bergmann

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.

 

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2013

Reading

Bergmann[2004] The Logic Book Section 10.1.

The Tutorial

Existential Elimination (often called '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.

2013

Reading

Bergmann[2004] The Logic Book Section 10.1.

The Tutorial

There is a rule for adding a Existential Quantifier, Existential Introduction (also commonly known as 'Existential Generalization'). This permits the step illustrated by the following proof fragments.

2013

Skill to be acquired:

To learn the rule for adding a Universal Quantifier.

Reading

Bergmann[2004] The Logic Book Section 10.1.

The Tutorial

There also is a rule for adding a Universal Quantifier (often called 'Universal Generalization'). This permits the step illustrated by the following proof fragments.

The rule is

2013

Skill to be acquired:

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

Reading

Bergmann[2004] The Logic Book Section 10.1.

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.

Reading

Bergmann[2008] The Logic Book Sections 7.4

2013

The Tutorial

The sentential rules of derivation carry over unchanged into Predicate Logic

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1/10/20

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}

1/12/20

Skill to be acquired:

To learn how to interpret simple predicate logic formulas as being true or false.

Why this is useful:

This helps in proving invalidity by the technique of displaying a counter-example.

Reading

Bergmann[2008] The Logic Book Section 8.1

The Tutorial:

In sentential logic, we just took it that each of the atomic sentences either is true or is false-- we did not look into the structure of the sentences.

Truth can be discussed in more detail in predicate logic.