Version 1.5 - Trent University
... and determine their truth. The real fun lies in the relationship between interpretation of statements, truth, and reasoning. This volume develops the basics of two kinds of formal logical systems, propositional logic and first-order logic. Propositional logic attempts to make precise the relationshi ...
... and determine their truth. The real fun lies in the relationship between interpretation of statements, truth, and reasoning. This volume develops the basics of two kinds of formal logical systems, propositional logic and first-order logic. Propositional logic attempts to make precise the relationshi ...
Informal proofs
... • The steps of the proofs are not expressed in any formal language as e.g. propositional logic • Steps are argued less formally using English, mathematical formulas and so on • One must always watch the consistency of the argument made, logic and its rules can often help us to decide the soundness o ...
... • The steps of the proofs are not expressed in any formal language as e.g. propositional logic • Steps are argued less formally using English, mathematical formulas and so on • One must always watch the consistency of the argument made, logic and its rules can often help us to decide the soundness o ...
Modal_Logics_Eyal_Ariel_151107
... Let be a set of propositions. These propositions describe facts about the system as “the system is deadlocked” or “the value of variable x is 5”. An interpreted system is a tuple (S,V), where ...
... Let be a set of propositions. These propositions describe facts about the system as “the system is deadlocked” or “the value of variable x is 5”. An interpreted system is a tuple (S,V), where ...
1 Deductive Reasoning and Logical Connectives
... of the object. For example, to express that a number is prime, we can use x to represent the number. To express the statement “x is prime”, so far we have been using some Boolean variable such as p. However, such a notation does not stress that p is a statement about x. Therefore, we sometimes use p ...
... of the object. For example, to express that a number is prime, we can use x to represent the number. To express the statement “x is prime”, so far we have been using some Boolean variable such as p. However, such a notation does not stress that p is a statement about x. Therefore, we sometimes use p ...
Proof analysis beyond geometric theories: from rule systems to
... and in general for results that involve the simultaneous analysis of derivability in different theories. The starting point of the investigation is given by an analytic proof system for pure logic, taken as the basis of the extension procedure. Natural deduction is often the privileged proof system ...
... and in general for results that involve the simultaneous analysis of derivability in different theories. The starting point of the investigation is given by an analytic proof system for pure logic, taken as the basis of the extension procedure. Natural deduction is often the privileged proof system ...
Logic and Resolution - Institute for Computing and Information
... Logic has a very long and rich tradition, going back to the ancient Greeks: its roots can be traced to Aristotle. However, it took until the present century before the mathematical foundations of modern logic were laid, amongst others by T. Skolem, J. Herbrand, K. Gödel, and G. Gentzen. The work of ...
... Logic has a very long and rich tradition, going back to the ancient Greeks: its roots can be traced to Aristotle. However, it took until the present century before the mathematical foundations of modern logic were laid, amongst others by T. Skolem, J. Herbrand, K. Gödel, and G. Gentzen. The work of ...
CHAPTER 1 The main subject of Mathematical Logic is
... • Inductive predicates are defined by their clauses and a least-fixedpoint (or induction) axiom. Their witnesses are generation trees. • Dually coductive predicates are defined by a single clause and a greatest-fixed-point (or coinduction) axiom. Their witnesses are destruction trees. • It could be ...
... • Inductive predicates are defined by their clauses and a least-fixedpoint (or induction) axiom. Their witnesses are generation trees. • Dually coductive predicates are defined by a single clause and a greatest-fixed-point (or coinduction) axiom. Their witnesses are destruction trees. • It could be ...
Math 318 Class notes
... g is called a left inverse of f if g f = id A ; g is called a right inverse of f if f g = id B . Proposition 3.11. f is injective iff it has a left inverse. Proof. ⇒ Write B = range( f ) and for b ∈ B0 , let a = g(b) be the unique element such that f ( a) = b. Next, let a ∈ A be arbitrary and define ...
... g is called a left inverse of f if g f = id A ; g is called a right inverse of f if f g = id B . Proposition 3.11. f is injective iff it has a left inverse. Proof. ⇒ Write B = range( f ) and for b ∈ B0 , let a = g(b) be the unique element such that f ( a) = b. Next, let a ∈ A be arbitrary and define ...
First-Order Proof Theory of Arithmetic
... This chapter discusses the proof-theoretic foundations of the first-order theory of the non-negative integers. This first-order theory of numbers, also called ‘first-order arithmetic’, consists of the first-order sentences which are true about the integers. The study of first-order arithmetic is imp ...
... This chapter discusses the proof-theoretic foundations of the first-order theory of the non-negative integers. This first-order theory of numbers, also called ‘first-order arithmetic’, consists of the first-order sentences which are true about the integers. The study of first-order arithmetic is imp ...
Chapter 1: The Foundations: Logic and Proofs
... Tautology: A compound proposition that is always true. Contradiction: A compound proposition that is always false. Contingency: A compound proposition that is neither a tautology nor a contradiction. ...
... Tautology: A compound proposition that is always true. Contradiction: A compound proposition that is always false. Contingency: A compound proposition that is neither a tautology nor a contradiction. ...
Topological Completeness of First-Order Modal Logic
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
