Our journey into the world of abstraction has dealt some dividends, so now we shall go a little bit further down the path of abstraction. What we are going to do is to define the propositional calculus as a formal language, rather like a computer programming language. We will use English, together with one or two Greek letters, to define and talk about the formal language of propositional calculus. Thus English is the metalanguage, and propositional calculus is the object language.
Put together, these symbols can be used to represent each of the four major forms of categorical proposition: Thus, for example, we can easily demonstrate the converse of an E proposition: Quantification Rules In order to prove the validity of syllogisms, however, we first need to strip the quantifiers from each statement, apply the appropriate rules of inference, and then restore quantifiers to each statement.
The four quantification rules dictate the conditions under which it is permissible to delete or add a quantifier: In similar fashion, Universal Generalization UG has the form: Only then can we be sure that what holds of it is not some special feature but something that would hold equally well of all " x.
Thus, it is usually best to employ EI as soon as possible certainly, before any application of UI in a proof. Finally, Existential Generalization EG is an argument of the form: Using these quantification rules together with the inference and replacement rules of the propositional calculus, it is possible to prove the validity of any of the valid categorical syllogisms.
For EIO-1, for example: In fact, Russell and Whitehead showed less than a hundred years ago that a higher-order predicate calculus one that fills in the chart above by including variables and quantifiers for predicates as well as for individuals is sufficient to provide logical demonstrations of simple arithmetical truths.
Permissions beyond the scope of this license may be available at http: Last modified 12 November Questions, comments, and suggestions may be sent to:The Foundations: Logic and Proofs Chapter 1, Part III: Proofs. Rules of Inference Section Section Summary predicate logic.
The rules of inference are the essential building block in the construction of valid arguments.
1. Propositional Logic 2. Inference Rules 3. Predicate Logic 4. PHI Introductory Logic Spring 2. Prove the following theorem of the predicate calculus.
You may use any of the rules of inference. // ((∃x)Fx → P) → (x)(Fx → P).
a) Predicate calculus formulas can easily be represented using the programming languages widely used in AI (LISP and Prolog). b) In fact, predicate calculus is the formal basis of Prolog.
In general, a predicate variable followed by any number of individual variables is a wff of the predicate calculus. Such a wff is known as an atomic formula, and the predicate variable in it is said to be of degree n, if n is the number of individual variables following it.
Insofar as inference rules in one way or another express the relation of logical necessity, and since there is a close link between this relation and the operation of implication in the majority of logical calculi, the same link exists between the inference rules and theorems of any calculus, in particular between the primitive inference rules.
Computation in Predicate Logic. In the process of proving the goal to be true, using substitution and the other rules of inference, Prolog substitutes values for the variables in the goal, thereby "computing" an answer.
A structure represents an atomic proposition of predicate calculus, and has the form "atom(parameter list)".