CS 399: Constructive Logic Final Exam (Sample Solution) Name Instructions
... once. If we relax this restriction and say that each hypothesis must be used at most once, we obtain what is called affine logic. The hypothesis rules become: Γ; ∆, A ` A ...
... once. If we relax this restriction and say that each hypothesis must be used at most once, we obtain what is called affine logic. The hypothesis rules become: Γ; ∆, A ` A ...
page 135 LOGIC IN WHITEHEAD`S UNIVERSAL ALGEBRA
... There are various ways of ‘being together’, that is, various ways of building classes, and this makes ‘togetherness’ an ambiguous notion. The extensionalist Whitehead notes that the particular mode of ‘togetherness’ is already an intension that infects the composite entity. Therefore, in mathematics ...
... There are various ways of ‘being together’, that is, various ways of building classes, and this makes ‘togetherness’ an ambiguous notion. The extensionalist Whitehead notes that the particular mode of ‘togetherness’ is already an intension that infects the composite entity. Therefore, in mathematics ...
The Complexity of Local Stratification - SUrface
... nondeterministic finite-register machine programs. Operationally, the choice instruction nondeterministically assigns a natural number to X in one step. This introduces infinitely branching nondeterministic computations that cannot be simulated in any suitable sense by finitely branching nondetermin ...
... nondeterministic finite-register machine programs. Operationally, the choice instruction nondeterministically assigns a natural number to X in one step. This introduces infinitely branching nondeterministic computations that cannot be simulated in any suitable sense by finitely branching nondetermin ...
Unary negation: ϕ1 ¬ϕ1 T F F T
... sentence letter or the negation of a sentence letter. Every formula can be put into DNF (and there is a corresponding CNF). This shows that every one of the infinite number of truth functions can be written using only the unary ¬ and the binary ∧ and ∨. Alternatively put: The set of connectives, {¬, ...
... sentence letter or the negation of a sentence letter. Every formula can be put into DNF (and there is a corresponding CNF). This shows that every one of the infinite number of truth functions can be written using only the unary ¬ and the binary ∧ and ∨. Alternatively put: The set of connectives, {¬, ...
1 Introduction - Institute for Logic, Language and Computation
... topological spaces have two different, and still equivalent, faces: Idempotent closure operations and filter fields, one and the same object. G. CHOQUET, in 1947, dispensing with transitivity, introduced pretopological spaces whose associated closure operations are no longer idempotent, of course. ...
... topological spaces have two different, and still equivalent, faces: Idempotent closure operations and filter fields, one and the same object. G. CHOQUET, in 1947, dispensing with transitivity, introduced pretopological spaces whose associated closure operations are no longer idempotent, of course. ...
duality of quantifiers ¬8xA(x) 9x¬A(x) ¬9xA(x) 8x¬A(x)
... There are problems that cannot be solved by computer programs (i.e. algorithms) even assuming unlimited time and space. What is an “algorithm”? The following are all equivalent: - C programs, scheme programs, Java programs . . . - Turing machines (Turing’s idea of an “algorithm”) ...
... There are problems that cannot be solved by computer programs (i.e. algorithms) even assuming unlimited time and space. What is an “algorithm”? The following are all equivalent: - C programs, scheme programs, Java programs . . . - Turing machines (Turing’s idea of an “algorithm”) ...
PDF
... practical theorem proving. Most, if not all theorems in practical mathematics do not involve the issue of self-application and will therefore remain provable in a formal system. Undecidability, however, is a bigger problem, as it shows the limitations of fully automated proof systems. While it is po ...
... practical theorem proving. Most, if not all theorems in practical mathematics do not involve the issue of self-application and will therefore remain provable in a formal system. Undecidability, however, is a bigger problem, as it shows the limitations of fully automated proof systems. While it is po ...
STANDARD COMPLETENESS THEOREM FOR ΠMTL 1
... The main result of this paper is a strengthening of the latter theorem. We will show that ΠMTL is complete with respect to a smaller class of algebras. Instead of all ΠMTL-chains, we will prove that it is sufficient to consider only ΠMTL-chains in the real unit interval [0, 1] with finitely many Arc ...
