Symbolic Logic I

With Chapter 9, we move from (mainly) propositional logic to predicate logic. We will need to extend and complicate our language.

**1. Primitive Symbols**

We now add two new pieces to our language: variables and quantifiers.

Up until now, the primitive symbols of our language have all been constants, predicates, and connectives.

Now, however, we will introduce two new syntactic categories. First, we need the
notion of a **term**. This is a general category for expressions that can
occupy the argument places of predicates (and function symbols, but we have
mostly been considering languages without function symbols). Constants are
terms, but they are no longer the only kind of term. Variables are
also terms. (So are function symbols with terms in their argument places.)

We use letters early in the alphabet for constants: a, b, c, d, e, f, g.

We use letters late in the alphabet for variables: t, u, v, w, x, y, z

Variables can appear anywhere constants do. So our syntax is now more complex.

Our second new syntactic category is the quantifier. We will have two
quantifiers,
∃
and
∀.
∃
is known as the *existential quantifier* and
∀
is known as the universal quantifier.

**2. Syntax**

We will define the notion of a well-formed formula, or wff for short. ("Well-formed" basically means "grammatically correct" or "legal.") Up until now the only wffs we have been able to construct are sentences. With the introduction of variables, however, we will be able to build wffs that are not sentences.

So, here is our new definition of a wff:

0. If F is a predicate of arity n, and t_{1}, . . ., t_{n}are terms, then F(t_{1}, . . ., t_{n}) is anatomic wff.

1. If P is a wff, then so is ¬P.

2. If P and Q are wffs, then so is (P ∧ Q).

3. If P and Q are wffs, then so is (P ∨ Q).

4. If P and Q are wffs, then so is (P → Q).

5. If P and Q are wffs, then so is (P ↔ Q).

6. If P is a wff, and v is a variable, then ∃v P is a wff.

7. If P is a wff, and v is a variable, the ∀v P is a wff.

8. Nothing else is a wff.

The formula ∃x Tet(x) is read: "There is an x such that x is a tetrahedron," or "There exists an x such that x is a tetrahedron," or, more colloquially, "There is a tetrahedron." (You can think of the backward E as standing for "exists.")

The formula ∀x Tet(x) is read: "For all x, x is a tetrahedron" or "For every x, x is a tetrahedron," or, more colloquially, "Everything is a tetrahedron." (You can think of the upside-down A as standing for "all.")

The *scope* of a quantifier is the first complete wff that follows it.

A quantifier *binds* any occurrence of a variable which (a) is within its scope, (b)
matches the quantifier's variable, and (c) isn't already bound by another
quantifier.

An occurrence of a variable that is not bound by any quantifier is *free*.

A *sentence* is a wff with no free variables.

Examples: (are they wffs? are they sentences? [remember that a sentence is a wff with no free variables])

Tet(x)

Ex Tet(x)

Ex Tet(y)

Ex Tet(x) & Ex Cube(x)

Ex Tet(x) & Cube(x)

Ex (Tet(x) & Cube(x))

Ex (Tet(x) & Cube(y))

**3. Semantics**

A formula with *n* free variables is
**satisfied by** a sequence of *n* objects from the domain iff the result of replacing the
variables by names of the objects, in order, yields a true sentence.

A universal sentence ∀x P(x) is true iff the formula P(x) is satisfied by every object in the domain.

An existential sentence ∃x P(x) is true iff the formula P(x) is satisfied by at least one object in the domain.

**
3. Translation Issues**

The four Aristotelian forms and their standard translations are extremely helpful in translating many English sentences containing quantifiers into symbolic notation. My summary of the Aristotelian forms is here (TLEARN link).

General strategy for translating from English into symbolic notation:

- Identify the overall form of the sentence (usually one of the Aristotelian forms: universal affirmative, universal negative, particular affirmative, particular negative).
- Identify the subject-property and predicate-property. (For instance, in a universal affirmative, All As are Bs, being an A is the subject property and being a B is the predicate property.)
- Translate subject and predicate as properties of an arbitrary, unknown object x (or y or z). For instance, in the sentence "All large tetrahedra are either left of b or right of c," A(x) is (Large(x) ∧ Tet(x)), and B(x) is (LeftOf(x, b) ∨ RightOf(x, c)).
- Plug the translations for A(x) and B(x) into the general form for that type of sentence. For instance, universal affirmatives have the form ∀x (A(x) → B(x)), so the translation for the example in 3 is: ∀x ((Large(x) ∧ Tet(x)) → (LeftOf(x, b) ∨ RightOf(x, c))).

A few issues that sometimes cause problems:

**"A", "an", "some", and "something"** most often indicate that you
need an existential quantifier. "A frog is happy," for instance, would get
translated
∃x (Frog(x) ∧ Happy(x)). But
not always! Sometimes we use these terms when making a universal claim. "An
elephant is a large mammal with a trunk" is most likely intended to be a
general claim about all elephants. So it would be translated by something
like ∀x (Elephant(x)
→ (Large(x) ∧ Mammal(x) ∧ HasTrunk(x))).
Similarly, "someone who studies hard will get an A" is most likely intended
to be a claim about anyone who studies hard, not about some particular
person, so it would be translated ∀x
(StudiesHard(x)
→ GetsA(x)).

The difference between **"any" and "every"** can be confusing. They
both "feel" universal, and in the absence of additional operators, there
doesn't seem to be much difference. "Brown will eat anything" and "Brown
will eat everything" both seem translatable by
∀x WillEat(b, x).

When combined with other logical operators, though, a difference emerges. Compare: "Brown will not eat everything": ¬∀x WillEat(b, x), but "Brown will not eat anything": ∀x ¬WillEat(b, x). One way to think about what is going on here: if "any" is translated by a universal quantifier, it takes "wide scope." That is, when it is combined with another logical connective, that other connective will be inside the scope of the universal quantifier. [However, note that the translation for "Brown will not eat anything" is equivalent, by the DeMorgan's quantifier equivalences, to ¬∃x WillEat(b, x). So a different way to think about things would be to say that when combined with other operators, "any" can be construed as existential instead of universal.]

Another example, this time combining "any" with the conditional. Situation 1: Brown is at a buffet with a lot of fatty food. He would like to try everything. But if he eats everything, he will get sick. So we have: "If Brown eats everything, he will get sick." One way to translate this would be: ∀x Eats(b, x) → GetSick(b). Notice that the universal quantifier only binds the antecedent of the sentence: the overall form of the sentence is a conditional, not a universal.

Contrast that with a situation in which the food is poisoned, so that if Brown eats any of it he will get sick. "If Brown eats anything, he will get sick": ∀x (Eats(b, x) → GetSick(b)). This says that for every piece of food on the buffet, if Brown eats that piece of food, he will get sick. [A different way to look at it: this quantifier translation is actually equivalent to ∃x Eats(b, x) → GetSick(b). So instead of translating the sentence as a universally quantified conditional, we could translate it as a conditional with an existential antecedent.]

One more issue: not every English sentence will fit one of the four Aristotelian forms. This includes some of the sentences in the exercises. Consider "There is something that is neither left of c nor back of d." This is an existential sentence, but it does not quite fit the particular affirmative or particular negative patterns. We want: Ex ~(LeftOf(x, c) v BackOf(c, d)). Similarly, "every object is a large cube" is not a universal affirmative or negative: it says Ax (Large(x) & Cube(x)).

Curtis Brown | Symbolic Logic | Philosophy Department | Trinity University

cbrown@trinity.edu