Propositional Logic What is logic? Propositions Negation
... Let F(x) be the predicate “x is female”. Let P(x) be the predicate “x is a parent”. Let M(x,y) be the predicate “x is the mother of y”. Let the universe of discourse be the set of all people. We can express the statement “If a person is female and is a parent, then this person is someone’s mother”. ...
... Let F(x) be the predicate “x is female”. Let P(x) be the predicate “x is a parent”. Let M(x,y) be the predicate “x is the mother of y”. Let the universe of discourse be the set of all people. We can express the statement “If a person is female and is a parent, then this person is someone’s mother”. ...
Theories.Axioms,Rules of Inference
... What do axioms do for us? That is where a logic comes in, with rules of inference, which allow us to derive theorems from axioms and other theorems. This is the alternate characterization of theorems, instead of saying a theorem is a valid(true in all possible assignments to free variables) formula ...
... What do axioms do for us? That is where a logic comes in, with rules of inference, which allow us to derive theorems from axioms and other theorems. This is the alternate characterization of theorems, instead of saying a theorem is a valid(true in all possible assignments to free variables) formula ...
S. P. Odintsov “REDUCTIO AD ABSURDUM” AND LUKASIEWICZ`S
... with algebras unit element of which corresponds to the only distinguished value. As usually, for an L-formulae ϕ and algebra A of the language L∪{1}, ϕ is an identity of A if V (ϕ) = 1 for any A-valuation V . A logic of A, LA, is defined as a set of all its identities and a logic of a class K of alge ...
... with algebras unit element of which corresponds to the only distinguished value. As usually, for an L-formulae ϕ and algebra A of the language L∪{1}, ϕ is an identity of A if V (ϕ) = 1 for any A-valuation V . A logic of A, LA, is defined as a set of all its identities and a logic of a class K of alge ...
Lecture 23 Notes
... refinement type of computational type theory to specify the classical computational content. The refinement type, {U nit|P }, is critical. If P is known by constructive evidence p, then the refinement type has ?, the one and only element of U nit, as its explicit element; and p is hidden. We imagine ...
... refinement type of computational type theory to specify the classical computational content. The refinement type, {U nit|P }, is critical. If P is known by constructive evidence p, then the refinement type has ?, the one and only element of U nit, as its explicit element; and p is hidden. We imagine ...
Propositional/First
... • Logical inference creates new sentences that logically follow from a set of sentences (KB) • An inference rule is sound if every sentence X it produces when operating on a KB logically follows from the KB –i.e., inference rule creates no contradictions • An inference rule is complete if it can pro ...
... • Logical inference creates new sentences that logically follow from a set of sentences (KB) • An inference rule is sound if every sentence X it produces when operating on a KB logically follows from the KB –i.e., inference rule creates no contradictions • An inference rule is complete if it can pro ...
EVALUATING INDUCTIVE ARGUMENTS
... selection that is thought to be random such that each member of the population from which it has been chosen has an equal chance of being chosen. ...
... selection that is thought to be random such that each member of the population from which it has been chosen has an equal chance of being chosen. ...
Document
... The rule UG is where the flagging method differs most from the standard system. In the flagging system UG involves as subproof. However, the first step consists not of an assumption but of a flagging step. ...
... The rule UG is where the flagging method differs most from the standard system. In the flagging system UG involves as subproof. However, the first step consists not of an assumption but of a flagging step. ...
mathematical logic: constructive and non
... 1. Mathematical logic Early in the century, especially in connection with Hubert's treatment of geometry (1899), it was being said t h a t the theorems of an axiomatic theory express truths about whatever systems of objects make the axioms true. I n the simplest case, a system S consists of a non-em ...
... 1. Mathematical logic Early in the century, especially in connection with Hubert's treatment of geometry (1899), it was being said t h a t the theorems of an axiomatic theory express truths about whatever systems of objects make the axioms true. I n the simplest case, a system S consists of a non-em ...
hilbert systems - CSA
... S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. ...
... S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. ...
