This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes α ⇒ (β ⇒ α),(α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)),(¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β). Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies are presented.Faculty of Economics and Informatics, University of Białystok, Kalvariju 135, LT-08221 Vilnius, LithuaniaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Leszek Borys. On paracompactness of metrizable spaces. Formalized Mathematics, 3(1): 81–84, 1992.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Mariusz Giero. Propositional linear temporal logic with initial validity semantics. Formalized Mathematics, 23(4):379–386, 2015. doi:10.1515/forma-2015-0030.Witold Pogorzelski. Dictionary of Formal Logic. Wydawnictwo UwB - Bialystok, 1992.Witold Pogorzelski. Notions and theorems of elementary formal logic. Wydawnictwo UwB - Bialystok, 1994.Piotr Rudnicki and Andrzej Trybulec. On same equivalents of well-foundedness. Formalized Mathematics, 6(3):339–343, 1997.Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics, 8(1):133–137, 1999.Anita Wasilewska. An Introduction to Classical and Non-Classical Logics. SUNY Stony Brook, 2005.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990

Giero, Mariusz

Formalized Mathematics

Summary
               This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes

                     
                        
                           α ⇒ (β ⇒ α),
                     
                     
                        (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)),
                     
                     
                        (¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β).
                     
                  
Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies are presented.</jats:p

Mariusz Giero

Crossref

The Axiomatization of Propositional Logic

Repozytorium Uniwersytetu w Białymstoku (University of Bialystok Repository)

