this PDF file
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
Introduction to Formal Logic - Web.UVic.ca
... Definition 1.6. Sentences α and β are consistent iff it is possible for them both to be true. In other words, two sentences are consistent if there is at least one conceivable situation in which they are both true. For example, the sentences ‘All mammals are carnivorous’ and ‘All mice are mammals’ a ...
... Definition 1.6. Sentences α and β are consistent iff it is possible for them both to be true. In other words, two sentences are consistent if there is at least one conceivable situation in which they are both true. For example, the sentences ‘All mammals are carnivorous’ and ‘All mice are mammals’ a ...
Modal_Logics_Eyal_Ariel_151107
... that it is possible in B’s knowledge that A’s forehead is muddy! Remember that: [i]A A ...
... that it is possible in B’s knowledge that A’s forehead is muddy! Remember that: [i]A A ...
On Linear Inference
... may never pick up a given block, even if the rule pickup would permit us to do so. This is more important in this new setting because inferences may be irreversible, so making an inference may constitute a real commitment. If all truths are persistent (and hence inference is monotonic) we can always ...
... may never pick up a given block, even if the rule pickup would permit us to do so. This is more important in this new setting because inferences may be irreversible, so making an inference may constitute a real commitment. If all truths are persistent (and hence inference is monotonic) we can always ...
Sets, Numbers, and Logic
... preliminary sentence: “If a is a real number such that 2a6 + a4 + 3a2 = 0, then a = 0.” However, the problem with all of this is that in ordinary mathematical writing, the hypothesis and conclusion may not be spelled out so clearly; it is you that has to extract them from the prose sentence. For ins ...
... preliminary sentence: “If a is a real number such that 2a6 + a4 + 3a2 = 0, then a = 0.” However, the problem with all of this is that in ordinary mathematical writing, the hypothesis and conclusion may not be spelled out so clearly; it is you that has to extract them from the prose sentence. For ins ...
Hierarchical Introspective Logics
... it was constructed. However an acceptable mathematical proof, in verbal form, was given by Goedel of the truth of his constructed formula provided that it could be assumed that the formal system at issue was ideally consistent. Thus the Goedel formula or assertion is not really of itself mysterious; ...
... it was constructed. However an acceptable mathematical proof, in verbal form, was given by Goedel of the truth of his constructed formula provided that it could be assumed that the formal system at issue was ideally consistent. Thus the Goedel formula or assertion is not really of itself mysterious; ...
CS389L: Automated Logical Reasoning Lecture 1
... Formulas F1 and F2 are equivalent (written F1 ⇔ F2 ) iff for all interpretations I , I |= F1 ↔ F2 F1 ⇔ F2 iff F1 ↔ F2 is valid ...
... Formulas F1 and F2 are equivalent (written F1 ⇔ F2 ) iff for all interpretations I , I |= F1 ↔ F2 F1 ⇔ F2 iff F1 ↔ F2 is valid ...
notes
... Cook’s proof of relative completeness depends on the notion of weakest liberal preconditions. Given a command c and a postcondition Q the weakest liberal precondition is the weakest assertion P such that {P } c {Q} is a valid triple. Here, “weakest” means that any other valid precondition implies P ...
... Cook’s proof of relative completeness depends on the notion of weakest liberal preconditions. Given a command c and a postcondition Q the weakest liberal precondition is the weakest assertion P such that {P } c {Q} is a valid triple. Here, “weakest” means that any other valid precondition implies P ...
Unary negation: ϕ1 ¬ϕ1 T F F T
... (b) Each of the disjuncts is a conjunction, so their main connectives are ∧ (c) Each of the items thus conjoined is a literal – either a 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 infini ...
... (b) Each of the disjuncts is a conjunction, so their main connectives are ∧ (c) Each of the items thus conjoined is a literal – either a 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 infini ...
pdf
... negation (¬), and the modal operator K. Call the resulting language LK 1 (Φ). (We often omit the Φ if it is clear from context or does not play a significant role.) As usual, we define ϕ∨ψ and ϕ ⇒ ψ as abbreviations of ¬(¬ϕ ∧ ¬ψ) and ¬ϕ ∨ ψ, respectively. The intended interpretation of Kϕ varies dep ...
