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# 20.2 Predicates: Properties and Relations

Terms refer to objects. Predicates are how we say things about those objects.

They are sort of like verbs and verb phrases in English. For example, when we say in English "Pia is guilty", "Pia" is the name and "...is guilty" is the predicate.

You already know how to write Pia: pia or p.

Now here's how we say the whole thing in long form or short form:

Guilty(pia)

G(p)

Now you try.

Unlike names, predicates must always start with a capital letter. The rest of the letters can be mixture of upper and lower case letters, like this:

DrinkingCoffee(pia)

RunningToClass(quinn)

Writing the long form in that way is called CamelCase, which makes it very easy to parse the words and read.

Short form for those sentences would just be:

D(p)

R(q)

Short form and long form have their own advantages and disadvantages. Long form makes it very clear what the symbols mean but it takes a lot more writing. Short form is very fast and convenient, if context makes it clear what the meaning is.

Now let's add more complexity. Here's how we write: Pia is taller than Quinn.

TallerThan(pia,quinn)

Just like BOOL and PROP, there are no spaces in FOL: a formula is just a string of symbols.

Now you try these.

You might have noticed that predicates have gaps in them, Guilty(...), just like the truth-functional connectives have gaps: ... & ... .

Argument places: the gaps in a predicate where we insert terms.

Another name for those gaps is argument places.

The difference is that we insert terms into the argument places of predicates, whereas we insert whole sentences into the argument places of connectives.

Like with connectives, though, we call the number of argument places its arity.

The order of the argument places matters.

The order of the argument places matters. Saying TallerThan(pia,quinn) is different than saying TallerThan(quinn,pia).

As a rule of thumb, we will have the order of the argument places in FOL copy the order of the nouns in English. That is why "Pia is between Quinn and Raquel" is

Between(pia,quinn,raquel)

Lastly, note that it is possible to mix long and short forms. We could write Between(p,q,r) or B(pia,quinn,raquel).

So when we ask you to use long form or short form, we mean both for names and predicates.

Properties: one-place predicates
Relations: two (or more)-place predicates

Lastly, there's some important terminology we'll use for different predicates.

Predicates with one argument place are called properties.

Predicates with two or more argument places are called relations.

20.2 Predicates: Properties and Relations