Lindenbaum lemma for infinitary logics
... Lindenbaum lemma says that for any finitary logic ` (i.e., a finitary substitution-invariant consequence relation over the set of formulas of a given language) each theory (i.e., a set of formulas closed under `) not containing a formula ϕ can be extended into a maximal theory not containing ϕ. The ...
... Lindenbaum lemma says that for any finitary logic ` (i.e., a finitary substitution-invariant consequence relation over the set of formulas of a given language) each theory (i.e., a set of formulas closed under `) not containing a formula ϕ can be extended into a maximal theory not containing ϕ. The ...
Modal logic and the approximation induction principle
... Hennessy-Milner logic [9] is a modal logic for specifying properties of states in a labelled transition system (LTS). Rob van Glabbeek [7] uses this logic to characterize a wide range of process semantics in terms of observations. That is, a process semantics is captured by means of a sublogic of He ...
... Hennessy-Milner logic [9] is a modal logic for specifying properties of states in a labelled transition system (LTS). Rob van Glabbeek [7] uses this logic to characterize a wide range of process semantics in terms of observations. That is, a process semantics is captured by means of a sublogic of He ...
the theory of form logic - University College Freiburg
... categories and formulate rules which take account of this classification? Such a syntax may result in a more restrictive system than predicate logic, perhaps ruling out the two arguably meaningless sentences from above. On the other extreme, why not shoehorn all terms into a single syntactical categ ...
... categories and formulate rules which take account of this classification? Such a syntax may result in a more restrictive system than predicate logic, perhaps ruling out the two arguably meaningless sentences from above. On the other extreme, why not shoehorn all terms into a single syntactical categ ...
A(x)
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
A(x)
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
Propositional Logic - faculty.cs.tamu.edu
... You should very carefully inspect this table! It is critical that you memorize and fully understand the meaning of each connective. The semantics of the language Prop is given by assigning truth values to each proposition in Prop. Clearly, an arbitrary assignment of truth values is not interesting, ...
... You should very carefully inspect this table! It is critical that you memorize and fully understand the meaning of each connective. The semantics of the language Prop is given by assigning truth values to each proposition in Prop. Clearly, an arbitrary assignment of truth values is not interesting, ...
logica and critical thinking
... Be the master of your own life To ask “why?” is to ask for the reasons for some beliefs and opinions (that is, to find out the premises for a certain conclusion) To evaluate how the reasons successfully support the claim (that is, to evaluate how the premises support the conclusion ) A critical thin ...
... Be the master of your own life To ask “why?” is to ask for the reasons for some beliefs and opinions (that is, to find out the premises for a certain conclusion) To evaluate how the reasons successfully support the claim (that is, to evaluate how the premises support the conclusion ) A critical thin ...
Sequent calculus for predicate logic
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
The Importance of Being Earnest: Scepticism and the Limits of
... inquiry is not progressive nor cumulative. Peirce can reject (2), although he accepts to qualify truth as the regulative limit towards which knowledge is constantly proceeding, because he abandons an imagist conception of it. Ultimate truth is indeed unattainable but is not unapproachable. On the co ...
... inquiry is not progressive nor cumulative. Peirce can reject (2), although he accepts to qualify truth as the regulative limit towards which knowledge is constantly proceeding, because he abandons an imagist conception of it. Ultimate truth is indeed unattainable but is not unapproachable. On the co ...
pdf
... validity problem for modal logics (see [7; 9; 14; 15] for some examples). In particular, Ladner [9] showed that the validity (and satisfiability) problem for every modal logic between K and S4 is PSPACE-hard; and is PSPACE-complete for the modal logics K, T, and S4. He also showed that the satisfiab ...
... validity problem for modal logics (see [7; 9; 14; 15] for some examples). In particular, Ladner [9] showed that the validity (and satisfiability) problem for every modal logic between K and S4 is PSPACE-hard; and is PSPACE-complete for the modal logics K, T, and S4. He also showed that the satisfiab ...
Propositional Logic What is logic? Propositions Negation
... Let F(x) be the predicate “x is female”. Let P(x) be the predicate “x is a parent”. Let M(x,y) be the predicate “x is the mother of y”. Let the universe of discourse be the set of all people. We can express the statement “If a person is female and is a parent, then this person is someone’s mother”. ...
... Let F(x) be the predicate “x is female”. Let P(x) be the predicate “x is a parent”. Let M(x,y) be the predicate “x is the mother of y”. Let the universe of discourse be the set of all people. We can express the statement “If a person is female and is a parent, then this person is someone’s mother”. ...
Lecture 11 Artificial Intelligence Predicate Logic
... appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new statements from old ones. ...
... appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new statements from old ones. ...
An Axiomatization of G'3
... to a designated value. The most simple example of a multivalued logic is classical logic where: D = {0, 1}, 1 is the unique designated value, and connectives are defined through the usual basic truth tables. Note that in a multivalued logic, so that it can truly be a logic, the implication connectiv ...
... to a designated value. The most simple example of a multivalued logic is classical logic where: D = {0, 1}, 1 is the unique designated value, and connectives are defined through the usual basic truth tables. Note that in a multivalued logic, so that it can truly be a logic, the implication connectiv ...
Chapter 4. Logical Notions This chapter introduces various logical
... representing the form of m-formulas. Thus (p1Zp2) (viewed now as a metaformula) represents a form whose only instance is the formula (p1Zp2) itself, while (AZB) represents a form whose instances are all the disjunctive formulas. Of course, these formula instances will themselves have 'ordinary' sen ...
... representing the form of m-formulas. Thus (p1Zp2) (viewed now as a metaformula) represents a form whose only instance is the formula (p1Zp2) itself, while (AZB) represents a form whose instances are all the disjunctive formulas. Of course, these formula instances will themselves have 'ordinary' sen ...
