Tutorial 14: Some Terminology for the Semantics of Predicate Logic

Logical System
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}

It is usual for the Universe to be non-empty. So, for example, Universe = {a,b} or Universe = {a} are in order, but Universe = {} is not.

Then there needs to be an account of how the various symbols in the predicate logic formulas apply in this Universe. There are the constant terms a,b,c... -- at an introductory level we can just let a name a, b name b etc. There are the predicates -- here we just have to say which members of the Universe a predicate applies to, and this can be done by writing, for example, F={a,c} to mean that a and c have the property F; many properties will not apply to any of the objects in the Universe, if, for example, G applies to nothing, this could be written G={} or nothing need be written at all.

Then there is the notion of a formula being true (or false) in an interpretation. For example, consider the formula Fa and

Interpretation 1

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

Interpretation 2

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

the formula is true under Interpretation 1 and false under Interpretation 2. An alternative piece of terminology is often used here-- the formula is said to be satisfied by Interpretation 1 and not satisfied by Interpretation 2.

Formulas containing propositional logical connectives (∼,&,∨,⊃,≡) are handled in the now familiar truth-table way-- for example, the formula Fa∨Fb is satisfied by both Interpretation 1 and Interpretation 2 and the formula Fa&Fb is satisfied by neither of these Interpretations.

If a sentence is satisfied by all interpretations (i.e. it is true in all interpretations) then it is logically true. For example, the formula Fa ∨ ∼Fa is logically true.

(You can see that this simple approach will not be suitable for more advanced work because, for instance, it limits the size of the Universes that we can discuss. There are only twelve constant terms in use (a..l) and yet we may need to discuss larger Universes. Consider, for example, the invalid argument 'There is at least one swan.' therefore 'There are not thirteen swans.' -- to produce a counter-example to this we would need (at least) a thirteen element Universe all of which were swans.)

 


Exercises to accompany Predicate Tutorial 4

In the Interpretation Widget ...

Individuals are circles, properties are rectangles which may or may not surround a particular individual, and, later, we will come to relations which are lines which connect individuals.

If an individual is within a property rectangle; it is considered to have that property. If it is not, it does not. So if you have an individual 'a', and no property 'P' , then a is taken to lack P and so the formula Pa is false.

Formulas written in the Journal Panel are taken to be about the Interpretation Panel.

Try doing some drawing. Click on the circle in the drawing palette, click on the point you want it in the window, you will be guided as to what you need to do. Up in the top left corner is a description of the drawing. You should see that the new entity is now part of the Universe.

Try drawing some properties. Click on the square in the palette, click on the drawing. Again you will be guided.

An individual may or may not have a property. So you can draw (or move) individuals almost anywhere. And similarly the properties can be moved around.

You can colour or shade your properties as you wish.

Exercise 1 (of 3)

Exercise 2 (of 3)

Exercise 3 (of 3)