gentzen

2013

The Tutorial

The propositional rules of derivation carry over unchanged into Predicate Logic

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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}

Introduction to Semantics

12/25/06

Introduction to Semantics

This video illustrates use of the downloadable application (and the symbol ∧ for 'and'). But, what it depicts and explains is equally good if you happen to be using the web pages applets (or a different symbol for 'and' eg '.').

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8/4/13

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.

The Tutorial:

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

Truth can be discussed in more detail in predicate logic.

Review of new material

A start can be made in predicate logic by taking apart 'atomic' propositions and by re-phrasing what they have to say in a 'entity-has-property' way.

The constant terms a,b,c...h are used to denote entities, the predicates A,B,C...Z are used to denote properties that these entities have, and these are put together by writing the predicate first followed by the term, for example Gb.

10 Software

In predicate logic, many different styles of expression in English get cast into the same 'property-is-had-by-entity' form. For example,

2013

Skills to be acquired in this tutorial:

To start learning how to symbolize propositions using predicate logic.

The Tutorial:

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

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

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

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

Roll your own derivations

2013

You may have derivations of your own that you wish to try. Just type, paste, or drag and drop, them into the panel, select your derivation, and click 'Start from selection'.

[Often copy-and-paste won't work directly from a Web Page; however, usually drag-and-drop will work!]

You will need to use the correct logical symbols. Here they are

F ∴ F ∧ G ∼ ∧ ∨ ⊃ ≡ ∀ ∃ ∴

And the right syntax (the premises separated by commas and then a 'therefore' followed by the conclusion).

Review of Propositional Logic

12/23/05

You now have to tools to appraise propositional arguments.

Let us run through how these might be used with two examples.

Example 1.

Consider the argument

If no human action is free, then no one is responsible for what they do.
If no one is responsible for what they do, no one should be punished.
Therefore
If no human action is free, no one should be punished.

First it should be symbolized