Sentential Logic Primer
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Ling 130 Notes: a Guide to Natural Deduction Proofs
Ling 130 Notes: A guide to Natural Deduction proofs Sophia A. Malamud February 7, 2018 1 Strategy summary Given a set of premises, ∆, and the goal you are trying to prove, Γ, there are some simple rules of thumb that will be helpful in finding ND proofs for problems. These are not complete, but will get you through our problem sets. Here goes: Given Delta, prove Γ: 1. Apply the premises, pi in ∆, to prove Γ. Do any other obvious steps. 2. If you need to use a premise (or an assumption, or another resource formula) which is a disjunction (_-premise), then apply Proof By Cases (Disjunction Elimination, _E) rule and prove Γ for each disjunct. 3. Otherwise, work backwards from the type of goal you are proving: (a) If the goal Γ is a conditional (A ! B), then assume A and prove B; use arrow-introduction (Conditional Introduction, ! I) rule. (b) If the goal Γ is a a negative (:A), then assume ::A (or just assume A) and prove con- tradiction; use Reductio Ad Absurdum (RAA, proof by contradiction, Negation Intro- duction, :I) rule. (c) If the goal Γ is a conjunction (A ^ B), then prove A and prove B; use Conjunction Introduction (^I) rule. (d) If the goal Γ is a disjunction (A _ B), then prove one of A or B; use Disjunction Intro- duction (_I) rule. 4. If all else fails, use RAA (proof by contradiction). 2 Using rules for conditionals • Conditional Elimination (modus ponens) (! E) φ ! , φ ` This rule requires a conditional and its antecedent. -
Tt-Satisfiable
CMPSCI 601: Recall From Last Time Lecture 6 Boolean Syntax: ¡ ¢¤£¦¥¨§¨£ © §¨£ § Boolean variables: A boolean variable represents an atomic statement that may be either true or false. There may be infinitely many of these available. Boolean expressions: £ atomic: , (“top”), (“bottom”) § ! " # $ , , , , , for Boolean expressions Note that any particular expression is a finite string, and thus may use only finitely many variables. £ £ A literal is an atomic expression or its negation: , , , . As you may know, the choice of operators is somewhat arbitary as long as we have a complete set, one that suf- fices to simulate all boolean functions. On HW#1 we ¢ § § ! argued that is already a complete set. 1 CMPSCI 601: Boolean Logic: Semantics Lecture 6 A boolean expression has a meaning, a truth value of true or false, once we know the truth values of all the individual variables. ¢ £ # ¡ A truth assignment is a function ¢ true § false , where is the set of all variables. An as- signment is appropriate to an expression ¤ if it assigns a value to all variables used in ¤ . ¡ The double-turnstile symbol ¥ (read as “models”) de- notes the relationship between a truth assignment and an ¡ ¥ ¤ expression. The statement “ ” (read as “ models ¤ ¤ ”) simply says “ is true under ”. 2 ¡ ¤ ¥ ¤ If is appropriate to , we define when is true by induction on the structure of ¤ : is true and is false for any , £ A variable is true iff says that it is, ¡ ¡ ¡ ¡ " ! ¥ ¤ ¥ ¥ If ¤ , iff both and , ¡ ¡ ¡ ¡ " ¥ ¤ ¥ ¥ If ¤ , iff either or or both, ¡ ¡ ¡ ¡ " # ¥ ¤ ¥ ¥ If ¤ , unless and , ¡ ¡ ¡ ¡ $ ¥ ¤ ¥ ¥ If ¤ , iff and are both true or both false. 3 Definition 6.1 A boolean expression ¤ is satisfiable iff ¡ ¥ ¤ there exists . -
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. -
2001 Boston Marathon, Overall Results 1 - 100
2001 Boston Marathon, Overall Results 1 - 100 Have you run this race? More Results: Then tell us about it . Last Name, First Name Time OverAll Sex Place DIV Net City, State, (Sex/Age) Place / Time Country Div Place Bong-Ju Lee (M30) 2:09:43 1 1 / 1 Open 2:09:43 Seoul, KOR Silvio Guerra (M32) 2:10:07 2 2 / 2 Open 2:10:07 Quito, ECU Joshua Chelang'a (M28) 2:10:29 3 3 / 3 Open 2:10:29 Baringo, KEN David Kiptum Busienei (M26) 2:11:47 4 4 / 4 Open 2:11:47 Kabiet, KEN Mbarek Hussein (M36) 2:12:01 5 5 / 5 Open 2:12:01 Kapsabet, KEN Rod De Haven (M34) 2:12:41 6 6 / 6 Open 2:12:41 Madison, WI, USA Laban Nkete (M30) 2:12:44 7 7 / 7 Open 2:12:44 Port Elizabeth, RSA Fedor V. Ryjov (M41) 2:13:54 8 8 / 1 Masters 2:13:54 Acoteias, Albe, POR Makhosonke Fika (M29) 2:14:13 9 9 / 8 Open 2:14:13 Cape Town, RSA Timothy Cherigat (M24) 2:14:21 10 10 / 9 Open 2:14:21 Chepkorio, KEN Joshua Kipkemboi (M42) 2:14:47 11 11 / 2 Masters 2:14:47 Concord, MA, USA Moses Tanui (M35) 2:15:05 12 12 / 10 Open 2:15:05 Eldoret, KEN Joao N'Tyamba (M33) 2:16:00 13 13 / 11 Open 2:16:00 Bogota, ANG Josh Cox (M25) 2:16:17 14 14 / 12 Open 2:16:17 El Cajon, CA, USA Shem Kororia (M28) 2:17:02 15 15 / 13 Open 2:17:02 Kapsokwong, Kitale, KEN Gezahegne Abera (M22) 2:17:04 16 16 / 14 Open 2:17:04 Addis Ababa, ETH Elijah Lagat (M34) 2:17:59 17 17 / 15 Open 2:17:59 Nandi District, KEN Motsehi Moeketsana (M31) 2:18:13 18 18 / 16 Open 2:18:13 Colleen Glen, RSA Mark Coogan (M34) 2:18:58 19 19 / 17 Open 2:18:58 Attleboro, MA, USA Makoto Ogura (M28) 2:20:24 20 20 / 18 Open 2:20:24 Hiroshima-Shi, -