FORMALIZED MATHEMATICSVol. 24, No. 4, Pages 281–290, 2016DOI: 10.1515/forma-2016-0024 degruyter.com/view/j/formaThe Axiomatization of Propositional Logic1Mariusz GieroFaculty of Economics and InformaticsUniversity of BiałystokKalvariju 135, LT-08221 VilniusLithuaniaSummary. This article introduces propositional logic as a formal system([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p |φ→ φ.Other connectives are introduced as abbrevations. The notions of model andsatisfaction in model are defined. The axioms are all the formulae of the followingschemes• α⇒ (β ⇒ α),• (α⇒ (β ⇒ γ))⇒ ((α⇒ β)⇒ (α⇒ γ)),• (¬β ⇒ ¬α)⇒ ((¬β ⇒ α)⇒ β).Modus ponens is the only derivation rule. The soundness theorem and the strongcompleteness theorem are proved. The proof of the completeness theorem is car-ried out by a counter-model existence method. In order to prove the completenesstheorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies arepresented.MSC: 03B05 03B35Keywords: completeness; formal system; Lindenbaum’s lemmaMML identifier: PL AXIOM, version: 8.1.05 5.39.12821This work was supported by the University of Bialystok grants: BST447 Formalization oftemporal logics in a proof-assistant. Application to System Verification, and BST225 Databaseof mathematical texts checked by computer.c© 2016 University of BiałystokCC-BY-SA License ver. 3.0 or laterISSN 1426–2630(Print), 1898-9934(Online)281Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PM282 mariusz giero1. PreliminariesNow we state the propositions:(1) Let us consider functions f , g. Suppose dom f ⊆ dom g and for every setx such that x ∈ dom f holds f(x) = g(x). Then rng f ⊆ rng g.(2) Let us consider Boolean objects p, q. Then p ∧ q ⇒ p = true.(3) Let us consider a Boolean object p. Then ¬¬p⇔ p = true.Let us consider Boolean objects p, q. Now we state the propositions:(4) ¬(p ∧ q)⇔ ¬p ∨ ¬q = true.(5) ¬(p ∨ q)⇔ ¬p ∧ ¬q = true.(6) p⇒ q ⇒ (¬q ⇒ ¬p) = true.Let us consider Boolean objects p, q, r. Now we state the propositions:(7) p⇒ q ⇒ (p⇒ r ⇒ (p⇒ q ∧ r)) = true.(8) p⇒ r ⇒ (q ⇒ r ⇒ (p ∨ q ⇒ r)) = true.Let us consider Boolean objects p, q. Now we state the propositions:(9) p ∧ q ⇔ q ∧ p = true.(10) p ∨ q ⇔ q ∨ p = true.Let us consider Boolean objects p, q, r. Now we state the propositions:(11) (p ∧ q) ∧ r ⇔ p ∧ (q ∧ r) = true.(12) (p ∨ q) ∨ r ⇔ p ∨ (q ∨ r) = true.(13) Let us consider Boolean objects p, q. Then ¬q ⇒ ¬p⇒ (¬q ⇒ p⇒ q) =true.Let us consider Boolean objects p, q, r. Now we state the propositions:(14) p ∧ (q ∨ r)⇔ p ∧ q ∨ p ∧ r = true.(15) p ∨ q ∧ r ⇔ (p ∨ q) ∧ (p ∨ r) = true.(16) Let us consider a finite set X, and a set Y. Suppose Y is ⊆-linear andX ⊆⋃Y and Y 6= ∅. Then there exists a set Z such that(i) X ⊆ Z, and(ii) Z ∈ Y.2. The SyntaxLet D be a set. We say that D has propositional variables if and only if(Def. 1) for every element n of N, 〈3 + n〉 ∈ D.We say that D is PL-closed if and only if(Def. 2) D ⊆ N∗ and D has FALSUM, implication and propositional variables.Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PMThe axiomatization of propositional logic 283Let us note that every set which is PL-closed is also non empty and hasalso FALSUM, implication, and propositional variables and every subset of N∗which has FALSUM, implication, and propositional variables is also PL-closed.The functor PL-WFF yielding a set is defined by(Def. 3) it is PL-closed and for every set D such that D is PL-closed holds it ⊆ D.Observe that PL-WFF is PL-closed and there exists a set which is PL-closedand non empty and PL-WFF is functional and every element of PL-WFF is finitesequence-like.The functor ⊥PL yielding an element of PL-WFF is defined by the term(Def. 4) 〈0〉.Let p, q