ELTE 2013. September
Today Past 1 Universal Algebra 1 Algebra 2 Transforming Algebras... Past 1 Homomorphism 2 Subalgebras 3 Direct products 3 Varieties 1 Algebraic Model Theory 1 Term Algebras 2 Meanings 3 Substitutions 4 Truth and Validity 5 Birkhoff-Theorem Today 1 Algebraization of Traditional Syntax 1 Lindenbaum Tarski Algebras 2 Boolean Algebras 2 Algebraization of Traditional Semantics 1 2-valued Semantics 2 Modal semantics 3 Generalized semantics 3 Representation Today 1 Modal Logic 2 Algebraic Proof System
Algebraization of Traditional Syntax
Reminder Language: Φ is the set of atomic sentences (sometimes called proposition letters ). The set of well-formed formulas based on Φ is ined by the rule ϕ ::= p ϕ ϕ ψ. where p ranges over Φ. We denote this set by FORM(Φ). Abbreviations: ϕ ψ ϕ ψ ϕ ψ ( ϕ ψ) (ϕ ψ) (ϕ ψ) (ψ ϕ)
Reminder The Frege Hilbert axiom system: ϕ (ψ ϕ) ( ϕ ψ) (ψ ϕ) (ϕ (ψ χ)) ((ϕ ψ) (ϕ χ)) ϕ, ϕ ψ ψ A proof of p p: p ((q p) p) 1st axiom (p ((q p) p)) (p (q p)) (p p) 2nd axiom (p (q p)) (p p) modus ponens (p (q p)) 1st axiom p p modus ponens
Algebraization of Traditional Syntax Note that traditional syntax is unstructurized : Definition (Formula algebra) Form(Φ) = FORM(Φ), +,, where ϕ + ψ ϕ = (ϕ ψ) = ϕ. Note that while + is a function + : FORM 2 FORM, is just a symbol. Note that + is not commutative: p + q = (p q) (q p) = q + p The reason: they are different formulas, even if they are seem to be equivalent. However, we have tools in the syntax to introduce a notion for this equvalence.
Algebraization of Traditional Syntax However, we have tools in the syntax to introduce a notion for this equvalence. Definition (Provable equivalency) Let denote the derivability in the Frege Hilbert calculus. ϕ ψ ϕ ψ Now that we have a relation on the algebra Form, an algebraist will surely ask the following questions: First question: Is this an equivalence relation, i.e., does it partition the carrier? Second question: Is this a congruence relation, i.e., is this relation closed to the operations, +?
Algebraization of Traditional Syntax Theorem ( is a Congruence Relation) is a congruence relation on Form, i.e., it is an equivalence relation closed to the functions of Form: (Equivalence relation) ϕ ϕ ϕ ψ ψ ϕ ϕ ψ ψ χ ϕ χ (Congruence relation) ϕ ψ ϕ ψ ϕ 1 ψ 1 ϕ 2 ψ 2 ϕ 1 ϕ 2 ψ 1 ψ 2
Lindenbaum Tarski Algebra Definition (Lindenbaum Tarski Algebra) The set of equivalence classes of FORM will be denoted by FORM/ = {P FORM : ( ϕ, ψ P) ϕ ψ} Let us denote the equivalence class containg a formula ϕ by [ϕ]. Then the Lindenbaum Tarski algebra of FORM(Φ) and is the algebra Form/ = FORM/, +,, 0, where the functions ined on the equivalence classes (instead of their elements): [ϕ] [ϕ] + [ψ] 0 = [ ] = [ ϕ] = [ϕ + ψ]
Lindenbaum Tarski Algebra The Lindenbaum-Tarski algebra is a canonical model, a universal refuting model : It gives counterexamples for every non-theorem of the Frege Hilbert calculus. Theorem ( Canonical Algebra Theorem ) ϕ Form/ = ϕ =
Lindenbaum Tarski Algebra Now commutativity became true: FORM/ = x + y = y + x The situation is the same with all usual algebraic rules of classical logic. Are we able to ine these logical equations recursively as well, i.e., can we axiomatize the set of valid logical equations?
Boolean Algebra Definition (Boolean Algebra) The Boolean equations are the following set of equations: x + y = y + x x y = y x (x + y) + z = x + (y + z) (x y) z = y (x z) x + 0 = x x 1 = 1 x + ( x) = 1 x ( x) = 0 x + (y z) = (x + y) (x + z) x (y + z) = (x y) + (x z) An algebraic structure A = A, + A, A, 0 A is a Boolean algebra iff it satisfies the Boolean equations. The variety of Boolean algebras will be denoted by BA.