... negation (¬), and the modal operator K. Call the resulting language LK 1 (Φ). (We often omit the Φ if it is clear from context or does not play a significant role.) As usual, we define ϕ∨ψ and ϕ ⇒ ψ as abbreviations of ¬(¬ϕ ∧ ¬ψ) and ¬ϕ ∨ ψ, respectively. The intended interpretation of Kϕ varies dep ...
Universally true assertions
... It is OK to say “ x ¹ 0 is almost always true,” because it is true with only one exception. Here I am using “almost always” in its ordinary sense in conversational English. In fact, “almost always” has a mathematical definition for which the statement “ x ¹ 0 is almost always true” is correct (for ...
... It is OK to say “ x ¹ 0 is almost always true,” because it is true with only one exception. Here I am using “almost always” in its ordinary sense in conversational English. In fact, “almost always” has a mathematical definition for which the statement “ x ¹ 0 is almost always true” is correct (for ...
Propositional Logic
... is true not only of resolution, but also for many other calculi. Of the calculi for automated deduction, resolution plays an exceptional role. Thus we wish to work a bit more closely with it. In contrast to other calculi, resolution has only two inference rules, and it works with formulas in conjunc ...
... is true not only of resolution, but also for many other calculi. Of the calculi for automated deduction, resolution plays an exceptional role. Thus we wish to work a bit more closely with it. In contrast to other calculi, resolution has only two inference rules, and it works with formulas in conjunc ...
notes
... Relative completeness follows by a simple argument: Proof Sketch. Let c be a command and let P and Q be assertions such that the partial correctness specification {P } c {Q} is valid. By Lemma 1 we have ⊨ P =⇒ wlp(c, Q). By Lemma 2 we have ⊢ {wlp(c, Q)} c {Q}. We conclude ⊢ {P } c {Q} using the C ON ...
... Relative completeness follows by a simple argument: Proof Sketch. Let c be a command and let P and Q be assertions such that the partial correctness specification {P } c {Q} is valid. By Lemma 1 we have ⊨ P =⇒ wlp(c, Q). By Lemma 2 we have ⊢ {wlp(c, Q)} c {Q}. We conclude ⊢ {P } c {Q} using the C ON ...
INTERMEDIATE LOGIC – Glossary of key terms
... proposition can be written using only those logical operators. In digital logic, a truth-functionally complete gate is a logic gate that by itself can be used to design any basic logic circuit. Truth table Lesson 2, page 15 A listing of the possible truth values for a set of one or more propositions ...
... proposition can be written using only those logical operators. In digital logic, a truth-functionally complete gate is a logic gate that by itself can be used to design any basic logic circuit. Truth table Lesson 2, page 15 A listing of the possible truth values for a set of one or more propositions ...
What is...Linear Logic? Introduction Jonathan Skowera
... relation between ⊗ and ⊕ is U ⊗ (V ⊕ W ) ∼ = U ⊗ V ⊕ U ⊗ W . The choice of symbols reflects this relation, but proving it requires a complete list of rules of inference, and we aren’t quite there yet. Implication Using the negative, implication can be defined in analogy with the classical case where ...
... relation between ⊗ and ⊕ is U ⊗ (V ⊕ W ) ∼ = U ⊗ V ⊕ U ⊗ W . The choice of symbols reflects this relation, but proving it requires a complete list of rules of inference, and we aren’t quite there yet. Implication Using the negative, implication can be defined in analogy with the classical case where ...
Problems on Discrete Mathematics1
... T rue. We will use T to denote T rue and F to denote F alse for the rest of this book. Comment: The statement s in the above example is a proposition, even though we do not know for sure that there exists a 100-digit prime number, but we are sure that s is either T or F . However, not every mathemat ...
... T rue. We will use T to denote T rue and F to denote F alse for the rest of this book. Comment: The statement s in the above example is a proposition, even though we do not know for sure that there exists a 100-digit prime number, but we are sure that s is either T or F . However, not every mathemat ...
