W. Hamilton. Elements of Quaternions. Cambridge University Press, Cambridge, 1899.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....L, then ff is a theorem of S1 . The theorem is proved by classical Henkin s construction as the proof for S by Dutertre [3] Some of the essential technical mechanisms developed by Dutertre can still be used. Below we give a schema of the proof. Step 1: As for classic logics (see for example, [4]) Theorem 3 is equivalent to For any sentence ff in an infinite ITL language L, if :ff is not a theorem of S1 , then ff can be satisfied in an infinite possible worlds model of L. The notions of consistent set of sentences and maximal consistent set of sentences are defined as usual. To prove ....

A.G. Hamilton. Logic for Mathematicians. Cambridge University Press, 1988.

....from A will be called the set of basic operations of A. For each A 2 A, the concrete operation corresponding to an index f will be called the A interpretation of f and denoted f A . We will also use terms of A, constructed from basic operations of A in the standard way (see, for example, [16, 11]) Again, for each A 2 A, the A interpretation of a term t is a concrete operation derived by the usual rules; it will be called a term operation and denoted t A . We say that an (n ary) term t preserves an (m ary) multi sorted relation over the collection of the universes of algebras from A ....

A. G. Hamilton. Logic for mathematicians. Cambridge University Press, 1988.

....set of all propositions P R for every proposition P 2 P. The notation [V 1 7 E 1 ; V 2 7 E 2 ; V n 7 E n ]P means the set of all propositions [V 1 7 E 1 ; V 2 7 E 2 ; V n 7 E n ]P for every proposition P 2 P. 13 See books on mathematical logic by Kleene[Kle67] Hamilton[Ham88], or Ebbinghaus[EFT94] 5 GTS MODELS 86 Note that we are using sets of variables and propositions because in this section, where we are dealing with modeling and validity, the order of variables and propositions in their various lists does not matter. De nition 81 (Logical Equivalence) Two GTS ....

A. G. Hamilton. Logic for Mathematicians. Cambridge University Press, 1988.

....relations, hence it is an element of T r h2i T r 2 ; RELATION, BINARY, TRANSITIVE, REFLEXIVE, ANTI SYM, SYMMETRIC, PARTIAL ORDER 2 T rc 2 because there is a second order relation type. 1 The completeness proof consists mainly in showing how the axioms and inference rules given in [Hamilton, 1988] can be translated to conceptual graphs. Simply put, a higher order type denotes a set of lower order types, and if t 1 is a subtype of t 2 then the denotation of t 1 must be a subset of t 2 s denotation. More specifically, relational concept types denote relation types, non relational concept ....

A. G. Hamilton. Logic for Mathematicians. Cambridge University Press, 1988. Revised edition.

....transformers, some extensions to the guardedcommand language, and Morgan s CSP semantics for action systems. 2.1 Predicates Let Sigma be some non empty set of states. We interpret a predicate OE to be a subset of Sigma and not, for instance a formula in a fixed langauge like first order logic [Ham78]. Our view of predicates is found, for example, in [Nel89] and avoids the necessity of higher order logic when quantifying over predicates: with our approach, the statement for all predicates OE is interpreted as for all subsets OE of Sigma . Two predicates OE and are equivalent, written OE ....

....OE, where s 0 is a state that coincides with s in all coordinates except x , where it has the value y(s) Nel89] For expression E , OE[x nE ] is short for (9 y ffl y = E OE[x ny] Substitutions may be composed and distributed in the usual manner. Standard laws of the predicate calculus [Ham78] may be used to that OE j , or OE V . 2.2 Predicate Transformers A predicate transformer is a function from predicates to predicates. Variable substitution is an example of a predicate transformer: consider the function sub, where for any predicate OE, sub(OE) b = OE[x nE ] Predicate ....

A.G. Hamilton. Logic for Mathematicians. Cambridge University Press, 1978.

....which shares these characteristics, RTL is still useful. The remainder of this section will prove the incompleteness of RTL, and then address the question whether it possess interesting decidable subsets. Presburger arithmetic is complete[38] Goedel showed that integer arithmetic is not [30]. The difference between the two is the absence of multiplication, except by constants, in Presburger arithmetic. If the occurrence relation can be used to introduce multiplication into RTL, then Goedel s incompleteness proof could be carried out for RTL, and RTL would be incomplete (and ....

A. G. Hamilton. Logic For Mathematicians. Cambridge University Press, New York, 1978.

....the temporal operators U (until) and S (since) the operator #, and the quantifier # to atomic formulae. The well known connectives of propositional calculus, such as #, #, and #, are defined in terms of and # in the usual way, whereas # can be obtained by combining and # (cf. Ham78] The other temporal 9 operators, like next X, sometime in the future F , always in the future G, and sometime in the past P , can be derived from U and S as well (cf. ECSD98] for details) The new operator # is a concurrency operator whose intention is clear later on, when the semantics ....

