lecture notes in Mathematical Logic
... formal proof. Finding a proof, on the other hand, is usually far from being routine, and to decide provability is in general not even possible. It became a question, then, what exactly should we consider a mechanical procedure; which integer functions, for instance, can we consider to be effectively ...
... formal proof. Finding a proof, on the other hand, is usually far from being routine, and to decide provability is in general not even possible. It became a question, then, what exactly should we consider a mechanical procedure; which integer functions, for instance, can we consider to be effectively ...
Hoare Logic, Weakest Liberal Preconditions
... semantics. It would also be possible to prove the validity of the triple using Hoare logic rules, but that would need some auxiliary results. The proof is performed by induction on the structure of statement s. We detail the proof only for the case of the while loop; the other cases are straightforw ...
... semantics. It would also be possible to prove the validity of the triple using Hoare logic rules, but that would need some auxiliary results. The proof is performed by induction on the structure of statement s. We detail the proof only for the case of the while loop; the other cases are straightforw ...
relevant reasoning as the logical basis of
... extensional notion of material implication (denoted by → in this paper) which is defined as A→B =df ¬(A∧¬B) or A→B =df ¬A∨B. However, the material implication is just a truth-function of its antecedent and consequent but not requires that there must exist a necessarily relevant and/or conditional re ...
... extensional notion of material implication (denoted by → in this paper) which is defined as A→B =df ¬(A∧¬B) or A→B =df ¬A∨B. However, the material implication is just a truth-function of its antecedent and consequent but not requires that there must exist a necessarily relevant and/or conditional re ...
Plausibility structures for default reasoning
... not be satisfied are known (e.g. when using defaults for expressing expected results of indeterministic actions, a common application of default reasoning). However, against the main traditions in default reasoning, the inference by plausibility defined in [4, 5] must satisfy a special case of the r ...
... not be satisfied are known (e.g. when using defaults for expressing expected results of indeterministic actions, a common application of default reasoning). However, against the main traditions in default reasoning, the inference by plausibility defined in [4, 5] must satisfy a special case of the r ...
Justification logic with approximate conditional probabilities
... us the connection between justification formulas and probabilistic formulas. Rules 3 and 4 are infinitary rules of inference and Rule 3 states that the probability of any formula belongs to S. Rule 5 formalizes that in neat models only the empty set has probability zero. This rule makes it possible ...
... us the connection between justification formulas and probabilistic formulas. Rules 3 and 4 are infinitary rules of inference and Rule 3 states that the probability of any formula belongs to S. Rule 5 formalizes that in neat models only the empty set has probability zero. This rule makes it possible ...
A Partially Truth Functional Approach to
... when A is false. We use U to denote the latter case, understood as the absence of a truth value rather ...
... when A is false. We use U to denote the latter case, understood as the absence of a truth value rather ...
Symbolic Logic II
... that P → Q, Q → R 2LP P → R, we need to show that there is some interpretation in which KVI (P → Q) 6= 0, KVI (Q → R) 6= 0 and KVI (P → R) = 0. If KVI (P → R) = 0, then KVI (P) = 1 and KVI (R) = 0. But if KVI (R) = 0 and KVI (Q → R) 6= 0, then KVI (Q) 6= 1. If KVI (Q) 6= 1 and KVI (P) = 1 and KVI (P ...
... that P → Q, Q → R 2LP P → R, we need to show that there is some interpretation in which KVI (P → Q) 6= 0, KVI (Q → R) 6= 0 and KVI (P → R) = 0. If KVI (P → R) = 0, then KVI (P) = 1 and KVI (R) = 0. But if KVI (R) = 0 and KVI (Q → R) 6= 0, then KVI (Q) 6= 1. If KVI (Q) 6= 1 and KVI (P) = 1 and KVI (P ...
Second-Order Logic of Paradox
... Truth values of compound formulas are derived from those of their subformulas by the familiar “truth tables” of Kleene’s (strong) 3-valued logic [9, §64], but whereas for Kleene (thinking of the “middle value” as truth-valuelessness) only the top value (True) is designated, for Priest the top two va ...
