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Introduction to Predicate Logic: Syntax, Semantics, and Inference

This presentation introduces predicate logic, including its special symbols, syntax, semantics, and procedures for deductive inference. It also explores truth values in logic and how they are determined for formulas. Additionally, the presentation discusses representing states at different times and the lack of built-in mechanisms in standard logic to account for changes in truth value and changing one's mind. The rationale behind proposing logic as a general-purpose representation and reasoning approach is also examined.

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Introduction to Predicate Logic: Syntax, Semantics, and Inference

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  1. Introduction to AI &AI Principles (Semester 1)WEEK 8 (07/08) [Barnden’s slides only] John Barnden Professor of Artificial Intelligence School of Computer Science University of Birmingham, UK

  2. Predicate Logic—The Meat • Predicate logic itself just consists of special symbols such as: •       ( ) • and the syntax (grammar)—how to structure expressions … • and general rules about the semantics of expressions (their meanings) … • and general procedures for doing deductive inference. • The particular symbols for entities, properties and relationships (e.g., TheodosiaKirkbride,happy, taller-than), and their meanings, • are up to the particular representation-developer.

  3. Truth Values in Logic(Part of the Semantics) • Each formula (i.e., expression that makes a statement) is considered to be eitherTRUE (T)or FALSE (F). • This is the formula's truth value(sometimes called its valuation). • Formulas with standard propositional structure (by conjunction, disjunction, implication, negation and related constructors) have their truth values determined rigorously by the truth values of their subformulas. • Related principles handle generalization. • There is no middle ground between TRUE and FALSE. • Clearly, this definiteness is a problem, in the case of many types of statement.

  4. Representing States at Different Times • Standard logic has no inbuilt facility for changes of truth value because of changes in the represented world. • So, the formulas are either about matters that are unchanging by their very nature (“eternal” matters): • 91 is prime • (NB: this is an enternal falsehood) • or are implicitlyabout some particular “time-slice” of the represented world: • Mary has three cars • or are explicitlyabout some particular time-slice: • On 22 Feb 2005 at 10am, Mary had three cars.

  5. Changing One’s Mind • Equally, standard logic does not itself contain any mechanism for a system changing its mind about the truth of something, • e.g changing its mind about whether 91 is a prime number or not, • or about whether Mary has three cars at 10am on 22 Feb 2005. • However, in an AI system, formulas can be made to change in truth-value (either because of world change or system changing its mind, or both), if special, extra mechanisms are added.

  6. Why Logic Has Been Proposed • Desire to capture human rationality. • Desire for general-purpose representation/reasoning approach. • General purpose in terms of both subject matter and role in cognition (info from vision, sentence meanings, internal memory, …) • Desire for common format for explaining what is going on in other representation/reasoning approaches.

  7. Rationality • Much concentration in history on people as (in part) rational beings. • Rationality as involving sound reasoning (deduction) • i.e. reasoning where truth of outputs is guaranteed by truth of inputs. • Analysis of deduction, leading to development of standard logics as (supposedly) important descriptors of human thought. • Spillover of the results, and some of the thinking behind it, into AI. • Standard logic does not have inbuilt facilities for unsound reasoning (involving uncertainty, assumptions, abduction, induction, analogy, …) even though this is crucial in real life … • It’s quite difficult to add such facilities, but there have been many proposals.

  8. “General-Purpose” Aim • Reaction to: completely ad hoc, special-purpose representations, and representation styles, created for specific tasks, specific types of task or specific types of information. Consequence of such special-purpose representations: • Duplication of representational design effort when approaching a new problem. • Difficulty of learning transferrable lessons about representational design. • Need for creating tailored reasoning methods to cope with the specialized representations. • A single AI system may need to deal with a wide variety of tasks and types of information, perhaps all mixed up together. Having disparate representation styles for different types of information causes problems ...... ...

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