aa r X i v : . [ m a t h . L O ] A ug From Intuitionism to Many-Valued Logics throughKripke Models ⋆ Saeed Salehi [0000 − − − Research Institute for Fundamental Sciences, University of Tabriz, 29 Bahman Boulevard,P.O.Box 51666-16471, Tabriz,
Iran .School of Mathematics, Institute for Research in Fundamental Sciences,P.O.Box 19395–5746, Tehran, Iran.
Abstract.
Intuitionistic Propositional Logic is proved to be an infinitely manyvalued logic by Kurt G¨odel (1932), and it is proved by Stanis law Ja´skowski (1936)to be a countably many valued logic. In this paper, we provide alternative proofsfor these theorems by using models of Saul Kripke (1959). G¨odel’s proof gaverise to an intermediate propositional logic (between intuitionistic and classical),that is known nowadays as G¨odel or the G¨odel-Dummet Logic, and is studied byfuzzy logicians as well. We also provide some results on the inter-definability ofpropositional connectives in this logic.
Keywords:
Intuitionistic Propositional Logic · Many Valued Logics · KripkeModels · G¨odel-Dummet Logic · Inter-Definability of Propositional Connectives.
Intuitionism grew out of some of the philosophical ideas of its founding father, LuitzenEgbertus Jan Brouwer (see e.g. [1]); what is known nowadays as intuitionistic logic isa formalization given by his student Arend Heyting [4]. Kripke models (originatingfrom [6]) provided an interesting mathematical interpretation for this formalization.Let us review some preliminaries about these models:
Definition 1 (Kripke Frames). A Kripke frame is a partially ordered set; i.e., an ordered pair h K, < i where < ⊆ K is a reflexive, transitive and anti-symmetric binary relation on K . ✧ Definition 2 (Atoms, Formulas, Languages).
Let At be the set of all the propositional atoms; atoms are usually denoted by letters p or q . Let ⊤ denote the verum (truth) constant.The language of propositional logics studied here is L = {¬ , ∧ , ∨ , → , ⊤ } . ⋆ Dedicated to Professor
Mohammad Ardeshir with high appreciation and admiration.
Saeed Salehi
For any A ⊆ At and B ⊆ L , the set of all the formulas constructed from A by meansof B is denoted by L ( B, A ) .Let Fm denote the set of all the formulas; i.e., L ( L , At ) . ✧ Definition 3 (Kripke Models). A Kripke model is a triple K = h K, < , (cid:13) i , where h K, < i is a Kripke frame equippedwith a persistent binary (satisfaction) relation (cid:13) ⊆ K × At ; persistency (of the relation (cid:13) with respect to < ) means that for all k, k ′ ∈ K and p ∈ At , if k ′ < k (cid:13) p then k ′ (cid:13) p .The satisfaction relation can be extended to all the (propositional) formulas, i.e., to (cid:13) ⊆ K × Fm , as follows: ◦ k (cid:13) ⊤ . ◦ k (cid:13) ( ϕ ∧ ψ ) ⇐⇒ k (cid:13) ϕ and k (cid:13) ψ . ◦ k (cid:13) ( ϕ ∨ ψ ) ⇐⇒ k (cid:13) ϕ or k (cid:13) ψ . ◦ k (cid:13) ( ¬ ϕ ) ⇐⇒ ∀ k ′ < k ( k ′ ϕ ) . ◦ k (cid:13) ( ϕ → ψ ) ⇐⇒ ∀ k ′ < k ( k ′ (cid:13) ϕ ⇒ k ′ (cid:13) ψ ) . ✧ Remark 1 (
On Persistency and its Converse ). It can be shown that the persistency conditions is inherited by the formulas; i.e., forany k, k ′ ∈ K in any Kripke model K = h K, < , (cid:13) i and for any formula ϕ , if k ′ < k (cid:13) ϕ then k ′ (cid:13) ϕ .Obviously, the converse may not hold ( k ′ (cid:13) ψ and k ′ < k do not necessarily implythat k (cid:13) ψ ); however, a partial converse holds for negated formulas:if k ′ < k and k ′ (cid:13) ¬ ϕ , then k ϕ . ✧ By the soundness and completeness of the intuitionistic propositional logic (IPL)with respect to finite Kripke models, the tautologies of IPL are the formulas (in Fm ) thatare satisfied in all the elements of any finite Kripke model. A super-intuitionistic andsub-classical logic is the so-called G¨odel-Dummet logic (see [2]), whose tautologiesare the formulas that are satisfied in all the elements of all the connected finite Kripkemodels. A kind of Kripke model theoretic characterization for this logic is given in [8]. Definition 4 (Connectivity).