Computing Default Extensions by Reductions on OR
... the authors state a modal reduction theorem to the effect that a formula O Rϕ is logically equivalent to a disjunction Oϕ1 ∨ · · · ∨ Oϕn , where each ϕk is a propositional formula. Because each such disjunct Oϕ k has a unique model, it is possible, within the logic itself, to break down a formula O ...
... the authors state a modal reduction theorem to the effect that a formula O Rϕ is logically equivalent to a disjunction Oϕ1 ∨ · · · ∨ Oϕn , where each ϕk is a propositional formula. Because each such disjunct Oϕ k has a unique model, it is possible, within the logic itself, to break down a formula O ...
How to tell the truth without knowing what you are talking about
... representation, is a branch of algebra, thus freeing logic from the tradition of philosophy. ...
... representation, is a branch of algebra, thus freeing logic from the tradition of philosophy. ...
Logic and Proof
... rise to contradiction (someone loves and does not loves Jill). • We must demonstrate that our specification does not draw the wrong inferences. • We must demonstrate that what we claim holds in the specification does hold. • The demonstration should be given as a proof, which is a systematic way to ...
... rise to contradiction (someone loves and does not loves Jill). • We must demonstrate that our specification does not draw the wrong inferences. • We must demonstrate that what we claim holds in the specification does hold. • The demonstration should be given as a proof, which is a systematic way to ...
sentential logic
... INTRODUCTION TO LOGIC Introduction: The term "logic" from the Greek word logos(reasoning ,discourse).The branch of philosophy that deals with forms of reasoning and thinking, especially inference or argument and scientific method. Logic may be defined as the study of the principles of correct reason ...
... INTRODUCTION TO LOGIC Introduction: The term "logic" from the Greek word logos(reasoning ,discourse).The branch of philosophy that deals with forms of reasoning and thinking, especially inference or argument and scientific method. Logic may be defined as the study of the principles of correct reason ...
Lectures on Laws of Supply and Demand, Simple and Compound
... Introduction Logic(word comes from the Greek word “logos” meaning word, speech or reason) is the mathematics of reasoning. The study of logic is the effort to determine the conditions under which one is justified in passing from given statements, called premises, to conclusions that is claimed to fo ...
... Introduction Logic(word comes from the Greek word “logos” meaning word, speech or reason) is the mathematics of reasoning. The study of logic is the effort to determine the conditions under which one is justified in passing from given statements, called premises, to conclusions that is claimed to fo ...
slides1
... In Heyting semantics, we are interested in witnesses to truth. Instead of asking “when is A true?”, we ask “what is the proof of A?” For atomic sentences, the proofs are intrinsic. For example, the proof of 27 × 37 = 999 is by calculation. A proof of A ∧ B is a pair (p, q) where p and q are proofs o ...
... In Heyting semantics, we are interested in witnesses to truth. Instead of asking “when is A true?”, we ask “what is the proof of A?” For atomic sentences, the proofs are intrinsic. For example, the proof of 27 × 37 = 999 is by calculation. A proof of A ∧ B is a pair (p, q) where p and q are proofs o ...
Logic seminar
... Propositional logic • If a formula F is true under an interpretation I, then we say that I satisfies F, or F is satisfied by I. • On the other hand, if a formula F is false under an interpretation I, then we say that I falsifies F or F is falsified by I. • For example, the formula (P(~Q)) is satis ...
... Propositional logic • If a formula F is true under an interpretation I, then we say that I satisfies F, or F is satisfied by I. • On the other hand, if a formula F is false under an interpretation I, then we say that I falsifies F or F is falsified by I. • For example, the formula (P(~Q)) is satis ...
Quantified Equilibrium Logic and the First Order Logic of Here
... programs. It also provides a useful logical foundation for answer set programming (ASP), the fast developing paradigm for declarative programming based on the answer set semantics [1]. A costly component of the computation of answer sets is the process of grounding a program containing variables by ...
... programs. It also provides a useful logical foundation for answer set programming (ASP), the fast developing paradigm for declarative programming based on the answer set semantics [1]. A costly component of the computation of answer sets is the process of grounding a program containing variables by ...
Sequentiality by Linear Implication and Universal Quantification
... Sequentialization is achieved in linear logic by a controlled form of backchaining, whose non-determinism is eliminated by the linearity of the calculus (linear implication) and a declarative way of producing unique identifiers (universal quantification). In our case study these two mechanisms, toge ...
... Sequentialization is achieved in linear logic by a controlled form of backchaining, whose non-determinism is eliminated by the linearity of the calculus (linear implication) and a declarative way of producing unique identifiers (universal quantification). In our case study these two mechanisms, toge ...
Justification logic with approximate conditional probabilities
... 1 whenever the condition has probability 0. Axiom 10 is the formula that states the standard definition of the conditional probability. Finally, the Axioms 11 and 12 (together with Inference Rule 4) give us the relationship between the conditional probability infinitesimally close to the some ration ...
... 1 whenever the condition has probability 0. Axiom 10 is the formula that states the standard definition of the conditional probability. Finally, the Axioms 11 and 12 (together with Inference Rule 4) give us the relationship between the conditional probability infinitesimally close to the some ration ...
Robot Morality and Review of classical logic.
... to win a medal in the ’04 Olympic games. Her husband assures her that she will surely place in one of the two events she has qualified for: ‘It’s ok,’ he says. “Either you’ll medal in the long jump or in the 400m relay.” ...
... to win a medal in the ’04 Olympic games. Her husband assures her that she will surely place in one of the two events she has qualified for: ‘It’s ok,’ he says. “Either you’ll medal in the long jump or in the 400m relay.” ...
CA208ex1 - DCU School of Computing
... Intutively, are the inferences above logically valid (i.e. is the conclusion true in all situations where the premises are true)? Is the following inference logically valid? ...
... Intutively, are the inferences above logically valid (i.e. is the conclusion true in all situations where the premises are true)? Is the following inference logically valid? ...
Proof theory of witnessed G¨odel logic: a
... or analytic calculus1 ; for example proofs in the Calculus of Structures [17, 13] or display logic [10] might contain logical or structural connectives that do not appear in the formulas to be proved and are not universally considered “well-behaved”, e.g. [23]. In this paper we propose an operationa ...
... or analytic calculus1 ; for example proofs in the Calculus of Structures [17, 13] or display logic [10] might contain logical or structural connectives that do not appear in the formulas to be proved and are not universally considered “well-behaved”, e.g. [23]. In this paper we propose an operationa ...
The semantics of predicate logic
... each atom. In predicate logic, the smallest unit to which we can assign a truth value is a predicate P (t1 , t2 , . . . , tn ) applied to terms. But we cannot arbitrarily assign a truth value, as we did for propositional atoms. There needs to be some consistency. We need to assign values to variable ...
... each atom. In predicate logic, the smallest unit to which we can assign a truth value is a predicate P (t1 , t2 , . . . , tn ) applied to terms. But we cannot arbitrarily assign a truth value, as we did for propositional atoms. There needs to be some consistency. We need to assign values to variable ...
Propositional Logic: Part I - Semantics
... “If pigs could fly then I’d enjoy brussel sprouts!” p : Pigs fly; b : Enjoy sprouts This (p |= b) is an invalid argument. Why use it? The real argument is: p, ¬p |= b which is a valid argument. Why is it valid? There is no counter example where p ∧ ¬p is true and b is false. Ex falso quod libet! i. ...
... “If pigs could fly then I’d enjoy brussel sprouts!” p : Pigs fly; b : Enjoy sprouts This (p |= b) is an invalid argument. Why use it? The real argument is: p, ¬p |= b which is a valid argument. Why is it valid? There is no counter example where p ∧ ¬p is true and b is false. Ex falso quod libet! i. ...
Dialetheic truth theory: inconsistency, non-triviality, soundness, incompleteness
... Abstract. The best-known application of dialetheism is to semantic paradoxes such as the Liar. In particular, Graham Priest has advocated the adoption of an axiomatic truth theory in which contradictions arising from the Liar paradox can be accepted as theorems, while the Liar sentence itself is eva ...
... Abstract. The best-known application of dialetheism is to semantic paradoxes such as the Liar. In particular, Graham Priest has advocated the adoption of an axiomatic truth theory in which contradictions arising from the Liar paradox can be accepted as theorems, while the Liar sentence itself is eva ...