Logic - CM0845 - · PDF file2011-06 K computer 8.16 8.1 sec 66.2 y 1.7 × 108 c Logic -...

Logic - CM0845

Andrés Sicard Ramírez

EAFIT University

Semester 2016-1

Introduction

Administrative Information

Master CoordinatorFreddy Hernán Marín Sánchez

Head of the Department of Mathematical SciencesMyladis Rocío Cogollo Flórez

Course web pagehttp://www1.eafit.edu.co/asr/courses/logic-CM0845/

Exams, programming labs, bibliography, etc.See course web page.

Logic - CM0845. Introduction 3/90

http://www1.eafit.edu.co/asr/courses/logic-CM0845/

Exams

Exam Subjects1st Propositional Logic

• Syntax [van Dalen 2013, § 2.1]• Semantics [van Dalen 2013, § 2.2]• Satisfiability [Ben-Ari 2012, § 2.5]• Semantic Tableaux [Ben-Ari 2012, § 2.6]

2nd Propositional Logic• Natural Deduction [van Dalen 2013, § 2.4]

First-Order Logic• Syntax [van Dalen 2013, § 3.1, § 3.2 and § 3.3]

3rd First-Order Logic• Semantics [van Dalen 2013, § 3.4]• Identidad [van Dalen 2013, § 3.6]• Examples of first-order languages [van Dalen 2013, § 3.7]• Natural Deduction [van Dalen 2013, § 3.8 and § 3.9]


Some Related Subjects

Logic monism or logic pluralism (Logic or logics?)Time matters (algorithmic complexity)When you only have a hammer, everything looks like a nail(paradigms of programming)Help from the machines (theorem provers)


Logic or Logics?

See slides in the course web page.


Algorithmic Complexity

Comparison of several time complexity functions

𝑓(𝑛) 10 50 100log 𝑛 2.3 sec 3.9 sec 4.6 sec𝑛 10 sec 50 sec 1.7 min𝑛2 1.7 min 41.7 min 2.8 h2𝑛 17.1 min 358.001 c 4 × 1020 c3𝑛 16.4 h 2.3 × 1014 c 1.6 × 1038 c𝑛! 42 d 9.7 × 1054 c 3 × 10148 c



Example (3-SAT: An intractable problem)To determine the satisfiability of a propositional formula in conjunctivenormal form where each disjunction of literals is limited to at most threeliterals.The problem was proposed in Karp’s 21 NP-complete problems.∗

∗Karp, Richard M. (1972). Reducibility Among Combinatorial Problems.Logic - CM0845. Introduction 8/90

Algorithmic Complexity∗

3-SAT better deterministic algorithmic complexity

Year Complexity Reference2010 𝑂(1.439𝑛) Kutzkov, Konstantin and Scheder, Dominik (2010). Using

CSP to Improve Deterministic 3-SAT (unpublish).2008 𝑂(1.465𝑛) Scheder, Dominik (2008). Guided Search and a Faster De-

terministic Algorithm for 3-SAT.2004 𝑂(1.473𝑛) Brueggemann, Tobias and Kern, Walter (2004). An Im-

proved Deterministic Local Search Algorithm for 3-SAT.

∗Moser, Robin A. and Scheder, Dominik (2011). A Full Derandomization ofSchöning’s 𝑘-SAT Algorithm.



3-SAT simulationMachines from: www.top500.orgPetaflops (PFs): 1015 floating-point operations per second

Date Machine PFs 100 150 2002013-06 2015-10 Tianhe-2 33.86 2.0 sec 16.0 y 4.1 × 107 c2012-06 Blue Gene/Q 16.32 4.1 sec 33.1 y 8.5 × 107 c2011-06 K computer 8.16 8.1 sec 66.2 y 1.7 × 108 c


www.top500.org

Some Paradigms of Programming

Imperative: Describe computation in terms of state-transformingoperations such as assignment. Programming is done with statements.Logic: Predicate calculus as a programming language. Programming isdone with sentences.Functional: Describe computation in terms of (mathematical) functions.Programming is done with expressions.

Examples

Imperative⎧{⎨{⎩

CC + +Java

Logic {CLP(R)Prolog Functional

⎧{{{⎨{{{⎩

MLErlang

Pure⎧{⎨{⎩

CleanHaskellIdris


Theorem Provers

See slides in the course web page.


Program Course

Propositional/First-order logicSyntax for propositional/first-order logicSemantics for propositional/first-order logicDeductive systemsNormal formsResolution

Intuitionistic logicConstructivist reasoningPropositional intuitionistic logicFirst-order intuitionistic logic


The Central Problem of the Logic

“One of the popular definitions of logic is that is the analysis of methodsof reasoning.”∗

“It is easy to find answers to the question ‘What is Logic?’ According toCharles Peirce (1925) ‘Nearly a hundred definitions of it have been given’.But Pierce goes on to write: ‘It, will, however, generally conceded that itscentral problem is the classification of arguments, so that all those that arebad are thrown into one division, and those which are good intoanother...’.”†

∗Mendelson, Elliott (1997). Introduction to Mathematical Logic, p. 1.†Copi, Irving M. (1974). Symbolic Logic, p. 15.


Preliminary Definitions

Definition (Proposition or statement)A sentence that can be assigned a truth value of true or false.

RemarkThe propositions do not include questions, exclamations or commands.

Definition ((Deductive) argument)Finite set of propositions of which one (called the conclusion) is claimedto follow only from the others (called the premises), which are regards asgrounds for the truth of that one.∗

∗Copi, Irving M. (1974). Symbolic Logic.Logic - CM0845. Introduction 16/90


Definition (Proposition or statement)A sentence that can be assigned a truth value of true or false.