Famous Jci Members and Alumni
FAMOUS JCI MEMBERS AND ALUMNI JCI (Junior Chamber International) provides leadership training to individuals throughout the world. The impact and importance of this training is demonstrated by the large number of JCI members who are holding or have held high positions in their respective countries and international bodies. Although incomplete, here is a list of members whom we would like to recognize: Australia BOND, Alan One of Australia's best-known corporate entrepreneurs and head of the syndicate that won the America's Cup in 1973; past member of JCI Fremantle, Australia. COURT, Hon. Charles, O.B.E., M.L.A. Premier of Western Australia (1978). HAYDEN, William Governor-General of Australia; past member of JCI Innisfail. LOWE, Hon. Doug, M.L.A. Premier of Tasmania (1978). LYNCH, Phillip Minister of Australia, National President of Australia Junior Chamber (JCI Australia) (1966). Belgium BRIL, Louis Secretary of State (Belgium), President of a JCI local organization (1978), past member of the Roeselare-Izegem Jaycees (JCI Roeselare-Izegem). DE CLERCK, Willy Commissioner of the European Common Market; JCI Senator No. 8412. HANSENNE, Michel Director-General of the International Labor Organization (ILO) (1989-1999), former Minister of Labor and Employment in Belgium; past member of the Liege Jaycees (JCI Liege), JCI Senator No. 17228. Bolivia BANZER-SUAREZ, Hugo President of Bolivia (1971-1978), JCI Senator No.15094, past member of the Cochabamba Jaycees (JCI Cochabamba). Famous JCI Members and Alumni Page 1 Bolivia, cont. HOZ DE VILA, Tito Congressman (1989-2002), Minister of Education (1997-2001), Senator of the Republic of Bolivia (2005-2009); JCI Vice President (1976), JCI Executive Vice President (1980), JCI General Legal Counsel (1982), JCI Senator 22425. -
Chapter 9: Initial Theorems About Axiom System
Initial Theorems about Axiom 9 System AS1 1. Theorems in Axiom Systems versus Theorems about Axiom Systems ..................................2 2. Proofs about Axiom Systems ................................................................................................3 3. Initial Examples of Proofs in the Metalanguage about AS1 ..................................................4 4. The Deduction Theorem.......................................................................................................7 5. Using Mathematical Induction to do Proofs about Derivations .............................................8 6. Setting up the Proof of the Deduction Theorem.....................................................................9 7. Informal Proof of the Deduction Theorem..........................................................................10 8. The Lemmas Supporting the Deduction Theorem................................................................11 9. Rules R1 and R2 are Required for any DT-MP-Logic........................................................12 10. The Converse of the Deduction Theorem and Modus Ponens .............................................14 11. Some General Theorems About ......................................................................................15 12. Further Theorems About AS1.............................................................................................16 13. Appendix: Summary of Theorems about AS1.....................................................................18 2 Hardegree, -
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 . -
Inference Versus Consequence* Göran Sundholm Leyden University
