How To Do Proofs (for 1B Semantics of Programming Languages)
... The focus here is on doing informal but rigorous proofs. These are rather different from the formal proofs, in Natural Deduction or Sequent Calculus, that were introduced in the Logic and Proof course. Formal proofs are derivations in one of those proof systems – they are in a completely well-define ...
... The focus here is on doing informal but rigorous proofs. These are rather different from the formal proofs, in Natural Deduction or Sequent Calculus, that were introduced in the Logic and Proof course. Formal proofs are derivations in one of those proof systems – they are in a completely well-define ...
Guarded negation
... Several answers to these questions have been proposed. The first one is to consider the two variable fragment of first-order logic, which is decidable and has the finite model property [12]. Unfortunately, this observation does not go very far towards explaining the robust decidability of modal logi ...
... Several answers to these questions have been proposed. The first one is to consider the two variable fragment of first-order logic, which is decidable and has the finite model property [12]. Unfortunately, this observation does not go very far towards explaining the robust decidability of modal logi ...
- Horn-Representation of a Concept Lattice,
... Relationships between FCA and Horn formulas have been pointed out by several other authors during the recent years. In [3] Horn functions of general closure systems are constructed (for example the closure systems of concept intents) in order to achieve a translation of certain notions of FCA (in ca ...
... Relationships between FCA and Horn formulas have been pointed out by several other authors during the recent years. In [3] Horn functions of general closure systems are constructed (for example the closure systems of concept intents) in order to achieve a translation of certain notions of FCA (in ca ...
Syllogistic Logic with Complements
... other. The only purpose of the axioms All X are X is to derive these sentences from all sets; otherwise, the axioms are invisible. The names “Barbara” and “Darii” are traditional from Aristotelian syllogisms. But the (Antitone) rule is not part of traditional syllogistic reasoning. It is possible to ...
... other. The only purpose of the axioms All X are X is to derive these sentences from all sets; otherwise, the axioms are invisible. The names “Barbara” and “Darii” are traditional from Aristotelian syllogisms. But the (Antitone) rule is not part of traditional syllogistic reasoning. It is possible to ...
The Pure Calculus of Entailment Author(s): Alan Ross Anderson and
... In order to introduce terminology and to exemplify a pattern of argument which we shall have further occasion to use, we shall reproduce Fitch's proof that the two formulations are equivalent. To see that the subproof formulation HI* contains the axiomatic formulation HI, we deduce the axioms of HI ...
... In order to introduce terminology and to exemplify a pattern of argument which we shall have further occasion to use, we shall reproduce Fitch's proof that the two formulations are equivalent. To see that the subproof formulation HI* contains the axiomatic formulation HI, we deduce the axioms of HI ...
Transfinite progressions: A second look at completeness.
... than an extension by n-reflection, unless the same formula is used to define the axioms of T in both extensions. (This is a consequence of the fact, which will emerge below, that definitions φ and of the axioms of T can be chosen so that T + REF0 (φ) proves the consistency of T + REFn ().) In the ca ...
... than an extension by n-reflection, unless the same formula is used to define the axioms of T in both extensions. (This is a consequence of the fact, which will emerge below, that definitions φ and of the axioms of T can be chosen so that T + REF0 (φ) proves the consistency of T + REFn ().) In the ca ...
File
... judicious combination of all the above methods of naming techniques is used throughout this course to describe the language structure. Now we are in a convenient position to write the statement. (11) “ Haritha is clever “is true. (12) (7) is true. The above representation is confirmed with accepted ...
... judicious combination of all the above methods of naming techniques is used throughout this course to describe the language structure. Now we are in a convenient position to write the statement. (11) “ Haritha is clever “is true. (12) (7) is true. The above representation is confirmed with accepted ...
Semantic Tableau Proof System for First-Order Logic
... Let U be a nonempty set, and let S be a set of (signed or unsigned) L-U-sentences. Definition 7: S is closed if S contains a conjugate pair of L-U-sentences. In other words, for some L-U-sentence A, S contains TA, FA. Definition 8: S is U-replete if S “obeys the tableau rules” with respect to U. We ...
... Let U be a nonempty set, and let S be a set of (signed or unsigned) L-U-sentences. Definition 7: S is closed if S contains a conjugate pair of L-U-sentences. In other words, for some L-U-sentence A, S contains TA, FA. Definition 8: S is U-replete if S “obeys the tableau rules” with respect to U. We ...
