The four Aristotelian forms are essential knowledge not just in their own right, but also because they are the key to translating many harder claims into FOL as well.

For example, try this, using short form.

The basic structure here is All Ps are Q. So you should first think:

Ax(P(x)->Q(x))

But in this case, the antecedent, the P(x) part, is complex: All happy dogs, namely H(x)&D(x).

That’s how we arrive at

Ax((H(x)&D(x))->R(x))

Similarly, “All happy, barking dogs are running” would just be:

Ax((H(x)&B(x)&D(x))->R(x))

Now try this.

As you can see in these examples, the basic structure is a universal quantifier wide scope around a conditional. In each case we just add some complexity to the antecedent or consequent.

Now try this.

Once we start translating complex forms, the formulas can get pretty messy.

That’s why we ask you to drop parentheses when you can. There’s no need to group the noun complex “happy dog”, as in:

Ex((H(x)&D(x))&R(x))

Since all the connectives are &, those parentheses aren’t needed. And from a logical point of view, all we need to know is that there is some object that is happy, a dog, and running.

You also know from commutativity that an equivalent translation is:

Ex(R(x)&H(x)&D(x))

But in order to all be on the same page, we’ll follow the order in English:

Ex(H(x)&D(x)&R(x))

Here’s another one.

Likes is a binary relation, so it’s important to get the order correct.

L(x,y) means “x likes y”. Since we’re trying to say that Pia likes the dog, we need the name for Pia first: L(p,x). Hence:

Ex(H(x)&D(x)&L(p,x))

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