Formalizing first-order logic Joris Roos University of Wisconsin-Madison Sommerakademie Leysin 2018 August 15, 2018 Joris Roos First-order logic August 15, 2018 1 / 30 Overview 1 Introduction 2 Syntax 3 Semantics 4 A simple theorem prover Joris Roos First-order logic August 15, 2018 2 / 30 They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 more complicated atomic formulas, consisting of: variables: x; y; z; ··· functions: f (x); g(x; y); :: predicates: P(x); x = y; ··· ; First-order logic is powerful enough to formalize \all of mathematics". Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) Joris Roos First-order logic August 15, 2018 3 / 30 First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 more complicated atomic formulas, consisting of: variables: x; y; z; ··· functions: f (x); g(x; y); :: predicates: P(x); x = y; ··· ; First-order logic is powerful enough to formalize \all of mathematics". Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. Joris Roos First-order logic August 15, 2018 3 / 30 more complicated atomic formulas, consisting of: variables: x; y; z; ··· functions: f (x); g(x; y); :: predicates: P(x); x = y; ··· ; First-order logic is powerful enough to formalize \all of mathematics". Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 Joris Roos First-order logic August 15, 2018 3 / 30 variables: x; y; z; ··· functions: f (x); g(x; y); :: predicates: P(x); x = y; ··· ; First-order logic is powerful enough to formalize \all of mathematics". Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 more complicated atomic formulas, consisting of: Joris Roos First-order logic August 15, 2018 3 / 30 functions: f (x); g(x; y); :: predicates: P(x); x = y; ··· ; First-order logic is powerful enough to formalize \all of mathematics". Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 more complicated atomic formulas, consisting of: variables: x; y; z; ··· Joris Roos First-order logic August 15, 2018 3 / 30 predicates: P(x); x = y; ··· ; First-order logic is powerful enough to formalize \all of mathematics". Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 more complicated atomic formulas, consisting of: variables: x; y; z; ··· functions: f (x); g(x; y); :: Joris Roos First-order logic August 15, 2018 3 / 30 First-order logic is powerful enough to formalize \all of mathematics". Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 more complicated atomic formulas, consisting of: variables: x; y; z; ··· functions: f (x); g(x; y); :: predicates: P(x); x = y; ··· ; Joris Roos First-order logic August 15, 2018 3 / 30 Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 more complicated atomic formulas, consisting of: variables: x; y; z; ··· functions: f (x); g(x; y); :: predicates: P(x); x = y; ··· ; First-order logic is powerful enough to formalize \all of mathematics". Joris Roos First-order logic August 15, 2018 3 / 30 Introduction Last time we considered propositional formulas: ((A _ D) ! (D _:B)) ^ :(A (B _ C ^ D) They consist of constants (>, ?), literals (A; B;::: ), logical connectives and punctuation. First-order logic (FOL) extends this in two different ways: quantifiers: 8; 9 more complicated atomic formulas, consisting of: variables: x; y; z; ··· functions: f (x); g(x; y); :: predicates: P(x); x = y; ··· ; First-order logic is powerful enough to formalize \all of mathematics". Example. Continuity 8x:8":" > 0 ! (9δ.δ > 0 ! (8y:jx − yj < δ ! jf (x) − f (y)j < ")) Joris Roos First-order logic August 15, 2018 3 / 30 1 Introduction 2 Syntax 3 Semantics 4 A simple theorem prover Joris Roos First-order logic August 15, 2018 4 / 30 The alphabet consists of Propositional constants: > (true), ? (false) Logical operators: :; ^; _ Punctuation: brackets ( ) and ; Quantifiers: 8; 9 Function symbols: f , g, ::: Predicate symbols: P, Q, R, ::: Variable symbols: x, y, z, ::: We will define FOL formulas in three steps: terms −! atomic formulas −! formulas We often also want to fix the allowed function and predicate symbols (this is called a \language"). Admissible symbols A FOL formula is a string of symbols from a certain alphabet. Joris Roos First-order logic August 15, 2018 5 / 30 Logical operators: :; ^; _ Punctuation: brackets ( ) and ; Quantifiers: 8; 9 Function symbols: f , g, ::: Predicate symbols: P, Q, R, ::: Variable symbols: x, y, z, ::: We will define FOL formulas in three steps: terms −! atomic formulas −! formulas We often also want to fix the allowed function and predicate symbols (this is called a \language"). Admissible symbols A FOL formula is a string of symbols from a certain alphabet. The alphabet consists of Propositional constants: > (true), ? (false) Joris Roos First-order logic August 15, 2018 5 / 30 Punctuation: brackets ( ) and ; Quantifiers: 8; 9 Function symbols: f , g, ::: Predicate symbols: P, Q, R, ::: Variable symbols: x, y, z, ::: We will define FOL formulas in three steps: terms −! atomic formulas −! formulas We often also want to fix the allowed function and predicate symbols (this is called a \language"). Admissible symbols A FOL formula is a string of symbols from a certain alphabet. The alphabet consists of Propositional constants: > (true), ? (false) Logical operators: :; ^; _ Joris Roos First-order logic August 15, 2018 5 / 30 Quantifiers: 8; 9 Function symbols: f , g, ::: Predicate symbols: P, Q, R, ::: Variable symbols: x, y, z, ::: We will define FOL formulas in three steps: terms −! atomic formulas −! formulas We often also want to fix the allowed function and predicate symbols (this is called a \language"). Admissible symbols A FOL formula is a string of symbols from a certain alphabet. The alphabet consists of Propositional constants: > (true), ? (false) Logical operators: :; ^; _ Punctuation: brackets ( ) and ; Joris Roos First-order logic August 15, 2018 5 / 30 Function symbols: f , g, ::: Predicate symbols: P, Q, R, ::: Variable symbols: x, y, z, ::: We will define FOL formulas in three steps: terms −! atomic formulas −! formulas We often also want to fix the allowed function and predicate symbols (this is called a \language"). Admissible symbols A FOL formula is a string of symbols from a certain alphabet. The alphabet consists of Propositional constants: > (true), ? (false) Logical operators: :; ^; _ Punctuation: brackets ( ) and ; Quantifiers: 8; 9 Joris Roos First-order logic August 15, 2018 5 / 30 Predicate symbols: P, Q, R, ::: Variable symbols: x, y, z, ::: We will define FOL formulas in three steps: terms −! atomic formulas −! formulas We often also want to fix the allowed function and predicate symbols (this is called a \language"). Admissible symbols A FOL formula is a string of symbols from a certain alphabet. The alphabet consists of Propositional constants: > (true), ? (false) Logical operators: :; ^; _ Punctuation: brackets ( ) and ; Quantifiers: 8; 9 Function symbols: f , g, ::: Joris Roos First-order logic August 15, 2018 5 / 30 Variable symbols: x, y, z, ::: We will define FOL formulas in three steps: terms −! atomic formulas −! formulas We often also want to fix the allowed function and predicate symbols (this is called a \language"). Admissible symbols A FOL formula is a string of symbols from a certain alphabet. The alphabet consists of Propositional constants: > (true), ? (false) Logical operators: :; ^; _ Punctuation: brackets ( ) and ; Quantifiers: 8; 9 Function symbols: f , g, ::: Predicate symbols: P, Q, R, ::: Joris Roos First-order logic August 15, 2018 5 / 30 We will define FOL formulas in three steps: terms −! atomic formulas −! formulas We often also want to fix the allowed function and predicate symbols (this is called a \language"). Admissible symbols A FOL formula is a string of symbols from a certain alphabet.