DOCSLIB.ORG

AI Principles, Semester 2, Week 3, Lecture 6 Inference Rules For

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
AI Principles, Semester 2, Week 3, Lecture 6 Inference Rules For AI Principles, Semester 2, Week 3, Lecture 6 Inference rules for Propositional Logic Inference rules, validity and tautologies in Truth Tables Lots of example proofs Equivalences in Propositional Logic Limitations of Propositional Logic First Order Predicate Logic Predicates and quantification Equivalences in Predicate Logic English to Predicate Logic 1 Review of logic from last lecture Propositional logic Logical connectives Rules for wffs Formally representing English sentences Truth tables for the connectives Using Truth Tables to evaluate expressions Tautologies Entailment Soundness and completeness of inference rules 2 Inference rules should be sound, and even better complete We saw from last lecture that we want our inference rules to always produce entailed sentences This means that all inference rules should always be sound If an inference system can produce all entailed sentences then we say it is complete 3 Inference rules can be used manually or automated Doing manual inference you are free to use a large number of different inference rules Automated reasoning can be done by the automated computation of Truth Tables or by using inference rules. However, the automated use of inference rules has to be done differently to how humans use inference rules Automated reasoning and other forms of logic next lecture 4 Modus Ponens (MP) A →B A therefore B Conjunction Introduction (∧ I) A B therefore A ∧ B Conjunction Elimination (∧ E) A ∧ BA ∧ B therefore A and therefore B 5 Double Negation (DN) ¬¬P therefore P Disjunction Introduction (∨ I) AB therefore A ∨ B and therefore A ∨ B Disjunction Elimination (∨ E) A → C B → C A ∨ B therefore C 6 Valid Inferences and tautologies P →Q P therefore Q can be seen to be a valid inference because the conditional statement ((P →Q) ∧ P) →Q is a tautology 1 2 3 4 5 P Q (P→Q) (P→Q)∧ P ((P→Q) ∧ P) →Q) T T T T T T F F F T F T T F T F F T F T 7 Example proof 1 Given: 1 A → B 2 B→ C 3 A prove: C 8 Example proof 1 part 2 Given: 1 A → B 2 B → C 3 A 4 B (Modus Ponens on 1 and 3) 9 Example proof 1 part 3 Given: 1 A → B 2 B → C 3 A 4 B (Modus Ponens on 1 and 3) 5 C (Modus Ponens on 2 and 4) 10 Example proof 2 Prove C Given: 1 (A ∧ D) → B 2 A 3 D 4 B → C 11 Example proof 2 part 2 Prove C Given: 1 (A ∧ D) → B 2 A 3 D 4 B → C 5 (A ∧ D) ( ∧ I on 2 and 3) 12 Example proof 2 part 3 Prove C Given: 1 (A ∧ D) → B 2 A 3 D 4 B → C 5 (A ∧ D) ( ∧ I on 2 and 3) 6 B (MP on 1 and 5) 13 Example proof 2 part 4 Prove C Given: 1 (A ∧ D) → B 2 A 3 D 4 B → C 5 (A ∧ D) ( ∧ I on 2 and 3) 6 B (MP on 1 and 5) 7 C (MP on 4 and 6) 14 Example proof 3 Prove M Given: 1 M ∧ N 15 Example proof 4 Prove J Given: 1 ¬¬K 2 L 3 (K ∧ L) → M 4 M → J 16 Example proof 5 Prove G Given: 1 E → G 2 F → G 3 E ∨ F 17 Example proof 6 Prove G Given: 1 (E → G) ∧ (F → G) ∧ (E ∨ F) 18 Equivalences in Propositional Logic ¬¬ A ≡ A ¬(A ∨B)≡¬ A∧¬B ¬(A ∧B)≡¬ A∨¬B De Morgan’s Laws pages 24, 25, 26 in Callan, many more equivalences that you should learn 19 Limitations of Propositional Logic Formalise the following English sentences: Al is small Ted is small Someone is small Everyone is small No-one is not small Propositional Logic would represent each of these as a different Proposition, so the five propositions might be represented by P.Q, R, S and T What this representation is missing is the similarity between the propositions, they are all concerned with the relation ’small’ Predicate logic allows relations and quantification (which allows the representation of English descriptors like someone, everyone and no- one) 20 First Order Predicate Logic (FOPL) Sentence letters, connectives and parentheses are adopted from Propositional Logic In addition, FOPL allows predicate symbols to be used to represent claims about objects and relations between objects. smaller(Birmingham, London) Birmingham is smaller than London ¬country(Birmingham) Birmingham is not a country man(Charlton Heston) Charlton Heston is a man In general: predicate-name(a1, a2, ...) (The number of arguments in this atomic sentence (that is the number of an, terms) is the arity of the predicate) 21 Examples of predicates without quantifiers Formalise the following in FOPL: Kipper is not small Jo and Flo share a room Bob, Rob, Sam, Pam and Tom are in the same team Birmingham is west of London and Bristol is west of Birmingham 22 Quantifiers There are two kinds of quantification in FOPL The universal quantifier stands for all or everything and is represented by the symbol ∀ The existential quantifier stands for there exists or at least one and is represented by the symbol ∃ 23 Quantifiers, variables and constants Every student likes logic ∀ X.