First-order predicate logic provides a powerful tool for the representation of knowledge and reasoning. We know that there are correct and complete calculi and theorem provers. When proving a theorem, that is, a true statement, such a prover is very helpful because, due to completeness, one knows after finite time that the statement really is true. What if the statement is not true? The completeness theorem does not answer this question. Specifically, there is no process that can prove or refute any formula from PL1* in finite time, for it holds that
Theorem 4.1 The set of valid formulas in first-order predicate logic is semidecidable
This theorem implies that there are programs (theorem provers) which, given a true (valid) formula as input, determine its truth in finite time. If the formula is not valid, however, it may happen that the prover never halts. Propositional logic is decidable because the truth table method provides all models of a formula in finite time. Evidently predicate logic with quantifiers and nested function symbols is a language somewhat too powerful to be decidable.
On the other hand, predicate logic is not powerful enough for many purposes. One often wishes to make statements about sets of predicates or functions. This does not work in PL1 because it only knows quantifiers for variables, but not for predicates or functions.
Kurt Godel showed, shortly after his completeness theorem for PL1, that completeness is lost if we extend PL1 even minimally to construct a higher-order logic. A first-order logic can only quantify over variables. A second-order logic can also quantify over formulas of the first order, and a third-order logic can quantify over formulas of the second order. Even adding only the induction axiom for the natural numbers makes the logic incomplete. The statement "If a predicate p(n) holds for n, then p(n + 1) also holds", or
∀p p(n) ↠ p(n + 1)
is a second-order proposition because it quantifies over a predicate. Godel proved the following theorem:
Theorem 4.2 (Godel's incompleteness theorem) Every axiom system for the natural numbers with addition and multiplication (arithmetic) is incomplete. That is, there are true statements in arithmetic that are not provable.
Godel's proof works with what is called Godelization, in which every arithmetic formula is encoded as a number. It obtains a unique Godel number. Godelization is now used to formulate the proposition
F = "I am not provable."
in the language of arithmetic. This formula is true for the following reason. Assume F is false. Then we can prove F and therefore show that F is not provable. This is a contradiction. Thus F is true and therefore not provable.
The deeper background of this theorem is that mathematical theories (axiom systems) and, more generally, languages become incomplete if the language becomes too powerful. A similar example is set theory. This language is so powerful that one can formulate paradoxes with it. These are statements that contradict themselves, such as the statement about the barbers who all shave those who do not shave themselves.
The dilemma consists therein that with languages which are powerful enough to describe mathematics and interesting applications, we are smuggling contradictions and incompleteness through the back door. This does not mean, however, that higher-order logics are wholly unsuited for formal methods. There are certainly formal systems as well as provers for higher-order logics.
* PL1 Formal language as a convenient interface between man and machine lends itself to the representation of knowledge in the knowledge base. Such languages. There is propositional calculus and first-order predicate logic (PL1).
About the Author
Dr. Wolfgang Ertel is a professor at the Institute for Artificial Intelligence at the Ravensburg-Weingarten University of Applied Sciences, Germany.
This accessible and engaging textbook presents a concise introduction to the exciting field of artificial intelligence (AI). The broad-ranging discussion covers the key subdisciplines within the field, describing practical algorithms and concrete applications in the areas of agents, logic, search, reasoning under uncertainty, machine learning, neural networks, and reinforcement learning. Fully revised and updated, this much-anticipated second edition also includes new material on deep learning.
Topics and features:
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• Includes chapters on predicate logic, PROLOG, heuristic search, probabilistic reasoning, machine learning and data mining, neural networks and reinforcement learning
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• Examines performance evaluation of clustering algorithms, and presents two practical examples explaining Bayes theorem and its relevance in everyday life
• Discusses search algorithms, analyzing the cycle check, explaining route planning for car navigation systems, and introducing Monte Carlo Tree Search
• Includes a section in the introduction on AI and society, discussing the implications of AI on topics such as employment and transportation
Ideal for foundation courses or modules on AI, this easy-to-read textbook offers an excellent overview of the field for students of computer science and other technical disciplines, requiring no more than a high-school level of knowledge of mathematics to understand the material.
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