... The main result of this paper is a strengthening of the latter theorem. We will show that ΠMTL is complete with respect to a smaller class of algebras. Instead of all ΠMTL-chains, we will prove that it is sufficient to consider only ΠMTL-chains in the real unit interval [0, 1] with finitely many Arc ...
pdf
... Proof: Assume S has a closed tableau T and consider the set Sp of premises of T . By König’s lemma, T must be finite and so is Sp . Sp must be unsatisfiable, since otherwise every branch containing Sp would be open. Theorem: If all finite subsets of a denumerable set S of pure formulas are satisfia ...
... Proof: Assume S has a closed tableau T and consider the set Sp of premises of T . By König’s lemma, T must be finite and so is Sp . Sp must be unsatisfiable, since otherwise every branch containing Sp would be open. Theorem: If all finite subsets of a denumerable set S of pure formulas are satisfia ...
Redundancies in the Hilbert-Bernays derivability conditions for
... of Hilbert and Bernays [3] , gives a "best possiblerr result in terms of the general class of logics treated in Theorem 1. To see this is the case, one need only consider Kreisel's example of a logic P* on page 154 of [6]. Here, a proof in P = Peano arithmetic is accepted as a proof in P* provided o ...
... of Hilbert and Bernays [3] , gives a "best possiblerr result in terms of the general class of logics treated in Theorem 1. To see this is the case, one need only consider Kreisel's example of a logic P* on page 154 of [6]. Here, a proof in P = Peano arithmetic is accepted as a proof in P* provided o ...
A proposition is any declarative sentence (including mathematical
... Definition: A tautology, or a law of propositional logic, is a statement which is always true A contradiction is a statement whose truth function has all Fs as outputs (in other words, it’s a statement whose negation is a tautology). Two statements are called propositionally equivalent if a tautolog ...
... Definition: A tautology, or a law of propositional logic, is a statement which is always true A contradiction is a statement whose truth function has all Fs as outputs (in other words, it’s a statement whose negation is a tautology). Two statements are called propositionally equivalent if a tautolog ...
Equality in the Presence of Apartness: An Application of Structural
... whenever their premisses are derivable, then so is their conclusion. Weakening and contraction are in addition height-preserving admissible, that is, whenever their premisses are derivable with derivation height bounded by n, then also their conclusion is derivable, with the same bound on the deriva ...
... whenever their premisses are derivable, then so is their conclusion. Weakening and contraction are in addition height-preserving admissible, that is, whenever their premisses are derivable with derivation height bounded by n, then also their conclusion is derivable, with the same bound on the deriva ...
The logic and mathematics of occasion sentences
... ABSTRACT. The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of ...
... ABSTRACT. The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of ...
Predicate Logic
... • Let H(x): “x is a human being.” • Let M(x): “x is mortal.” • The domain of discourse U is all human beings. • “All human beings are mortal.” translates to x (H(x) M(x)) “Sachin is a human being.” translates to H(Sachin) • Therefore, for H(Sachin) M(Sachin) to be true it must be the case that ...
... • Let H(x): “x is a human being.” • Let M(x): “x is mortal.” • The domain of discourse U is all human beings. • “All human beings are mortal.” translates to x (H(x) M(x)) “Sachin is a human being.” translates to H(Sachin) • Therefore, for H(Sachin) M(Sachin) to be true it must be the case that ...
An Automata Theoretic Decision Procedure for the Propositional Mu
... greatest fixpoints (both of which share the fixpoint property). Least fixpoint sentences must have terminating evaluations, while nontermination is consistent with the semantics for greatest fixpoints (this explains why pX. X - false and vX. X E true). Disjunctions p v q and existential program sent ...
... greatest fixpoints (both of which share the fixpoint property). Least fixpoint sentences must have terminating evaluations, while nontermination is consistent with the semantics for greatest fixpoints (this explains why pX. X - false and vX. X E true). Disjunctions p v q and existential program sent ...
