A gentle introduction to Abstract Algebraic Logic I Logics, logical matrices, and completeness theorems Petr Cintula Institute of Computer Science, Academy of Sciences of the Czech Republic Prague, Czech Republic www.cs.cas.cz/cintula/AAL Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 1 / 36
Logic throughout the history Logic is the science that studies correct reasoning Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 2 / 36
Logic throughout the history Logic is the science that studies correct reasoning Logic started as a part of philosophy All men are mortal and Socrates in a man, therefore Socrates is mortal Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 2 / 36
Logic throughout the history Logic is the science that studies correct reasoning Since 19th century a part of logic has evolved into mathematical logic PA �⊢ ¬∃ w Proof ( w , ‘ ¯ 0 = ¯ 1 ’ ) Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 2 / 36
Logic throughout the history Logic is the science that studies correct reasoning Nowadays logic is applied mainly in computer science ⊢ [ α ]( x = 4 ) → [ α ; ( x := 2 x )]( x = 8 ) Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 2 / 36
Logic throughout the history Logic is the science that studies correct reasoning However its role in the study of (human) reasoning has been diminished Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 2 / 36
Logic throughout the history Logic is the science that studies correct reasoning However its role in the study of (human) reasoning has been diminished Stenning (psychologist) and van Lambalgen (logician) advocate the indispensability of logic for the study of (human) reasoning Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 2 / 36
Logic throughout the history Logic is the science that studies correct reasoning However its role in the study of (human) reasoning has been diminished Stenning (psychologist) and van Lambalgen (logician) advocate the indispensability of logic for the study of (human) reasoning The key component: the transformation of natural reasoning scenarios into formalized ones where various logics are directly applicable Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 2 / 36
Various logic(s) Formal logical systems are many and vary in expressive powers: propositional logics and its modal extensions first- and higher- order logics various type theories dynamic and non-monotonic logics . . . Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 3 / 36
Various logic(s) Formal logical systems are many and vary in expressive powers: propositional logics and its modal extensions first- and higher- order logics various type theories dynamic and non-monotonic logics . . . This tutorial focuses on truth-functional context-independent transitive monotonic propositional cores of these systems: Classical logic Intuitionistic logic Linear logic Superintuitionistic logics Fuzzy logics Modal logics Substructural logics Paraconsistent logics . . . Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 3 / 36
Algebraic logic Algebraic logic: a field of mathematical logic studying logics using Universal Algebra (a field of mathematics studying classes of algebraic structures) Logic Algebraic counterpart Classical logic Boolean algebras Modal logics Modal algebras Intuitionistic logic Heyting algebras Linear logics Commutative residuated lattices Fuzzy logics Semilinear residuated lattices Relevance logics Commutative contractive residuated lattices . . . . . . Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 4 / 36
Abstract Algebraic Logic AAL is the evolution of algebraic logic that wants to: understand the several ways by which a logic can be given an algebraic semantics build a general and abstract theory of non-classical logics based on their relation to algebras understand the role of connectives in (non-)classical logics classify non-classical logics find general results connecting logical and algebraic properties (bridge theorems) generalize various properties from syntax to semantics (transfer theorems) advance the study of particular (families of) non-classical logics by using the abstract notions and results It works best, by far, when restricted to propositional logics Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 5 / 36