be elements of PL-WFF. The functor p ⇒ q yielding an element ofPL-WFF is defined by the term(Def. 5) (〈1〉 a p) a q.Let n be an element of N. The functor Propn yielding an element of PL-WFFis defined by the term(Def. 6) 〈3 + n〉.The functor AP yielding a subset of PL-WFF is defined by(Def. 7) for every set x, x ∈ it iff there exists an element n of N such thatx = Propn.From now on p, q, r, s, A, B denote elements of PL-WFF, F , G, H denotesubsets of PL-WFF, k, n denote elements of N, and f , f1, f2 denote finitesequences of elements of PL-WFF.Let D be a subset of PL-WFF. Observe that D has implication if and onlyif the condition (Def. 8) is satisfied.(Def. 8) for every p and q such that p, q ∈ D holds p⇒ q ∈ D.The scheme PLInd deals with a unary predicate P and states that(Sch. 1) For every r, P[r]provided• P[⊥PL] and• for every n, P[Propn] and• for every r and s such that P[r] and P[s] holds P[r ⇒ s].Now we state the proposition:(17) PL-WFF ⊆ HP-WFF.Proof: Define P[element of PL-WFF] ≡ $1 ∈ HP-WFF. For every n,P[Propn]. For every r and s such that P[r] and P[s] holds P[r ⇒ s]. Forevery A, P[A] from PLInd. Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PM284 mariusz gieroLet us consider p. The functor ¬p yielding an element of PL-WFF is definedby the term(Def. 9) p⇒ ⊥PL.The functor >PL yielding an element of PL-WFF is defined by the term(Def. 10) ¬⊥PL.Let us consider p and q. The functors: p ∧ q and p ∨ q yielding elements ofPL-WFF are defined by terms(Def. 11) ¬(p⇒ ¬q),(Def. 12) ¬p⇒ q,respectively. The functor p ⇔ q yielding an element of PL-WFF is defined bythe term(Def. 13) (p⇒ q) ∧ (q ⇒ p).3. The SemanticsA PL-model is a subset of AP . From now on M denotes a PL-model.LetM be a PL-model. The functor SATM yielding a function from PL-WFFinto Boolean is defined by(Def. 14) it(⊥PL) = 0 and for every k, it(Prop k) = 1 iff Prop k ∈M and for everyp and q, it(p⇒ q) = it(p)⇒ it(q).Now we state the propositions:(18) SATM (A⇒ B) = 1 if and only if SATM (A) = 0 or SATM (B) = 1.(19) SATM (¬p) = ¬(SATM (p)).(20) SATM (¬A) = 1 if and only if SATM (A) = 0. The theorem is a consequ-ence of (19).(21) SATM (A ∧ B) = SATM (A) ∧ SATM (B). The theorem is a consequenceof (19).(22) SATM (A∧B) = 1 if and only if SATM (A) = 1 and SATM (B) = 1. Thetheorem is a consequence of (21).(23) SATM (A ∨ B) = SATM (A) ∨ SATM (B). The theorem is a consequenceof (19).(24) SATM (A ∨ B) = 1 if and only if SATM (A) = 1 or SATM (B) = 1. Thetheorem is a consequence of (23).(25) SATM (A ⇔ B) = SATM (A) ⇔ SATM (B). The theorem is a consequ-ence of (21).(26) SATM (A⇔ B) = 1 if and only if SATM (A) = SATM (B). The theoremis a consequence of (25).Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PMThe axiomatization of propositional logic 285Let us consider M and p. We say that M |= p if and only if(Def. 15) SATM (p) = 1.Let us consider F . We say that M |= F if and only if(Def. 16) for every p such that p ∈ F holds M |= p.Let us consider p. We say that F |= p if and only if(Def. 17) for every M such that M |= F holds M |= p.Let us consider A. We say that A is a tautology if and only if(Def. 18) for every M , SATM (A) = 1.Now we state the propositions:(27) A is a tautology if and only if ∅PL-WFF |= A.(28) p⇒ (q ⇒ p) is a tautology.(29) p⇒ (q ⇒ r)⇒ (p⇒ q ⇒ (p⇒ r)) is a tautology.(30) ¬q ⇒ ¬p⇒ (¬q ⇒ p⇒ q) is a tautology. The theorem is a consequenceof (19) and (13).(31) p ⇒ q ⇒ (¬q ⇒ ¬p) is a tautology. The theorem is a consequence of(19) and (6).(32) p ∧ q ⇒ p is a tautology. The theorem is a consequence of (21) and (2).(33) p ∧ q ⇒ q is a tautology. The theorem is a consequence of (21) and (2).(34) p⇒ p ∨ q is a tautology. The theorem is a consequence of (23).