Discrete Event Calculus Deduction using First
... the one-to-one unification of atoms’ arguments. Note, however, that while this prevents the deduction of anomalies such as holdsAt(tapOn, waterLevel(3)), it does not allow deduction of the negations of such anomalies, e.g., it is not possible to deduce ...
... the one-to-one unification of atoms’ arguments. Note, however, that while this prevents the deduction of anomalies such as holdsAt(tapOn, waterLevel(3)), it does not allow deduction of the negations of such anomalies, e.g., it is not possible to deduce ...
Answer Sets for Propositional Theories
... this note, we propose a new definition of equilibrium logic, equivalent to Pearce’s definition, which uses the concept of a reduct, as in the one used in the standard definition of an answer sets. Second, we apply the generalized concept of an answer set to the problem of defining the semantics of a ...
... this note, we propose a new definition of equilibrium logic, equivalent to Pearce’s definition, which uses the concept of a reduct, as in the one used in the standard definition of an answer sets. Second, we apply the generalized concept of an answer set to the problem of defining the semantics of a ...
Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory
... In this section we describe the logic BL∀∆ and the theory FST, which is a fuzzy set theory over BL∀∆. See [8] for details on the logic and [11] for development of the set theory; some degree of familiarity with both is an advantage in technical details to follow. The logic BL∀∆ is the first-order Ba ...
... In this section we describe the logic BL∀∆ and the theory FST, which is a fuzzy set theory over BL∀∆. See [8] for details on the logic and [11] for development of the set theory; some degree of familiarity with both is an advantage in technical details to follow. The logic BL∀∆ is the first-order Ba ...
PREDICATE LOGIC
... St A only affects the free occurrence of the variable x. For example, Syx ∀ x P (x) is still ∀ x P (x), that is, the variable x is not free. However„ Syx (Q(x)∧∀ x P (x)) yields Q(y) ∧ ∀ x P (x). Hence, instantiation treats the variable x differently, depending on whether it is free or bound, even i ...
... St A only affects the free occurrence of the variable x. For example, Syx ∀ x P (x) is still ∀ x P (x), that is, the variable x is not free. However„ Syx (Q(x)∧∀ x P (x)) yields Q(y) ∧ ∀ x P (x). Hence, instantiation treats the variable x differently, depending on whether it is free or bound, even i ...
CS 399: Constructive Logic Final Exam (Sample Solution) Name Instructions
... Hyp Hyp Γ, ¬A, A ` Γ, ¬A, A ` Γ, ¬A, A ` ⊥ ...
... Hyp Hyp Γ, ¬A, A ` Γ, ¬A, A ` Γ, ¬A, A ` ⊥ ...
Quantified Equilibrium Logic and the First Order Logic of Here
... introduced in [25, 26], and its monotonic base logic, here-and-there. We present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternati ...
... introduced in [25, 26], and its monotonic base logic, here-and-there. We present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternati ...
Interactive Theorem Proving in Coq and the Curry
... statement is true, we rely on a proof. Normally, proofs are given in natural languages, which are full of ambiguities. A proof of a proposition is an argument, that can convince anyone of the correctness of the proposition. To be of any use, a proof should always be finite. We often omit many small ...
... statement is true, we rely on a proof. Normally, proofs are given in natural languages, which are full of ambiguities. A proof of a proposition is an argument, that can convince anyone of the correctness of the proposition. To be of any use, a proof should always be finite. We often omit many small ...
Second-order Logic
... are always interpreted as ranging over the entire domain. But, crucially, quantification is only allowed over elements of the domain, and so only variables are allowed to follow a quantifier. In second-order logic, both the language and the definition of satisfaction are extended to include free and ...
... are always interpreted as ranging over the entire domain. But, crucially, quantification is only allowed over elements of the domain, and so only variables are allowed to follow a quantifier. In second-order logic, both the language and the definition of satisfaction are extended to include free and ...
Algebraizing Hybrid Logic - Institute for Logic, Language and
... on at least one branch of the tableau will be satisfiable by label too. Remark 2.4.2. The systematic tableau construction is defined in [5]. Roughly speaking, this construction is needed in order to prove strong completeness. Theorem 2.4.1. ([5]) Any consistent set of formulas in countable language ...
... on at least one branch of the tableau will be satisfiable by label too. Remark 2.4.2. The systematic tableau construction is defined in [5]. Roughly speaking, this construction is needed in order to prove strong completeness. Theorem 2.4.1. ([5]) Any consistent set of formulas in countable language ...
pdf - at www.arxiv.org.
