CHAPTER 0: WELCOME TO MATHEMATICS A Preface of Logic
... met. Thus, anyone playing the game of mathematics should agree on the truth value of a statement asserting that an object satisfies a definition. The fact that definitions and axioms are accepted without proof does not mean that anything goes. A good definition should both clarify the meaning of a t ...
... met. Thus, anyone playing the game of mathematics should agree on the truth value of a statement asserting that an object satisfies a definition. The fact that definitions and axioms are accepted without proof does not mean that anything goes. A good definition should both clarify the meaning of a t ...
Assumption Sets for Extended Logic Programs
... In logic, the notion of strong negation was introduced by Nelson [12] in 1949. Nelson’s logic N is known as constructive logic with strong negation. N can be regarded as an extension of intuitionistic logic, H, in which the language of intuitionistic logic is extended by adding a new, strong negatio ...
... In logic, the notion of strong negation was introduced by Nelson [12] in 1949. Nelson’s logic N is known as constructive logic with strong negation. N can be regarded as an extension of intuitionistic logic, H, in which the language of intuitionistic logic is extended by adding a new, strong negatio ...
A General Proof Method for ... without the Barcan Formula.*
... This paper generalizes the proof method for modal predicate logic first described in Jackson [1987] and axiomatized in Jackson & Reichgelt [1987]. As before, the inference rules are identical for each system; different systems differ only with respect to the definition of complementarity between for ...
... This paper generalizes the proof method for modal predicate logic first described in Jackson [1987] and axiomatized in Jackson & Reichgelt [1987]. As before, the inference rules are identical for each system; different systems differ only with respect to the definition of complementarity between for ...
Computing Default Extensions by Reductions on OR
... equivalence-preserving reduction of the O R-formula to a disjunction of modalized propositional formulae of the form Oϕk . The O R-formula in the example reduces to Op ∨ Oq. The third step is to determine the set of extensions of the default theory from the simpler formula obtained in the second ste ...
... equivalence-preserving reduction of the O R-formula to a disjunction of modalized propositional formulae of the form Oϕk . The O R-formula in the example reduces to Op ∨ Oq. The third step is to determine the set of extensions of the default theory from the simpler formula obtained in the second ste ...
A puzzle about de rebus beliefs
... are critics who admire only one another’ as asserting that Ralph believes one of the de rebus or plural propositions that can be obtained by supplying some critics as argument to the propositional function described by the English open sentence ‘that they admire only one another’; there are some ind ...
... are critics who admire only one another’ as asserting that Ralph believes one of the de rebus or plural propositions that can be obtained by supplying some critics as argument to the propositional function described by the English open sentence ‘that they admire only one another’; there are some ind ...
A Revised Concept of Safety for General Answer Set Programs
... by saying that a rule is safe if any variable in the rule also appears in its positive body – this condition will be referred here as DLP safety. Programs are safe if all their rules are safe. The safety of a program ensures that its answer sets coincide with the answer sets of its ground version a ...
... by saying that a rule is safe if any variable in the rule also appears in its positive body – this condition will be referred here as DLP safety. Programs are safe if all their rules are safe. The safety of a program ensures that its answer sets coincide with the answer sets of its ground version a ...
An Introduction to Lower Bounds on Formula
... I am going to present a number of recent results and open questions related to the problem of proving lower bounds on the size of modal formulae defining properties of Kripke frames and models. To put things in perspective, I am going to start by giving an informal overview of some techniques used f ...
... I am going to present a number of recent results and open questions related to the problem of proving lower bounds on the size of modal formulae defining properties of Kripke frames and models. To put things in perspective, I am going to start by giving an informal overview of some techniques used f ...
Natural deduction for predicate logic
... In the next module, we will describe the semantics of predicate logic, and discuss soundness and completeness without proof. Still to come are other proof systems for predicate logic, and a discussion of how to ensure that specific mathematical situations (such as number theory or set theory) are pr ...
... In the next module, we will describe the semantics of predicate logic, and discuss soundness and completeness without proof. Still to come are other proof systems for predicate logic, and a discussion of how to ensure that specific mathematical situations (such as number theory or set theory) are pr ...