(35) q ⇒ p ∨ q is a tautology. The theorem is a consequence of (23).(36) p∧ q ⇔ q ∧ p is a tautology. The theorem is a consequence of (25), (21),and (9).(37) p∨ q ⇔ q ∨ p is a tautology. The theorem is a consequence of (25), (23),and (10).(38) (p∧ q)∧ r ⇔ p∧ (q ∧ r) is a tautology. The theorem is a consequence of(25), (21), and (11).(39) (p∨ q)∨ r ⇔ p∨ (q ∨ r) is a tautology. The theorem is a consequence of(25), (23), and (12).(40) p ∧ (q ∨ r)⇔ p ∧ q ∨ p ∧ r is a tautology. The theorem is a consequenceof (25), (21), (23), and (14).(41) p∨ q ∧ r ⇔ (p∨ q)∧ (p∨ r) is a tautology. The theorem is a consequenceof (25), (23), (21), and (15).(42) ¬¬p⇔ p is a tautology. The theorem is a consequence of (25), (19), and(3).(43) ¬(p∧ q)⇔ ¬p∨¬q is a tautology. The theorem is a consequence of (25),(19), (21), (23), and (4).Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PM286 mariusz giero(44) ¬(p∨ q)⇔ ¬p∧¬q is a tautology. The theorem is a consequence of (25),(19), (23), (21), and (5).(45) p ⇒ q ⇒ (p ⇒ r ⇒ (p ⇒ q ∧ r)) is a tautology. The theorem isa consequence of (21) and (7).(46) p ⇒ r ⇒ (q ⇒ r ⇒ (p ∨ q ⇒ r)) is a tautology. The theorem isa consequence of (23) and (8).(47) If F |= A and F |= A⇒ B, then F |= B.4. The Axioms. Derivability.Let D be a set. We say that D is with axioms of PL if and only if(Def. 19) for every p, q, and r holds p⇒ (q ⇒ p), p⇒ (q ⇒ r)⇒ (p⇒ q ⇒ (p⇒r)), ¬q ⇒ ¬p⇒ (¬q ⇒ p⇒ q) ∈ D.The functor PL-axioms yielding a subset of PL-WFF is defined by(Def. 20) it is with axioms of PL and for every subset D of PL-WFF such that Dis with axioms of PL holds it ⊆ D.One can check that PL-axioms is with axioms of PL.Let us consider p, q, and r. We say that MP(p, q, r) if and only if(Def. 21) q = p⇒ r.Observe that PL-axioms is non empty.Let us consider A. We say that A is the simplification axiom if and only if(Def. 22) there exists p and there exists q such that A = p⇒ (q ⇒ p).We say that A is Frege axiom if and only if(Def. 23) there exists p and there exists q and there exists r such that A = p ⇒(q ⇒ r)⇒ (p⇒ q ⇒ (p⇒ r)).We say that A is the explosion axiom if and only if(Def. 24) there exists p and there exists q such that A = ¬q ⇒ ¬p⇒ (¬q ⇒ p⇒q).Now we state the propositions:(48) Every element of PL-axioms is the simplification axiom or Frege axiomor the explosion axiom.(49) If A is the simplification axiom or Frege axiom or the explosion axiom,then F |= A. The theorem is a consequence of (28), (29), and (30).Let i be a natural number. Let us consider f and F . We say that prc(f, F, i)if and only if(Def. 25) f(i) ∈ PL-axioms or f(i) ∈ F or there exist natural numbers j, k suchthat 1 ¬ j < i and 1 ¬ k < i and MP(fj , fk, fi).Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PMThe axiomatization of propositional logic 287Let us consider p. We say that F ` p if and only if(Def. 26) there exists f such that f(len f) = p and 1 ¬ len f and for every naturalnumber i such that 1 ¬ i ¬ len f holds prc(f, F, i).Now we state the propositions:(50) Let us consider natural numbers i, n. Suppose n+ len f ¬ len f2 and forevery natural number k such that 1 ¬ k ¬ len f holds f(k) = f2(k + n)and 1 ¬ i ¬ len f . If prc(f, F, i), then prc(f2, F, i+ n).(51) Suppose f2 = f a f1 and 1 ¬ len f and 1 ¬ len f1 and for every naturalnumber i such that 1 ¬ i ¬ len f holds prc(f, F, i) and for every naturalnumber i such that 1 ¬ i ¬ len f1 holds prc(f1, F, i). Let us considera natural number i. If 1 ¬ i ¬ len f2, then prc(f2, F, i). The theorem isa consequence of (50).(52) Suppose f = f1 a 〈p〉 and 1 ¬ len f1 and for every natural number i suchthat 1 ¬ i ¬ len f1 holds prc(f1, F, i) and prc(f, F, len f). Then(i) for every natural number i such that 1 ¬ i ¬ len f holds prc(f, F, i),and(ii) F ` p.The theorem is a consequence of (50).