gentzen

2013

Tutorial 22 Symbolizing Relations.

The Tutorial

Thus far we have considered only 'monadic' predicates-- our atomic formulas consist of a predicate followed by only one term-- for example, Fx. But in English we regularly encounter dyadic predicates or relations. For example, 'Arthur is taller than Bert' cannot be symbolized with the tools we have used so far; what is needed is a relation to represent '...is taller than ...' Txy, say, and then the proposition would be symbolized Tab.

2013

The Tutorial

Existential Instantiation permits you to remove an existential quantifier from a formula which has an existential quantifier as its main connective. It is one of those rules which involves the adoption and dropping of an extra assumption (like ∼I,⊃I,∨E, and ≡I).

The circumstance that Existential Instantiation gets invoked looks like this.

8/11/2026

The Tutorial

There is a rule for adding a Existential Quantifier. This permits the step illustrated by the following proof fragments.

8/12/26

The Tutorial

There also is a rule for adding a Universal Quantifier. This permits the step illustrated by the following proof fragments.

8/12/26

Skill to be acquired:

To understand the concepts of scope, free, bound. To meet substitution and a simplified version of the rule for removing a Universal Quantifier.

Quiz 7 Invalidity

10/30/06 Quiz 7 was formerly known as Quiz 4

Quiz 7 Invalidity

[This has been done in the downloadable application, but the principles apply to the applet version also.]

Under revision 2013

The Tutorial

We will certainly wish to discuss the truth and falsity of formulas with quantifiers in them.

Let us start with an Interpretation

Interpretation 1

Universe= {a,b}
F={a}

Tutorial 16 Symbolization using the quantifiers.

2013

Skill to be acquired in this tutorial:

To learn how to use the Universal and Existential Quantifiers in symbolizing propositions.

The Tutorial

In Predicate Logic there are two new logical connectives, the Universal Quantifier (∀x) and the Existential Quantifier (∃x). These are used for symbolizing certain English constructions (they also have their own rules of inference and their own semantics, which we will learn about later).

8/4/13

The Tutorial

A few concepts are needed give a simple portrayal of the truth and falsity of predicate logic formulas.

There is the notion of an Interpretation which consists of a Universe together with an account of how the various symbols in the predicate logic formulas apply in this Universe.

There should be a Universe, which is the collection of the objects that the formulas is about. We write, for example,

Universe = {a,b,c}