slides - Computer and Information Science
... proposition by prefixing it with: It is true that . . . and seeing whether the result makes grammatical sense. • Atomic propositions. Intuitively, these are the set of smallest propositions. • Definition: An atomic proposition is one whose truth or falsity does not depend on the truth or falsity of ...
... proposition by prefixing it with: It is true that . . . and seeing whether the result makes grammatical sense. • Atomic propositions. Intuitively, these are the set of smallest propositions. • Definition: An atomic proposition is one whose truth or falsity does not depend on the truth or falsity of ...
PDF
... Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgements of the form φ1 , . . . , φn ` φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ` φ without assumptions, then φ is a theorem of intuitionistic logic. The system is cal ...
... Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgements of the form φ1 , . . . , φn ` φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ` φ without assumptions, then φ is a theorem of intuitionistic logic. The system is cal ...
Conditional and Indirect Proofs
... • Use CP if your conclusion is equivalent to a conditional • Every proof can be solved using IP. So, if all else fails, try ...
... • Use CP if your conclusion is equivalent to a conditional • Every proof can be solved using IP. So, if all else fails, try ...
4 slides/page
... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...
... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...
Propositions as types
... Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgements of the form φ1 , . . . , φn ` φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ` φ without assumptions, then φ is a theorem of intuitionistic logic. The system is cal ...
... Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgements of the form φ1 , . . . , φn ` φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ` φ without assumptions, then φ is a theorem of intuitionistic logic. The system is cal ...
74.419 Artificial Intelligence 2002 Description Logics
... Richard A. Frost, Introduction to Knowledge-Base Systems, Collins, 1986 (out of print) Comments: one of my favourite books; contains (almost) everything you need w.r.t. foundations of classical and non-classical logic; very compact, comprehensive and relatively easy to understand. Allan Ramsay, Form ...
... Richard A. Frost, Introduction to Knowledge-Base Systems, Collins, 1986 (out of print) Comments: one of my favourite books; contains (almost) everything you need w.r.t. foundations of classical and non-classical logic; very compact, comprehensive and relatively easy to understand. Allan Ramsay, Form ...
Set Theory (MATH 6730) HOMEWORK 1 (Due on February 6, 2017
... 8. Prove that {Pair] , Fnd} ` ∀x ¬ x ∈ x by formalizing our informal proof for this statement. 9. As in Russell’s Paradox, consider the class S of all sets A such that A ∈ / A. The L-sentence σ ≡ ¬∃s ∀x (x ∈ s ↔ ¬ x ∈ x) expresses that S is not a set. (i) Prove ` σ by formalizing our informal proof ...
... 8. Prove that {Pair] , Fnd} ` ∀x ¬ x ∈ x by formalizing our informal proof for this statement. 9. As in Russell’s Paradox, consider the class S of all sets A such that A ∈ / A. The L-sentence σ ≡ ¬∃s ∀x (x ∈ s ↔ ¬ x ∈ x) expresses that S is not a set. (i) Prove ` σ by formalizing our informal proof ...
Disjunctive Normal Form
... A direct proof of a conditional statement p q is constructed when the first step is the assumption that p is true, subsequent steps using rules of inference, with the final step showing q must also be true. Indirect proof – if we prove the theorem without starting with the premises and end with th ...
... A direct proof of a conditional statement p q is constructed when the first step is the assumption that p is true, subsequent steps using rules of inference, with the final step showing q must also be true. Indirect proof – if we prove the theorem without starting with the premises and end with th ...
Predicate Logic Review
... The strategy for defining truth in a model that we have just outlined is due in its essentials to Alfred Tarski. It was one of his great achievements, because it showed logicians how to use semantic notions rigorously. Once we’ve defined truth in a model, of course, we can define logical consequence ...
... The strategy for defining truth in a model that we have just outlined is due in its essentials to Alfred Tarski. It was one of his great achievements, because it showed logicians how to use semantic notions rigorously. Once we’ve defined truth in a model, of course, we can define logical consequence ...
PDF
... Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgements of the form φ1 , . . . , φn ` φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ` φ without assumptions, then φ is a theorem of intuitionistic logic. The system is cal ...
... Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgements of the form φ1 , . . . , φn ` φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ` φ without assumptions, then φ is a theorem of intuitionistic logic. The system is cal ...
Mathematical Logic Deciding logical consequence Complexity of
... axioms and inference rules from which it was possible to derive all the tautologies. Unnatural Proofs and deductions in Hilbert axiomatization are awkward and unnatural. Other proof styles, such as Natural Deductions, are more intuitive. As a matter of facts, nobody is practically using Hilbert calc ...
... axioms and inference rules from which it was possible to derive all the tautologies. Unnatural Proofs and deductions in Hilbert axiomatization are awkward and unnatural. Other proof styles, such as Natural Deductions, are more intuitive. As a matter of facts, nobody is practically using Hilbert calc ...
Theories.Axioms,Rules of Inference
... Here is a pictorial proof, that starts from the axioms or other known theorems and applies Rules of Inference to eventually obtain the formula we need to prove. And hence by definition, the above is a theorem. I show this proof format only to explain the formal definition of a proof, but its sometim ...
... Here is a pictorial proof, that starts from the axioms or other known theorems and applies Rules of Inference to eventually obtain the formula we need to prove. And hence by definition, the above is a theorem. I show this proof format only to explain the formal definition of a proof, but its sometim ...
Horseshoe and Turnstiles
... that is, if φ’s truth is not dependent on any other proposition(s). It is always true. We can express this as, (2) ⊧ φ. In this case, φ is also sometimes called a ‘logical truth’. More controversially, we could say that (2) says that φ is an axiom or a self-evident truth. Now, (3) Γ ⊧ φ iff there is ...
