Classical first-order predicate logic This is a powerful extension of
... Some other quantifiers can be expressed with these. (They can also express each other.) But quantifiers like infinitely many and more than cannot be expressed in first-order logic in general. (They can in, e.g., second-order logic.) ...
... Some other quantifiers can be expressed with these. (They can also express each other.) But quantifiers like infinitely many and more than cannot be expressed in first-order logic in general. (They can in, e.g., second-order logic.) ...
Intuitionistic and Modal Logic
... • Let C0, C1, C2, · · · be a sequence of disjoint countably infinite sets of new constants. It suffices to consider theories in the languages Ln obtained by adding C0 ∪ C1 · · · ∪ Cn to the original language L. We consider theories containing ∃xϕ(x) → ϕ(cϕ) as in the classical Henkin proof. That wil ...
... • Let C0, C1, C2, · · · be a sequence of disjoint countably infinite sets of new constants. It suffices to consider theories in the languages Ln obtained by adding C0 ∪ C1 · · · ∪ Cn to the original language L. We consider theories containing ∃xϕ(x) → ϕ(cϕ) as in the classical Henkin proof. That wil ...
TR-14-06 - Ynot - Harvard University
... monad, HTT already provides the underlying basis of first-class modules with specifications. However, truly reusable components require that their internal invariants are appropriately abstracted. That is, the interfaces of components and objects need to include not only abstract types, but also abs ...
... monad, HTT already provides the underlying basis of first-class modules with specifications. However, truly reusable components require that their internal invariants are appropriately abstracted. That is, the interfaces of components and objects need to include not only abstract types, but also abs ...
x - Homepages | The University of Aberdeen
... Applications of Predicate Logic It is one of the most-used formal notations for writing mathematical definitions, axioms, and theorems. For example, in linear algebra, a partial order is introduced saying that a relation R is reflexive and transitive – and these notions are defined using predicate l ...
... Applications of Predicate Logic It is one of the most-used formal notations for writing mathematical definitions, axioms, and theorems. For example, in linear algebra, a partial order is introduced saying that a relation R is reflexive and transitive – and these notions are defined using predicate l ...
Equivalence for the G3'-stable models semantics
... which two programs are strongly G03 -equivalent also guarantee that two disjunctive programs are strongly equivalent in the p-stable semantics. We present two main results that guarantee G03 strong equivalence, one for two arbitrary programs and another one for a couple of programs of the form P , P ...
... which two programs are strongly G03 -equivalent also guarantee that two disjunctive programs are strongly equivalent in the p-stable semantics. We present two main results that guarantee G03 strong equivalence, one for two arbitrary programs and another one for a couple of programs of the form P , P ...
Modal Languages and Bounded Fragments of Predicate Logic
... What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has b ...
... What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has b ...
Advanced Logic —
... P ROOF. Suppose IND is true, but for a reductio, suppose that LNP is not. Then there is some subset B ⊆ N such that: (1) B is not empty; and (2) B has no least element. Let A = N\B. Clearly 0 ∈ / B (for then 0 would be the least element), so 0 ∈ A. Moreover, since B has no least element, this just m ...
... P ROOF. Suppose IND is true, but for a reductio, suppose that LNP is not. Then there is some subset B ⊆ N such that: (1) B is not empty; and (2) B has no least element. Let A = N\B. Clearly 0 ∈ / B (for then 0 would be the least element), so 0 ∈ A. Moreover, since B has no least element, this just m ...
Higher Order Logic - Theory and Logic Group
... us agree to use the phrase theory for a set of formulas, and V -theory for a set of V -formulas. A rst order theory is then simply a set of rst order formulas, and a second order theory a set of second order formulas. A structure S is a model of a theory ? if every formula in ? is true in S . A co ...
... us agree to use the phrase theory for a set of formulas, and V -theory for a set of V -formulas. A rst order theory is then simply a set of rst order formulas, and a second order theory a set of second order formulas. A structure S is a model of a theory ? if every formula in ? is true in S . A co ...
Higher Order Logic - Indiana University
... us agree to use the phrase theory for a set of formulas, and V -theory for a set of V -formulas. A rst order theory is then simply a set of rst order formulas, and a second order theory a set of second order formulas. A structure S is a model of a theory , if every formula in , is true in S . A co ...
... us agree to use the phrase theory for a set of formulas, and V -theory for a set of V -formulas. A rst order theory is then simply a set of rst order formulas, and a second order theory a set of second order formulas. A structure S is a model of a theory , if every formula in , is true in S . A co ...
Mathematical Logic
... Definition 1.1.2. A propositional language L is a set of propositional atoms p, q, r, . . .. An atomic L-formula is an atom of L. Definition 1.1.3. The set of L-formulas is generated inductively according to the following rules: 1. If p is an atomic L-formula, then p is an L-formula. 2. If A is an L ...
... Definition 1.1.2. A propositional language L is a set of propositional atoms p, q, r, . . .. An atomic L-formula is an atom of L. Definition 1.1.3. The set of L-formulas is generated inductively according to the following rules: 1. If p is an atomic L-formula, then p is an L-formula. 2. If A is an L ...
Fuzzy Control
... involves finding the centroid of the net output fuzzy set L' shown in Figures E.10 and E.11. Although we have used the MIN-MAX rule in the previous section we will begin by deriving the centroid equation for the sum rule shown in Figure E.11. This will illuminate the assumptions made in deriving the ...
... involves finding the centroid of the net output fuzzy set L' shown in Figures E.10 and E.11. Although we have used the MIN-MAX rule in the previous section we will begin by deriving the centroid equation for the sum rule shown in Figure E.11. This will illuminate the assumptions made in deriving the ...
Logic in Nonmonotonic Reasoning
... serve as a basis for thinking. Due to this deficiency, traditional logic cannot discuss what ought to be deduced under ordinary circumstances. Minsky was also one of the first who pointed to monotonicity of logical systems as a source of the problem: Monotonicity: ... In any logistic system, all the ...
... serve as a basis for thinking. Due to this deficiency, traditional logic cannot discuss what ought to be deduced under ordinary circumstances. Minsky was also one of the first who pointed to monotonicity of logical systems as a source of the problem: Monotonicity: ... In any logistic system, all the ...
Characterizations of stable model semantics for logic programs with
... Gelfond–Lifschitz transformation are minimal models. Thus, by the known relationships among different definitions of stable models, the stable model semantics defined in this paper extends the semantics of conventional disjunctive logic programs (Gelfond and Lifschitz 1991), the semantics defined by Mar ...
... Gelfond–Lifschitz transformation are minimal models. Thus, by the known relationships among different definitions of stable models, the stable model semantics defined in this paper extends the semantics of conventional disjunctive logic programs (Gelfond and Lifschitz 1991), the semantics defined by Mar ...
x - Loughborough University Intranet
... The semantic conception of truth (Tarski, 1933, 1944) • The main problem is to give a definition of truth materially adequate and formally correct (Tarski, 1944, 1974, p.269)*. • In this study, I only look for grasping the intuitions expressed by the so named « classical » theory of truth, i.e. this ...
... The semantic conception of truth (Tarski, 1933, 1944) • The main problem is to give a definition of truth materially adequate and formally correct (Tarski, 1944, 1974, p.269)*. • In this study, I only look for grasping the intuitions expressed by the so named « classical » theory of truth, i.e. this ...