RemarkThe propositions do not include questions, exclamations or commands.

Definition ((Deductive) argument)Finite set of propositions of which one (called the conclusion) is claimedto follow only from the others (called the premises), which are regards asgrounds for the truth of that one.∗

∗Copi, Irving M. (1974). Symbolic Logic.Logic - CM0845. Introduction 17/90


Definition (Validity and invalidity)El lógico responde a la pregunta: ¿Se sigue la conclusión de las premisasque se han supuesto? Si afirmar la verdad de las premisas constituye unaverdadera garantía para afirmar la verdad de la conclusión entonces el ar-gumento es válido, de lo contrario es inválido.∗

RemarkNote that a proposition can be true or false, and an argument can be validor invalid.

∗Sierra A., Manuel (2010). Argumentación deductiva con diagramas y árboles deforzamiento.



Definition (Validity and invalidity)El lógico responde a la pregunta: ¿Se sigue la conclusión de las premisasque se han supuesto? Si afirmar la verdad de las premisas constituye unaverdadera garantía para afirmar la verdad de la conclusión entonces el ar-gumento es válido, de lo contrario es inválido.∗

RemarkNote that a proposition can be true or false, and an argument can be validor invalid.



Truth and Validity∗

Manuel Sierra A.

66

La verdad y la falsedad se predican de proposiciones, nunca de argumentos. Los atri-

butos de validez e invalidez pueden pertenecer solamente a los argumentos deductivos, nunca a las proposiciones. Hay una conexión entre la validez o invalidez de un argumento y la verdad o falsedad de sus premisas y de su conclusión, indicada por el condicional asocia-do al argumento. Los argumentos pueden mostrar diferentes combinaciones de verdad y falsedad de las premisas y conclusiones.

Argumentos válidos

Conclusión verdadera Conclusión falsa

Premisas verdaderas

[1] [Todos los números naturales son números enteros], [2] [todos los núme-ros enteros son números racionales]. Por lo tanto, [3] [todos los números naturales son números racionales].

Imposible

Premisas falsas

[1] [Todos los presidentes son depre-dadores], [2] [todos los depredadores son humanos]. Por lo que, [3] [Todos los presidentes son humanos].

[1] [Algunos caballos vuelan], [2] [todo el que vuela es un gran empre-sario]. Luego, [3] [algunos caballos son grandes empresarios].

Se observa que la verdad o falsedad de la conclusión de un argumento no determina por

sí misma la validez o invalidez del argumento. Y el hecho de que un argumento sea válido no garantiza la verdad de su conclusión.

Un punto de importancia fundamental: sí un argumento es válido y su conclusión es falsa, no todas sus premisas pueden ser verdaderas. Y también: Si un argumento es válido y sus premisas son verdaderas, con toda certeza la conclusión debe ser también verdadera.

Determinar la verdad o falsedad de las premisas es tarea de la ciencia en general, puesto que las premisas pueden referirse a cualquier tema. Algunos argumentos perfecta-mente válidos tienen conclusiones falsas, pero tal género de argumentos debe al menos te-ner alguna premisa falsa. Cuando un argumento es válido y todas sus premisas son verdade-ras, le llamamos sólido o bien fundado. La conclusión de un argumento sólido tiene que ser verdadera. Si un argumento deductivo no es sólido, lo cual significa o que no es válido o que no todas sus premisas son verdaderas, entonces no sirve para establecer la verdad de la conclusión. El lógico no está interesado en la verdad o falsedad de las proposiciones, sino en las relaciones lógicas entre ellas, es decir, el lógico está interesado en las relaciones que determinan la corrección o incorrección (validez o invalidez) de los argumentos en los que pueden aparecer tales proposiciones. Determinar la corrección o incorrección de los argu-mentos es una labor que corresponde enteramente a la lógica. El lógico está interesado in-cluso en la corrección de los argumentos cuyas premisas podrían ser falsas.



Truth and Validity∗

Argumentación deductiva con diagramas y árboles de forzamiento

65

bicondicional es aceptado, es decir (A ∧ B) → (A ↔ B), si los dos componentes de un bi-condicional son rechazados entonces el bicondicional es aceptado, es decir (~A ∧ ~B) → (A ↔ B), si los componentes de un bicondicional tienen diferentes valores de verdad, uno es aceptado y el otro es rechazado entonces el bicondicional es rechazado es decir (A ∧ ~B) → ~(A ↔ B) y (~A ∧ B) → ~(A ↔ B). De la tabla de verdad de la disyunción exclusiva se tiene que una disyunción exclusiva es aceptada si y solamente si un disyunto es aceptado y el otro es rechazado, es decir (A ∨ B) ↔ ((~A ∧ B) ∨ (A ∧ ~B)), si una disyunción exclusiva es aceptada y un disyunto es acep-tado entonces el otro disyunto es rechazado, es decir (A ∨ B) → (A → ~B) y (A ∨ B) → (B → ~A), si una disyunción exclusiva es aceptada y un disyunto es rechazado entonces el otro disyunto es aceptado, es decir (A ∨ B) → (~A → B) y (A ∨ B) → (~B → A), si una disyunción exclusiva es rechazada y un disyunto es aceptado entonces el otro disyunto es aceptado, es decir ~(A ∨ B) → (A → B) y ~(A ∨ B) → (B → A), si una disyunción exclu-siva es rechazada y un disyunto es rechazado entonces el otro disyunto es rechazado, es decir ~(A ∨ B) → (~A → ~B) y ~(A ∨ B) → (~B → ~A), si los dos disyuntos de una dis-yunción exclusiva son aceptados entonces la disyunción exclusiva es rechazada, es decir (A ∧ B) → ~(A ∨ B), si los dos disyuntos de una disyunción exclusiva son rechazados enton-ces la disyunción exclusiva es rechazada, es decir (~A ∧ ~B) → ~(A ∨ B), si los disyuntos de una disyunción exclusiva tienen diferentes valores de verdad, uno es aceptado y el otro es rechazado entonces la disyunción exclusiva es aceptada es decir (A ∧ ~B) → (A ∨ B) y (~A ∧ B) → (A ∨ B), una disyunción exclusiva es aceptada si y solamente si el bicondicio-nal es rechazado, es decir (A ∨ B) ↔ ~(A ↔ B), una disyunción exclusiva es aceptada si y solamente si al menos uno de los disyuntos es aceptado pero no ambos, es decir (A ∨ B) ↔ ((A ∨ B) ∧ ~(A ∧ B)).