A binary relation R ⊆ K × K is called connected , when for any k, k ′ , k ′′ ∈ K , if k ′ < k and k ′′ < k , then we have either k ′ < k ′′ or k ′′ < k ′ (cf. [10]). ✧ The logic IPL is perhaps the most famous non-classical logic. A natural question(that according to Kurt G¨odel [3] was asked by his supervisor Hans Hahn) was whetherIPL is a finitely many valued logic or not. G¨odel [3] showed in 1932 that IPL is notfinitely many valued. Stanis law Ja´skowski [5] showed in 1936 that IPL is indeed acountably (infinite) many valued logic. In Section 2 we give alternative proofs forthese theorems by using Kripke models [6] which were invented later in 1959. G¨odel’sproof gave birth to an intermediate logic, that today is called the G¨odel-Dummet logic(GDL). Finally, in Section 3 we study the problem of inter-definability of propositionalconnectives in GDL and IPL. rom Intuitionism to Many-Valued Logics through Kripke Models 3 ω − Many Values for Intuitionistic Propositional Logic
Let us begin with a formal definition of a many-valued logic. Throughout the paper,we are dealing with propositional logics only.
Definition 5 (Many Valued Logics). A many valued logic is h V , τ , ∽∽ , Λ , V , = > i , where V is a set of values with a designated element τ ∈ V (interpreted as the truth ) and the functions ∽∽ : V → V , Λ : V → V , V : V → V , and = > : V → V constitute a truth table on V .A valuation function is any mapping ν : At → V , which can be extended to all theformulas, denoted also by ν : Fm → V , as follows: ◦ ν ( ¬ ϕ ) = ∽∽ ν ( ϕ ) . ◦ ν ( ϕ ∧ ψ ) = ν ( ϕ ) Λ ν ( ψ ) . ◦ ν ( ϕ ∨ ψ ) = ν ( ϕ ) V ν ( ψ ) . ◦ ν ( ϕ → ψ ) = ν ( ϕ ) = > ν ( ψ ) .A formula θ is called tautology , when it is mapped to the designated value under anyvaluation function; i.e., ν ( θ ) = τ for any valuation ν . ✧ Theorem 1 appears in [7] and [9]. In the following, the disjunction operation ( ∨ )is assumed to be commutative and associative. Lemma 1 (A Tautology in n -Valued Logics). For any n > , the formula WW i Intuitionistic propositional logic is not finitely many valued.Proof. By Lemma 1 it suffices to show that for any n > , the formula WW i Definition 6 (Monotone Functions). For a Kripke frame ( K, < ) , a function f : K → { , } is called monotone , when forany k, k ′ ∈ K , if k ′ < k , then f ( k ′ ) > f ( k ) . We indicate the monotonicity of f bywriting f : ( K, < ) → { , } . ✧ Example 1 ( f ψ K ). For any Kripke model K = ( K, < , (cid:13) ) and any formula ψ , the function f ψ K : K → { , } , f ψ K ( k ) = ( if k (cid:13) ψ if k ψ is monotone. ✧ Definition 7 ( ∽∽ , Λ , V and = > ). For a Kripke frame ( K, < ) and monotone functions f, g : ( K, < ) → { , } , let ∽∽ f : K → { , } be defined by ( ∽∽ f )( k ) = ( if ∀ k ′ < k ( f ( k ′ ) = 0)0 if ∃ k ′ < k ( f ( k ′ ) = 1) , f Λ g : K → { , } be defined by ( f Λ g )( k ) = min { f ( k ) , g ( k ) } , f V g : K → { , } be defined by ( f V g )( k ) = max { f ( k ) , g ( k ) } , f = > g : K → { , } be defined by ( f = > g )( k ) = ( if ∀ k ′ < k ( f ( k ′ ) = 1 ⇒ g ( k ′ ) = 1)0 if ∃ k ′ < k ( f ( k ′ ) = 1 & g ( k ′ ) = 0) ,for all k ∈ K . ✧ Definition 8 (Constant Functions). Let K : K → { , } be the constant function, i.e., K ( k ) = 1 for all k ∈ K ; and let K : K → { , } be the constant function: K ( k ) = 0 for all k ∈ K . ✧ rom Intuitionism to Many-Valued Logics through Kripke Models 5 It is easy to see that the