This document provides an overview of first-order logic and languages. It discusses how objects, names, predicates, functions, and equality can be represented. Examples of representing relational databases, numbers, geometry, sets, and graphs as first-order languages are also given. Various concepts are defined, such as terms, atomic sentences, and functions. Applications to fields like databases, geometry, and sets are demonstrated through sample expressions in those domains.

1. Truth, Deduction,
Computation
Lecture 2
First Order Languages
Vlad Patryshev
SCU
2013

2. In chapter 1 of LPL book...
● the world consists of existing objects
○ some of them have names
○ some of them have more than one name
● objects can participate in n-ary
relationships
○ relationships have names and arity
○ equality relationship
● functions
○ given an object, provide another object

3. Objects
Intuitively, they are something we know exists
Examples:
● Max, Claire, Scruffy, 1, 2, Saturday
Examples of non-objects:
● Pegasus, Zeus, Santa Claus :(

4. Names
Symbols that refer to objects
<name> ::= <number>|<id>
<number> ::= <digit>|<number><digit>
<letterOrDigit> ::= <letter>|<digit>
<id> ::= <letter>|<id><letterOrDigit>
<letters> ::= <letter>|<letters><letter>
E.g. max, claire, scruffy, 42, line1

5. Predicate
Informally, describes a feature of an object.
English: Max likes Claire
FOL: Likes(max, claire)

6. Examples (used throughout the book)
Tet(a)
a is a tetrahedron
Cube(a)
a is a cube
Small(a)
a is small
Medium(a)
a is medium size
Large(a)
a is large
Larger(a,b)
a is larger than b
Smaller(a,b)
a is smaller than b
SameSize(a,b)
a is the same size as b
SameShape(a,b)
a is the same shape as b
LeftOf(a,b)
a is to the left of b
FrontOf(a,b)
I is in front of b
Between(a, b, c)
a, b, c are in the same line and a is between b
and c

7. Atomic Sentences
<atomic> ::= <predicateName>(<arguments>)
<predicateName> ::= <Capital>
<predicateName> ::= <Capital><letters>
<arguments> ::= <name>|<names>,<name>
They take a value which can be either TRUE or FALSE.

8. Equality Predicate
Equals(a,b)
aka*
a = b (infix form)
aka ≡ Also Known As

9. Function Symbols
<functionTerm> ::= <id>(<arguments>)
<arguments> ::= <term>|<arguments>,<term>
<term> ::= <name>|<functionTerm>
Examples
● mother(father(father(max)))
● son(father(I))
● brother(I)

10. Combining All These
Now redefine atomic formula:
<atomic> ::= <predicate>(<arguments>)
Examples:
●
●
●
●
Taller(father(claire), father(max))
john=father(max)
Taller(clair, mother(mother(claire)))
mother(melanie)=mother(claire)

11. Alternative to Function
Can do without:
father(a) → FatherOf(b, a)
In general:
f(x1,x2,...,xn) → Is_f(y,x1,x2,...,
xn)

12. Applications. Relational DB
--CREATE TABLE users…
--CREATE TABLE NSA…
CREATE TABLE NSA_is_watching (
user_id character varying(30) NOT NULL,
bigbrother_id character varying(30) NOT NULL,
last_checked timestamp
CONSTRAINT bigbrother_id_fkey FOREIGN KEY (bigbrother_id)
REFERENCES NSA (id),
CONSTRAINT user_id_fkey FOREIGN KEY (user_id)
REFERENCES users (id)
)

13. Applications. Relational DB
insert into NSA_is_watching values
(‘John Doe’,
‘Agent Smith’, now()),
(‘Jane Dough’,
‘Agent Smith’, now()),
(‘Joe the Plumber’, ‘Agent Smith’, now())

14. Numbers as First Order Language
1.
2.
3.
4.
0,
a,
a,
a,
1
b
b
b
are terms (names, actually)
terms => a + b is a term (aka +(a,b))
terms => a * b is a term(aka *(a,b))
terms => a < b is a formula (aka <(a,b))
Question:
Can there be infinitely many terms referring to number 1?

15. Geometry as a First Order Language
1.
2.
3.
4.
5.
6.
7.
numbers, points, lines, circles, angles
line(p1, p2)
circle(p, r)
LiesOn(p, l)
LiesOn(p, c)
Between(p1, p2, p3)
Parallel(l1, l2)

16. Geometry as a First Order Language
Sample formulas:
● Parallel(line(p1, p2), line(p3, p4))
● LiesOn(p1, circle(p2, r))

17. Sets as First Order Language
1. sets; elements (may not be sets)
2. a ∈ b, where a is an element, b must be a
set
3. a = b
4. {a1,...an} is a set, where a1...an are
elements
5. s1 ∩ s2 is a set, where s1 and s2 are sets
6. s1 ∪ s2 is a set, where s1 and s2 are sets

18. Sets as First Order Language
Sample formulas
● 5 ∈ {1,2,42}
● 0 ∈ {0, {0, {0}} ∩ {1, {1}}}

19. Oriented Graphs
1. nodes and edges
2. Between(n1, n2, e)

20. Reference(s)
List of first-order-theories (wikipedia)