The Axiomatization of Propositional Logic

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.


mariusz giero 1. Preliminaries
Now we state the propositions: (1) Let us consider functions f , g. Suppose dom f ⊆ dom g and for every set x 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, r.Now we state the propositions: (7) 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) (16) Let us consider a finite set X, and a set Y. Suppose Y is ⊆-linear and X ⊆ Y and Y = ∅.Then there exists a set Z such that (i) X ⊆ Z, and

The Syntax
Let 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.
Let us note that every set which is PL-closed is also non empty and has also 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-closed and non empty and PL-WFF is functional and every element of PL-WFF is finite sequence-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 of PL-WFF is defined by the term (Def.5) ( 1 p) q.
Let n be an element of N. The functor Prop n yielding an element of PL-WFF is 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 that x = Prop n.From now on p, q, r, s, A, B denote elements of PL-WFF, F , G, H denote subsets of PL-WFF, k, n denote elements of N, and f , f 1 , f 2 denote finite sequences of elements of PL-WFF.
Let D be a subset of PL-WFF.Observe that D has implication if and only if 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 Let us consider p.The functor ¬p yielding an element of PL-WFF is defined by 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 of PL-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 by the term (Def.13) (p ⇒ q) ∧ (q ⇒ p).
(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 is a consequence of ( 21) and ( 7).(46) p ⇒ r ⇒ (q ⇒ r ⇒ (p ∨ q ⇒ r)) is a tautology.The theorem is a consequence of ( 23) and ( 8). ( 47 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 D is 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 axiom or 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 such that 1 j < i and 1 k < i and MP(f j , f k , f i ).
Let 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 natural number 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 f 2 and for every natural number k such that 1 k (51) Suppose f 2 = f f 1 and 1 len f and 1 len f 1 and for every natural number i such that 1 i len f holds prc(f, F, i) and for every natural number i such that 1 The theorem is a consequence of (50).
(52) Suppose f = f 1 p and 1 len f 1 and for every natural number i such that 1 i len f 1 holds prc(f 1 , 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 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 1 and 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 holds prc(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 every natural number k such that 1 For every natural number k, P[k] from [1,Sch. 4].
(65) If F is not consistent, then there exists G such that G is finite and G is not 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 theorem is a consequence of (55).(67) If F is consistent and F ∪ {A} is not consistent, then F ∪ {¬A} is consistent.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 consistent and maximal.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 .The theorem 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 G is maximal.Set M = {Prop n, where n is an element of N : Prop n ∈ G}.
Proof: Set L = PL-WFF.Consider R being a binary relation such that R well orders L. Reconsider R 2 = R | 2 L as a binary relation on L. Reconsider R 1 = L, R 2 as a non empty relational structure.Set c = the carrier of R 1 .Define H[object, object, object] ≡ for every p for every partial functionf from c to 2 L such that $ 1 = p and $ 2 = f holds if ( rng(f qua (2 L )-valued binary relation) ∪ F ) ∪ {p} is consistent, then $ 3 = ( rng f ∪ F ) ∪ {p} and if ( rng(f qua (2 L )-valued binary relation) ∪ F ) ∪ {p} is not consistent, then $ 3 = rng f ∪ F .For every objects x, y such that x ∈ c and y ∈ c →2 L there exists an object z such that z ∈ 2 L and H[x, y, z] by [8, (46)].Consider h being a function from c × (c →2 L ) into 2 L such that for every objects x, y such that x ∈ c and y ∈ c →2 L holds H[x, y, h(x, y)] from [5, Sch.1].Consider f being a function from c into 2 L such that f is recursively expressed by h.Reconsider G = rng(f qua (2 L )-valued binary relation) as a subset of PL-WFF.Set i 1 = the internal relation of R 1 .For every A and B such thatA, B ∈ R 2 holds f (A) ⊆ f (B) by[4, (1)], [2, (4), (29), (9)].rng f is ⊆-linear.Define S[element of R 1 ] ≡ f ($ 1 ) is consistent.For every element x of R 1 such that for every element y of R 1 such that y = x and y, x ∈ i 1 holds S[y] holds S[x] by[2, (9)],[7, (32)],[2, (1)],[15, (42)].For every element A of R 1 , S[A] from[12, Sch.3].F ⊆ G by[6, (3)].G is consistent by (65), ( Let us consider A. We say that A is a tautology if and only if (Def.18) for every M , SAT M (A) = 1.Now we state the propositions: (27) A is a tautology if and only if ∅ PL-WFF |= A. ( The theorem is a consequence of (25).Let us consider M and p.We say that M |= p if and only if (Def.15) SAT M (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.