functions K and K obey the rules of the classicalpropositional logic with the operations ∽∽ , Λ , V and = > . For example, ( ∽∽ K ) = K , ( K Λ K ) = K , ( K V K ) = K and ( K = > K ) = K . We omit the proof ofthe following straightforward observation. Lemma 2 (Monotonicity of K , K , ∽∽ f, f Λ g, f V g and f = > g ). For any Kripke frame ( K, < ) , the constant functions K and K are monotone, and if f, g : ( K, < ) → { , } are monotone, then so are ∽∽ f, f Λ g, f V g and f = > g . ❑ Finally, we can provide the following countably many values for IPL: Definition 9 (Countably Many Values for IPL). Enumerate all the finite Kripke frames as ( K , < ) , ( K , < ) , ( K , < ) , · · · , where K n ⊂ N for all n ∈ N . Let V = {h f , f , f , · · · i | ∀ n [ f n : ( K n , < n ) → { , } ] & ∃ N ∈ N [( ∀ n > N f n = K n ) or ( ∀ n > N f n = K n )] } . In the other words, the set of values V consists of all the sequences h f , f , f , · · · i such that for each n , f n is a monotone function on ( K n , < n ) , and the sequences areultimately constant (from a step onward, f n ’s are either all K n or all K n ).Let τ = h K , K , K , · · · i be the designated element (for truth).For f = h f , f , f , · · · i ∈ V and g = h g , g , g , · · · i ∈ V , let (cf. Definition 7) ∽∽ f = h ∽∽ f , ∽∽ f , ∽∽ f , · · · i , f Λ g = h f Λ g , f Λ g , f Λ g , · · · i , f V g = h f V g , f V g , f V g , · · · i , and f = > g = h f = > g , f = > g , f = > g , · · · i . ✧ It can be immediately seen that V is a countable set, and Lemma 2 implies that V is closed under the operations ∽∽ , Λ , V and = > . Before proving the main theorem, wemake a further definition and prove an auxiliary lemma. Definition 10 ( hh α ii n , (cid:13) ν n and ν (cid:13) m ). For a sequence α , let hh α ii n denote its n -th element (if any), for any n ∈ N .(1) Let a valuation ν : At → V be given. The satisfaction relation (cid:13) ν n is defined on anyfinite Kripke frame ( K n , < n ) , with K n ⊂ N (see Definition 9), by the following forany atom p ∈ At and any k ∈ K n : k (cid:13) ν n p ⇐⇒ hh ν ( p ) ii n ( k ) = 1 .(2) Let a Kripke model K = ( K m , < m , (cid:13) ) on the Kripke frame ( K m , < m ) be given(see Definition 9). Define the valuation ν (cid:13) m by ν (cid:13) m ( p ) = h K , · · · , K m − , f p K , K m +1 , · · · i for any p ∈ At , where f p K : K m → { , } is the function that was defined in Example 1: f p K ( k ) = 1 if k (cid:13) p , and f p K ( k ) = 0 if k p , for any k ∈ K m . ✧ It is clear that the relation (cid:13) ν n ⊆ K n × At is persistent. Saeed Salehi Lemma 3 (On (cid:13) ν n and ν (cid:13) m ). (1) Let a valuation ν : At → V be given, and the satisfaction relation (cid:13) ν n be defined on ( K n , < n ) as in Definition 10. Then for any formula ϕ ∈ Fm and any k ∈ K n , we have k (cid:13) ν n ϕ ⇐⇒ hh ν ( ϕ ) ii n ( k ) = 1 . (2) Let a Kripke model K = ( K m , < m , (cid:13) ) be given on the frame ( K m , < m ) , and thevaluation ν (cid:13) m be defined as in Definition 10. Then for any formula ϕ ∈ Fm and any k ∈ K m , we have k ϕ ⇐⇒ hh ν (cid:13) m ( ϕ ) ii m ( k ) = 0 .Proof. Both assertions can be proved by induction on ϕ . They are clear for ϕ = ⊤ and holdfor atomic ϕ ∈ At by Definition 10. The inductive cases follow immediately fromDefinitions 3, 5, 7, and 9. ❑ Theorem 2 (Ja´skowski 1936: IPL Is Countably Many Valued). Intuitionistic propositional logic is countably infinite many valued.Proof. We show that a formula ϕ ∈ Fm is satisfied in all the elements of all the finite Kripkemodels if and only if it is mapped to the designated element under all the valuationfunctions:(1) If ϕ is satisfied in any element of any finite Kripke model, then for any valuation ν by Lemma 3(1) we have hh ν ( ϕ ) ii n = K n for any n ∈ N , so ν ( ϕ ) = τ .