Boolean Algebra Theorem Lindenbaum-Tarski algebras are Boolean algebras.
Boolean Algebra Theorem Lindenbaum-Tarski algebras are Boolean algebras. Proof. Check that the Boolean equations are derivable from the Frege Hilbert Calculus
Boolean Algebra Theorem Lindenbaum-Tarski algebras are Boolean algebras. Theorem (BA algebraizes classical theoremhood) ϕ BA = ϕ =
Boolean Algebra Theorem Lindenbaum-Tarski algebras are Boolean algebras. Theorem (BA algebraizes classical theoremhood) ϕ BA = ϕ = Proof. : ϕ Form/ = ϕ = BA = ϕ =
Reminder : Proof by induction... ϕ (ψ ϕ) BA = ϕ + ( ψ + ϕ) ( ϕ ψ) (ψ ϕ) BA = (ϕ + ψ) + ( ψ + ϕ) (ϕ (ψ χ)) ((ϕ ψ) (ϕ χ)) BA = ( ϕ + ( ψ + χ)) + ( ( ϕ + ψ) + ( ϕ + χ)) ϕ, ϕ ψ ψ ϕ, ϕ + ψ ψ (Or maybe Natürliche Kalkül... )
Algebraization of Traditional Semantics
Algebraization of 2-valued Semantics Remember [ ] M? [ ] M = 0 [p ] M = v(p) [ ϕ ] M = 1 [ϕ ] M [ϕ ψ ] M = max ( [ϕ ] M, [ψ ] M )
Algebraization of 2-valued Semantics Remember [ ] M? [ ] M = 0 [p ] M = v(p) [ ϕ ] M = 1 [ϕ ] M = [ϕ ] M [ϕ ψ ] M = max ( [ϕ ] M, [ψ ] M ) = [ϕ ] M + [ψ ] M
Algebraization of 2-valued Semantics Remember [ ] M? [ ] M = 0 [p ] M = v(p) [ ϕ ] M = 1 [ϕ ] M = [ϕ ] M [ϕ ψ ] M = max ( [ϕ ] M, [ψ ] M ) = [ϕ ] M + [ψ ] M This is a homomorphism between Form and Definition (Algebra of truth values) 2 = 2, +,, 0 where a = 1 a and a + b = max(a, b). What is more, trivially Theorem (2 Algebraizes Classical Validity) = ϕ 2 = ϕ =
Algebraization of Traditional Semantics
Algebraization of Modal Semantics Remember [ ] M? [ ] M = [p ] M = v(p) [ ϕ ] M = W [ϕ ] M [ϕ ψ ] M = [ϕ ] M [ψ ] M
Algebraization of Modal Semantics Remember [ ] M? [ ] M = [p ] M = v(p) [ ϕ ] M = W [ϕ ] M = [ϕ ] M [ϕ ψ ] M = [ϕ ] M [ψ ] M = [ϕ ] M + [ψ ] M
Algebraization of Modal Semantics Remember [ ] M? [ ] M = [p ] M = v(p) [ ϕ ] M = W [ϕ ] M = [ϕ ] M [ϕ ψ ] M = [ϕ ] M [ψ ] M = [ϕ ] M + [ψ ] M This is a homomorphism between Form and Definition (Powerset Algebras) The power set algebra of A: What is more, P(A) = P(A),,, Theorem (Powersets algebraize classical validity) = ϕ {P(A) : A is a set} = ϕ =
Algebraization of Modal Semantics Theorem Every power set algebra is isomorphic to a power of 2 = {, { }}, and conversely. Proof. First we show that ( A)( I)P(A) = 2 I. Brilliant idea: Let I be A. Remember that P(A) = 2 A = A 2. What are these? Characteristic functions: functions deciding whether an element a of A is in the subset X or not: 1, if a X χ X (a) 0, if a / X
Algebraization of Modal Semantics Theorem Every power set algebra is isomorphic to a power of 2 = {, { }}, and conversely. Proof. 1, if a X χ X (a) 0, if a / X
Algebraization of Modal Semantics Theorem Every power set algebra is isomorphic to a power of 2 = {, { }}, and conversely. Proof. 1, if a X χ X (a) 0, if a / X A P(A) a b c d e X χ α f : A 2 a 1 b 1 c 1 d 0 e 0 HW: Verify that χ : P(A) A 2.