Sample pages 2 PDF
... In computer science, an expression denoted the computation of a value from other values; for example, 2 ∗ 9 + 5. In propositional logic, the term formula is used instead. The formal definition will be in terms of trees, because our the main proof technique called structural induction is easy to unde ...
... In computer science, an expression denoted the computation of a value from other values; for example, 2 ∗ 9 + 5. In propositional logic, the term formula is used instead. The formal definition will be in terms of trees, because our the main proof technique called structural induction is easy to unde ...
The Anti-Foundation Axiom in Constructive Set Theories
... Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-well ...
... Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-well ...
Formal deduction in propositional logic
... ’Contrariwise,’ continued Tweedledee, ’if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’ (Lewis Caroll, “Alice in Wonderland”) Formal deduction in propositional logic ...
... ’Contrariwise,’ continued Tweedledee, ’if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’ (Lewis Caroll, “Alice in Wonderland”) Formal deduction in propositional logic ...
Document
... Towards Completeness of SLD-Resolution - To assess completeness of SLD-Resolution it is convenient to characterise the set of sintactically derivable queries. For this purpose the concepts of term models and implication trees are defined. - The main idea is to interpret the set of functional symbol ...
... Towards Completeness of SLD-Resolution - To assess completeness of SLD-Resolution it is convenient to characterise the set of sintactically derivable queries. For this purpose the concepts of term models and implication trees are defined. - The main idea is to interpret the set of functional symbol ...
Using linear logic to reason about sequent systems
... Dual to the initial rule, these clauses have an empty head and two bodies. Other variations on the cut rule also seem possible: namely, one or both of the −◦ can be replaced with ⇒. The four displayed possibilities for the cut rule, however, entail these other variations, for example: ?dBe −◦ bBc−◦ ...
... Dual to the initial rule, these clauses have an empty head and two bodies. Other variations on the cut rule also seem possible: namely, one or both of the −◦ can be replaced with ⇒. The four displayed possibilities for the cut rule, however, entail these other variations, for example: ?dBe −◦ bBc−◦ ...
Reasoning about Action and Change
... fragment of PDL is a definitional extension of the underlying stratified multimodal logic containing only atomic programs terms. (Fine & Schurz speak of atomic program terms rather than action variables). All frame completeness transfer theorems proved there apply to this fragment. This means that i ...
... fragment of PDL is a definitional extension of the underlying stratified multimodal logic containing only atomic programs terms. (Fine & Schurz speak of atomic program terms rather than action variables). All frame completeness transfer theorems proved there apply to this fragment. This means that i ...
Reasoning without Contradiction
... Adding or subtracting a tautology to its premises will have no effect on the validity of an argument, so it is reasonable to believe that tautologies are not required for reasoning. But contradictions, it seems, feature in tried and trusted proof procedures, so one might suppose that, were contradic ...
... Adding or subtracting a tautology to its premises will have no effect on the validity of an argument, so it is reasonable to believe that tautologies are not required for reasoning. But contradictions, it seems, feature in tried and trusted proof procedures, so one might suppose that, were contradic ...
Propositional Logic
... might want to say that it is false or that it is true if some other proposition is true. In this chapter, we first look at the syntactic rules for the language of Propositional Logic. We then look at semantic interpretation for the expressions specified by these rules. Given this semantics, we defin ...
... might want to say that it is false or that it is true if some other proposition is true. In this chapter, we first look at the syntactic rules for the language of Propositional Logic. We then look at semantic interpretation for the expressions specified by these rules. Given this semantics, we defin ...
Propositional logic - Cheriton School of Computer Science
... Note that in natural deduction, these and all subsequent rules apply to whole formulas. They cannot be used to selectively rewrite subformulas. For instance, we cannot rewrite (p ∧ q) as p if it appears buried within a larger formula. More concretely, the rule does not permit the following deduction ...
... Note that in natural deduction, these and all subsequent rules apply to whole formulas. They cannot be used to selectively rewrite subformulas. For instance, we cannot rewrite (p ∧ q) as p if it appears buried within a larger formula. More concretely, the rule does not permit the following deduction ...