(2) If ϕ is not satisfied in some element of some finite Kripke model, then for some m ∈ N there is a Kripke model K = ( K m , < m , (cid:13) ) such that k 1 ϕ for some k ∈ K m .So, by Lemma 3(2) we have hh ν (cid:13) m ( ϕ ) ii m ( k ) = 0 , thus ν (cid:13) m ( ϕ ) = τ . ❑ In classical propositional logic (which is a two valued logic), all the connectives canbe defined by (the so-called complete set of connectives) {¬ , ∧} , {¬ , ∨} or {¬ , →} only. In this last section we will see that no propositional connective is definable fromthe others in IPL, and in GDL only the disjunction operation ( ∨ ) can be defined bythe conjunction ( ∧ ) and implication ( → ) operations. Most of these facts are alreadyknown (they appear in e.g. [9] and [10]). Theorem 3 is from [10] with a slightly differentproof; Theorem 4 is from [10] with the same proof. All of our proofs are Kripke modeltheoretic, as usual. Theorem 3 ( ∧ Is Not Definable From the Others in GDL). In G¨odel-Dummet Logic, the conjunction connective ( ∧ ) is not definable from the otherpropositional connectives. rom Intuitionism to Many-Valued Logics through Kripke Models 7 Proof. Consider the Kripke model K = h K, < , (cid:13) i where K = { a, b, c } , < is the reflexiveclosure of { ( a, b ) , ( c, b ) } , and (cid:13) = { ( a, p ) , ( b, p ) , ( b, q ) , ( c, q ) } , for atoms p , q ∈ At . • b [[ p , q ]] • a [[ p ]] ✲ • c [[ q ]] ✛ We show that for all formulas θ ∈ L ( ¬ , ∨ , → , ⊤ , p , q ) we have: ( ∗ ) b (cid:13) θ = ⇒ a (cid:13) θ or c (cid:13) θ .This will prove the desired conclusion, since b (cid:13) p ∧ q but a, c p ∧ q , and so p ∧ q cannot belong to L ( ¬ , ∨ , → , ⊤ , p , q ) . We prove ( ∗ ) by induction on θ . The cases of θ = ⊤ , p , q are trivial, and the induction step of ¬ ϕ follows from Remark 1, and thecase of ϕ ∨ ψ is rather easy. So, only the non-trivial case of θ = ϕ → ψ remains.Suppose that ( ∗ ) holds for ϕ and ψ , and assume (for the sake of a contradiction) that b (cid:13) ϕ → ψ but a, c ϕ → ψ . So, a (cid:13) ϕ and a ψ ; and also c (cid:13) ϕ and c ψ .Whence, by persistency, we should have also b (cid:13) ϕ , thus b (cid:13) ψ . So, by the inductionhypothesis ( ∗ for θ = ψ ) we should have either a (cid:13) ψ or c (cid:13) ψ ; a contradiction. ❑ Theorem 4 ( → Is Not Definable From the Others in GDL). In G¨odel-Dummet Logic, the implication connective ( → ) is not definable from the otherpropositional connectives.Proof. For the Kripke model K = h K, < , (cid:13) i , where K = { a, b, c } , < is the reflexive closureof { ( a, b ) , ( c, b ) } , and (cid:13) = { ( a, p ) , ( b, p ) , ( b, q ) } , for p , q ∈ At , • b [[ p , q ]] • a [[ p ]] ✲ • c [[]] ✛ we show that for all the formulas θ ∈ L ( ¬ , ∨ , ∧ , ⊤ , p , q ) , the following holds: ( ∗ ) b, c (cid:13) θ = ⇒ a (cid:13) θ .This completes the proof since b, c (cid:13) p → q but a p → q (by a (cid:13) p , a q ); thuswe have ( p → q ) 6∈ L ( ¬ , ∨ , ∧ , ⊤ , p , q ) . The proof of ( ∗ ) is by induction on θ ; the onlynon-trivial cases to consider