A. G. Hamilton. Logic for Mathematicians. Cambridge University Press, Cambridge, 1978.

....and first order (predicate) logic 1 . Some familiarity with universal algebra [99] will also be useful, so we provide a brief and simplified review of the basic concepts. 1 Three texts which the author recommends as particularly gentle, pleasing to read introductions to sets and logic are [44, 133, 59] 14 1.5.1 Algebras, signatures and terms A multi sorted algebra is a collection A of sets, called the carriers, together with functions of the form f : A 1 . A n # Am (where A k # A) called operations, and possibly some distinguished elements c i called constants. A ....

A. Hamilton. Logic for Mathematicians. Cambridge University Press, 1978.

....r.e. provability predicate and 4 a strong provability predicate) Japaridze s logic is decidable and has a reasonable Kripke semantics. An extensive treatment of Japaridze s logic is given in Boolos [1993b] Bimodal analysis of other unusual provability concepts has been undertaken by Visser [1989,1995] and Shavrukov [1991,1994] Using the work of Guaspari and The Logic of Provability 495 Solovay [1979] Shavrukov [1991] found a complete axiomatization of the bimodal logic of the usual and Rosser s provability predicate for Peano arithmetic (see also section 9) It is worth noting that Rosser s ....

....different modal behaviour; e.g. Rosser consistency of PA is a provable fact, but on the other hand, Rosser s provability predicate is not provably closed under modus ponens. Shavrukov [1994] characterizes the logic of the so called Feferman provability predicate. This work was preceded by Visser [1989,1995], where the concept of provability in PA from nonstandardly finitely many axioms and some other unusual provability concepts were bimodally characterized. These systems were motivated by their connections with interpretability logic, but another motivation originates with Jeroslow and Putnam who ....

[Article contains additional citation context not shown here]

Computability and Logic, 3rd ed., Cambridge University Press. G. Boolos and V. McGee

....P) x:T R . P) We also have the rules monotonicity 1 and monotonicity 2 by replacing with in the above, respectively. Instantiation ( x:T . P) P All of these derived inference rules remain valid in E3. The deduction theorem continues to hold in E3; standard proofs (see for example [9]) rely only on theorems which are valid in E3. With the quantifiers, there are some additional ways of justifying steps in proof presentations. Briefly, when a proof step (i.e. a pair of lines in a proof, together with their connecting or ) is an instance of the conclusion of one of the ....

A. G. Hamilton, Logic for Mathematicians (Cambridge University Press, Cambridge, 1988). 14

....variable. Essentially generators indicate the range of a tuple variable i.e. its possible values. All other atoms are filters and are used to select those tuples which have the desired properties. A generative formula has three properties ffl It is in prenex form. This is a well known normal form[5] in which the quantifiers occur on the left of the expression, with the remainder of the formula quantifier free (i.e. a proposition) This resolves variable scoping issues, and Date[4] recommends it as a natural way of expressing queries. ffl The quantifier free part of the formula is in ....

....of the expression, with the remainder of the formula quantifier free (i.e. a proposition) This resolves variable scoping issues, and Date[4] recommends it as a natural way of expressing queries. ffl The quantifier free part of the formula is in disjunctive normal form, another well known form [5]. ffl All generators in each conjunct occur before, i.e. to the left of, all filters. Essentially the generative property ensures that the range of each tuple variable is known before the desired values are filtered out. Note that safety and generativity are independent, an unsafe formula may be ....

Hamilton A.G. Logic for Mathematicians. Cambridge University Press, 1978.

....with general objects are defined and their conceptual structure and relation to other contexts is analyzed with methods of Formal Concept Analysis. Contents 1. Introduction 2. Class Contexts 3. Extent Contexts 4. Generalized Extent Contexts 1 Introduction In his book Logic and Information [De91], Keith Devlin discusses the problem of individuation, i.e. the question, how we come to see objects in the world as objects of consideration. He points out that it is always a matter of purpose what parts of the world are individuated as objects. These objects are not necessarily atomic entities, ....

K. Devlin: Logic and Information, Cambridge University Press, 1991.

....given an existentially quantified predicate, and the Instantiate rule discharges an existentially quantified obligation given an instance. The concept behind skolemizing a universally quantified expression is to choose a constant which can be of any arbitrary value to substitute the quantifier [17]. This constant is called a skolem constant. To avoid clashes between the representation of a skolem constant and previously defined variables, the skolem constant cannot be a free variable in the target obligation, or any of the hypotheses on the hypothesis list. It is sufficient to check the ....

A.G. Hamilton. Logic for Mathematicians. Cambridge University Press, Cambrige, 1988.

....L, then ff is a theorem of S1 . The theorem is proved by classical Henkin s construction as the proof for S by Dutertre [3] Some of the essential technical mechanisms developed by Dutertre can still be used. Below we give a schema of the proof. Step 1: As for classic logics (see for example, [4]) Theorem 3 is equivalent to For any sentence ff in an infinite ITL language L, if :ff is not a theorem of S1 , then ff can be satisfied in an infinite possible worlds model of L. The notions of consistent set of sentences and maximal consistent set of sentences are defined as usual. To prove ....