(student(X)→likes(X, logic) Every loyal subject likes Elizabeth Windsor ∀ X.(loyal-subject(X)→likes(X, Elizabeth-Windsor) 24 wffs An important aspect of forming wffs in Predicate Logic is using the correct arity for predicates the predicate large might have arity = 1, whereas the predicate bigger-than might have arity two large(elephant) bigger-than(elephant, mouse) more details in page 39 and 40 in Callan 25 Functions Certain kinds of relationships are best considered as functions. This occurs when a given object, (such as a human body or a car or some other artifact), must be related to exactly one other object, (such as, respectively, a leg (of the body), or a wheel (of the car) or some other part of the artifact in question) Russel and Norvig 245, 246 No questions on functions in class test 26 Scoping Scoping of variables is important when variables are replaced with other variables or terms. Only free occurences of variables can be replaced In the expresssions ∀ X.Φ and ∃ X.Φ the expression Φ is said to be the scope of the quantifier more details in page 40 and 41 in Callan 27 English to Predicate Logic Translate the following sentences into Predicate Logic: All students are human Some students are aliens No students are children 28 English to Predicate Logic Translate the following sentences into Predicate Logic: All students are human ∀ x(Sx → Hx) Some students are aliens ∃ x(Sx ∧ Ax) No students are children ∀ x(Sx → ¬ Cx) ¬∃ x(Sx ∧ Cx) 29 English to Predicate Logic Translate the following sentences into Predicate Logic: Children behave badly if not supervised Children behave badly only if not supervised No children behave badly if supervised 30 English to Predicate Logic Translate the following sentences into Predicate Logic: Children behave badly if not supervised ∀ x(Cx → (¬ Sx → Bx)) Children behave badly only if not supervised ∀ x(Cx → ( Bx → ¬ Sx)) No children behave badly if supervised ¬∃ x(Cx ∧ Bx ∧ Sx) 31 Equivalences in Predicate Logic De Morgan’s Rules ∀x¬P ≡ ¬∃xP ¬∀xP ≡ ∃x¬P ∀xP ≡ ¬∃x¬P ∃xP ≡ ¬∀x¬P page 38 Callan 32 Conclusion Inference rules for Propositional Logic Inference rules, validity and tautologies in Truth Tables Lots of example proofs Equivalences in Propositional Logic Limitations of Propositional Logic First Order Predicate Logic Predicates and quantification Equivalences in Predicate Logic English to Predicate Logic 33.
Load more

Recommended publications
  • Chapter 5: Methods of Proof for Boolean Logic
    Chapter 5: Methods of Proof for Boolean Logic § 5.1 Valid inference steps Conjunction elimination Sometimes called simplification. From a conjunction, infer any of the conjuncts. • From P ∧ Q, infer P (or infer Q). Conjunction introduction Sometimes called conjunction. From a pair of sentences, infer their conjunction. • From P and Q, infer P ∧ Q. § 5.2 Proof by cases This is another valid inference step (it will form the rule of disjunction elimination in our formal deductive system and in Fitch), but it is also a powerful proof strategy. In a proof by cases, one begins with a disjunction (as a premise, or as an intermediate conclusion already proved). One then shows that a certain consequence may be deduced from each of the disjuncts taken separately. One concludes that that same sentence is a consequence of the entire disjunction. • From P ∨ Q, and from the fact that S follows from P and S also follows from Q, infer S. The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from it. You repeat this process for each disjunct. So it doesn’t matter which disjunct is true—you get the same conclusion in any case. Hence you may infer that it follows from the entire disjunction. In practice, this method of proof requires the use of “subproofs”—we will take these up in the next chapter when we look at formal proofs.