2.5 Verdad y validez

Argumentos inválidos

Conclusión verdadera Conclusión falsa Premisas

verdaderas

[1] [Cuando el sol agote su combusti-ble entonces no irradiará calor]. [2] [el sol no agotó su combustible]. Por lo tanto, [3] [el sol irradia calor].

[1] [Cuando el sol agote su combusti-ble entonces no irradiará calor], [2] [el sol irradia calor]. Por lo tanto, [3] [él sol agotó su combustible].

Premisas falsas

[1] [Todos los presidentes son depre-dadores], [2] [todos los depredadores son humanos]. Por lo que, [3] [algunos depredadores no son presidentes].

[1] [Todos los presidentes son depre-dadores], [2] [todos los depredadores son humanos]. Por lo que, [3] [algu-nos presidentes no son humanos].



Propositional Logic: Syntax∗

∗The reference for this section is [van Dalen 2013, § 2.1].

Propositions and Logical Constants

Alphabet1 proposition symbols: 𝑝0, 𝑝1, 𝑝2, …,2 logical constants (connectives): ∧, ∨, →, ¬, ↔ and ⊥,3 auxiliary symbols: (, ),

where 𝑝𝑖 and ⊥ represent atomic propositions or atoms∗ and the logicalconstants represent∧ (and) conjunction∨ (or) (inclusive) disjunction→ (if…, then…) (material) implication¬ (not) negation↔ (iff) (material) equivalence, (material) bi-implication⊥ (falsity) bottom, falsum

∗Propositions that cannot be further decomposed in other propositions.Logic - CM0845. Propositional Logic: Syntax 23/90


Natural language versus logical languageSome examples:

John drove on and hit a pedestrian.John hit a pedestrian and drove on.

If I open the window then we’ll have fresh air.If I open the window then 1 + 3 = 4.If 1 + 2 = 4, then we’ll have fresh air.John is working or he is at home.Euclid was a Greek or a mathematician.

Logic - CM0845. Propositional Logic: Syntax 24/90



John drove on and hit a pedestrian.John hit a pedestrian and drove on.If I open the window then we’ll have fresh air.

If I open the window then 1 + 3 = 4.If 1 + 2 = 4, then we’ll have fresh air.John is working or he is at home.Euclid was a Greek or a mathematician.




John drove on and hit a pedestrian.John hit a pedestrian and drove on.If I open the window then we’ll have fresh air.If I open the window then 1 + 3 = 4.If 1 + 2 = 4, then we’ll have fresh air.

John is working or he is at home.Euclid was a Greek or a mathematician.




John drove on and hit a pedestrian.John hit a pedestrian and drove on.If I open the window then we’ll have fresh air.If I open the window then 1 + 3 = 4.If 1 + 2 = 4, then we’ll have fresh air.John is working or he is at home.Euclid was a Greek or a mathematician.


The Set of Propositions

Definition (van Dalen [2013], Definition 2.1.2)The set of propositions∗, denoted PROP, is the smallest set 𝑋 with theproperties:

1 ⊥ ∈ 𝑋,2 𝑝𝑖 ∈ 𝑋, for all 𝑖,3 𝜑, 𝜓 ∈ 𝑋 ⇒ (𝜑 ∧ 𝜓), (𝜑 ∨ 𝜓), (𝜑 → 𝜓) and (𝜑 ↔ 𝜓) ∈ 𝑋, and4 𝜑 ∈ 𝑋 ⇒ (¬𝜑) ∈ 𝑋.

ExamplesWhiteboard.

ExerciseProve that ¬¬⊥ ∉ PROP.

∗Also called the set of well-formed formulae.Logic - CM0845. Propositional Logic: Syntax 28/90




ExamplesWhiteboard.




RemarkThe definition of PROP is an inductive definition.

RemarkIn the definition of PROP, the Greek letters 𝜑 and 𝜓 are propositionalmeta-variables and the symbol ⇒ is the implication in the meta-language.


Induction Principle and Recursive Definitions

NotationThe symbol ‘□’ represents any binary connective.

Theorem (Induction principle)Let 𝐴 be a property, then 𝐴(𝜑) holds for all 𝜑 ∈ PROP if

1 𝐴(⊥),2 𝐴(𝑝𝑖), for all 𝑖,3 𝐴(𝜑) and 𝐴(𝜓) ⇒ 𝐴((𝜑 □ 𝜓)),4 𝐴(𝜑) ⇒ 𝐴((¬𝜑)).

ProofWhiteboard.






ProofWhiteboard.



Example (Proof by induction)Prove that each proposition has an even number of brackets.



Example (Recursive definition)Given 𝜑 ∈ PROP, the number 𝑏(𝜑) of brackets of 𝜑 can be defined by:

𝑏 ∶ PROP → ℕ𝑏(⊥) = 0,𝑏(𝑝𝑖) = 0, for all 𝑖

𝑏((𝜑 □ 𝜓)) = 𝑏(𝜑) + 𝑏(𝜓) + 2,𝑏((¬𝜑)) = 𝑏(𝜑) + 2.