A.G. Hamilton. Logic for Mathematicians. Cambridge University Press, 1988.

....Hence, by modus ponens, Delta [ fffg C fi. Proposition 3.6 The property Conditionalization succeeds for the C consequence relation. Proof 3.6 Use ff (fi ff) and (ff fi) ff (fi fl) ff fl) as for classical logic. For more complete proof, see for example proposition 2. 8 in [Ham88]. Proposition 3.7 The property Consistency preservation succeeds for the C consequence relation. Proof 3.7 For all Delta; ff, Delta C ff implies Delta ff. Proposition 3.8 The property Or succeeds for the C consequence relation. Proof 3.8 Assume Delta [ fffg C fl and Delta [ ....

A Hamilton. Logic for Mathematicians. Cambridge University Press, 1988.

No context found.

Computability and Logic, 3rded.,Cambridge University Press. G. Boolos and V. McGee

....usual r.e. provability predicate and 4 a strong provability predicate) Japaridze s logic is decidable and has a reasonable Kripke semantics. An extensive treatment of Japaridze s logic is given in Boolos [1993b] Bimodal analysis of other unusual provability concepts has been undertaken by Visser [1989,1995] and Shavrukov [1991,1994] Using the work of Guaspari and Solovay [1979] Shavrukov [1991] found a complete axiomatization of the bimodal Draft April 17, 378 Giorgi Japaridze and Dick de Jongh logic of the usual and Rosser s provability predicate for Peano arithmetic (see also section 9) It is ....

....different modal behaviour; e.g. Rosser consistency of PA is a provable fact, but on the other hand, Rosser s provability predicate is not provably closed under modus ponens. Shavrukov [1994] characterizes the logic of the so called Feferman provability predicate. This work was preceded by Visser [1989,1995], where the concept of provability in PA from nonstandardly finitely many axioms and some other unusual provability concepts were bimodally characterized. These systems were motivated by their connections with interpretability logic, but another motivation originates with Jeroslow and Putnam who ....

[Article contains additional citation context not shown here]

Computability and Logic, 3rd ed., Cambridge University Press. G. Boolos and V. McGee

....this presentation. Synthesis theorems can be uniformly represented in all decent higher order algebraic specification formalisms or logics. We chose SPECTRUM as logical object language in the context of the KORSO project (see [Broy , 94] which is essentially a three valued version of LCF (see [Paul 87] This decision is partly due to the fact that logical encoding situations are significantly easier to handle in higher order languages, partly due to our belief that LCF like languages are most adequate to formalise transformations for (functional) programs. We would like to emphasise here, ....

.... in case of classical refinement (the input pattern has to follow from the output, or in algebraic jargon: the model class of the output specification is included in the model class of the input specification) or f the Scott definedness ordering in case of robust implementations (see [Paul 87] 3 The general scheme of a synthesis theorem can be defined as follows (cf. HK 93] SYN = param sort S 1 , S m ; F 1 : Typ 1 , Fm :Typ m parameters of the trafo endparam body enrich predefined operation symbols axioms matching variables in verification condition ....

L.C. Paulson: Logic and computation: Interactive proof with Cambridge LCF. Cambridge University Press, 1987.

....this presentation. Synthesis theorems can be uniformly represented in all decent higher order algebraic specification formalisms or logics. We chose SPECTRUM as logical object language in the context of the KORSO project (see [Broy , 94] which is essentially a three valued version of LCF (see [Paul 87] This decision is partly due to the fact that logical encoding situations are significantly easier to handle in higher order languages, partly due to our belief that LCF like languages are most adequate to formalise transformations for (functional) programs. We would like to emphasise here, ....

.... in case of classical refinement (the input pattern has to follow from the output, or in algebraic jargon: the model class of the output specification is included in the model class of the input specification) or f the Scott definedness ordering in case of robust implementations (see [Paul 87] 3 The general scheme of a synthesis theorem can be defined as follows (cf. HK 93] SYN = param sort S 1 , S m ; F 1 : Typ 1 , Fm :Typ m parameters of the trafo endparam body enrich predefined operation symbols axioms matching variables in verification condition ....

L.C. Paulson: Logic and Computation: Interactive proof with Cambridge LCF. Cambridge University Press, 1987.

No context found.

W. Hamilton. Elements of Quaternions. Cambridge University Press, Cambridge, 1899.

No context found.

Logic of Mathematical Discovery. Cambridge University Press, Cambridge, England, 1976.

No context found.

Logic of Mathematical Discovery. Cambridge University Press, Cambridge, England, 1976.

No context found.

Hamilton, A. G. Logic for Mathematicians. Cambridge University Press, New York, NY, USA, 1978 (revised 1988).

No context found.

Logic of Matemathical Discovery, Imre Lakatos, Cambridge University Press, 1979

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC