (( P ⇒ Q ) ∨ ( Q ⇒ P ) ⇒ Z ) ⇒ Z . This gives us the following initial goal: P , P ⇔ P ⇒ Z, P ⇔ P ∨ P , P ⇔ P ⇒ Q, P ⇔ ( Q ⇒ P ) Z This is in Case (ii) and replacing P by ⊤ in P ⇔ P ⇒ Z and simplifying weget: P , P ⇒ Z, P ⇔ P ∨ P , P ⇔ P ⇒ Q, P ⇔ ( Q ⇒ P ) Z This is now in Case (iii) with d = 2. This leads to two subgoals: C : P , P ⇒ Z, P ⇔ P ∨ P , P ⇒ P ⇒ Q, P ⇔ ( Q ⇒ P ) , P, Q ⇒ P QC : P , P ⇒ Z, P ⇔ P ∨ P , P ⇔ P ⇒ Q, P ⇒ ( Q ⇒ P ) , Q, P ⇒ P P Either continuing to follow Theorem 7 or by inspection, we find the followingstrong refutations of these subgoals in B . C : ( { P, P , P , P , Z } × { t } ) ∪ ( { Q, P } × { f } ) C : ( { Q, P , P , P , Z } × { t } ) ∪ ( { P, P } × { f } )9ombining these we should obtain a refutation I = { P ( t , f ) , Q ( f , t ) } of A in Γ( B ) ⊆ J . And, indeed, in Γ( B ) we have:(( t , f ) → ( f , t )) ⊔ (( f , t ) → ( t , f )) = ( f , t ) ⊔ ( t , f )= α (( f , t ) ⊔ B ( t , f ))= α (( t , t )) = ∗ 6 = t ..