    [Show full text]
  • Notes on Proof Theory
    Notes on Proof Theory Master 1 “Informatique”, Univ. Paris 13 Master 2 “Logique Mathématique et Fondements de l’Informatique”, Univ. Paris 7 Damiano Mazza November 2016 1Last edit: March 29, 2021 Contents 1 Propositional Classical Logic 5 1.1 Formulas and truth semantics . 5 1.2 Atomic negation . 8 2 Sequent Calculus 10 2.1 Two-sided formulation . 10 2.2 One-sided formulation . 13 3 First-order Quantification 16 3.1 Formulas and truth semantics . 16 3.2 Sequent calculus . 19 3.3 Ultrafilters . 21 4 Completeness 24 4.1 Exhaustive search . 25 4.2 The completeness proof . 30 5 Undecidability and Incompleteness 33 5.1 Informal computability . 33 5.2 Incompleteness: a road map . 35 5.3 Logical theories . 38 5.4 Arithmetical theories . 40 5.5 The incompleteness theorems . 44 6 Cut Elimination 47 7 Intuitionistic Logic 53 7.1 Sequent calculus . 55 7.2 The relationship between intuitionistic and classical logic . 60 7.3 Minimal logic . 65 8 Natural Deduction 67 8.1 Sequent presentation . 68 8.2 Natural deduction and sequent calculus . 70 8.3 Proof tree presentation . 73 8.3.1 Minimal natural deduction . 73 8.3.2 Intuitionistic natural deduction . 75 1 8.3.3 Classical natural deduction . 75 8.4 Normalization (cut-elimination in natural deduction) . 76 9 The Curry-Howard Correspondence 80 9.1 The simply typed l-calculus . 80 9.2 Product and sum types . 81 10 System F 83 10.1 Intuitionistic second-order propositional logic . 83 10.2 Polymorphic types . 84 10.3 Programming in system F ...................... 85 10.3.1 Free structures .
    [Show full text]
  • Relevant and Substructural Logics
    Relevant and Substructural Logics GREG RESTALL∗ PHILOSOPHY DEPARTMENT, MACQUARIE UNIVERSITY [email protected] June 23, 2001 http://www.phil.mq.edu.au/staff/grestall/ Abstract: This is a history of relevant and substructural logics, written for the Hand- book of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods.1 1 Introduction Logics tend to be viewed of in one of two ways — with an eye to proofs, or with an eye to models.2 Relevant and substructural logics are no different: you can focus on notions of proof, inference rules and structural features of deduction in these logics, or you can focus on interpretations of the language in other structures. This essay is structured around the bifurcation between proofs and mod- els: The first section discusses Proof Theory of relevant and substructural log- ics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, Dunn's algebraic models [76, 77] Urquhart's operational semantics [267, 268] and Routley and Meyer's rela- tional semantics [239, 240, 241] arrived decades after the initial burst of ac- tivity from Alan Anderson and Nuel Belnap. The same goes for work on the Lambek calculus: although inspired by a very particular application in lin- guistic typing, it was developed first proof-theoretically, and only later did model theory come to the fore.