Example (Recursive definition)The parsing tree 𝑇 (𝜑) of a proposition 𝜑 is defined by∗

2.1 Propositions and Connectives 11

Here are some examples of definition by recursion.

1. The (parsing) tree of a proposition ϕ is defined by

T (ϕ) = �ϕ for atomic ϕ

T ((ϕ�ψ)) = � (ϕ�ψ)��

��

T (ϕ) T (ψ)

T ((¬ϕ))= � (¬ϕ)

T (ϕ)

Examples

A simpler way to exhibit the trees consists of listing the atoms at the bottom, andindicating the connectives at the nodes.

∗Figure from [van Dalen 2013, p. 11].Logic - CM0845. Propositional Logic: Syntax 38/90


Example (van Dalen [2013], Exercise 8.a, p. 14)Let 𝜑 be a proposition, #(𝑇 (𝜑)) be the number of nodes of the parsingtree 𝑇 (𝜑), #𝐶(𝜑) be the number of connectives in 𝜑 and #𝐴(𝜑) benumber of atoms in 𝜑. If 𝜑 does not contain ⊥ prove that

#𝐶(𝜑) + #𝐴(𝜑) ≤ #(𝑇 (𝜑)).



Definition (Sub-formula)Given 𝜑 ∈ PROP, the set of sub-formulae of 𝜑 is given by:

𝑆𝑢𝑏 ∶ PROP → 𝑃 (PROP)𝑆𝑢𝑏(⊥) = {⊥},𝑆𝑢𝑏(𝑝𝑖) = {𝑝𝑖}, for all 𝑖

𝑆𝑢𝑏((𝜑 □ 𝜓)) = 𝑆𝑢𝑏(𝜑) ∪ 𝑆𝑢𝑏(𝜓) ∪ {(𝜑 □ 𝜓)},𝑆𝑢𝑏((¬𝜑)) = 𝑆𝑢𝑏(𝜑) ∪ {(¬𝜑)},

where 𝑃(𝑋) represents the set of subsets of 𝑋.We say that 𝜓 is a sub-formula of 𝜑 if 𝜓 ∈ 𝑆𝑢𝑏(𝜑).

ExamplesWhiteboard.


Notational Conventions

We discard the outermost brackets and we discard brackets in thecase of negations.Precedence of the logical connectives: ¬ binds more strongly than theother connectives, and ∧ and ∨ bind more strongly than → and ↔.

ExamplesWhiteboard.


Propositional Logic: Some Properties∗

∗The reference for this section is [van Dalen 2013, § 2.3].

Finite Conjunctions and Disjunctions

Definitions

⋀𝑖≤0

𝜑𝑖 = 𝜑0 ⋁𝑖≤0

𝜑𝑖 = 𝜑0

⋀𝑖≤𝑛+1

𝜑𝑖 = ⋀𝑖≤𝑛

𝜑𝑖 ∧ 𝜑𝑛+1 ⋁𝑖≤𝑛+1

𝜑𝑖 = ⋁𝑖≤𝑛

𝜑𝑖 ∨ 𝜑𝑛+1

Logic - CM0845. Propositional Logic: Some Properties 44/90

Propositional Logic: Satisfiability∗

∗The reference for this section is [Ben-Ari 2012, § 2.5].

Satisfiability, Validity, Unsatisfiability and Falsifiability

Let 𝜑 ∈ PROP.

Definitions𝜑 is satisfiable iff ⟦𝜑⟧𝑣 = 1 for some interpretation 𝑣.In this case, 𝑣 is called a model for 𝜑.𝜑 is valid (a tautology), denoted ⊧ 𝜑, iff ⟦𝜑⟧𝑣 = 1 for allinterpretations 𝑣.𝜑 is unsatisfiable iff it is not satisfiable, that is, if ⟦𝜑⟧𝑣 = 0 for allinterpretations 𝑣.𝜑 is falsifiable, denoted ⊮ 𝜑, iff it is not valid, that is, if ⟦𝜑⟧𝑣 = 0 forsome interpretation 𝑣.

Logic - CM0845. Propositional Logic: Satisfiability 46/90

Satisfiability, Validity, Unsatisfiability and Falsifiability∗

30 2 Propositional Logic: Formulas, Models, Tableaux

Fig. 2.6 Satisfiability and validity of formulas

Proof Let I be an arbitrary interpretation. vI (A)= T if and only if vI (¬A)= F

by the definition of the truth value of a negation. Since I was arbitrary, A is true inall interpretations if and only if ¬A is false in all interpretations, that is, iff ¬A isunsatisfiable.

If A is satisfiable then for some interpretation I , vI (A) = T . By definition ofthe truth value of a negation, vI (¬A)= F so that ¬A is falsifiable. Conversely, ifvI (¬A)= F then vI (A)= T .

2.5.1 Decision Procedures in Propositional Logic

Definition 2.40 Let U ⊆F be a set of formulas. An algorithm is a decision pro-cedure for U if given an arbitrary formula A ∈F , it terminates and returns theanswer yes if A ∈U and the answer no if A �∈U .

If U is the set of satisfiable formulas, a decision procedure for U is called adecision procedure for satisfiability, and similarly for validity.

By Theorem 2.39, a decision procedure for satisfiability can be used as a decisionprocedure for validity. To decide if A is valid, apply the decision procedure forsatisfiability to ¬A. If it reports that ¬A is satisfiable, then A is not valid; if itreports that ¬A is not satisfiable, then A is valid. Such an decision procedure iscalled a refutation procedure, because we prove the validity of a formula by refutingits negation. Refutation procedures can be efficient algorithms for deciding validity,because instead of checking that the formula is always true, we need only search fora falsifying counterexample.