A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC–Part
... It is immediate by the conservation requirement that, conversely, a connected modal chain of type (3) can be replaced by a classical, possibly disconnected, chain of type (2). Modal extensions of propositional ([2], [8], [9], [10], [12], [17]) and predicate ([1], [3], [5], [9], [10], [11]) classical ...
... It is immediate by the conservation requirement that, conversely, a connected modal chain of type (3) can be replaced by a classical, possibly disconnected, chain of type (2). Modal extensions of propositional ([2], [8], [9], [10], [12], [17]) and predicate ([1], [3], [5], [9], [10], [11]) classical ...
Ethics of terminology
... • The object must be somehow beforehand or independently acquainted The interpreter must have some collateral observation about the object: ...
... • The object must be somehow beforehand or independently acquainted The interpreter must have some collateral observation about the object: ...
Speaking Logic - SRI International
... Interpretation ascribes an intended sense to these assertions by fixing the meaning of certain symbols, e.g., the logical connectives, equality, and delimiting the variation in the meanings of other symbols, e.g., variables, functions, and predicates. An assertion is valid if it holds in all interpr ...
... Interpretation ascribes an intended sense to these assertions by fixing the meaning of certain symbols, e.g., the logical connectives, equality, and delimiting the variation in the meanings of other symbols, e.g., variables, functions, and predicates. An assertion is valid if it holds in all interpr ...
slides
... In later lectures we shall see that it extends to predicate (i.e., first-order) logic (and beyond!) (Indeed Prolog implements predicate logic, not only propositional logic.) ...
... In later lectures we shall see that it extends to predicate (i.e., first-order) logic (and beyond!) (Indeed Prolog implements predicate logic, not only propositional logic.) ...
CHAPTER 5 SOME EXTENSIONAL SEMANTICS
... Many valued logics in general and 3-valued logics in particular is an old object of study which had its beginning in the work of Lukasiewicz (1920). He was the first to define a 3- valued semantics for a language L¬,∩,∪,⇒ of classical logic, and called it a three valued logic for short. He left the ...
... Many valued logics in general and 3-valued logics in particular is an old object of study which had its beginning in the work of Lukasiewicz (1920). He was the first to define a 3- valued semantics for a language L¬,∩,∪,⇒ of classical logic, and called it a three valued logic for short. He left the ...
A Paedagogic Example of Cut-Elimination
... In his groundbreaking 1934 paper, “Untersuchungen über das logische Schließen” [1934], Gerhard Genzen introduced a new calculus for first-order logic, the so-called sequent calculus LK. In contrast to the then (and now) predominant Hilbert-style calculi, LK’s power lies in its rules, not in its axi ...
... In his groundbreaking 1934 paper, “Untersuchungen über das logische Schließen” [1934], Gerhard Genzen introduced a new calculus for first-order logic, the so-called sequent calculus LK. In contrast to the then (and now) predominant Hilbert-style calculi, LK’s power lies in its rules, not in its axi ...
Boolean unification with predicates
... Instead of working modulo an equational theory, the standard (i.e. syntactic first-order) unification problem has also been extended to allow for higher-order variables. Second-order unification permits, in addition to individual variables, also function variables and asks for a substitution resulti ...
... Instead of working modulo an equational theory, the standard (i.e. syntactic first-order) unification problem has also been extended to allow for higher-order variables. Second-order unification permits, in addition to individual variables, also function variables and asks for a substitution resulti ...
A systematic proof theory for several modal logics
... a theorem. Logical systems are schematic, that is they allow schematic letters that may stand in for arbitrary propositions, so the theory of a logical system for a more restrictive language can naturally be projected to richer languages by application of substitution. Two logical systems are equiva ...
... a theorem. Logical systems are schematic, that is they allow schematic letters that may stand in for arbitrary propositions, so the theory of a logical system for a more restrictive language can naturally be projected to richer languages by application of substitution. Two logical systems are equiva ...
Deciding Intuitionistic Propositional Logic via Translation into
... As in the previous example we refute the main implication by adding a refuting knowledge stage w1 accessible from w0 with w1 I1 ∧ I2 (and thus w1 I1 as well as w1 I2 ) but w1 6 c. From w1 I1 , i.e. w1 (a ⇒ b) ⇒ c and w1 6 c we obtain the refinement w1 6 a ⇒ b which is indicated by the arrow at w1 in ...
... As in the previous example we refute the main implication by adding a refuting knowledge stage w1 accessible from w0 with w1 I1 ∧ I2 (and thus w1 I1 as well as w1 I2 ) but w1 6 c. From w1 I1 , i.e. w1 (a ⇒ b) ⇒ c and w1 6 c we obtain the refinement w1 6 a ⇒ b which is indicated by the arrow at w1 in ...
Logic and Proof
... • We must demonstrate that our specification does not give rise to contradiction (someone loves and does not loves Jill). • We must demonstrate that our specification does not draw the wrong inferences. • We must demonstrate that what we claim holds in the specification does hold. • The demonstratio ...
... • We must demonstrate that our specification does not give rise to contradiction (someone loves and does not loves Jill). • We must demonstrate that our specification does not draw the wrong inferences. • We must demonstrate that what we claim holds in the specification does hold. • The demonstratio ...
Document
... were not previously given at all. What we shall be able to infer from it, cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it. The conclusions we draw from it extend our knowledge, and ought therefore, on KANT’s view, to be regarded as sy ...
... were not previously given at all. What we shall be able to infer from it, cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it. The conclusions we draw from it extend our knowledge, and ought therefore, on KANT’s view, to be regarded as sy ...