Algebraization of Modal Semantics Theorem Every power set algebra is isomorphic to a power of 2 = {, { }}, and conversely. Proof. A P(A) a b c d e X χ α f : A 2 a 1 b 1 c 1 d 0 e 0 Now we show that ( I)( A)P(A) = 2 I. Again, I := A. Then subsets = those elements whose image is 1 by f
Algebraization of Modal Semantics Theorem Every power set algebra is isomorphic to a power of 2 = {, { }}, and conversely. Proof. A P(A) a b c d e X χ α f : A 2 a 1 b 1 c 1 d 0 e 0 HW: Verify that α : I 2 P(A). α(f) = {i I : f(i) = 1}.
Algebraization of Modal Semantics Theorem (Powersets algebraize classical validity) = ϕ {P(A) : A is a set} = ϕ = Proof. Validity of equations is preserved under, and every power set algebra is isomorphic to a power of 2.
Algebraization of Generalized Semantics
Algebraization of Generalized Semantics Definition (Set Algebras) A set algebra or field of sets is a subalgebra of a power set algebra. The class of set algebras will be denoted by Set. Theorem (Set algebraizes classical validity) = ϕ Set = ϕ = Proof. Validity of equations is preserved under.
Summary Traditional Algebraic Language Propositional Equational Calculus Natural Deduction Frege Hilbert BA+ Equational logic Semantics 2-valued 2 Modal {P(A) : A is a set} Generalized Set
How do we use Algebraization? Theorem (Stone) Any boolean algebra is isomorphic to a set algebra. Or BA = ({2}). Theorem (Weak completeness) The Frege-Hilbert calculus is sound and weakly complete with respect to the 2-valued semantics, modal semantics and generalized semantics. Proof. ϕ BA = ϕ = Set = ϕ = = ϕ
Today Past 1 Universal Algebra 1 Algebra 2 Transforming Algebras... Past 1 Homomorphism 2 Subalgebras 3 Direct products 3 Varieties 1 Algebraic Model Theory 1 Term Algebras 2 Meanings 3 Substitutions 4 Truth and Validity 5 Birkhoff-Theorem Today 1 Algebraization of Traditional Syntax 1 Lindenbaum Tarski Algebras 2 Boolean Algebras 2 Algebraization of Traditional Semantics 1 2-valued Semantics 2 Modal semantics 3 Generalized semantics 3 Representation Today 1 Modal Logic 2 Algebraic Proof System
Modal Logic
Language Φ is the set of atomic sentences (sometimes called proposition letters ). The set of well-formed formulas based on Φ is ined by the rule ϕ ::= p ϕ ϕ ψ. ϕ where p ranges over Φ. We denote this set by FORM(Φ). Abbreviations: ϕ ψ ϕ ψ ϕ ψ ϕ ( ϕ ψ) (ϕ ψ) (ϕ ψ) (ψ ϕ) ϕ
Reminder The minimal normal modal system K: ϕ (ψ ϕ) ( ϕ ψ) (ψ ϕ) (ϕ (ψ χ)) ((ϕ ψ) (ϕ χ)) (K) : (ϕ ψ) ( ϕ ψ) ϕ, ϕ ψ ψ (MP) ϕ ϕ (RN) A proof of the Lemmon-rule: ϕ ψ ϕ ϕ : ϕ ψ Theorem by assumption (ϕ ψ) (RN) (ϕ ψ) ( ϕ ψ) (K) ϕ ψ (MP)
Semantics Definition (Frame) A frame/relatitional structure F is a pair of a nonempty set W, called the set of possible worlds, and the accessibility/alternative relation R W 2.
Semantics Definition (Valuations, Models, Truth) Let F = W, R be a frame. The valuations on F are the functions v : Φ P(W). The normal modal models M are the pairs M = F, v = W, R, v The satisfiability / truth / forcing relation : M, w M, w p M, w ϕ = W w v(p) M, w ϕ M, w ϕ ψ M, w ϕ and M, w ψ M, w ϕ ( w w)m, w ϕ R
Intension Definition (Intension) The semantic value or intension induced by the valuation v is the function [ ] M : FORM(Φ) P(W) ined in the following way: [ϕ ] M = {w W : M, w ϕ}, i.e., [ ] M = [p ] M = v(p) [ ϕ ] M = W [ϕ ] M [ϕ ψ ] M = [ϕ ] M [ψ ] M [ ϕ ] M = {HW} [ ϕ ] M = {HW}