... Truth values of compound formulas are derived from those of their subformulas by the familiar “truth tables” of Kleene’s (strong) 3-valued logic [9, §64], but whereas for Kleene (thinking of the “middle value” as truth-valuelessness) only the top value (True) is designated, for Priest the top two va ...
A Resolution-Based Proof Method for Temporal Logics of
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
Logic: Introduction - Department of information engineering and
... Computer scientists design and study systems through the use of formal languages that can themselves be interpreted by a formal system. Formal languages of modern logic serve as a working tool for computer science. Some of the most basic applications of this tool are: • Boolean circuits: The design ...
... Computer scientists design and study systems through the use of formal languages that can themselves be interpreted by a formal system. Formal languages of modern logic serve as a working tool for computer science. Some of the most basic applications of this tool are: • Boolean circuits: The design ...
Propositional Logic - Department of Computer Science
... • Only one person can come in first, etc: represent this using Q, where Q = (¬(L1 ∧ R1) ∧ ¬(L2 ∧ R2) ∧ ¬(L3 ∧ R3) ∧ (R1 ∧ J 1) · · · ) Any interpretation I with I(J ∧ A ∧ P1 ∧ P2 ∧ Q) = 1 corresponds to a possible placing of the three contestants. Logic in Computer Science ...
... • Only one person can come in first, etc: represent this using Q, where Q = (¬(L1 ∧ R1) ∧ ¬(L2 ∧ R2) ∧ ¬(L3 ∧ R3) ∧ (R1 ∧ J 1) · · · ) Any interpretation I with I(J ∧ A ∧ P1 ∧ P2 ∧ Q) = 1 corresponds to a possible placing of the three contestants. Logic in Computer Science ...
On the specification of sequent systems
... to specify and reason about a variety of proof systems. Since the encodings of such logical systems are natural and direct, the meta-theory of linear logic can be used to draw conclusions about the object-level proof systems. More specifically, in [MP02], the authors present a decision procedure for ...
... to specify and reason about a variety of proof systems. Since the encodings of such logical systems are natural and direct, the meta-theory of linear logic can be used to draw conclusions about the object-level proof systems. More specifically, in [MP02], the authors present a decision procedure for ...
Lesson 1
... Examples of deductively valid arguments All agarics (mushrooms) have a strong toxic effect. This apple is an agaric. ---------------------------------------------------------------------Hence This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (fa ...
... Examples of deductively valid arguments All agarics (mushrooms) have a strong toxic effect. This apple is an agaric. ---------------------------------------------------------------------Hence This apple has a strong toxic effect. The argument is valid. But the conclusion is evidently not true (fa ...
Redundancies in the Hilbert-Bernays derivability conditions for
... they are closed under cut. It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with foundations, for it treats logics with quantifiers, and in that case one can dispose entirely of the first and second derivability conditions. re ...
... they are closed under cut. It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with foundations, for it treats logics with quantifiers, and in that case one can dispose entirely of the first and second derivability conditions. re ...
(formal) logic? - Departamento de Informática
... Much of standard mathematics can be done within the framework of intuitionistic logic, but the task is very difficult, so mathematicians use methods of classical logic (as proofs by contradiction). However the philosophy behind intuitionistic logic is appealing for a computer scientist. For an intuiti ...
... Much of standard mathematics can be done within the framework of intuitionistic logic, but the task is very difficult, so mathematicians use methods of classical logic (as proofs by contradiction). However the philosophy behind intuitionistic logic is appealing for a computer scientist. For an intuiti ...
Notions of locality and their logical characterizations over nite
... logic, cf. [1]. Another area of application is descriptive complexity. It turns out that familiar logics capture complexity classes over classes of (ordered) nite structures, cf. [8, 18]. Since compactness fails in restriction to nite structures [15], to prove results about the limits of expressi ...
... logic, cf. [1]. Another area of application is descriptive complexity. It turns out that familiar logics capture complexity classes over classes of (ordered) nite structures, cf. [8, 18]. Since compactness fails in restriction to nite structures [15], to prove results about the limits of expressi ...
