USC Logic Web — Unit VI: Syntax

Exercises on the vocabulary of quantificational logic, formulas, free and bound variables, open and closed formulas, and construction trees.

Vocabulary

Is each expression part of the official vocabulary of quantificational logic?

1.1

α (alpha)

Yes No

1.2

P

Yes No

1.3

a

Yes No

Formulas

Which of the following are atomic formulas? Which are formulas? Which are neither?

2.a

(∀x Px ∨ ∃x Qx)

atomic formula formula neither

2.b

Qb

atomic formula formula neither

2.c

P(¬)

atomic formula formula neither

2.d

¬Pa

atomic formula formula neither

2.e

∃y ∀x Rxya

atomic formula formula neither

2.f

(x → y)

atomic formula formula neither

2.g

Rayc

atomic formula formula neither

Free and Bound Variables

Which variables occur free in each formula? You can enter any formula into the checker below to verify — it highlights free variables automatically.

3.a

∀x Rxy

x y None

3.b

∃y ∀x Ryx

x y None

3.c

∀x ∀y (Pxy ↔ (Qxz ∨ Qzy))

x y z None

3.d

(∃x Px ∨ ∃y Qxy)

x y None

3.e

(∀x Px → ∀y (Py → Rxy))

x y None

3.f

(∃x Px ∧ ∃y Qy)

x y None

Check a formula

Enter any formula to see its parse tree and free variables highlighted.

Open and Closed Formulas

Which of the following are closed formulas (sentences)?

a. ∃y (∀x Rxy → Py)
b. ∃y ∀x (Ryx ∨ Rxy)
c. (Pa → ∀x ∀y(Rxy → (Rax ∧ Ray)))
d. (Px → ∀y(Ray → Rxy))
e. (Rab ∧ ¬Rba)
f. ∃x ∃y(Rxy ∧ Ryx)

4

Select all closed formulas (sentences).

Construction Trees

Switch to the Build tab to construct the parse tree for each formula step by step.