    [Show full text]
  • Qvp) P :: ~~Pp :: (Pvp) ~(P → Q
    TEN BASIC RULES OF INFERENCE Negation Introduction (~I – indirect proof IP) Disjunction Introduction (vI – addition ADD) Assume p p Get q & ~q ˫ p v q ˫ ~p Disjunction Elimination (vE – version of CD) Negation Elimination (~E – version of DN) p v q ~~p → p p → r Conditional Introduction (→I – conditional proof CP) q → r Assume p ˫ r Get q Biconditional Introduction (↔I – version of ME) ˫ p → q p → q Conditional Elimination (→E – modus ponens MP) q → p p → q ˫ p ↔ q p Biconditional Elimination (↔E – version of ME) ˫ q p ↔ q Conjunction Introduction (&I – conjunction CONJ) ˫ p → q p or q ˫ q → p ˫ p & q Conjunction Elimination (&E – simplification SIMP) p & q ˫ p IMPORTANT DERIVED RULES OF INFERENCE Modus Tollens (MT) Constructive Dilemma (CD) p → q p v q ~q p → r ˫ ~P q → s Hypothetical Syllogism (HS) ˫ r v s p → q Repeat (RE) q → r p ˫ p → r ˫ p Disjunctive Syllogism (DS) Contradiction (CON) p v q p ~p ~p ˫ q ˫ Any wff Absorption (ABS) Theorem Introduction (TI) p → q Introduce any tautology, e.g., ~(P & ~P) ˫ p → (p & q) EQUIVALENCES De Morgan’s Law (DM) (p → q) :: (~q→~p) ~(p & q) :: (~p v ~q) Material implication (MI) ~(p v q) :: (~p & ~q) (p → q) :: (~p v q) Commutation (COM) Material Equivalence (ME) (p v q) :: (q v p) (p ↔ q) :: [(p & q ) v (~p & ~q)] (p & q) :: (q & p) (p ↔ q) :: [(p → q ) & (q → p)] Association (ASSOC) Exportation (EXP) [p v (q v r)] :: [(p v q) v r] [(p & q) → r] :: [p → (q → r)] [p & (q & r)] :: [(p & q) & r] Tautology (TAUT) Distribution (DIST) p :: (p & p) [p & (q v r)] :: [(p & q) v (p & r)] p :: (p v p) [p v (q & r)] :: [(p v q) & (p v r)] Conditional-Biconditional Refutation Tree Rules Double Negation (DN) ~(p → q) :: (p & ~q) p :: ~~p ~(p ↔ q) :: [(p & ~q) v (~p & q)] Transposition (TRANS) CATEGORICAL SYLLOGISM RULES (e.g., Ǝx(Fx) / ˫ Fy).
    [Show full text]
  • Natural Deduction with Propositional Logic
    Natural Deduction with Propositional Logic Introducing Natural Natural Deduction with Propositional Logic Deduction Ling 130: Formal Semantics Some basic rules without assumptions Rules with assumptions Spring 2018 Outline Natural Deduction with Propositional Logic Introducing 1 Introducing Natural Deduction Natural Deduction Some basic rules without assumptions 2 Some basic rules without assumptions Rules with assumptions 3 Rules with assumptions What is ND and what's so natural about it? Natural Deduction with Natural Deduction Propositional Logic A system of logical proofs in which assumptions are freely introduced but discharged under some conditions. Introducing Natural Deduction Introduced independently and simultaneously (1934) by Some basic Gerhard Gentzen and Stanis law Ja´skowski rules without assumptions Rules with assumptions The book & slides/handouts/HW represent two styles of one ND system: there are several. Introduced originally to capture the style of reasoning used by mathematicians in their proofs. Ancient antecedents Natural Deduction with Propositional Logic Aristotle's syllogistics can be interpreted in terms of inference rules and proofs from assumptions. Introducing Natural Deduction Some basic rules without assumptions Rules with Stoic logic includes a practical application of a ND assumptions theorem. ND rules and proofs Natural Deduction with Propositional There are at least two rules for each connective: Logic an introduction rule an elimination rule Introducing Natural The rules reflect the meanings (e.g. as represented by Deduction Some basic truth-tables) of the connectives. rules without assumptions Rules with Parts of each ND proof assumptions You should have four parts to each line of your ND proof: line number, the formula, justification for writing down that formula, the goal for that part of the proof.
    [Show full text]
  • Logical Verification Course Notes
    Logical Verification Course Notes Femke van Raamsdonk [email protected] Vrije Universiteit Amsterdam autumn 2008 Contents 1 1st-order propositional logic 3 1.1 Formulas . 3 1.2 Natural deduction for intuitionistic logic . 4 1.3 Detours in minimal logic . 10 1.4 From intuitionistic to classical logic . 12 1.5 1st-order propositional logic in Coq . 14 2 Simply typed λ-calculus 25 2.1 Types . 25 2.2 Terms . 26 2.3 Substitution . 28 2.4 Beta-reduction . 30 2.5 Curry-Howard-De Bruijn isomorphism . 32 2.6 Coq . 38 3 Inductive types 41 3.1 Expressivity . 41 3.2 Universes of Coq . 44 3.3 Inductive types . 45 3.4 Coq . 48 3.5 Inversion . 50 4 1st-order predicate logic 53 4.1 Terms and formulas . 53 4.2 Natural deduction . 56 4.2.1 Intuitionistic logic . 56 4.2.2 Minimal logic . 60 4.3 Coq . 62 5 Program extraction 67 5.1 Program specification . 67 5.2 Proof of existence . 68 5.3 Program extraction . 68 5.4 Insertion sort . 68 i 1 5.5 Coq . 69 6 λ-calculus with dependent types 71 6.1 Dependent types: introduction . 71 6.2 λP ................................... 75 6.3 Predicate logic in λP ......................... 79 7 Second-order propositional logic 83 7.1 Formulas . 83 7.2 Intuitionistic logic . 85 7.3 Minimal logic . 88 7.4 Classical logic . 90 8 Polymorphic λ-calculus 93 8.1 Polymorphic types: introduction . 93 8.2 λ2 ................................... 94 8.3 Properties of λ2............................ 98 8.4 Expressiveness of λ2........................