To appear in LOGICA Yearbook, 1998, Czech Acad. Sc., Prague. Inference versus Consequence* Göran Sundholm Leyden University The following passage, hereinafter "the passage", could have been taken from a modern textbook.1 It is prototypical of current logical orthodoxy: The inference (*) A1, …, Ak. Therefore: C is valid if and only if whenever all the premises A1, …, Ak are true, the conclusion C is true also. When (*) is valid, we also say that C is a logical consequence of A1, …, Ak. We write A1, …, Ak |= C. It is my contention that the passage does not properly capture the nature of inference, since it does not distinguish between valid inference and logical consequence. The view that the validity of inference is reducible to logical consequence has been made famous in our century by Tarski, and also by Wittgenstein in the Tractatus and by Quine, who both reduced valid inference to the logical truth of a suitable implication.2 All three were anticipated by Bolzano.3 Bolzano considered Urteile (judgements) of the form A is true where A is a Satz an sich (proposition in the modern sense).4 Such a judgement is correct (richtig) when the proposition A, that serves as the judgemental content, really is true.5 A correct judgement is an Erkenntnis, that is, a piece of knowledge.6 Similarly, for Bolzano, the general form I of inference * I am indebted to my colleague Dr. E. P. Bos who read an early version of the manuscript and offered valuable comments. 1 Could have been so taken and almost was; cf. -
Starting the Dismantling of Classical Mathematics
Australasian Journal of Logic Starting the Dismantling of Classical Mathematics Ross T. Brady La Trobe University Melbourne, Australia [email protected] Dedicated to Richard Routley/Sylvan, on the occasion of the 20th anniversary of his untimely death. 1 Introduction Richard Sylvan (n´eRoutley) has been the greatest influence on my career in logic. We met at the University of New England in 1966, when I was a Master's student and he was one of my lecturers in the M.A. course in Formal Logic. He was an inspirational leader, who thought his own thoughts and was not afraid to speak his mind. I hold him in the highest regard. He was very critical of the standard Anglo-American way of doing logic, the so-called classical logic, which can be seen in everything he wrote. One of his many critical comments was: “G¨odel's(First) Theorem would not be provable using a decent logic". This contribution, written to honour him and his works, will examine this point among some others. Hilbert referred to non-constructive set theory based on classical logic as \Cantor's paradise". In this historical setting, the constructive logic and mathematics concerned was that of intuitionism. (The Preface of Mendelson [2010] refers to this.) We wish to start the process of dismantling this classi- cal paradise, and more generally classical mathematics. Our starting point will be the various diagonal-style arguments, where we examine whether the Law of Excluded Middle (LEM) is implicitly used in carrying them out. This will include the proof of G¨odel'sFirst Theorem, and also the proof of the undecidability of Turing's Halting Problem. -
An Interpretationist Approach to the Thinking Mind DISSERTATION
Thought Without Language: an Interpretationist Approach to the Thinking Mind DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Michael Dean Jaworski Graduate Program in Philosophy The Ohio State University 2010 Dissertation Committee: Neil Tennant, Advisor William Taschek Ben Caplan Copyright by Michael Dean Jaworski 2010 Abstract I defend an account of thought on which non-linguistic beings can be thinkers. This result is significant in that many philosophers have claimed that the ability to think depends on the ability to use language. These opponents of my view note that our everyday understanding of our own cognitive activities qua thought bestows upon those activities the propositional structure of sentences and the inferential norms of public linguistic practice. They hold that our attributions of thought to non-linguistic beings project non-existent structure onto the cognitive activities of those beings, and assess the beings’ activities according to standards to which the beings bear no responsibility. So, despite the complex neural and behavioral activities of many non-linguistic beings, my opponents hold that those beings are not properly described as thinkers. To respond to my opponents successfully, one must not merely cite scientific and folk practices of thought attribution that permit thought to be attributed to some non- linguistic beings. My opponents’ insights might be taken to demonstrate a need to revise those practices, or to treat the attributions of thought to non-linguistic beings made within those practices as instrumentally valuable but technically false. Instead, my strategy is to acknowledge the language-like structure and norms of thought, and show that a non- linguistic being’s cognitive activities might nonetheless have that structure and be subject ii to those norms. -
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. -
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).