... that is, if φ’s truth is not dependent on any other proposition(s). It is always true. We can express this as, (2) ⊧ φ. In this case, φ is also sometimes called a ‘logical truth’. More controversially, we could say that (2) says that φ is an axiom or a self-evident truth. Now, (3) Γ ⊧ φ iff there is ...
CSE 321, Discrete Structures
... • Suppose we want to prove p is true. • Assume p is false, and derive a contradiction ...
... • Suppose we want to prove p is true. • Assume p is false, and derive a contradiction ...
chapter 16
... — A universal proof (or universal derivation) is an ordered list of sentences in which every sentence is either a premise or is derived from earlier lines (not within a completed subproof) using an inference rule. If we are able to prove Φ(xʹ) where xʹ does not appear free in any line above the univ ...
... — A universal proof (or universal derivation) is an ordered list of sentences in which every sentence is either a premise or is derived from earlier lines (not within a completed subproof) using an inference rule. If we are able to prove Φ(xʹ) where xʹ does not appear free in any line above the univ ...
Exam-Computational_Logic-Subjects_2016
... b) syntactic method (resolution, definition of deduction, the theorem of deduction and its reverse) c) direct method (truth table, conjunctive normal form, definition of deduction, the theorem of deduction and its reverse) d) refutation method (resolution, semantic tableau) prove the validity of som ...
... b) syntactic method (resolution, definition of deduction, the theorem of deduction and its reverse) c) direct method (truth table, conjunctive normal form, definition of deduction, the theorem of deduction and its reverse) d) refutation method (resolution, semantic tableau) prove the validity of som ...
(P Q). - Snistnote
... A statement formula which is true regardless of the truth values of the statements which replace the variables in it is called a universally valid formula or a tautology or a logical truth A statement formula which is false regardless of the truth values of the statements which replace the varia ...
... A statement formula which is true regardless of the truth values of the statements which replace the variables in it is called a universally valid formula or a tautology or a logical truth A statement formula which is false regardless of the truth values of the statements which replace the varia ...
PROVING UNPROVABILITY IN SOME NORMAL MODAL LOGIC
... Remark. In fact most of the above introduced rules are rather rule schemata, since every number k yields a rule R(k). One can be easily persuaded that these schemata cannot be restricted to some fixed k. For instance, take RK . An easy induction shows that if we restrict RK to some R(k) then all K a ...
... Remark. In fact most of the above introduced rules are rather rule schemata, since every number k yields a rule R(k). One can be easily persuaded that these schemata cannot be restricted to some fixed k. For instance, take RK . An easy induction shows that if we restrict RK to some R(k) then all K a ...
cs621-lect27-bp-applcation-logic-2009-10-15
... always speak the truth or always lie. A tourist T comes to a junction in the country and finds an inhabitant S of the country standing there. One of the roads at the junction leads to the capital of the country and the other does not. S can be asked only yes/no questions. • Question: What single yes ...
... always speak the truth or always lie. A tourist T comes to a junction in the country and finds an inhabitant S of the country standing there. One of the roads at the junction leads to the capital of the country and the other does not. S can be asked only yes/no questions. • Question: What single yes ...
Negative translation - Homepages of UvA/FNWI staff
... Negative translation It is natural to think of classical logic as an extension of intuitionistic logic as it can be obtained from intuitionistic logic by adding an additional axiom (for instance, the Law of Excluded Middle ϕ ∨ ¬ϕ). However, the opposite point of view makes sense as well: one could a ...
... Negative translation It is natural to think of classical logic as an extension of intuitionistic logic as it can be obtained from intuitionistic logic by adding an additional axiom (for instance, the Law of Excluded Middle ϕ ∨ ¬ϕ). However, the opposite point of view makes sense as well: one could a ...
Tautologies Arguments Logical Implication
... This is (yet another) hot area of computer science. • How do you prove that your program is correct? – You could test it on a bunch of instances. That runs the risk of not exercising all the features of the program. In general, this is an intractable problem. • For small program fragments, formal ve ...
... This is (yet another) hot area of computer science. • How do you prove that your program is correct? – You could test it on a bunch of instances. That runs the risk of not exercising all the features of the program. In general, this is an intractable problem. • For small program fragments, formal ve ...
Monadic Predicate Logic is Decidable
... These 2 sequences of domain objects do not match: c1,..,cn = a,b,a d1,..,dn = a,b,c If a and c are similar then clause 1 is fulfilled, but clause 2 is not (because c1=c3 but d1<>d3) Reason behind clause 2: equality statements in FOPL (e.g. in the sentence AxEy -(x=y)) ...
... These 2 sequences of domain objects do not match: c1,..,cn = a,b,a d1,..,dn = a,b,c If a and c are similar then clause 1 is fulfilled, but clause 2 is not (because c1=c3 but d1<>d3) Reason behind clause 2: equality statements in FOPL (e.g. in the sentence AxEy -(x=y)) ...
Logic in Proofs (Valid arguments) A theorem is a hypothetical
... hypothesis, a tautology, or a consequence of previous members of the chain by using an allowable rule of inference. In creating a formal proof we use Substitution Rules Names don’t matter in a tautology (only the form)! Equivalences do not change truth value! Consider a proof of [(p 6 q) v (q 6 r) v ...
... hypothesis, a tautology, or a consequence of previous members of the chain by using an allowable rule of inference. In creating a formal proof we use Substitution Rules Names don’t matter in a tautology (only the form)! Equivalences do not change truth value! Consider a proof of [(p 6 q) v (q 6 r) v ...