A Calculus for Belnap`s Logic in Which Each Proof Consists of Two
... This is the notion of entailment considered in Belnap [5, 6], but not that of Arieli & Avron [1], who use a single-barrelled notion. The two notions of entailment are coextensional on sets of formulas based on classical connectives only, but not on formulas based on a functionally complete set of co ...
... This is the notion of entailment considered in Belnap [5, 6], but not that of Arieli & Avron [1], who use a single-barrelled notion. The two notions of entailment are coextensional on sets of formulas based on classical connectives only, but not on formulas based on a functionally complete set of co ...
CSE 1400 Applied Discrete Mathematics Proofs
... proofs 12 Theorem 5 (Quotient-Remainder). Given an integer a ∈ Z and an integer n 6= 0, there exists integers q and r, called the quotient and remainder such that a = q·n+r and 0 ≤ r < |n|. Proof. Let us pretend a, n ∈ Z, n 6= 0 are integers and consider the non-empty set of natural numbers A = { a ...
... proofs 12 Theorem 5 (Quotient-Remainder). Given an integer a ∈ Z and an integer n 6= 0, there exists integers q and r, called the quotient and remainder such that a = q·n+r and 0 ≤ r < |n|. Proof. Let us pretend a, n ∈ Z, n 6= 0 are integers and consider the non-empty set of natural numbers A = { a ...
A Textbook of Discrete Mathematics
... Notes: 1. In ordinary language, it is customary to assume some kind of relationship between the antecedent and the consequent in using the conditional. But in logic, the antecedent and the consequent in a conditional statement are not required to refer to the same subject matter. For example, the st ...
... Notes: 1. In ordinary language, it is customary to assume some kind of relationship between the antecedent and the consequent in using the conditional. But in logic, the antecedent and the consequent in a conditional statement are not required to refer to the same subject matter. For example, the st ...
The Foundations
... occurs (happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now (the current situation), the statement “It_is_raining” is false (in the current situation). But if it were raining now, then I would say that “It_is_raining” is true. Fac ...
... occurs (happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now (the current situation), the statement “It_is_raining” is false (in the current situation). But if it were raining now, then I would say that “It_is_raining” is true. Fac ...
Truth and proof
... and logic must be playable by actual human beings, at least in principle. This playability of our "language games" is one of the most characteristic features of the thought of both Wittgenstein and Dummett.” (Hintikka 1996) ...
... and logic must be playable by actual human beings, at least in principle. This playability of our "language games" is one of the most characteristic features of the thought of both Wittgenstein and Dummett.” (Hintikka 1996) ...
mj cresswell
... everything w i l l b e 0 . A n d he thought th is was false because even i f everything now existing will always be 0 it does not follow that always it will be that everything then existing is 0 . But you don't have to interpret BF that way. (See Cresswell 1990, p.96) You can interpret v as ranging ...
... everything w i l l b e 0 . A n d he thought th is was false because even i f everything now existing will always be 0 it does not follow that always it will be that everything then existing is 0 . But you don't have to interpret BF that way. (See Cresswell 1990, p.96) You can interpret v as ranging ...
Chapter 1 Elementary Number Theory
... (iii) Since the steps are valid, it can only be the assumption that is false. (iv) If the negation is false, the original statement must be true. ...
... (iii) Since the steps are valid, it can only be the assumption that is false. (iv) If the negation is false, the original statement must be true. ...
A BRIEF INTRODUCTION TO MODAL LOGIC Introduction Consider
... existence is a tautology, but rather that in every conceivable world the proposition ‘God exists’ is true. Therefore, they argue, it is a necessary truth that God exists. 3The parentheses in this and the preceding rule, while crucial in some instances for rigor, will often be omitted when the meanin ...
... existence is a tautology, but rather that in every conceivable world the proposition ‘God exists’ is true. Therefore, they argue, it is a necessary truth that God exists. 3The parentheses in this and the preceding rule, while crucial in some instances for rigor, will often be omitted when the meanin ...