    [Show full text]
  • Isabelle for Philosophers∗
    Isabelle for Philosophers∗ Ben Blumson September 20, 2019 It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used. Liebniz [11] p. 181. This is an introduction to the Isabelle proof assistant aimed at philosophers and students of philosophy.1 1 Propositional Logic Imagine you are caught in an air raid in the Second World War. You might reason as follows: Either I will be killed in this raid or I will not be killed. Suppose that I will. Then even if I take precautions, I will be killed, so any precautions I take will be ineffective. But suppose I am not going to be killed. Then I won't be killed even if I neglect all precautions; so on this assumption, no precautions ∗This draft is based on notes for students in my paradoxes and honours metaphysics classes { I'm grateful to the students for their help, especially Mark Goh, Zhang Jiang, Kee Wei Loo and Joshua Thong. I've also benefitted from discussion or correspondence on these issues with Zach Barnett, Sam Baron, David Braddon-Mitchell, Olivier Danvy, Paul Oppenheimer, Bruno Woltzenlogel Paleo, Michael Pelczar, David Ripley, Divyanshu Sharma, Manikaran Singh, Neil Sinhababu and Weng Hong Tang. 1I found a very useful introduction to be Nipkow [8]. Another still helpful, though unfortunately dated, introduction is Grechuk [6]. A person wishing to know how Isabelle works might first consult Paulson [9]. For the software itself and comprehensive docu- mentation, see https://isabelle.in.tum.de/.
    [Show full text]
  • List of Rules of Inference 1 List of Rules of Inference
    List of rules of inference 1 List of rules of inference This is a list of rules of inference, logical laws that relate to mathematical formulae. Introduction Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules. Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation indicates such a subderivation from the temporary assumption to . Rules for classical sentential calculus Sentential calculus is also known as propositional calculus. Rules for negations Reductio ad absurdum (or Negation Introduction) Reductio ad absurdum (related to the law of excluded middle) Noncontradiction (or Negation Elimination) Double negation elimination Double negation introduction List of rules of inference 2 Rules for conditionals Deduction theorem (or Conditional Introduction) Modus ponens (or Conditional Elimination) Modus tollens Rules for conjunctions Adjunction (or Conjunction Introduction) Simplification (or Conjunction Elimination) Rules for disjunctions Addition (or Disjunction Introduction) Separation of Cases (or Disjunction Elimination) Disjunctive syllogism List of rules of inference 3 Rules for biconditionals Biconditional introduction Biconditional Elimination Rules of classical predicate calculus In the following rules, is exactly like except for having the term everywhere has the free variable . Universal Introduction (or Universal Generalization) Restriction 1: does not occur in .
    [Show full text]
  • Derivations in the Sentential Calculus Having Already Two Methods For
    Derivations in the Sentential Calculus Having already two methods for showing an SC argument to be valid, the method of truth tables and search-for-counterexamples, we now want to introduce a third, in which we show an argument to be valid by producing aprooJ which is a sequence of sentences formed according to certain rules that guarantee that any sentence that the rules permit you to write down is a logical consequence of a specified set of premisses. It may seem a little perverse to introduce a new technique when we have two already, but the actual situation is even worse than this, for the new procedure is, in a crucial respect, vastly inferior to the ones we have already. The old methods were decisionprocedures, which enabled us to test whether a given argument was valid. If the argument was valid, the procedure would show its validity, whereas is the argument was invalid, the procedure would show its invalidity. The new method is only aproofprocedure: If an argument is valid, the method will enable to show that it is valid, but the method won't provide us any way of showing that an invalid argument is invalid. If an argument is valid, we can show it's valid by providing a proof, but the fact that we haven't been able to find a proof for a given argument doesn't show that the argument is invalid; maybe we just haven't looked hard enough. The new method is a giant step backward. The reason for taking it will only appear when we start work on the predicate calculus.