... rewriting systems by introducing a transformation method that translates any logic program without existential variables into a term rewriting system (we call this process functionalisation). After functionalisation, standard term rewriting methods for detecting termination can be applied. We also g ...
... rewriting systems by introducing a transformation method that translates any logic program without existential variables into a term rewriting system (we call this process functionalisation). After functionalisation, standard term rewriting methods for detecting termination can be applied. We also g ...
Judgment and consequence relations
... βp pq 1. It is important to realise that valuations are just a matter of separating out the variables from the constants. A constant is a proposition that does not change truth value. It is either true or false; if it is true, it is always true. If it is false, it must always be so. Variables can ...
... βp pq 1. It is important to realise that valuations are just a matter of separating out the variables from the constants. A constant is a proposition that does not change truth value. It is either true or false; if it is true, it is always true. If it is false, it must always be so. Variables can ...
PROBLEM SOLVING THROUGH FIRST-ORDER LOGIC
... the construction of the parse tree, that is the most suitable representation for computer processing anyway. If we consider only non-existential formulas, then the resolution is completely trivial and does not require anything special. I decided to write the thesis in English, since I hope that it m ...
... the construction of the parse tree, that is the most suitable representation for computer processing anyway. If we consider only non-existential formulas, then the resolution is completely trivial and does not require anything special. I decided to write the thesis in English, since I hope that it m ...
Lectures on Proof Theory - Create and Use Your home.uchicago
... of the principles of logic to be used in deriving theorems from the axioms, so that logic itself could be axiomatized. In this case, too, the timing was just right: the analysis of logic in Frege’s Begriffsschrift was just what was required. (There was a remarkable meeting here of supply and demand. ...
... of the principles of logic to be used in deriving theorems from the axioms, so that logic itself could be axiomatized. In this case, too, the timing was just right: the analysis of logic in Frege’s Begriffsschrift was just what was required. (There was a remarkable meeting here of supply and demand. ...
Propositional Logic - Department of Computer Science
... P1 = ((L1 ∨ L2 ∨ L3) ∧ (R1 ∨ R2 ∨ R3) ∧ (J 1 ∨ J 2 ∨ J 3)) and P2 = (¬(L1 ∧ L2) ∧ ¬(L1 ∧ L3) ∧ ¬(L2 ∧ L3) ∧ ¬(R1 ∧ R2) · · · ) • 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 ...
... P1 = ((L1 ∨ L2 ∨ L3) ∧ (R1 ∨ R2 ∨ R3) ∧ (J 1 ∨ J 2 ∨ J 3)) and P2 = (¬(L1 ∧ L2) ∧ ¬(L1 ∧ L3) ∧ ¬(L2 ∧ L3) ∧ ¬(R1 ∧ R2) · · · ) • 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 ...
propositional logic extended with a pedagogically useful relevant
... All this will sound familiar to people acquainted with the work of Anderson and Belnap. The language being W 1 , there is no need for index sets; the star will be sufficient to recall whether the hypothesis of the subproof is or is not relevant to the conclusion of the subproof. If it is, an arrow c ...
... All this will sound familiar to people acquainted with the work of Anderson and Belnap. The language being W 1 , there is no need for index sets; the star will be sufficient to recall whether the hypothesis of the subproof is or is not relevant to the conclusion of the subproof. If it is, an arrow c ...
Introduction to Formal Logic - Web.UVic.ca
... This inference fulfils condition (i): there is no possible case where its premises could be true and its conclusion false. Hence the inference is valid. But the inference also fulfils condition (ii), because its premises are true: all whales are in fact mammals, and all mammals have spinal chords. N ...
... This inference fulfils condition (i): there is no possible case where its premises could be true and its conclusion false. Hence the inference is valid. But the inference also fulfils condition (ii), because its premises are true: all whales are in fact mammals, and all mammals have spinal chords. N ...
Topological Completeness of First-Order Modal Logic
... This is achieved by introducing two constructions that are general enough to be applicable to a wider range of logics. One is, essentially, to regard a first-order modal language as if it were a classical language; we call this “de-modalization” (Subsection 3.1). It enables us to apply the completen ...
... This is achieved by introducing two constructions that are general enough to be applicable to a wider range of logics. One is, essentially, to regard a first-order modal language as if it were a classical language; we call this “de-modalization” (Subsection 3.1). It enables us to apply the completen ...