This document introduces the basics of translating statements from natural language to the formal language of Quantified Logic (QL). It explains that QL uses constants to represent singular terms, predicates represented by capital letters, and variables represented by lowercase letters. Quantifiers like "for all" and "there exists" are used to represent statements about properties of individuals or groups. To translate a statement to QL, one must identify whether quantifiers are used, what the universe of discourse is, any singular terms, and the relevant predicates to determine the proper representation using constants, predicates, variables, quantifiers and logical connectives.

1. TRANSLATION
TO QL

2. Why do we need QL?
Consider this argument:
Willard is a logician. All logicians wear funny hats.
. ̇. Willard wears a funny hat.
How would we translate to SL?
L – Williard is a logician.
H – All logicians wear funny hats.
W – Williard wears a funny hat.
L. H. . ̇. W.

3. • L. H. . ̇. W.
That’s invalid.

4. Language of QL:
• 1. Singular terms (constants)
‘Bobbie’
‘New York’
‘Natalia Karablina’

5. • Singular Terms
‘Albany’ ‘The capital of NYS’
(Proper name) (Definite description)

6. • For singular terms we use lower case letters: a
–w, a1-w1

7. 2. Predicates
Predicates = properties
“… is a dog”
“… is cute”
“… is smart”
We use capital letters to symbolize Predicates:
A-Z, A1-Z1…An-Zn

8. EXAMPLES
• Ax – x is angry
T – x is as tall or taller than y
d – Donald
g- Gregor
m – Mary
1. Donald is angry
Ad
2. If Donald is angry, then so are Gregor and Mary.
Ad→ (Ag&Am)

9. • Ax – x is angry
Txy – x is as tall or taller than y
Oxy –x is as tough or tougher than y
d – Donald
g- Gregor
m – Mary
Mary is at least as tall and as tough as Gregor.
Tmg & Omg

10. Bxyz- y is between x and z
d – Donald
g- Gregor
m – Mary
Gregor is between Donald and Mary.
y=g x=d z=m
Bdgm

11. • What’s x?
x is a variable.
Variables is the third element of QL.
We use lower case letter x,y,z, x1-xn, y1-yn, z1-zn for
variables
Variables mean that we could put any letter (that
is in Domain of Discourse) instead of the variable

12. 4. Quantifiers
∀ -- “For all”
∃ -- “There are some”
There is always a variable after quantifier.
∀x – For all x
∃x – There are some x
x is a variable, so we could put different things on its
place

13. • For example,
All kitties are cute
Kx – x is a kitty
Cx – x is cute
∀x (Kx → Cx)
For all things, if a thing is a kitty, then this thing is
cute

14. • Some flowers are cute.
Fx – x is a flower
Cx – x is cute
∃x (Fx&Cx)
There are some things such that there are
flowers and they are cute.

15. • In both those cases we are saying “things”, that’s
because we assume the Universe of Discourse. The
Universe of Discourse in those cases are things.
∀x – For all x
∃x – There are some x
Are we saying for all things, people, places?
In different cases it’s different so we will have to clarify
what are we talking about.
What is our discourse/talk about? What scope does it
cover?

16. • Socrates is wise
Wx – x is wise
s – Socrates
Ws

17. • Socrates is happy unless Plato is unhappy
Hx –x is happy
s – Socrates
p- Plato
Hs v ¬ Hp

18. • Someone is wise
UD – people
Wx – x is wise
∃x Wx
There exists at least one person x such that x is
wise.

19. • Someone is happy and someone is wise.
UD – people
Wx – x is wise
Hx – x is happy
∃x Wx &∃x Hx

20. • S0meone is happy and wise.
UD – people
Wx – x is wise
Hx – x is happy
∃x (Wx & Hx)

21. • Everyone is happy
UD – people
Wx – x is wise
Hx – x is happy
∀x Hx
Everyone who is happy is wise.
∀x (Hx →Wx)

22. • Every coin in my pocket is a quarter.
UD- coins
Px – x is in my pocket
Qx – x is a quarter
∀x (Px→Qx)

23. • There is a dime on the table.
UD- coins
Px – x is in my pocket
Qx – x is a quarter
Tx – x is on the table
Dx – x is a dime
∃x (Tx&Dx)

24. • Not all the coins on the table are dimes.
UD- coins
Px – x is in my pocket
Qx – x is a quarter
Tx – x is on the table
Dx – x is a dime
∃x (Tx&¬Dx)
¬∀x (Tx→Dx)

25. • None of the coins in my pocket are dimes.
UD- coins
Px – x is in my pocket
Qx – x is a quarter
Tx – x is on the table
Dx – x is a dime
∀x (Px→¬Dx)
¬∃x (Px&Dx)

26. How to translate?
3 most important questions to identify:
Q1. whether there are any quantifiers that you will need to use
– looking by words “Some” or “All” in the sentences.
• If yes, use those quantifiers, if not – what other parts are in the sentence?
• Could it still be the case that there are no such words, but the sentences is still
about a properties of some individual or all individuals?
Q2. Are there any singular terms in the sentence? (Names or
def descriptions)
Q3. What are the predicates that the sentence is talking
about? (“is smart,” “is beautiful”, “is a dinosaur,” “is a reptile”

27. More on Q1. Are there quantifiers?
If you said ‘yes’ for question a) -- > You will need to start with the quantifier in the
sentence “For all x” and “Some x”:
∀x
∃x
b) Think about the universe of discourse. What is the sentence talking about? Things
in general, animals, people, places?
• Think about whether it’s possible to choose narrower UD than
‘ everything’ or ’the Universe’ for this sentence
c) Write down UD, all predicates and singular terms (if there are any) in the key
d) Think how to formulate the sentence in terms of using quantifiers (or what are
differnt possible ways a sentence could be formulated logically)? To answer that it’s
important to think about what’s the logical message that the sentence is giving
e) You can use all the connectives from SL. Connectives from SL are part of QL too.

28. More on Q2 and Q3. If it’s not about “all” and
“some” individuals within the UD, then there is
probably the singular term in the sentence
Then, we don’t need quantifiers, we need to
write down predicates and singular terms.