Abstract Algebraic Logic AAL is the evolution of algebraic logic that wants to: understand the several ways by which a logic can be given an algebraic semantics build a general and abstract theory of non-classical logics based on their relation to algebras understand the role of connectives in (non-)classical logics classify non-classical logics find general results connecting logical and algebraic properties (bridge theorems) generalize various properties from syntax to semantics (transfer theorems) advance the study of particular (families of) non-classical logics by using the abstract notions and results Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 6 / 36
Basic syntactical notions – formulas and consecutions Propositional language: a countable set of connective with arities we write � c , n � ∈ L if L has an n -ary connective c Formulas: build from infinite countable set Var of variables it the usual way . . . by Fm L we denote the set of formulas in L Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 7 / 36
Basic syntactical notions – formulas and consecutions Propositional language: a countable set of connective with arities we write � c , n � ∈ L if L has an n -ary connective c Formulas: build from infinite countable set Var of variables it the usual way . . . by Fm L we denote the set of formulas in L Consecution: a pair Γ ⊲ ϕ , where Γ ∪ { ϕ } ⊆ Fm L Note: A set L of consecutions can be seen as a relation between sets of formulas and formulas. We write ‘ Γ ⊢ L ϕ ’ instead of ‘ Γ ⊲ ϕ ∈ L ’ Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 7 / 36
Basic syntactical notions – formulas and consecutions Propositional language: a countable set of connective with arities we write � c , n � ∈ L if L has an n -ary connective c Formulas: build from infinite countable set Var of variables it the usual way . . . by Fm L we denote the set of formulas in L Consecution: a pair Γ ⊲ ϕ , where Γ ∪ { ϕ } ⊆ Fm L Note: A set L of consecutions can be seen as a relation between sets of formulas and formulas. We write ‘ Γ ⊢ L ϕ ’ instead of ‘ Γ ⊲ ϕ ∈ L ’ Substitution: a mapping σ : Fm L → Fm L , such that for each � c , n � ∈ L and every ϕ 1 , . . . , ϕ n ∈ Fm L : σ ( c ( ϕ 1 , . . . , ϕ n )) = c ( σ ( ϕ 1 ) , . . . , σ ( ϕ n )) Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 7 / 36
Basic syntactical notions – logics Definition A set L of consecutions in L is called a logic in L whenever If ϕ ∈ Γ , then Γ ⊢ L ϕ . (Reflexivity) If ∆ ⊢ L ψ for each ψ ∈ Γ and Γ ⊢ L ϕ , then ∆ ⊢ L ϕ . (Cut) If Γ ⊢ L ϕ , then σ [Γ] ⊢ L σ ( ϕ ) for each substitution σ . (Structurality) A logic L is finitary if furthermore If Γ ⊢ L ϕ , then there is finite Γ ′ ⊆ Γ and Γ ′ ⊢ L ϕ . (Finitarity) Observe that reflexivity and cut entail: If Γ ⊢ L ϕ , then Γ ∪ ∆ ⊢ L ϕ . (Monotonicity) Theorem of L : a formula ϕ such that ∅ ⊢ L ϕ Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 8 / 36
Basic syntactical notions – some (trivial) examples The least logic Dumb is described as (note that it has no theorems): Γ ⊢ Dumb ϕ iff ϕ ∈ Γ Inconsistent logic Inc : the set all consecutions (equivalently: a logic where all formulas are theorems) Almost Inconsistent logic AInc : the maximum logic without theorems (note that Γ , ϕ ⊢ AInc ψ ) Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 9 / 36
Basic syntactical notions – theories Theory: a set of formulas T such that if T ⊢ L ϕ then ϕ ∈ T Th ( L ) is the set of all theories of L Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 10 / 36
Basic syntactical notions – theories Theory: a set of formulas T such that if T ⊢ L ϕ then ϕ ∈ T Th ( L ) is the set of all theories of L Th ( Dumb ) = P ( Fm L ) Th ( Inc ) = { Fm L } Th ( AInc ) = {∅ , Fm L } Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 10 / 36
Basic syntactical notions – theories Theory: a set of formulas T such that if T ⊢ L ϕ then ϕ ∈ T Th ( L ) is the set of all theories of L Th ( Dumb ) = P ( Fm L ) Th ( Inc ) = { Fm L } Th ( AInc ) = {∅ , Fm L } Note that the set of all theorems is the least theory the set Th L (Γ) = { ϕ ∈ Fm L | Γ ⊢ L ϕ } is a theory Petr Cintula (ICS CAS) Abstract Algebraic Logic – 1st lesson www.cs.cas.cz/cintula/AAL 10 / 36
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