An Introduction to Formal Logic
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 . -
Gibbardian Collapse and Trivalent Conditionals
Gibbardian Collapse and Trivalent Conditionals Paul Égré* Lorenzo Rossi† Jan Sprenger‡ Abstract This paper discusses the scope and significance of the so-called triviality result stated by Allan Gibbard for indicative conditionals, showing that if a conditional operator satisfies the Law of Import-Export, is supraclassical, and is stronger than the material conditional, then it must collapse to the material conditional. Gib- bard’s result is taken to pose a dilemma for a truth-functional account of indicative conditionals: give up Import-Export, or embrace the two-valued analysis. We show that this dilemma can be averted in trivalent logics of the conditional based on Reichenbach and de Finetti’s idea that a conditional with a false antecedent is undefined. Import-Export and truth-functionality hold without triviality in such logics. We unravel some implicit assumptions in Gibbard’s proof, and discuss a recent generalization of Gibbard’s result due to Branden Fitelson. Keywords: indicative conditional; material conditional; logics of conditionals; triva- lent logic; Gibbardian collapse; Import-Export 1 Introduction The Law of Import-Export denotes the principle that a right-nested conditional of the form A → (B → C) is logically equivalent to the simple conditional (A ∧ B) → C where both antecedentsare united by conjunction. The Law holds in classical logic for material implication, and if there is a logic for the indicative conditional of ordinary language, it appears Import-Export ought to be a part of it. For instance, to use an example from (Cooper 1968, 300), the sentences “If Smith attends and Jones attends then a quorum *Institut Jean-Nicod (CNRS/ENS/EHESS), Département de philosophie & Département d’études cog- arXiv:2006.08746v1 [math.LO] 15 Jun 2020 nitives, Ecole normale supérieure, PSL University, 29 rue d’Ulm, 75005, Paris, France. -
1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
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. -
Glossary for Logic: the Language of Truth
Glossary for Logic: The Language of Truth This glossary contains explanations of key terms used in the course. (These terms appear in bold in the main text at the point at which they are first used.) To make this glossary more easily searchable, the entry headings has ‘::’ (two colons) before it. So, for example, if you want to find the entry for ‘truth-value’ you should search for ‘:: truth-value’. :: Ambiguous, Ambiguity : An expression or sentence is ambiguous if and only if it can express two or more different meanings. In logic, we are interested in ambiguity relating to truth-conditions. Some sentences in natural languages express more than one claim. Read one way, they express a claim which has one set of truth-conditions. Read another way, they express a different claim with different truth-conditions. :: Antecedent : The first clause in a conditional is its antecedent. In ‘(P ➝ Q)’, ‘P’ is the antecedent. In ‘If it is raining, then we’ll get wet’, ‘It is raining’ is the antecedent. (See ‘Conditional’ and ‘Consequent’.) :: Argument : An argument is a set of claims (equivalently, statements or propositions) made up from premises and conclusion. An argument can have any number of premises (from 0 to indefinitely many) but has only one conclusion. (Note: This is a somewhat artificially restrictive definition of ‘argument’, but it will help to keep our discussions sharp and clear.) We can consider any set of claims (with one claim picked out as conclusion) as an argument: arguments will include sets of claims that no-one has actually advanced or put forward. -
Physical and Mathematical Sciences 2014, № 1, P. 3–6 Mathematics ON
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2014, № 1, p. 3–6 Mathematics ON ZIGZAG DE MORGAN FUNCTIONS V. A. ASLANYAN ∗ Oxford University UK, Chair of Algebra and Geometry YSU, Armenia There are five precomplete classes of De Morgan functions, four of them are defined as sets of functions preserving some finitary relations. However, the fifth class – the class of zigzag De Morgan functions, is not defined by relations. In this paper we announce the following result: zigzag De Morgan functions can be defined as functions preserving some finitary relation. MSC2010: 06D30; 06E30. Keywords: disjunctive (conjunctive) normal form of De Morgan function; closed and complete classes; quasimonotone and zigzag De Morgan functions. 1. Introduction. It is well known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the lattice of monotone Boolean functions of n variables. Analogous to these facts we have introduced the concept of De Morgan functions and proved that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables [1]. The Post’s functional completeness theorem for Boolean functions plays an important role in discrete mathematics [2]. In the paper [3] we have established a functional completeness criterion for De Morgan functions. In this paper we show that zigzag De Morgan functions, which are used in the formulation of the functional completeness theorem, can be defined by a finitary relation. -
1 Lecture 13: Quantifier Laws1 1. Laws of Quantifier Negation There
Ling 409 lecture notes, Lecture 13 Ling 409 lecture notes, Lecture 13 October 19, 2005 October 19, 2005 Lecture 13: Quantifier Laws1 true or there is at least one individual that makes φ(x) true (or both). Those are also equivalent statements. 1. Laws of Quantifier Negation Laws 2 and 3 suggest a fundamental connection between universal quantification and There are a number of equivalences based on the set-theoretic semantics of predicate logic. conjunction and existential quantification and disjunction: Take for instance a statement of the form (∃x)~ φ(x)2. It asserts that there is at least one individual d which makes ~φ(x) true. That means that d makes ϕ(x) false. That, in turn, (∀x) ψ(x) is true iff ψ(a) and ψ(b) and ψ(c) … (where a, b, c … are the names of the means that (∀x) φ(x) is false, which means that ~(∀x) φ(x) is true. The same reasoning individuals in the domain.) can be applied in the reverse direction and the result is the first quantifier law: ∃ ψ ψ ψ ψ ( x) (x) is true iff (a) or (b) or c) … (where a, b, c … are the names of the (1) Laws of Quantifier Negation individuals in the domain.) Law 1: ~(∀x) φ(x) ⇐⇒ (∃x)~ φ(x) From that point of view, Law 1 resembles a generalized DeMorgan’s Law: (Possible English correspondents: Not everyone passed the test and Someone did not pass (4) ~(ϕ(a) & ϕ(b) & ϕ(c) … ) ⇐⇒ ~ϕ(a) v ~ϕ(b) v ϕ(c) … the test.) (5) Law 4: (∀x) ψ(x) v (∀x)φ(x) ⇒ (∀x)( ψ(x) v φ(x)) ϕ ⇐ ϕ Because of the Law of Double Negation (~~ (x) ⇒ (x)), Law 1 could also be written (6) Law 5: (∃x)( ψ(x) & φ(x)) ⇒ (∃x) ψ(x) & (∃x)φ(x) as follows: These two laws are one-way entailments, NOT equivalences. -
