CS173: Discrete Math
... • P q: “You can take the flight if and only if you buy a ticket” – This statement is true • If you buy a ticket and take the flight • If you do not buy a ticket and you cannot take the flight ...
... • P q: “You can take the flight if and only if you buy a ticket” – This statement is true • If you buy a ticket and take the flight • If you do not buy a ticket and you cannot take the flight ...
Functional programming and NLP
... what’s the issue with functions? It is possible to define functions in an imperative language like C. The main difference is that in a real functional language, the functions are first-class objects, which means that they can be given as arguments to other functions, and be returned as results. 1 As ...
... what’s the issue with functions? It is possible to define functions in an imperative language like C. The main difference is that in a real functional language, the functions are first-class objects, which means that they can be given as arguments to other functions, and be returned as results. 1 As ...
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 ...
8.1 Symbols and Translation
... If Rx means “x is a rabbit,” and Sx means “x is a snake,” then the premise translates as, “If everything in the universe is a rabbit, everything in the universe is snake.” ◦ The statement is true because the antecedent is false: not everything in the universe is a rabbit. However, the conclusion is ...
... If Rx means “x is a rabbit,” and Sx means “x is a snake,” then the premise translates as, “If everything in the universe is a rabbit, everything in the universe is snake.” ◦ The statement is true because the antecedent is false: not everything in the universe is a rabbit. However, the conclusion is ...
Section 1
... Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is Proposition 3 1. An implication and its contrapositive are logically equivalent 2. The converse a ...
... Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is Proposition 3 1. An implication and its contrapositive are logically equivalent 2. The converse a ...
An Independence Result For Intuitionistic Bounded Arithmetic
... in [M1] to show that certain apparently stronger extensions of S12 are actually stronger assuming the above mentioned complexity assumption. Here, we strengthen the first independence result mentioned above by showing that the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y| → x = |z|) is not provable in the intuitio ...
... in [M1] to show that certain apparently stronger extensions of S12 are actually stronger assuming the above mentioned complexity assumption. Here, we strengthen the first independence result mentioned above by showing that the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y| → x = |z|) is not provable in the intuitio ...
Resolution Algorithm
... • Entailment means that one thing follows from another: KB ╞ α • Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true – E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won” – E.g., x+y = 4 entails 4 = x+y ...
... • Entailment means that one thing follows from another: KB ╞ α • Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true – E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won” – E.g., x+y = 4 entails 4 = x+y ...
A Simple Exposition of Gödel`s Theorem
... numbers which was very strange. I suppose I had been trying to formulate my argument against materialism. Nine years later I was able to go to Princeton to study mathematical logic properly, and on my return tried out my argument on colleagues at Cambridge, then in a paper in 1959 to the Oxford Phil ...
... numbers which was very strange. I suppose I had been trying to formulate my argument against materialism. Nine years later I was able to go to Princeton to study mathematical logic properly, and on my return tried out my argument on colleagues at Cambridge, then in a paper in 1959 to the Oxford Phil ...
Comparing Constructive Arithmetical Theories Based - Math
... (term) a, and also consider the formula ∀z 6 a(x + z = |a| → ∀y 6 t¬A(z, y)) as B(x). To prove P V + ¬¬N P − LIN D `i P V + coN P − LIN D, make similar changes. (iii) This is an immediate consequence of Proposition 2.2 and part (ii). Recall that the theory CP V is the classical closure of IP V an ...
... (term) a, and also consider the formula ∀z 6 a(x + z = |a| → ∀y 6 t¬A(z, y)) as B(x). To prove P V + ¬¬N P − LIN D `i P V + coN P − LIN D, make similar changes. (iii) This is an immediate consequence of Proposition 2.2 and part (ii). Recall that the theory CP V is the classical closure of IP V an ...
com.1 The Compactness Theorem
... Problem com.2. In the standard model of arithmetic N, there is no element k ∈ |N| which satisfies every formula n < x (where n is 0...0 with n 0’s). Use the compactness theorem to show that the set of sentences in the language of arithmetic which are true in the standard model of arithmetic N are a ...
... Problem com.2. In the standard model of arithmetic N, there is no element k ∈ |N| which satisfies every formula n < x (where n is 0...0 with n 0’s). Use the compactness theorem to show that the set of sentences in the language of arithmetic which are true in the standard model of arithmetic N are a ...
PDF
... 3. A set S of formulas is called decidable if the set of Gödel numbers of S is decidable, i.e. if the characteristic function of that set is computable. 4. T is called axiomatizable, if there is a decidable subset of T whose logical consequences are exactly the theorems of T . T is finitely axiomat ...
... 3. A set S of formulas is called decidable if the set of Gödel numbers of S is decidable, i.e. if the characteristic function of that set is computable. 4. T is called axiomatizable, if there is a decidable subset of T whose logical consequences are exactly the theorems of T . T is finitely axiomat ...
Second-Order Logic and Fagin`s Theorem
... CHAPTER 7. SECOND-ORDER LOGIC AND FAGIN’S THEOREM The converse of Lynch’s Theorem is an open question: ...
... CHAPTER 7. SECOND-ORDER LOGIC AND FAGIN’S THEOREM The converse of Lynch’s Theorem is an open question: ...
Logic - Mathematical Institute SANU
... connectives and, or, if, if and only if and not, which are studied in propositional logic, and the quantifier expressions for every and for some, and identity (i.e. the relational expression equals), which, together with the connectives, are studied in predicate logic. If logic is indeed the theory ...
... connectives and, or, if, if and only if and not, which are studied in propositional logic, and the quantifier expressions for every and for some, and identity (i.e. the relational expression equals), which, together with the connectives, are studied in predicate logic. If logic is indeed the theory ...
Lindenbaum lemma for infinitary logics
... We use the existing disjunction connective to combine and ‘remember’ these special formulas. The resulting theory does not prove any of those special formulas and thus is of a ‘finitary’ nature. Then, with a bit more of work, the theorem follows. One may object that the new lemma gives less informat ...
... We use the existing disjunction connective to combine and ‘remember’ these special formulas. The resulting theory does not prove any of those special formulas and thus is of a ‘finitary’ nature. Then, with a bit more of work, the theorem follows. One may object that the new lemma gives less informat ...
Sentence (linguistics)
... From Wikipedia, the free encyclopedia Jump to:navigation, search In the field of linguistics, a sentence is an expression in natural language, often defined to indicate a grammatical and lexical unit consisting of one or more words that represent distinct concepts. A sentence can include words group ...
... From Wikipedia, the free encyclopedia Jump to:navigation, search In the field of linguistics, a sentence is an expression in natural language, often defined to indicate a grammatical and lexical unit consisting of one or more words that represent distinct concepts. A sentence can include words group ...