The existence of a decision procedure for satisfiability in propositional logic istrivial, because we can build a truth table for any formula. The truth table in Ex-ample 2.21 shows that p→ q is satisfiable, but not valid; Example 2.22 shows that(p → q)↔ (¬q →¬p) is valid. The following example shows an unsatisfiableformula.

∗Figure 2.6 of [Ben-Ari 2012].Logic - CM0845. Propositional Logic: Satisfiability 47/90

Satisfiability, Validity, Unsatisfiability and Falsifiability

Theorem (Ben-Ari [2012], Theorem 2.39)Let 𝜑 ∈ PROP.

The proposition 𝜑 is valid if and only if ¬𝜑 is unsatisfiable.The proposition 𝜑 is satisfiable if and only if ¬𝜑 is falsifiable.


Satisfiability of a Set of Propositions

Let Γ = {𝜑1, … } be a set of propositions.

DefinitionsΓ is satisfiable iff there exists an interpretation 𝑣 such that ⟦𝜑⟧𝑣 = 1for all 𝜑𝑖 ∈ Γ. In this case, 𝑣 is a model of Γ.Γ is unsatisfiable iff for every interpretation 𝑣, there exists an 𝜑𝑖 ∈ Γsuch that ⟦𝜑⟧𝑣 = 0.

Example (Ben-Ari [2012], Exercise 2.15, p. 46)Prove that if Γ is unsatisfiable and for some 𝑖, the proposition 𝜑𝑖 is valid,then Γ − {𝜑𝑖} is unsatisfiable.


Propositional Logic: Semantic Tableaux∗

∗The reference for this section is [Ben-Ari 2012, § 2.6].

Semantics Tableaux

Theorem (Ben-Ari [2012], Theorem 2.67)Let 𝜑 be a proposition and 𝔗 be a completed tableau for 𝜑. The proposi-tion 𝜑 is unsatisfiable if and only if 𝔗 is closed.

Example (Ben-Ari [2012], Exercise 2.9, p. 46)Prove that ⊧ (𝜑 ∧ 𝜓 → 𝜎) → ((𝜑 → 𝜎) ∨ (𝜓 → 𝜎)) using semantictableaux.

Logic - CM0845. Propositional Logic: Semantic Tableaux 52/90

Semantics Tableaux

Theorem (Ben-Ari [2012], Theorem 2.67)Let 𝜑 be a proposition and 𝔗 be a completed tableau for 𝜑. The proposi-tion 𝜑 is unsatisfiable if and only if 𝔗 is closed.

Example (Ben-Ari [2012], Exercise 2.9, p. 46)Prove that ⊧ (𝜑 ∧ 𝜓 → 𝜎) → ((𝜑 → 𝜎) ∨ (𝜓 → 𝜎)) using semantictableaux.

Logic - CM0845. Propositional Logic: Semantic Tableaux 53/90

Propositional Logic: Natural Deduction∗

∗The reference for this section is [van Dalen 2013, § 2.4 and § 2.6].

Derivations Rules for {∧, →, ⊥}

𝜑 𝜓 ∧I𝜑 ∧ 𝜓𝜑 ∧ 𝜓 ∧E𝜑

𝜑 ∧ 𝜓 ∧E𝜓

[𝜑]⋮𝜓 →I𝜑 → 𝜓

𝜑 𝜑 → 𝜓 →E𝜓

⊥ ⊥E𝜑

[¬𝜑]⋮⊥ RAA𝜑

where ¬𝜑 ∶= 𝜑 → ⊥.

Logic - CM0845. Propositional Logic: Natural Deduction 55/90

Derivations Rules for {∧, →, ⊥}

Example (van Dalen [2013], Exercise 3.(a), p. 37)Prove that 𝜑 ⊢ ¬(¬𝜑 ∧ 𝜓).

[𝜑][¬𝜑 ∧ 𝜓]1

∧E¬𝜑→E

⊥ →I1¬(¬𝜑 ∧ 𝜓)


Set of Derivations

Notation(Whiteboard)

Definition (van Dalen [2013], Definition 2.4.1)The set 𝒟 of derivations is the smallest set 𝑋 with the properties:(see next slide)


Set of Derivations

34 2 Propositional Logic

of the meaning of negation, we only would get ¬¬ϕ. It is by no means clear that¬¬ϕ is equivalent to ϕ (indeed, this is denied by the intuitionists), so it is an extraproperty of our logic. (This is confirmed in a technical sense: ¬¬ϕ → ϕ is notderivable in the system without RAA.)

We now return to our theoretical notions.

Definition 2.4.1 The set of derivations is the smallest set X such that

(1) The one-element tree ϕ belongs to X for all ϕ ∈ PROP.

(2∧) If Dϕ

, D′ϕ′ ∈X, then

Dϕ

D′ϕ′

ϕ∧ϕ′∈X.

If Dϕ∧ψ

∈X, thenD

ϕ∧ψ

ϕ

,D

ϕ∧ψ

ψ

∈X.

(2→) IfϕDψ

∈X, then[ϕ]Dψ

ϕ→ψ

∈X.

If Dϕ

, D′ϕ→ψ

∈X, thenDϕ

D′ϕ→ψ

ψ

∈X.

(2⊥) If D⊥ ∈X, then

D⊥ϕ

∈X.

If¬ϕD⊥∈X, then

[¬ϕ]D⊥ϕ

∈X.

The bottom formula of a derivation is called its conclusion. Since the class ofderivations is inductively defined, we can mimic the results of Sect. 2.1.