Section 3 - UCLA Department of Mathematics
... Case (i)(b) A is Pin t1 . . . tn for some i and n, and some variables or constants t, . . . , tn . Unique readability holds for L∗C as it does for L. Lemma 3.2. No proper initial segment of a formula of L∗C is a formula of L∗C . Theorem 3.3 (Unique Readability). Let A be a formula of L∗C . Then exac ...
... Case (i)(b) A is Pin t1 . . . tn for some i and n, and some variables or constants t, . . . , tn . Unique readability holds for L∗C as it does for L. Lemma 3.2. No proper initial segment of a formula of L∗C is a formula of L∗C . Theorem 3.3 (Unique Readability). Let A be a formula of L∗C . Then exac ...
Logic and Resolution - Institute for Computing and Information
... Satisfiability, validity, equivalence and logical consequence are semantic notions; these properties are generally established using truth tables. However, for deriving logical consequences from of a set of formulas for example, propositional logic provides other techniques than using truth tables a ...
... Satisfiability, validity, equivalence and logical consequence are semantic notions; these properties are generally established using truth tables. However, for deriving logical consequences from of a set of formulas for example, propositional logic provides other techniques than using truth tables a ...
Relational Calculus
... TRC: Variables range over (i.e., get bound to) tuples. DRC: Variables range over domain elements (= field values). Both TRC and DRC are simple subsets of first-order logic. ...
... TRC: Variables range over (i.e., get bound to) tuples. DRC: Variables range over domain elements (= field values). Both TRC and DRC are simple subsets of first-order logic. ...
22.1 Representability of Functions in a Formal Theory
... What remains to show is that this representation is in fact correct, that is that p(x)=y implies the validity of Rp (x,y) in Peano Arithmetic and that p(x)6=y implies the validity of ∼Rp (x,y). Fortunately, we are not required to give a formal proof for the validity of these formulas.2 Instead, we c ...
... What remains to show is that this representation is in fact correct, that is that p(x)=y implies the validity of Rp (x,y) in Peano Arithmetic and that p(x)6=y implies the validity of ∼Rp (x,y). Fortunately, we are not required to give a formal proof for the validity of these formulas.2 Instead, we c ...
Propositional Logic
... proposition is a possible “condition'” of the world about which we want to say something. The condition need not be true in order for us to talk about it. In fact, we might want to say that it is false or that it is true if some other proposition is true. In this chapter, we first look at the syntac ...
... proposition is a possible “condition'” of the world about which we want to say something. The condition need not be true in order for us to talk about it. In fact, we might want to say that it is false or that it is true if some other proposition is true. In this chapter, we first look at the syntac ...
2/TRUTH-FUNCTIONS
... I. Write MI/implication, BE/biconditional, and NOT for neither of the two. 1. R.Carnap's `logical involution' is false just in case the premises are true and the conclusion false. 2. “..a concept is clearer if and only if it is easier.” 3. Mathematics is not the `science of quantity', unless it dist ...
... I. Write MI/implication, BE/biconditional, and NOT for neither of the two. 1. R.Carnap's `logical involution' is false just in case the premises are true and the conclusion false. 2. “..a concept is clearer if and only if it is easier.” 3. Mathematics is not the `science of quantity', unless it dist ...
Game Theory: Logic, Set and Summation Notation
... However, what does the universal quantifier mean if the set X is empty? Thus, we clarify the interpretation to state “there is no x in X such that p(x) is false”: [(∀x ∈ X)p(x)] ¬[(∃x ∈ X)¬p(x)]. So, the original statement is vacuously (automatically) true if X is empty. For example, let X be the s ...
... However, what does the universal quantifier mean if the set X is empty? Thus, we clarify the interpretation to state “there is no x in X such that p(x) is false”: [(∀x ∈ X)p(x)] ¬[(∃x ∈ X)¬p(x)]. So, the original statement is vacuously (automatically) true if X is empty. For example, let X be the s ...
ON PRESERVING 1. Introduction The
... given above as necessary, is also sufficient. It is similarly easy to see that since every set is contained in its deductive closure by [R], and since inconsistency is preserved by supersets, given [Mon], every inference relation satisfying the three structural rules preserves consistency in the str ...
... given above as necessary, is also sufficient. It is similarly easy to see that since every set is contained in its deductive closure by [R], and since inconsistency is preserved by supersets, given [Mon], every inference relation satisfying the three structural rules preserves consistency in the str ...