(53) If p ∈ PL-axioms or p ∈ F , then F ` p.Proof: Define P[set, set] ≡ $2 = p. Consider f such that dom f = Seg 1and for every natural number k such that k ∈ Seg 1 holds P[k, f(k)] from[3, Sch. 5]. For every natural number j such that 1 ¬ j ¬ len f holdsprc(f, F, j). (54) If F ` p and F ` p⇒ q, then F ` q.Proof: Consider f such that f(len f) = p and 1 ¬ len f and for everynatural number i such that 1 ¬ i ¬ len f holds prc(f, F, i). Consider f1such that f1(len f1) = p⇒ q and 1 ¬ len f1 and for every natural numberi such that 1 ¬ i ¬ len f1 holds prc(f1, F, i). Set g = (f a f1) a 〈q〉. Forevery natural number i such that 1 ¬ i ¬ len f1 holds g(len f + i) = f1(i)by [3, (22), (39)], [1, (12)], [3, (65), (64)]. For every natural number i suchthat 1 ¬ i ¬ len(f a f1) holds prc(f a f1, F, i). (55) If F ⊆ G, then if F ` p, then G ` p.Proof: Consider f such that f(len f) = p and 1 ¬ len f and for eve-ry natural number k such that 1 ¬ k ¬ len f holds prc(f, F, k). DefineP[natural number] ≡ if 1 ¬ $1 ¬ len f , then G ` f$1 . For every naturalnumber k, P[k] from [1, Sch. 4]. Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PM288 mariusz giero5. Soundness Theorem. Deduction Theorem.Now we state the propositions:(56) If F ` A, then F |= A.Proof: Consider f such that f(len f) = A and 1 ¬ len f and for every na-tural number i such that 1 ¬ i ¬ len f holds prc(f, F, i). Define P[naturalnumber] ≡ if 1 ¬ $1 ¬ len f , then F |= f$1 . For every natural number isuch that for every natural number j such that j < i holds P[j] holds P[i]by [1, (14)], [9, (1)], (48), (49). For every natural number i, P[i] from [1,Sch. 4]. (57) F ` A⇒ A. The theorem is a consequence of (53) and (54).(58) Deduction theorem:If F ∪ {A} ` B, then F ` A⇒ B.Proof: Consider f such that f(len f) = B and 1 ¬ len f and for everynatural number i such that 1 ¬ i ¬ len f holds prc(f, F ∪ {A}, i). DefineP[natural number] ≡ if 1 ¬ $1 ¬ len f , then F ` A ⇒ f$1 . For everynatural number i such that for every natural number j such that j < iholds P[j] holds P[i] by [1, (14)], (53), [9, (1)], (54). For every naturalnumber i, P[i] from [1, Sch. 4]. (59) If F ` A⇒ B, then F ∪{A} ` B. The theorem is a consequence of (53),(55), and (54).(60) F ` ¬A ⇒ (A ⇒ B). The theorem is a consequence of (53), (54), and(58).(61) F ` ¬A ⇒ A ⇒ A. The theorem is a consequence of (53), (57), and(54).6. Strong Completeness TheoremLet us consider F . We say that F is consistent if and only if(Def. 27) there exists no p such that F ` p and F ` ¬p.Now we state the propositions:(62) F is consistent if and only if there exists A such that F 0 A. The theoremis a consequence of (60) and (54).(63) If F 0 A, then F ∪ {¬A} is consistent. The theorem is a consequence of(58), (62), (61), and (54).(64) F ` A if and only if there exists G such that G ⊆ F and G is finite andG ` A. The theorem is a consequence of (55).Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PMThe axiomatization of propositional logic 289(65) If F is not consistent, then there exists G such that G is finite and G isnot consistent and G ⊆ F . The theorem is a consequence of (64) and (55).Let us consider F . We say that F is maximal if and only if(Def. 28) for every p holds p ∈ F or ¬p ∈ F .Now we state the propositions:(66) If F ⊆ G and F is not consistent, then G is not consistent. The theoremis a consequence of (55).