For example, we have a principle of induction on D: let A be a property. If A(D)

holds for one-element derivations and A is preserved under the clauses (2∧), (2→)

and (2⊥), then A(D) holds for all derivations. Likewise we can define mappings onthe set of derivations by recursion (cf. Exercises 6, 7, 9).

Definition 2.4.2 The relation Γ � ϕ between sets of propositions and propositionsis defined as follows: there is a derivation with conclusion ϕ and with all (un-canceled) hypotheses in Γ . (See also Exercise 6.)

We say that ϕ is derivable from Γ . Note that by definition Γ may contain manysuperfluous “hypotheses”. The symbol � is called the turnstile.

If Γ = ∅, we write � ϕ, and we say that ϕ is a theorem.We could have avoided the notion of “derivation” and taken instead the notion

of “derivability” as fundamental, see Exercise 10. The two notions, however, areclosely related.


Derivations Rules for {∨, ¬, ↔}

𝜑 ∨I𝜑 ∨ 𝜓𝜓 ∨I𝜑 ∨ 𝜓 𝜑 ∨ 𝜓

[𝜑]⋮𝜎

[𝜓]⋮𝜎 ∨E𝜎

[𝜑]⋮

⊥ ¬I¬𝜑

𝜑 ¬𝜑 ¬E⊥

[𝜑]⋮𝜓

[𝜓]⋮𝜑 ↔I𝜑 ↔ 𝜓

𝜑 𝜑 ↔ 𝜓 ↔E𝜓𝜓 𝜑 ↔ 𝜓 ↔E𝜑


Derivations Rules for {∧, ∨, →, ⊥}𝜑 𝜓 ∧I𝜑 ∧ 𝜓

𝜑 ∧ 𝜓 ∧E𝜑𝜑 ∧ 𝜓 ∧E𝜓

𝜑 ∨I𝜑 ∨ 𝜓𝜓 ∨I𝜑 ∨ 𝜓 𝜑 ∨ 𝜓

[𝜑]⋮𝜎

[𝜓]⋮𝜎 ∨E𝜎

[𝜑]⋮𝜓 →I𝜑 → 𝜓

𝜑 𝜑 → 𝜓 →E𝜓

⊥ ⊥E𝜑

[¬𝜑]⋮⊥ RAA𝜑


Derivations Rules for {∧, ∨, →, ⊥}

Example (van Dalen [2013], example p. 49)Prove that ⊢ 𝜑 ∨ ¬𝜑.

[𝜑]1∨I𝜑 ∨ ¬𝜑 [¬(𝜑 ∨ ¬𝜑)]2

→E⊥ →I1¬𝜑

∨I𝜑 ∨ ¬𝜑 [¬(𝜑 ∨ ¬𝜑)]2→E

⊥ RAA2𝜑 ∨ ¬𝜑


Derivations Rules for {∧, ∨, →, ⊥}: Alternative Notation∗

Γ, 𝜑 ⊢ 𝜑 (Ax)

Γ ⊢ 𝜑 Γ ⊢ 𝜓 ∧IΓ ⊢ 𝜑 ∧ 𝜓Γ ⊢ 𝜑 ∧ 𝜓 ∧EΓ ⊢ 𝜑

Γ ⊢ 𝜑 ∧ 𝜓 ∧EΓ ⊢ 𝜓

Γ ⊢ 𝜑 ∨IΓ ⊢ 𝜑 ∨ 𝜓Γ ⊢ 𝜓 ∨IΓ ⊢ 𝜑 ∨ 𝜓

Γ ⊢ 𝜑 ∨ 𝜓 Γ, 𝜑 ⊢ 𝜎 Γ, 𝜓 ⊢ 𝜎 ∨EΓ ⊢ 𝜎

Γ, 𝜑 ⊢ 𝜓 →IΓ ⊢ 𝜑 → 𝜓Γ ⊢ 𝜑 Γ ⊢ 𝜑 → 𝜓 →EΓ ⊢ 𝜓

Γ ⊢ ⊥ ⊥EΓ ⊢ 𝜑Γ, ¬𝜑 ⊢ ⊥

RAAΓ ⊢ 𝜑

∗[Barendregt 1992, p. 36].Logic - CM0845. Propositional Logic: Natural Deduction 62/90

First-Order Logic: Syntax∗

∗The reference for this section is [van Dalen 2013, § 3.1, § 3.2 and § 3.3].

Introduction

Examples (van Dalen [2013], informal examples p. 53)∃𝑥𝑃 (𝑥) there is an 𝑥 with property 𝑃

∀𝑦𝑃(𝑦) for all 𝑦 𝑃 holds (all 𝑦 have the property 𝑃 )

∀𝑥∃𝑦(𝑥 = 2𝑦) for all 𝑥 there is a 𝑦 such that 𝑥 is two times 𝑦

∀𝜀(𝜀 > 0 → ∃𝑛(𝑛 < 𝜀)) for all positive there is an 𝑛 such that 𝑛 < 𝜀

𝑥 < 𝑦 → ∃𝑧(𝑥 < 𝑧 ∧ 𝑧 < 𝑦) if 𝑥 < 𝑦, then there is a 𝑧 such that 𝑥 < 𝑧and 𝑧 < 𝑦

∀𝑥∃𝑦(𝑥.𝑦 = 1) for each 𝑥 there exists an inverse 𝑦.

