compactness slides
... and F. By the unique readability theorem C ∗ is freely generated from the set of sentence symbols by the functions in F. This guarantees the uniqueness of the extension of truth assignment v to v̄ by the recursion theorem below. ...
... and F. By the unique readability theorem C ∗ is freely generated from the set of sentence symbols by the functions in F. This guarantees the uniqueness of the extension of truth assignment v to v̄ by the recursion theorem below. ...
• Use mathematical deduction to derive new knowledge. • Predicate
... • Predicate Logic is a powerful representation scheme used by many AI programs. • Propositional logic is much simpler (less ...
... • Predicate Logic is a powerful representation scheme used by many AI programs. • Propositional logic is much simpler (less ...
1-1
... • Acceptor – A finite state acceptor is used for languages (sets of strings) for which only a finite number of things need to be remembered. • Recursive methods – a finite basis set is given along with rules for forming the reset of the elements from existing elements. • Grammars – Languages are spe ...
... • Acceptor – A finite state acceptor is used for languages (sets of strings) for which only a finite number of things need to be remembered. • Recursive methods – a finite basis set is given along with rules for forming the reset of the elements from existing elements. • Grammars – Languages are spe ...
INTRODUCTION TO LOGIC Lecture 6 Natural Deduction Proofs in
... Proofs in Natural Deduction Proofs in Natural Deduction are trees of L2 -sentences ...
... Proofs in Natural Deduction Proofs in Natural Deduction are trees of L2 -sentences ...
The Compactness Theorem 1 The Compactness Theorem
... Theorem. This will play an important role in the second half of the course when we study predicate logic. This is due to our use of Herbrand’s Theorem to reduce reasoning about formulas of predicate logic to reasoning about infinite sets of formulas of propositional logic. Before stating and proving ...
... Theorem. This will play an important role in the second half of the course when we study predicate logic. This is due to our use of Herbrand’s Theorem to reduce reasoning about formulas of predicate logic to reasoning about infinite sets of formulas of propositional logic. Before stating and proving ...
Lecture 9. Model theory. Consistency, independence, completeness
... predicate logic. Note that the syntax is in each case autonomous in the sense that it is fully described independently of the semantics, and the semantics is inherently relational, involving a compositional mapping from syntactic expressions to objects that belong to a model structure. 1. Syntactic ...
... predicate logic. Note that the syntax is in each case autonomous in the sense that it is fully described independently of the semantics, and the semantics is inherently relational, involving a compositional mapping from syntactic expressions to objects that belong to a model structure. 1. Syntactic ...
THE SYNTAX-SEMANTICS INTERFACE
... question. The observed one-to-one correspondence has motivated an analysis of verbs as mathematical functions. A mathematical function maps an argument to a result. Crucially, a function must take exactly one argument to yield a result, and therefore the one-to-one property of predication is explain ...
... question. The observed one-to-one correspondence has motivated an analysis of verbs as mathematical functions. A mathematical function maps an argument to a result. Crucially, a function must take exactly one argument to yield a result, and therefore the one-to-one property of predication is explain ...
deductive system
... Given a language L, two deductive systems D1 and D2 are deductively equivalent if any theorem deducible from one system is deducible from another. In other words, `D1 A iff `D2 A. There is also a stronger notion of deductive equivalence: D1 is (strongly) deductively equivalent to D2 exactly when ∆ ` ...
... Given a language L, two deductive systems D1 and D2 are deductively equivalent if any theorem deducible from one system is deducible from another. In other words, `D1 A iff `D2 A. There is also a stronger notion of deductive equivalence: D1 is (strongly) deductively equivalent to D2 exactly when ∆ ` ...
Part 1: Propositional Logic
... Obviously, A(F ) depends only on the values of those finitely many variables in F under A. If F contains n distinct propositional variables, then it is sufficient to check 2n valuations to see whether F is satisfiable or not. ⇒ truth table. So the satisfiability problem is clearly decidable (but, by ...
... Obviously, A(F ) depends only on the values of those finitely many variables in F under A. If F contains n distinct propositional variables, then it is sufficient to check 2n valuations to see whether F is satisfiable or not. ⇒ truth table. So the satisfiability problem is clearly decidable (but, by ...
Lecture 7. Model theory. Consistency, independence, completeness
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