(67) If F is consistent and F ∪ {A} is not consistent, then F ∪ {¬A} isconsistent. The theorem is a consequence of (58), (62), (61), and (54).In the sequel x, y denote sets. Now we state the propositions:(68) Lindenbaum’s lemma:If F is consistent, then there exists G such that F ⊆ G and G is consistentand maximal.Proof: Set L = PL-WFF. Consider R being a binary relation suchthat R well orders L. Reconsider R2 = R |2 L as a binary relation onL. Reconsider R1 = 〈L,R2〉 as a non empty relational structure. Setc = the carrier of R1. Define H[object, object, object] ≡ for every p forevery partial function f from c to 2L such that $1 = p and $2 = fholds if (⋃rng(f qua (2L)-valued binary relation)∪F )∪{p} is consistent,then $3 = (⋃rng f ∪ F ) ∪ {p} and if (⋃rng(f qua (2L)-valued binaryrelation) ∪ F ) ∪ {p} is not consistent, then $3 =⋃rng f ∪ F . For everyobjects x, y such that x ∈ c and y ∈ c→̇2L there exists an object z suchthat z ∈ 2L and H[x, y, z] by [8, (46)]. Consider h being a function fromc × (c→̇2L) into 2L such that for every objects x, y such that x ∈ c andy ∈ c→̇2L holds H[x, y, h(x, y)] from [5, Sch. 1]. Consider f being a func-tion from c into 2L such that f is recursively expressed by h. ReconsiderG =⋃rng(f qua (2L)-valued binary relation) as a subset of PL-WFF. Seti1 = the internal relation of R1. For every A and B such that 〈A, B〉 ∈ R2holds f(A) ⊆ f(B) by [4, (1)], [2, (4), (29), (9)]. rng f is ⊆-linear. DefineS[element of R1] ≡ f($1) is consistent. For every element x of R1 suchthat for every element y of R1 such that y 6= x and 〈y, x〉 ∈ i1 holds S[y]holds S[x] by [2, (9)], [7, (32)], [2, (1)], [15, (42)]. For every element A ofR1, S[A] from [12, Sch. 3]. F ⊆ G by [6, (3)]. G is consistent by (65), (16),[15, (42)], (66). G is maximal by [6, (3)], (17), [13, (16)], (66). (69) If F is maximal and consistent, then for every p, F ` p iff p ∈ F . Thetheorem is a consequence of (53).(70) If F |= A, then F ` A.Proof: Consider G such that F ∪ {¬A} ⊆ G and G is consistent and Gis maximal. Set M = {Propn, where n is an element of N : Propn ∈ G}.Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PM290 mariusz gieroM ⊆ AP . Define P[element of PL-WFF] ≡ $1 ∈ G iff M |= $1. P[⊥PL].For every n, P[Propn]. For every r and s such that P[r] and P[s] holdsP[r ⇒ s]. For every B, P[B] from PLInd. M 6|= A. (71) A is a tautology if and only if ∅PL-WFF ` A.References[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathe-matics, 1(1):41–46, 1990.[2] Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129,1990.[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finitesequences. Formalized Mathematics, 1(1):107–114, 1990.[4] Leszek Borys. On paracompactness of metrizable spaces. Formalized Mathematics, 3(1):81–84, 1992.[5] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.[7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164,1990.[8] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.[9] Mariusz Giero. Propositional linear temporal logic with initial validity semantics. For-malized Mathematics, 23(4):379–386, 2015. doi:10.1515/forma-2015-0030.[10] Witold Pogorzelski. Dictionary of Formal Logic. Wydawnictwo UwB - Bialystok, 1992.[11] Witold Pogorzelski. Notions and theorems of elementary formal logic. Wydawnictwo UwB- Bialystok, 1994.[12] Piotr Rudnicki and Andrzej Trybulec. On same equivalents of well-foundedness. Forma-lized Mathematics, 6(3):339–343, 1997.[13] Andrzej Trybulec. Defining by structural induction in the positive propositional language.Formalized Mathematics, 8(1):133–137, 1999.[14] Anita Wasilewska. An Introduction to Classical and Non-Classical Logics. SUNY StonyBrook, 2005.[15] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.Received October 18, 2016Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 5/16/17 2:16 PM

The Axiomatization of Propositional Logic

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