# Truth, deduction, and computation: logic and semantics for computer science

## Master’s Programmes in Computer Science

For example, Access Limited Logic is restricted as I understand it to relations r a,b available when r is accessed, and uses inference rules which only chain forward along such links. There is also a "partitioning" of the Web by making partitioning the rules in order to limit complexity. Philosophically, the semantic web produces more than a set of rules for manipulation of formulae. It defines documents on the Web has having a socially significant meaning. Therefore it is not simply sufficient to demonstrate that one can constrain the semantic web so as to make it isomorphic to a particular algebra of a given system, but one must ensure that a particular mapping is defined so that the web representation of that particular system conveys is semantics in a way that it can meaningfully be combined with the rest of the semantic web.

- Logic for Computer Science/Propositional Logic.
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- Logic and Proof!
- Climb the Green Ladder: Make Your Company and Career More Sustainable.
- Platos Republic: A Dialogue in 16 Chapters.

Electronic commerce needs a solid foundation in this way, and the development of the semantic web is in essential to provide a rigid framework in which to define electronic commerce terms, before electronic commerce expands as a mass of vaguely defined semantics and ad hoc syntax which leaves no room for automatic treatment, and in which the court of law rather than a logical derivation settles arguments.

Practically, the meaning of semantic web data is grounded in non-semantic-web applications which are interfaced to the semantic web. For example, currency transfer or ecommerce applications, which accept semantic web input, define for practical purposes what the terms in the currency transfer instrument mean.

### mathematics and statistics online

I [DanC] think this section is outdated by recent thoughts [] on paradox and the excluded middle. To the level of first order logic, we don't really need to pick one set of axioms in that there are equivalent choices which lead to demonstrably the same results. If we add anything else we have to be careful that it should either be definable in terms of the first order set or that the resulting language is a subset of a well proven logical system -- or else we have a lot of work to do in establishing a new system!

These are two goals to which we explicitly do not aspire in the Semantic Web in order to get in return expressive power. We still require consistency! The world is full of undecidable statements, and intractable problems. The semantic web has to give the power to express such things. Do we need in practice to decide what an agent could deduce from its logic base? No, not in general. The agent may have various kinds of reasoning engine, and in practice also various amounts of connectivity, storage space, access to indexes, and processing power which will determine what it will actually deduce.

Knowing that a certain algorithm may be nondeterministic polynomial in the size of the entire Web may not be at all helpful, as even linear time would be quite impractical. Practical computability may be assured by topological properties of the web, or the existence of know shortcuts such as precompiled indexes and definitive exclusive lists. Keeping a language less powerful than first order predicate calculus is quite reasonable within an application, but not for the Web.

### "+_.G(b)+"

A dream of logicians in the last century to find languages in which all sentences were either true or false, and provably so. This involved trying to restrict the language so as to avoid the possibility of for example self-contradictory statements which can not be categorized as a true or not true.

On the Semantic Web, this looks like a very academic problem, when in fact one anyway operates with a mass of untrustworthy data at any point, and restricts what one uses to a limited subset of the web. Clearly one must not be able to derive a self-contradictory statement, but there is no harm in the language being powerful enough to express it.

Indeed, endorsement systems must give us the power to say "that statement is false" and so loops which if believed prove self-contradictory will arise by accident or design. A typical response of a system which finds a self-contradictory statement might be similar to the response to finding a contradiction, for example, to cease to trust information from the same source or public key.

There is also a fundamental niceness to having a system powerful enough to describe its own rules, of course, just as one expects to be able to write a compiler for a programming language in the same language need to study references from Hayes , esp "Tarski's results on meta-descriptions a consistent language can't be the same expressive power as its own metatheory , Montague's paradox showing that even quite weak languages can't consistently describe their own semantics ".

When Frege tried second-order logic, I understand, Russel showed that his logic was inconsistent. But can we make a language in which is consistent you can't derive a contradiction from its axioms and yet allows enough to for example The sort of rule it is tempting to write is such as to allow the inference of an RDF triple from a message whose semantic content one can algebraically derive that triple.

This breaks the boundary between the premises which deal with the mechanics of the language and the conclusion which is about the subject-matter of the language.

Do we really need to do this, or can we get by with several independent levels of machinery, letting one machine prepare a "believable" message stream and parse it into a graph, and then a second machine which shares no knowledge space with the first, do the reasoning on the result? To me this seems hopeless, as one will in practice want to direct the front end's search for new documents from the needs of the reasoning by the back end. But this is all hunch. Peregrin tries to categorize the needs for and problems with higher order logic HOL in [Peregrin].

His description of Henkinian Understanding of HOL in which predicates are are subclass of objects "individuals" seems to describe my current understanding of the mapping of RDF into logic, with RDF predicates, binary relations, being subclass of RDF nodes. It seems clear that FOL is insufficient in that some sort of induction seems necessary. I agree with Tait Finitism, J. Robert S. Reflective lambda-calculus which matches the power of proof polynomials. So far, we have succeeded in building the reflective lambda-calculus confluent and strongly normalizable for the core underlying system based on the intuitionistic propositions with implication and conjunction.

Next steps here: to capture universal quantifiers, disjunction, classical logic, second order abstraction. Building feasible reflection in Nuprl and related systems. An ability of a language to talk about its own objects sentences, types, proofs is an essential component of its expressive power. In order to catch up with the conciseness of human reasoning a formal deductive system should have a tractable reflection mechanism. Traditional Goedel numbering is notoriously inefficient and cannot be implemented without principal reconstruction which constitute a challenging theoretical and practical problem.

