1. Propositional Logic 1.1. Basic Definitions. Definition 1.1. The
... The linear structure of of Hilbert-style deductions, and the very simple list of cases (each step can be only an axiom or an instance of modus ponens) makes it very easy to prove some theorems about Hilbert systems. However these systems are very far removed from ordinary mathematics, and they don’t ...
... The linear structure of of Hilbert-style deductions, and the very simple list of cases (each step can be only an axiom or an instance of modus ponens) makes it very easy to prove some theorems about Hilbert systems. However these systems are very far removed from ordinary mathematics, and they don’t ...
On Elkan`s theorems: Clarifying their meaning
... In the following remark we argue that the last assumption of Definition 1 is unnatural for fuzzy logic and, in fact, makes Elkan’s logical system a “nonfuzzy logic.” Remark 1 (on assumptions of Theorem 1). Note first that assumption t~A ∨ B! ⫽ max$t~A!, t~B!% of Definition 1 was not used in our proo ...
... In the following remark we argue that the last assumption of Definition 1 is unnatural for fuzzy logic and, in fact, makes Elkan’s logical system a “nonfuzzy logic.” Remark 1 (on assumptions of Theorem 1). Note first that assumption t~A ∨ B! ⫽ max$t~A!, t~B!% of Definition 1 was not used in our proo ...
Knowledge Representation and Reasoning
... propositions — called premisses — which match certain patterns, we can deduce that some further proposition is true — this is called the conclusion. Thus we saw that from two propositions with the forms α → β and α we can deduce β. The inference from P → Q and P to Q is of this form. An inference ru ...
... propositions — called premisses — which match certain patterns, we can deduce that some further proposition is true — this is called the conclusion. Thus we saw that from two propositions with the forms α → β and α we can deduce β. The inference from P → Q and P to Q is of this form. An inference ru ...
Introduction to Mathematical Logic lecture notes
... Then S = n∈N Sn is the set of all formulae. We call Sn the set of formulae constructed in n steps. For example, (P → Q) and (R ∨ (P ∧ (¬Q))) are formulae, but (∧P ) and ¬Q) are not. With time we will allow ourselves to omit some parentheses if the meaning remains clear: for example, instead of (¬((¬ ...
... Then S = n∈N Sn is the set of all formulae. We call Sn the set of formulae constructed in n steps. For example, (P → Q) and (R ∨ (P ∧ (¬Q))) are formulae, but (∧P ) and ¬Q) are not. With time we will allow ourselves to omit some parentheses if the meaning remains clear: for example, instead of (¬((¬ ...
The Foundations: Logic and Proofs
... known theorems and construct a sequence of steps that end in the conclusion. Start with p and prove q, or start with ¬q and prove ¬p. 2. If this doesn’t work, try backward reasoning. When trying to prove q, find a statement p that we can prove with the property p → q. ...
... known theorems and construct a sequence of steps that end in the conclusion. Start with p and prove q, or start with ¬q and prove ¬p. 2. If this doesn’t work, try backward reasoning. When trying to prove q, find a statement p that we can prove with the property p → q. ...
The Computer Modelling of Mathematical Reasoning Alan Bundy
... theorem proving’ techniques could be readily brought into a Resolution framework, and how this helped us to relate the various techniques – creating coherence from confusion. In order to achieve this goal I have taken strong historical liberties in my descriptions of the work of Boyer and Moore, Gel ...
... theorem proving’ techniques could be readily brought into a Resolution framework, and how this helped us to relate the various techniques – creating coherence from confusion. In order to achieve this goal I have taken strong historical liberties in my descriptions of the work of Boyer and Moore, Gel ...
Systems of modal logic - Department of Computing
... Systems of modal logic In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ ...
... Systems of modal logic In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ ...
Bounded Proofs and Step Frames - Università degli Studi di Milano
... principle and elements from Γ as well as modus ponens, necessitation and inferences from Ax (again notice that uniform substitution cannot be applied to members of Γ ). We need some care when replacing a logic L with an inference system Ax, because we want global consequence relation to be preserve ...
... principle and elements from Γ as well as modus ponens, necessitation and inferences from Ax (again notice that uniform substitution cannot be applied to members of Γ ). We need some care when replacing a logic L with an inference system Ax, because we want global consequence relation to be preserve ...
Introduction to Formal Logic - Web.UVic.ca
... This inference fulfils condition (i): there is no possible case where its premises could be true and its conclusion false. Hence the inference is valid. But the inference also fulfils condition (ii), because its premises are true: all whales are in fact mammals, and all mammals have spinal chords. N ...
... This inference fulfils condition (i): there is no possible case where its premises could be true and its conclusion false. Hence the inference is valid. But the inference also fulfils condition (ii), because its premises are true: all whales are in fact mammals, and all mammals have spinal chords. N ...
Modal Reasoning
... The expressive power of any language can be measured by its ability to distinguish between two situations or–equivalently–the situations it considers to be indistinguishable. To capture the expressive power of a language, it’s necessary to to find an appropriate structural invariance between models. ...
... The expressive power of any language can be measured by its ability to distinguish between two situations or–equivalently–the situations it considers to be indistinguishable. To capture the expressive power of a language, it’s necessary to to find an appropriate structural invariance between models. ...
From Syllogism to Common Sense Normal Modal Logic
... logic intact. Whilst they are therefore`redundant’ in a sense, they can significantly shorten proofs, which is our main concern here. ‣ Example: Congruence rules. ‣ The general form of a rule is the following: ...
... logic intact. Whilst they are therefore`redundant’ in a sense, they can significantly shorten proofs, which is our main concern here. ‣ Example: Congruence rules. ‣ The general form of a rule is the following: ...
