semantics

11/9/2007 10Software

The Tutorial

The semantics of relations proceeds in much the way one would expect-- the new item that has to be taken account of is the order of the terms (because, for example, Tab is not at all the same thing as Tba -- Arthur being taller than Beryl is not the same as Beryl being taller than Arthur).

Let us start with an Interpretation

Interpretation 1

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

12/25/06

Introduction to Free Variables

This video illustrates use of the downloadable application (and the symbol ∧ for 'and' and (∀x) for the universal quantifier, some systems use (x) for this). But, what the film depicts and explains is equally good if you happen to be using the web pages applets (or different symbols for 'and' and the universal quantifier).

10Software

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}

10 Software

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}

6/6/09 10Software

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 propositional 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.

There are many valid arguments which cannot be shown to be valid using sentential logic alone. For example,

Beryl is a philosopher.
All philosophers are wise.
Therefore
Beryl is wise.

1/9/09

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[What is shown below is a Video, with sound. Your computer should be able to run this. If you have external speakers, or similar, you may need to turn them on and turn the volume up.]

The video shows an older implementation (but it should convey the principles).

12/26/13

Skills to be acquired in this tutorial:

To start to learn about identities, the semantics of identities, and typical expressions in English that might use identities.

Why this is useful:

Reasoning with identity is vital for mathematics, philosophy, and many other areas.

12/19/20

Welcome!

The tutorials presented here look at some topics in logic to the level of intermediate to advanced Predicate Calculus. They build on the Tutorials of Easy Deriver which provided the introductory material to Propositional and Predicate Calculus.

The program, widgets, or Notes, should be accompanied by a suitable textbook, such as:

Welcome!

These web pages provide an introduction to logic to the level of Propositional and Predicate Calculus.

The focus of the program is on arguments and the question of whether they are valid. Arguments have the form <list of premises> ∴<conclusion>. An argument is valid if and only if it is not possible for all its premises to be true and its conclusion false at one and the same time; an argument which is not valid is invalid.