Lecture 3
... all the ambiguities of English. So, we need a separate ‘logical language’ where many of the ambiguities are filtered out. • We start with propositional logic. • There are several mathematical systems (calculi) for reasoning about propositions: – Truth tables (semantic), Equational Logic (syntactic/s ...
... all the ambiguities of English. So, we need a separate ‘logical language’ where many of the ambiguities are filtered out. • We start with propositional logic. • There are several mathematical systems (calculi) for reasoning about propositions: – Truth tables (semantic), Equational Logic (syntactic/s ...
Exercise
... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
... P(x) it is not enough to show that P(a) is true for one or some a’s. 2. To show that a statement of the form x P(x) is FALSE, it is enough to show that P(a) is false for one a ...
Document
... Lemma: A minor theorem used as a stepping-stone to proving a major theorem. Corollary: A minor theorem proved as an easy consequence of a major theorem. Conjecture: A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.) Theory: The set of all ...
... Lemma: A minor theorem used as a stepping-stone to proving a major theorem. Corollary: A minor theorem proved as an easy consequence of a major theorem. Conjecture: A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.) Theory: The set of all ...
Completeness of Propositional Logic Truth Assignments and Truth
... Theorem (Reformulation of Completeness) Every formally consistent set of sentences is tt-satisfiable. The Completeness Theorem results from applying this to the set ...
... Theorem (Reformulation of Completeness) Every formally consistent set of sentences is tt-satisfiable. The Completeness Theorem results from applying this to the set ...
2015Khan-What is Math-anOverview-IJMCS-2015
... mathematician and perhaps the greatest logician since Aristotle. His famous “incompleteness theorem” was a fundamental result about axiomatic systems, showing that in any axiomatic mathematical system, there are propositions that cannot be proved or disproved within the axioms of the system. In part ...
... mathematician and perhaps the greatest logician since Aristotle. His famous “incompleteness theorem” was a fundamental result about axiomatic systems, showing that in any axiomatic mathematical system, there are propositions that cannot be proved or disproved within the axioms of the system. In part ...
Propositional Logic Proof
... propositional logic statement, (2) each statement is a premise or follows unequivocally by a previously established rule of inference from the truth of previous statements, and (3) the last statement is the conclusion. A very constrained form of proof, but a good starting point. Interesting proofs w ...
... propositional logic statement, (2) each statement is a premise or follows unequivocally by a previously established rule of inference from the truth of previous statements, and (3) the last statement is the conclusion. A very constrained form of proof, but a good starting point. Interesting proofs w ...
WRITING PROOFS Christopher Heil Georgia Institute of Technology
... For example, the contrapositive of the theorem “if it rains then there are clouds in the sky” is the theorem “if there are no clouds in the sky then it is not raining.” These are logically the SAME statement. So, if you want to prove Theorem I and don’t see how, you can try proving Theorem II instea ...
... For example, the contrapositive of the theorem “if it rains then there are clouds in the sky” is the theorem “if there are no clouds in the sky then it is not raining.” These are logically the SAME statement. So, if you want to prove Theorem I and don’t see how, you can try proving Theorem II instea ...
A(x)
... completeness of the 1st order predicate calculus, which was expected. He even proved the strong completeness: if SA |= T then SA |– T (SA – a set of assumptions). But Hilbert wanted more: he supposed that all the truths of mathematics can be proved in this mechanic finite way. That is, that a theory ...
... completeness of the 1st order predicate calculus, which was expected. He even proved the strong completeness: if SA |= T then SA |– T (SA – a set of assumptions). But Hilbert wanted more: he supposed that all the truths of mathematics can be proved in this mechanic finite way. That is, that a theory ...
A(x)
... completeness of the 1st order predicate calculus, which was expected. He even proved the strong completeness: if SA |= T then SA |– T (SA – a set of assumptions). But Hilbert wanted more: he supposed that all the truths of mathematics can be proved in this mechanic finite way. That is, that a theory ...
... completeness of the 1st order predicate calculus, which was expected. He even proved the strong completeness: if SA |= T then SA |– T (SA – a set of assumptions). But Hilbert wanted more: he supposed that all the truths of mathematics can be proved in this mechanic finite way. That is, that a theory ...