    [Show full text]
  • Rules of Inference
    Appendix I: Rules of Inference What follows are graphic representations for each of the 15 Rules of Inference, followed by a brief discussion, that will be used in this course. This supplement to the material in the Manual Part II, is intended to provide an alternative explanation of what each rule requires and what results from its application, in a more condensed, visual way that may be more easily accessible for some learners. Each rule is presented as an “input-output” machine. The “input(s)” represent the type of statement to which you can apply a given rule, or the type of statement or statements required to be “in play” in order to apply the rule. These statements could be premises that are given at the outset of the argument, derived statements or even assumptions. In the diagram the input is presented as a type of statement. Any substitution instance of this type can be a suitable input. The “output” is the result, the inference, or the conclusion you can draw forward when the rule is validly applied to the input(s). The output statement would be the statement you add to your left-hand derivation column, and justify with the rule you are applying. Within each box is an explanation of what the rule does to any given input to achieve the output. Each rule works on a main logical operator, or with a quantifier. Each rule is unique, reflecting the specific meaning of the logical sign (quantifier or operator) addressed by the rule. Quantifier rules Universal Elimination E Input 1) ELIMINATE the universal (x) Bx quantifer Output 2) Change the Ba variable (x, y, or z) to an individual Discussion: You can use this rule on ANY universally quantified statement, at any time in a proof.
    [Show full text]
  • Chapter 6: Formal Proofs and Boolean Logic
    Chapter 6: Formal Proofs and Boolean Logic The Fitch program, like the system F, uses “introduction” and “elimination” rules. The ones we’ve seen so far deal with the logical symbol =. The next group of rules deals with the Boolean connectives ∧, ∨, and ¬. § 6.1 Conjunction rules Conjunction Elimination (∧ Elim) P1 ∧…∧ Pi ∧ …∧ Pn ❺ Pi This rule tells you that if you have a conjunction in a proof, you may enter, on a new line, any of its conjuncts. (Pi here represents any of the conjuncts, including the first or the last.) Notice this important point: the conjunction to which you apply ∧ Elim must appear by itself on a line in the proof. You cannot apply this rule to a conjunction that is embedded as part of a larger sentence. For example, this is not a valid use of ∧ Elim: 1. ¬(Cube(a) ∧ Large(a)) ❺ 2. ¬Cube(a) x ∧ Elim: 1 The reason this is not valid use of the rule is that ∧ Elim can only be applied to conjunctions, and the line that this “proof” purports to apply it to is a negation. And it’s a good thing that this move is not allowed, for the inference above is not valid—from the premise that a is not a large cube it does not follow that a is not a cube. a might well be a small cube (and hence not a large cube, but still a cube). This same restriction—the rule applies to the sentence on the entire line, and not to an embedded sentence—holds for all of the rules of F, by the way.
    [Show full text]
  • Introduction
    Chapter 1 Introduction According to Wikipedia, logic is the study of the principles of valid inferences and demonstration. From the breadth of this definition it is immediately clear that logic constitutes an important area in the disciplines of philosophy and mathematics. Logical tools and methods also play an essential role in the de- sign, specification, and verification of computer hardware and software. It is these applications of logic in computer science which will be the focus of this course. In order to gain a proper understanding of logic and its relevance to computer science, we will need to draw heavily on the much older logical tra- ditions in philosophy and mathematics. We will discuss some of the relevant history of logic and pointers to further reading throughout these notes. In this introduction, we give only a brief overview of the contents and approach of this class. The course is divided into four parts: I. Proofs as Evidence for Truth II. Proofs as Programs III. Proofs as Computations IV. Proofs as Refutations Proofs are central in all parts of the course, and give it its constructive nature. In each part, we will exhibit connections between proofs and forms of compu- tations studied in computer science. These connections will take quite different forms, which shows the richness of logic as a foundational discipline at the nexus between philosophy, mathematics, and computer science. In Part I we establish the basic vocabulary and systematically study propo- sitions and proofs, mostly from a philosophical perspective. The treatment will be rather formal in order to permit an easy transition into computational appli- cations.
    [Show full text]