are θ = ϕ ∨ ψ and θ = ϕ ∧ ψ . Suppose that ( ∗ ) holds for ϕ and ψ ; and that b, c (cid:13) ϕ ∨ ψ . Then we have either c (cid:13) ϕ or c (cid:13) ψ ; by the persistency,the former implies b (cid:13) ϕ and the latter b (cid:13) ψ . So, in either case by the inductionhypothesis we have a (cid:13) ϕ ∨ ψ . The case of θ = ϕ ∧ ψ is even simpler. ❑ Saeed Salehi The following has been known for a long time; see e.g. [2]. Theorem 5 ( ∨ Is Definable From ∧ , → in GDL). In G¨odel-Dummet Logic, the disjunction connective ( ∨ ) is definable from some otherpropositional connectives.Proof. It is rather easy to see that IPL (cid:13) ( p ∨ q ) −→ [( p → q ) → q ] ∧ [( q → p ) → p ] . Now,we show that GDL (cid:13) [( p → q ) → q ] ∧ [( q → p ) → p ] −→ ( p ∨ q ) holds. Take anarbitrary connected Kripke model K = h K, < , (cid:13) i , and suppose that for an arbitrary a ∈ K we have a (cid:13) [( p → q ) → q ] ∧ [( q → p ) → p ] . We show that a (cid:13) p ∨ q . Assumenot; then a p , q . Therefore, a ( p → q ) and a ( q → p ) , by a (cid:13) [( p → q ) → q ] and a (cid:13) [( q → p ) → p ] , respectively. So, there should exist some b, c ∈ K with b, c < a suchthat b (cid:13) p , b q , c (cid:13) q , and c p . • b [[ p ]] • c [[ q ]] • a [[]] ✲ ✛ By the connectivity of < , we should have either b < c or c < b . Both cases lead to acontradiction, by the persistency condition. So, the following equivalence ( p ∨ q ) ≡ [( p → q ) → q ] ∧ [( q → p ) → p ] holds in GDL. ❑ The fact of the matter is that ( p ∨ q ) ≡ [( p → q ) → q ] ∧ [( q → p ) → p ] is the onlynon-trivial equivalence relation between the propositional connectives in GDL. Thefirst half of the following theorem was proved in [9]. Theorem 6 (In GDL ∨ Is Not Definable Without Both ∧ , → ). In G¨odel-Dummet Logic, the disjunction connective ( ∨ ) is not definable from the otherpropositional connectives, unless both the conjunction and the implication connectives arepresent. In the other words, ∨ is definable neither from the set {¬ , → , ⊤ } nor from theset {¬ , ∧ , ⊤ } .Proof. Take the Kripke model K = h K, < , (cid:13) i with K = { a, b, c, d } , < = the reflexive closureof { ( a, b ) , ( c, d ) } , and (cid:13) = { ( b, p ) , ( d, q ) } , for p , q ∈ At . • b [[ p ]] • d [[ q ]] • a [[]] ✻ • c [[]] ✻ rom Intuitionism to Many-Valued Logics through Kripke Models 9 We show that for all θ ∈ L ( ¬ , → , ⊤ , p , q ) we have ( ∗ ) b, d (cid:13) θ = ⇒ a (cid:13) θ or c (cid:13) θ .Since b, d (cid:13) p ∨ q but a, c p ∨ q , then it follows that p ∨ q 6∈ L ( ¬ , → , ⊤ , p , q ) .Now, ( ∗ ) can be proved by induction on θ ; the only non-trivial case is θ = ϕ → ψ . If ( ∗ ) holds for ϕ and ψ , then if b, d (cid:13) ϕ → ψ but a ϕ → ψ and c ϕ → ψ , then weshould have a (cid:13) ϕ and a ψ , and also c (cid:13) ϕ and c ψ . So, by persistency, b (cid:13) ϕ and d (cid:13) ϕ ; thus b (cid:13) ψ and d (cid:13) ψ . So, by the induction hypothesis ( ∗ for θ = ψ ) weshould have either a (cid:13) ψ or c (cid:13) ψ ; a contradiction.Now, for proving p ∨ q 6∈ L ( ¬ , ∧ , ⊤ , p , q ) , we show that for all the formulas θ in L ( ¬ , ∧ , ⊤ , p , q ) we have ( ‡ ) b, d (cid:13) θ = ⇒ a, c (cid:13) θ .Trivially, ( ‡ ) holds for θ = ⊤ , p , q ; so by Remark 1 it only suffices to show that ( ‡ ) holds for θ = ϕ ∧ ψ , when it holds for ϕ and ψ . Now, if b, d (cid:13) ϕ ∧ ψ then b, d (cid:13) ϕ and b, d (cid:13) ψ ; so the induction