This document discusses semantics and models in formal logic. It defines key terms like semantics, metalanguage, object language, logical symbols, non-logical symbols, interpretation, and models. Interpretation provides meaning for symbols and formulas, while models add factual information about how the interpreted symbols relate to the world. Truth and falsity of formulas depends on both interpretation and the state of the world. The document provides examples of assigning interpretations, constructing models that specify domains and extensions, and using models to evaluate formulas for truth. It concludes with practice problems assigning interpretations and models to evaluate formulas.

1. SEMANTICS AND
MODELS

2. 1. SEMANTICS
• Means “related to the meaning”
• We are going to talk about the meaning of the
formulas in SL and QL

3. 2. METALNAGUAGE AND
OBJECT L.
• We are going to talk about the meanings of
the formulas of SL and QL in metalanguage.
• Metalanguage is the language that we use to
talk about object language
• Here object languages are: SL and QL
• Metalanguage here is primarily English

4. Example:
Яблоко
Those symbols together mean “apple.”
The word and sentences: made out of letters (symbols) and different rules.
When we talk about this word we talk about it in English.
What’s metalanguage here? And what is the object language?

5. 3. LOGICAL SYMBOLS VS. NON-
LOGICAL SYMBOLS
• Logical symbols: their meaning is specified within the
formal language
Quantifiers (∃, ∀) and connectives (v, &, →) are logical
symbols
• Non-logical symbols: sentence letters – meaning not
specified in the formal language

6. What we did in SL and QL before:
We had sentences and arguments → we translated them to the language → looked at the meaning of the parts of the
sentence or an argument → represented it as a letter or the whole formula (we just focused on the logical structure of
the sentences) → we worked with what we got in the language itself (and we used different mechanisms that we had to
analyze the formula).
For example,
If it’s raining, then it’s cloudy. It’s not raining. Therefore, it’s not cloudy.
1. We see the logical structure in it. And we translate it to SL.
P →Q. ¬P. ∴ ¬Q.
2. Now we can set up truth table to show that it’s invalid.
What 0 and 1 mean in a truth table?
We set up a function v to assign a value to a sentence in SL.
For any sentences A, v(A) = 1 if A is true, and v(A) = 0 if A is false.
We set up the definition of 0 and 1 in metalanguage (in SL).
In truth tables, then we abstract from their meaning True and False, and strictly speaking, what we mean
by 0 or 1 is set up in SL by definitions of truth connectives.

7. What we did: from the sentences in English we went to formulas in
SL and QL. And then we analyzed them.
We can do the reverse procedure.
We can start with the formula itself. And give it interpretation
(meaning).

8. 5. INTERPRETATION AND TRUTH
• Consider sentence letter M.
• Is it true?
• It depends on what M means.
• If it means “Mars exists”, then it’s true.
• If it means “Mars is triangular”., then it’s false
• So the meaning that we will ascribe to M allows for M
to have a truth value. It ascribes the truth value to it.
But just interpretation (the meaning that we ascribe) is
not enough

9. 4. TRUTH
• We also need to know the facts of the world. And we need to know how
the formula and the meaning that we ascribe to it correlate to the facts of
the world.
• For example, could people say what is the truth value of the sentence
“Mars exists” in Middle Ages?
TRUTH/ FALSITY = INTERPRETATION + STATE OF THE WORLD

10. • For example,
Fa
• 1. INTERPRETATION
If we set up the following interpretation:
UD: people
Fx: x is a human
a: Socrates
Fa is true

11. • Another interpretation:
Fx: x is potato
A: Socrates
On this interpretation Fa is false.
But notice, that not just because of interpretation
that we give, it’s also because of 2. THE FACTS OF
THE WORLD

12. 5. MODELS
• Models give us information about the facts of the world. And how they are related to the
meaning.
Let’s consider following interpretation:
UD: main female actors in the Sex and the City
Fx: x has curly hair
a: Sarah Jessica-Parker
Fa is true. But what if you haven’t watched this show and don’t know anything about it?

13. UD: main female actors in the Sex and the City
Fx: x has curly hair
a: Sarah Jessica-Parker
MODEL for this interpretation:
UD= {Sarah Jessica Parker, Kim Catrall, Cynthia Nixon, Kristin Davis}
Extension F = {Sarah Jessica Parker}
Referent(a)= Sarah Jessica Parker
You can find of from the model that Fs is true, because Sarah Jessica Parker (there referent for
a) is in the extension of F
Would ∃xFx be true in this model?
What about ∀xFx?

14. • So in order to construct a model we need:
1. UD
2. Extension of each predicate
3. A referent for each constant
5.1. MODEL: WHAT WE NEED

15. 5.2. MODELS: EXAMPLE
Let’s say we have an interpretation:
UD: whole numbers less than 10
Ex: x is even
Nx: x is negative
Lxy: x is less than y
What model goes with this interpretation?
UD = {1,2,3,4,5,6,7,8,9}
Ext (E) = {2,4,6,8}
Ext (N) = {0}
Ext (L) = { <1,2>, <1,3>, <1,4>, <1,5>, ….<8,9>}
collection of ordered pairs where
the first number is lower than
second

16. 6. PRACTICE
Ch. 5 PART A
UD= {Corwin, Benedict}
Extension (A) = {Corwin, Benedict}
Extension (B) = {Benedict}
Extension (N) = {0}
Referent (c) = Corwin
1. Bc
2. Ac ↔ ¬Nc
4. ∀xFx

17. HW FOR TUE:
1. Reread (5.2., 5.3 and 5.4)
2. Do the rest of exercises in Part A (p. 103-104)
3. And Part C (p. 104): #1-7