Explicit extensibility mechanism for verification systems.

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A typical problem of verification is to establish a fact F by means of a given formal system V by verifying in V that a certain construction D is a proof of F. It defines documents on the Web has having a socially significant meaning. Therefore it is not simply sufficient to demonstrate that one can constrain the semantic web so as to make it isomorphic to a particular algebra of a given system, but one must ensure that a particular mapping is defined so that the web representation of that particular system conveys is semantics in a way that it can meaningfully be combined with the rest of the semantic web.

Electronic commerce needs a solid foundation in this way, and the development of the semantic web is in essential to provide a rigid framework in which to define electronic commerce terms, before electronic commerce expands as a mass of vaguely defined semantics and ad hoc syntax which leaves no room for automatic treatment, and in which the court of law rather than a logical derivation settles arguments. Practically, the meaning of semantic web data is grounded in non-semantic-web applications which are interfaced to the semantic web. For example, currency transfer or ecommerce applications, which accept semantic web input, define for practical purposes what the terms in the currency transfer instrument mean.

I [DanC] think this section is outdated by recent thoughts [] on paradox and the excluded middle. To the level of first order logic, we don't really need to pick one set of axioms in that there are equivalent choices which lead to demonstrably the same results. If we add anything else we have to be careful that it should either be definable in terms of the first order set or that the resulting language is a subset of a well proven logical system -- or else we have a lot of work to do in establishing a new system!

These are two goals to which we explicitly do not aspire in the Semantic Web in order to get in return expressive power. We still require consistency! The world is full of undecidable statements, and intractable problems. The semantic web has to give the power to express such things. Do we need in practice to decide what an agent could deduce from its logic base?

No, not in general. The agent may have various kinds of reasoning engine, and in practice also various amounts of connectivity, storage space, access to indexes, and processing power which will determine what it will actually deduce.

www.modernylekarnik.sk/modules/boqafef/ Knowing that a certain algorithm may be nondeterministic polynomial in the size of the entire Web may not be at all helpful, as even linear time would be quite impractical. Practical computability may be assured by topological properties of the web, or the existence of know shortcuts such as precompiled indexes and definitive exclusive lists.

- Many-Valued First-Order Logics with Probabilistic Semantics.
- Arts and Aesthetics in a Globalizing World!
- The Infrared & Electro-Optical Systems Handbook. Atmospheric Propagation of Radiation;

Keeping a language less powerful than first order predicate calculus is quite reasonable within an application, but not for the Web. A dream of logicians in the last century to find languages in which all sentences were either true or false, and provably so. This involved trying to restrict the language so as to avoid the possibility of for example self-contradictory statements which can not be categorized as a true or not true. On the Semantic Web, this looks like a very academic problem, when in fact one anyway operates with a mass of untrustworthy data at any point, and restricts what one uses to a limited subset of the web.

Clearly one must not be able to derive a self-contradictory statement, but there is no harm in the language being powerful enough to express it. Indeed, endorsement systems must give us the power to say "that statement is false" and so loops which if believed prove self-contradictory will arise by accident or design. A typical response of a system which finds a self-contradictory statement might be similar to the response to finding a contradiction, for example, to cease to trust information from the same source or public key.

There is also a fundamental niceness to having a system powerful enough to describe its own rules, of course, just as one expects to be able to write a compiler for a programming language in the same language need to study references from Hayes , esp "Tarski's results on meta-descriptions a consistent language can't be the same expressive power as its own metatheory , Montague's paradox showing that even quite weak languages can't consistently describe their own semantics ". When Frege tried second-order logic, I understand, Russel showed that his logic was inconsistent.

But can we make a language in which is consistent you can't derive a contradiction from its axioms and yet allows enough to for example The sort of rule it is tempting to write is such as to allow the inference of an RDF triple from a message whose semantic content one can algebraically derive that triple.

This breaks the boundary between the premises which deal with the mechanics of the language and the conclusion which is about the subject-matter of the language. Do we really need to do this, or can we get by with several independent levels of machinery, letting one machine prepare a "believable" message stream and parse it into a graph, and then a second machine which shares no knowledge space with the first, do the reasoning on the result? To me this seems hopeless, as one will in practice want to direct the front end's search for new documents from the needs of the reasoning by the back end.

- The Infrared & Electro-Optical Systems Handbook. Atmospheric Propagation of Radiation.
- Quark Matter: Proceedings of the Sixth International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions — Quark Matter 1987 Nordkirchen, FRG, 24–28 August 1987!
- Learning outcomes;
- Aspects of Aristotles Logic of Modalities.
- The Pemberley Chronicles: A Companion Volume to Jane Austens Pride and Prejudice: Book 1.
- Same, Different, Equal: Rethinking Single-Sex Schooling.
- Logic for Computer Science/Propositional Logic - Wikibooks, open books for an open world.

But this is all hunch. Peregrin tries to categorize the needs for and problems with higher order logic HOL in [Peregrin]. His description of Henkinian Understanding of HOL in which predicates are are subclass of objects "individuals" seems to describe my current understanding of the mapping of RDF into logic, with RDF predicates, binary relations, being subclass of RDF nodes. It seems clear that FOL is insufficient in that some sort of induction seems necessary.