Logic - CM0845. First-Order Logic: Syntax 64/90

Structures

DefinitionA structure is an ordered sequence

⟨𝐴, 𝑅1, … , 𝑅𝑛, 𝐹1, … , 𝐹𝑚, {𝑐𝑖 ∣ 𝑖 ∈ 𝐼}⟩,

where𝐴 is a non-empty set (the universe of the structure),𝑅1, … , 𝑅𝑛 are relations on 𝐴,𝐹1, … , 𝐹𝑚 are functions on 𝐴, andthe 𝑐𝑖, where 𝑖 ∈ 𝐼 , are elements of 𝐴 (constants).

Notation• Structures are denoted by Gothic capitals: 𝔄, 𝔅, ℭ, …• 𝐴 = |𝔄|


Structures

ExamplesWhiteboard.


Structures

DefinitionThe similarity type (or signature or non-logical constants) of a structure

⟨𝐴, 𝑅1, … , 𝑅𝑛, 𝐹1, … , 𝐹𝑚, {𝑐𝑖 ∣ 𝑖 ∈ 𝐼}⟩,

is a sequence,

⟨𝑟1, … , 𝑟𝑛; 𝑎1, … , 𝑎𝑚; 𝜅⟩,

where𝑅𝑖 ⊆ 𝐴𝑟𝑖 ,𝐹𝑗 ∶ 𝐴𝑎𝑗 → 𝐴, and𝜅 = |{𝑐𝑖 ∣ 𝑖 ∈ 𝐼}| (cardinality of 𝐼).


Structures

ExamplesWhiteboard.

Limiting cases0-ary relations and 0-ary functions.

ConventionAll the structures are equipped implicitly with the identity relation.


Structures

ExamplesWhiteboard.




Alphabet

Alphabet1 Predicate symbols: 𝑃1, … , 𝑃𝑛, ≐2 Function symbols: 𝑓1, … , 𝑓𝑚3 Constant symbols: 𝑐𝑖 for 𝑖 ∈ 𝐼4 Variables: 𝑥0, 𝑥1, 𝑥2, … (countably many)5 Connectives: ∨, ∧, →, ¬, ↔, ⊥, ∀, ∃6 Auxiliary symbols: (, )

RemarkThe equality symbol.


The Set of Terms

DefinitionThe set of terms, denoted TERM, is the smallest set 𝑋 with the properties:

1 𝑥𝑖 ∈ 𝑋, where 𝑖 ∈ ℕ,2 𝑐𝑖 ∈ 𝑋, where 𝑖 ∈ 𝐼 , and3 𝑡1, … , 𝑡𝑎𝑖

∈ 𝑋 ⇒ 𝑓𝑖(𝑡1, … , 𝑡𝑎𝑖) ∈ 𝑋, for 1 ≤ 𝑖 ≤ 𝑚.

ExamplesWhiteboard.


The Set of Formulae

DefinitionThe set of formulae, denoted FORM, is the smallest set 𝑋 with the prop-erties:

1 ⊥ ∈ 𝑋,2 𝑃𝑖 ∈ 𝑋 if 𝑟𝑖 = 0,3 𝑡1, … , 𝑡𝑟𝑖

∈ TERM ⇒ 𝑃𝑖(𝑡1, … , 𝑡𝑟𝑖) ∈ 𝑋,

4 𝑡1, 𝑡2 ∈ TERM ⇒ 𝑡1 ≐ 𝑡2 ∈ 𝑋,5 𝜑, 𝜓 ∈ 𝑋 ⇒ (𝜑 □ 𝜓) ∈ 𝑋, where □ ∈ {∧, ∨, →, ↔},6 𝜑 ∈ 𝑋 ⇒ (¬𝜑) ∈ 𝑋,7 𝜑 ∈ 𝑋 ⇒ ((∀𝑥𝑖)𝜑) ∈ 𝑋 and8 𝜑 ∈ 𝑋 ⇒ ((∃𝑥𝑖)𝜑) ∈ 𝑋.

The formulae defined in the four first items are called atomic formulae oratoms.



We use the conventions of propositional logic.We delete the outer brackets and the brackets round ∀𝑥 and ∃𝑥whenever possible.Quantifiers bind more strongly than binary connectives.Join strings of quantifiers, e.g. ∀𝑥1𝑥2∃𝑥3𝑥4𝜑 stands for∀𝑥1∀𝑥2∃𝑥3∃𝑥4𝜑.

ExamplesWhiteboard.


Induction Principles and Recursive Definitions

RemarkGiven that TERM and FORM are set inductively defined, we have inductionprinciples and recursive definition on them.


Set of Free Variables of a Term

DefinitionThe set of free variables of a term 𝑡, denoted 𝐹𝑉 (𝑡), is defined by

𝐹𝑉 ∶ TERM → {𝑥𝑖 ∣ 𝑖 ∈ ℕ}𝐹𝑉 (𝑥𝑖) = {𝑥𝑖},𝐹𝑉 (𝑐𝑖) = ∅,

𝐹𝑉 (𝑓(𝑡1, … , 𝑡𝑛)) = 𝐹𝑉 (𝑡1) ∪ ⋯ ∪ 𝐹𝑉 (𝑡𝑛).


Closed Terms

DefinitionA term 𝑡 is called closed if 𝐹𝑉 (𝑡) = ∅. The set of closed terms is denotedby TERMc.

ExamplesWhiteboard.


Set of Free Variables of a Formula

DefinitionThe set of free variables of a formula 𝜑, denoted 𝐹𝑉 (𝜑), is defined by

𝐹𝑉 ∶ FORM → {𝑥𝑖 ∣ 𝑖 ∈ ℕ}𝐹𝑉 (⊥) = ∅𝐹𝑉 (𝑃 ) = ∅, for 𝑃 propositional symbol

𝐹𝑉 (𝑃(𝑡1, … , 𝑡𝑛)) = 𝐹𝑉 (𝑡1) ∪ ⋯ ∪ 𝐹𝑉 (𝑡𝑛),𝐹𝑉 (𝑡1 ≐ 𝑡2) = 𝐹𝑉 (𝑡1) ∪ 𝐹𝑉 (𝑡2),

𝐹𝑉 (𝜑 □ 𝜓) = 𝐹𝑉 (𝜑) ∪ 𝐹𝑉 (𝜓),𝐹𝑉 (¬𝜑) = 𝐹𝑉 (𝜑),

𝐹𝑉 (∀𝑥𝑖𝜑) = 𝐹𝑉 (∃𝑥𝑖𝜑) = 𝐹𝑉 (𝜑) − {𝑥𝑖}.


Sentences

DefinitionA formula 𝜑 is called closed if 𝐹𝑉 (𝜑) = ∅. A closed formula is also calleda sentence. The set of sentences is denoted by SENT.

ExamplesWhiteboard.


Free Terms for a Variable in in Formula

DefinitionA term 𝑡 is free for a variable 𝑥 in a formula 𝜑 if

1 𝜑 is atomic,2 𝜑 ∶= ¬𝜓 and 𝑡 is free for 𝑥 in 𝜓,3 𝜑 ∶= 𝜑1 □ 𝜑2 and 𝑡 is free for 𝑥 in 𝜑1 and 𝜑2,4 𝜑 ∶= ∀𝑦𝜓 and if 𝑥 ∈ 𝐹𝑉 (𝜑), then 𝑦 ∉ 𝐹𝑉 (𝑡) and 𝑡 is free for 𝑥

in 𝜓, or5 𝜑 ∶= ∃𝑦𝜓 and if 𝑥 ∈ 𝐹𝑉 (𝜑), then 𝑦 ∉ 𝐹𝑉 (𝑡) and 𝑡 is free for 𝑥 in 𝜓.


The Extended Language

DefinitionThe extended language, 𝐿(𝔄), of 𝔄 is obtained from the language 𝐿, of thetype of 𝔄, by adding constant symbols for all elements of |𝔄|. We denotethe constant symbol, belonging to 𝑎 ∈ |𝔄|, by 𝑎.

ExamplesWhiteboard.


First-Order Logic: Natural Deduction∗

∗The reference for this section is [van Dalen 2013, § 3.8 and § 3.9].

Derivations

Example (van Dalen [2013], p. 92)Prove that ⊢ ∃𝑥(𝜑(𝑥) ∨ 𝜓(𝑥)) → ∃𝑥𝜑(𝑥) ∨ ∃𝑥𝜓(𝑥).

[∃𝑥(𝜑(𝑥) ∨ 𝜓(𝑥))]3[𝜑(𝑥) ∨ 𝜓(𝑥)]2

[𝜑(𝑥)]1∃I

∃𝑥𝜑(𝑥)∨I

∃𝑥𝜑(𝑥) ∨ ∃𝑥𝜓(𝑥)

[𝜓(𝑥)]1∃I

∃𝑥𝜓(𝑥)∨I

∃𝑥𝜑(𝑥) ∨ ∃𝑥𝜓(𝑥)∨E1∃𝑥𝜑(𝑥) ∨ ∃𝑥𝜓(𝑥)

∃E2∃𝑥𝜑(𝑥) ∨ ∃𝑥𝜓(𝑥)→I3∃𝑥(𝜑(𝑥) ∨ 𝜓(𝑥)) → ∃𝑥𝜑(𝑥) ∨ ∃𝑥𝜓(𝑥)

Logic - CM0845. First-Order Logic: Natural Deduction 87/90

References

References

Barendregt, Henk (1992). Lambda Calculi with Types. In: Handbook of Logic inComputer Science. Ed. by S. Abramsky, Dov M. Gabbay andT. S. E. Maibaum. Vol. 2. Clarendon Press, pp. 117–309.

Ben-Ari, Mordechai (2012). Mathematical Logic for Computer Science. 3rd ed.Springer.

Brueggemann, Tobias and Kern, Walter (2004). An Improved Deterministic LocalSearch Algorithm for 3-SAT. Theoretical Computer Science 329.1–3,pp. 303–313. doi: 10.1016/j.tcs.2004.08.002.

Copi, Irving M. (1974). Symbolic Logic. 4th ed. Macmillan Publishing Co.Karp, Richard M. (1972). Reducibility Among Combinatorial Problems. In:

Complexity of Computer Computations. Ed. by Raymond E. Miller andJames W. Thatcher. Plenum Press, pp. 85–103.

Kutzkov, Konstantin and Scheder, Dominik (2010). Using CSP to ImproveDeterministic 3-SAT. CoRR abs/1007.1166.

Mendelson, Elliott (1997). Introduction to Mathematical Logic. 4th ed. Chapman& Hall.

Logic - CM0845. References 89/90

https://doi.org/10.1016/j.tcs.2004.08.002

ReferencesMoser, Robin A. and Scheder, Dominik (2011). A Full Derandomization of

Schöning’s 𝑘-SAT Algorithm. In: Proceedings of the Forty-third Annual ACMSymposium on Theory of Computing (STOC 2011), pp. 245–252.

Scheder, Dominik (2008). Guided Search and a Faster Deterministic Algorithm for3-SAT. In: Proc. of the 8th Latin American Symposium on TheoreticalInformatic (LATIN 2008). Ed. by Eduardo Sany Laber et al. Vol. 4957. LectureNotes in Computer Science. Springer, pp. 60–71.

Sierra A., Manuel (2010). Argumentación deductiva con diagramas y árboles deforzamiento. Fondo Editorial Universidad EAFIT.

van Dalen, Dirk (2013). Logic and Structure. 5th ed. Springer.

Logic - CM0845. References 90/90

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