Intuitionistic modal logic made explicit
... the -modality in families of so-called justification terms. Instead of formulas A, meaning that A is known, justification logics include formulas t : A, meaning that A is known for reason t. Artemov’s original semantics for the first justification logic, the Logic of Proofs LP, was a provability s ...
... the -modality in families of so-called justification terms. Instead of formulas A, meaning that A is known, justification logics include formulas t : A, meaning that A is known for reason t. Artemov’s original semantics for the first justification logic, the Logic of Proofs LP, was a provability s ...
Deductive Reasoning
... alternative views that deny this claim. One view is that humans do not possess a generalpurpose mechanism for deductive reasoning, but rather a different kind of generalpurpose reasoning mechanism, for example one devoted to probabilistic or explanatory reasoning. A different view is that humans lac ...
... alternative views that deny this claim. One view is that humans do not possess a generalpurpose mechanism for deductive reasoning, but rather a different kind of generalpurpose reasoning mechanism, for example one devoted to probabilistic or explanatory reasoning. A different view is that humans lac ...
Integrating Logical Reasoning and Probabilistic Chain Graphs
... where the predicates of the atoms D and Bi are at least unary and the atoms Rj , called templates, express relationships among variables, where at least one variable appearing in the atoms D and Bi occurs in at least one template Rj . An example illustrating this representation is shown below (Examp ...
... where the predicates of the atoms D and Bi are at least unary and the atoms Rj , called templates, express relationships among variables, where at least one variable appearing in the atoms D and Bi occurs in at least one template Rj . An example illustrating this representation is shown below (Examp ...
Introduction to Formal Logic - Web.UVic.ca
... need not be sound. An inference may have one or more false premises, and yet still be valid. Here is an example: All flowers are blue. Roses are flowers. ∴ Roses are blue. The first premise is false, and hence by definition the inference is not sound. However, this inference is still valid: if the p ...
... need not be sound. An inference may have one or more false premises, and yet still be valid. Here is an example: All flowers are blue. Roses are flowers. ∴ Roses are blue. The first premise is false, and hence by definition the inference is not sound. However, this inference is still valid: if the p ...
Beginning Deductive Logic
... This is rather a deep and tricky question, as of course are relatives like: “What is physics?” and “What is economics?” and “What is art?”. Perhaps one has to be either brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to cr ...
... This is rather a deep and tricky question, as of course are relatives like: “What is physics?” and “What is economics?” and “What is art?”. Perhaps one has to be either brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to cr ...
Polarizing Double-Negation Translations
... we must constrain our classical sequent calculus to forbid arbitrary proofs, and in particular to impose that once a rule has been applied on some formula of the right-hand side, the next rule must apply on the corresponding subformula of the premiss. Working on the same formula up to some well-chos ...
... we must constrain our classical sequent calculus to forbid arbitrary proofs, and in particular to impose that once a rule has been applied on some formula of the right-hand side, the next rule must apply on the corresponding subformula of the premiss. Working on the same formula up to some well-chos ...
Turner`s Logic of Universal Causation, Propositional Logic, and
... McCain and Turner’s causal action theories have been the basis for the semantics of several expressive action languages, such as C and C+ [11,5]. They have been translated to propositional logic and logic programming. Ferraris [2] provided a translation from causal theories to disjunctive logic prog ...
... McCain and Turner’s causal action theories have been the basis for the semantics of several expressive action languages, such as C and C+ [11,5]. They have been translated to propositional logic and logic programming. Ferraris [2] provided a translation from causal theories to disjunctive logic prog ...
Introduction to Predicate Logic
... • Problem with the semantic rule having to do with quantifiers in Trial 1: In the semantic rule in Trial 1, quantifiers are ranging over constants, individuals that have names. But it could be that there is an individual that does not have a name. Then, this individual will be excluded from being co ...
... • Problem with the semantic rule having to do with quantifiers in Trial 1: In the semantic rule in Trial 1, quantifiers are ranging over constants, individuals that have names. But it could be that there is an individual that does not have a name. Then, this individual will be excluded from being co ...