hypothesis ( ‡ for θ = ϕ, ψ ) implies that a, c (cid:13) ϕ and a, c (cid:13) ψ , therefore a, c (cid:13) ϕ ∧ ψ . ❑ We end the paper with a Kripke model theoretic proof of a known fact. Proposition 1 (No Connective Is Definable From the Others in IPL). In IPL, no propositional connective is definable from the others.Proof. By Theorems 3 and 4, ∧ and → are not definable from the other connectives even inGDL. The statement ¬ p 6∈ L ( ∧ , ∨ , → , ⊤ , p ) can be easily verified by noting that allthe operations on the righthand side are positive. So, all it remains is to show that wehave p ∨ q 6∈ L ( ¬ , ∧ , → , ⊤ , p , q ) in IPL (cf. Theorem 5). Consider the Kripke model K = h K, < , (cid:13) i with K = { a, b, c } , < = the reflexive closure of { ( a, b ) , ( a, c ) } , and (cid:13) = { ( b, p ) , ( c, q ) } , for p , q ∈ At . • b [[ p ]] • c [[ q ]] • a [[]] ✲ ✛ We show that for all formulas θ ∈ L ( ¬ , ∧ , → , ⊤ , p , q ) we have: ( ∗ ) b, c (cid:13) θ = ⇒ a (cid:13) θ .This will prove the theorem, since b, c (cid:13) p ∨ q but a p ∨ q , and so p ∨ q is not in L ( ¬ , ∧ , → , ⊤ , p , q ) in IPL. Indeed, ( ∗ ) can be proved by induction on θ ; for which weconsider the case of θ = ϕ → ψ only. So, suppose that ( ∗ ) holds for ϕ and ψ andthat b, c (cid:13) ϕ → ψ but a ϕ → ψ . Then we should have a (cid:13) ϕ and a ψ ; but bypersistency we should have that b, c (cid:13) ϕ , and so b, c (cid:13) ψ holds. Now, the inductionhypothesis ( ∗ for θ = ψ ) implies that a (cid:13) ψ , a contradiction. ❑ Saeed Salehi References Brouwer, Luitzen Egbertus Jan ; Intuitionism and Formalism , Bulletin of the AmericanMathematical Society doi : 10.1090/S0273-0979-99-00802-22. Dummet, Michael ; A Propositional Calculus with Denumerable Matrix , The Journal ofSymbolic Logic doi : 10.2307/29647533. G¨odel, Kurt ; Zum Intuitionistischen Aussagenkalk¨ul , Anzeiger Akademie der Wis-senschaften Wien 69 (1932) 65–66. Translated as “ On the Intuitionistic PropositionalCalculus ”, in: Kurt C¨odel Collected Works (Volume I) Publications 1929-1936 ( isbn : 9780195147209) eds.: S. Feferman, et al. (Oxford University Press 1986), pp. 222–225.4. Heyting, Arend ; Die Formalen Regeln der Intuitionistischen Logik , Sitzungsberichte derPreussischen Akademie von Wissenschaften, Physikalisch Mathematische Klasse (1930) 42–56. Die Formalen Regeln der Intuitionistischen Mathematik : I , ibid. II , ibid. Ja´skowski, Stanis law ; Recherches sur le Systme de la Logique Intuitioniste , Actes du Con-grs International de Philosophie Scientifique, VI. Philosophie des Mathmatiques,Actualits Scientifiques et Industrielles 393 , Hermann & C ie (Parsi, 1936) pp. 58–61.6. Kripke, Saul A. ; A Completeness Theorem in Modal Logic , The Journal of Symbolic Logic doi : 10.2307/29645687. Safari, Parvin ; Investigating Kripke Semantics for Fuzzy Logics , Ph.D. Dissertation,under the supervision of Saeed Salehi , University of Tabriz (2017).8. Safari, Parvin & Salehi, Saeed ; Kripke Semantics for Fuzzy Logics , Soft Computing doi : 10.1007/s00500-016-2387-49. Safari, Parvin & Salehi, Saeed ; Truth Values and Connectives in Some Non-Classical Logics , Journal of New Researches in Mathematics , 5:19 (2019) 31–36 (in Farsi). Available onthe net at: http://jnrm.srbiau.ac.ir/article 14450.html vejdar, Vtzslav & Bendov´a, Kamila ; On Inter-Expressibility of Logical Connectives in GdelFuzzy Logic , Soft Computing doidoi