Expressive Completeness
Truth-Functional Completeness 1. A set of truth-functional operators is said to be truth-functionally complete (or expressively adequate) just in case one can take any truth-function whatsoever, and construct a formula using only operators from that set, which represents that truth-function. In what follows, we will discuss how to establish the truth-functional completeness of various sets of truth-functional operators. 2. Let us suppose that we have an arbitrary n-place truth-function. Its truth table representation will have 2n rows, some true and some false. Here, for example, is the truth table representation of some 3- place truth function, which we shall call $: Φ ψ χ $ T T T T T T F T T F T F T F F T F T T F F T F T F F T F F F F T This truth-function $ is true on 5 rows (the first, second, fourth, sixth, and eighth), and false on the remaining 3 (the third, fifth, and seventh). 3. Now consider the following procedure: For every row on which this function is true, construct the conjunctive representation of that row – the extended conjunction consisting of all the atomic sentences that are true on that row and the negations of all the atomic sentences that are false on that row. In the example above, the conjunctive representations of the true rows are as follows (ignoring some extraneous parentheses): Row 1: (P&Q&R) Row 2: (P&Q&~R) Row 4: (P&~Q&~R) Row 6: (~P&Q&~R) Row 8: (~P&~Q&~R) And now think about the formula that is disjunction of all these extended conjunctions, a formula that basically is a disjunction of all the rows that are true, which in this case would be [(Row 1) v (Row 2) v (Row 4) v (Row6) v (Row 8)] Or, [(P&Q&R) v (P&Q&~R) v (P&~Q&~R) v (P&~Q&~R) v (~P&Q&~R) v (~P&~Q&~R)] 4. -
Three Ways of Being Non-Material
Three Ways of Being Non-Material Vincenzo Crupi, Andrea Iacona May 2019 This paper presents a novel unified account of three distinct non-material inter- pretations of `if then': the suppositional interpretation, the evidential interpre- tation, and the strict interpretation. We will spell out and compare these three interpretations within a single formal framework which rests on fairly uncontro- versial assumptions, in that it requires nothing but propositional logic and the probability calculus. As we will show, each of the three intrerpretations exhibits specific logical features that deserve separate consideration. In particular, the evidential interpretation as we understand it | a precise and well defined ver- sion of it which has never been explored before | significantly differs both from the suppositional interpretation and from the strict interpretation. 1 Preliminaries Although it is widely taken for granted that indicative conditionals as they are used in ordinary language do not behave as material conditionals, there is little agreement on the nature and the extent of such deviation. Different theories tend to privilege different intuitions about conditionals, and there is no obvious answer to the question of which of them is the correct theory. In this paper, we will compare three interpretations of `if then': the suppositional interpretation, the evidential interpretation, and the strict interpretation. These interpretations may be regarded either as three distinct meanings that ordinary speakers attach to `if then', or as three ways of explicating a single indeterminate meaning by replacing it with a precise and well defined counterpart. Here is a rough and informal characterization of the three interpretations. According to the suppositional interpretation, a conditional is acceptable when its consequent is credible enough given its antecedent. -
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, -
Completeness
Completeness The strange case of Dr. Skolem and Mr. G¨odel∗ Gabriele Lolli The completeness theorem has a history; such is the destiny of the impor- tant theorems, those for which for a long time one does not know (whether there is anything to prove and) what to prove. In its history, one can di- stinguish at least two main paths; the first one covers the slow and difficult comprehension of the problem in (what historians consider) the traditional development of mathematical logic canon, up to G¨odel'sproof in 1930; the second path follows the L¨owenheim-Skolem theorem. Although at certain points the two paths crossed each other, they started and continued with their own aims and problems. A classical topos of the history of mathema- tical logic concerns the how and the why L¨owenheim, Skolem and Herbrand did not discover the completeness theorem, though they proved it, or whe- ther they really proved, or perhaps they actually discovered, completeness. In following these two paths, we will not always respect strict chronology, keeping the two stories quite separate, until the crossing becomes decisive. In modern pre-mathematical logic, the notion of completeness does not appear. There are some interesting speculations in Kant which, by some stretching, could be realized as bearing some relation with the problem; Kant's remarks, however, are probably more related with incompleteness, in connection with his thoughts on the derivability of transcendental ideas (or concepts of reason) from categories (the intellect's concepts) through a pas- sage to the limit; thus, for instance, the causa prima, or the idea of causality, is the limit of implication, or God is the limit of disjunction, viz., the catego- ry of \comunance". -
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.