Tutorial 14: Some Terminology for the Semantics of Predicate Logic

Logical System

10Software

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 Applet ...

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, move it slightly (to show your clicks are really intended), then release. This will leave you a little circle, probably with four blobs on it. The blobs mean that the individual is still selected. Selected items are not considered to be genuine parts of the drawing. Click down somewhere else in the drawing. This will stop the circle being selected, and thus add it. 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.

If you wish to choose a name for the an entity that you add just type the constant term you want. Try this. Click on the circle in the palette, type lower case 'd', draw your circle. This will add 'd' to the Universe. If you do not wish to choose, the program will automatically choose names for the entities.

Try drawing some properties. Again if you want to give one a name just type the Predicate you want. Try this. Click on the square in the palette, type upper case 'K', draw a square around an existing circle. The drawing is constrained to be well presented.

An individual may or may not have a property. So you can draw (or move) individuals almost anywhere. But properties must apply to at least one individual-- you can draw a property around an individual, but not in empty space.

You can colour or shade your properties as you wish.

Exercise 1 (of 3)

Interpretations Applet

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Exercise 2 (of 3)

Interpretations Applet

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Exercise 3 (of 3)

Interpretations Applet

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