1. Introduction to Computational Linguistics
Eleni Miltsakaki
AUTH
Fall 2005-Lecture 7

2. What’s the plan for today?
Basic concepts of logic
Propositional logic (aka statement logic)
Predicate logic (aka first order logic)

3. Natural language and formal languages
Natural languages: acquired as first language in childhood to serve any communicative goal we have
Formal languages: designed by people for a clear, particular purpose
One of the most important uses linguists make of formal languages is to represent meaning in natural languages.

4. Logic
Logical languages can be used as meta-languages to formalize reasoning from axioms and theorems in a specific domain.
Here, we’ll develop the background for applications of logic and formal languages to natural languages

5. Language of logic Syntax-semantics
Syntax: primitives, axioms, well-formedness, rules of inference or rewrite rules, theorems
Semantics: relation of syntax to interpretation

6. Semantics
For our purposes: semantics is the study of the relations of formal systems to their interpretations

7. About propositional and predicate logic
Each is a formal language with its own vocabulary, rules of syntax and semantics (or system of interpretation)
Both are much simpler than natural languages
Only declarative statement (no questions, imperatives)
Connectives: and, or, not, if…then, if and only if but no because, while, after, although etc
Determiners: some, all, no, every but not most, many, a few, several, one half etc
Logic is the study of reasoning (the product, not the process) with the objective of finding correct, or valid, instances and distinguishing them from those that are invalid.

8. Example All men are mortals Socrates is a man
_______________________________________________________
Therefore, Socrates is mortal
All cats are mammals
All dogs are mammals
________________________________________
Therefore, all cats are dogs
 A systematic account of what underlies our intuitions about the validity of our inferences

9. Argument form  Systematic replacements All rabbits are rodents
Peter is a rabbit
_______________________________________
Therefore, Peter is a rodent
All X’s are Y’s
a is an X
Therefore, a is a Y

10. Propositional logic Vocabulary Syntax
Infinite vocabulary of atomic statements, p, q, r, s, etc
Syntax
Any atomic statement itself is a sentence or well-formed formula (wff)
Any wff preceded by ~ is also a wff
Any two wffs can be made into another wff by conjunction, disjunction, conditional or biconditional

11. Propositional logic Semantics Truth values: 1 or 0 Truth tables
Negation
Conjunction
Disjunction
Conditional
Bi-conditional

12. Laws of propositional logic
Idempotent laws
Associative laws
Commutative laws
Distributive laws
Identity laws
Complement laws
DeMorgan’s laws
Conditional laws
Bi-conditional laws

13. Rules of inference Modus Ponens Modus Tollens Hypothetical syllogism
Disjunctive syllogism
Simplification
Conjunction
Addition

14. Some simple exercises
Example: If John is at the party, then Mary is too.
Translation: (pq)
Key: p=John is at the party, q=Mary is at the party
Translate the following sentences in propositional logic
Either John is in that room or Mary is, or possibly they both are
The fire was set by an arsonist, or there was an accidental explosion in the boiler room
When it rains, it pours
Sam wants a dog but Alice prefers cats

15. Predicate logic Vocabulary Individual constants: j, m,...
Individual variables: x, y, z, …
Predicates: P, Q, R, … each with a fixed number of arguments called its arity
The five connectives of propositional logic (negation, and, or, if-then, if and only if)
Two quantifiers: universal and existential
Auxiliary symbols: parentheses and brackets

16. Syntax of predicate logic
See handout

17. Semantics of predicate logic
Truth values: 1, 0
Truth value is determined by the semantic value of the components of a formula
Set D of individuals, discourse domain
Example
H(s)
s has as its semantic value some individual chosen from a set D of individuals presumed to be fixed in advance. Say D includes Socrates and Aristotle. The statement H(s) gets the truth value true by the fact that the individual corresponding to s is a member of the set corresponding to H.
A two place predicate has as its semantic value a set of ordered pairs of individuals from D and so on and so forth.

18. Some simple exercises
Translate the following English sentences into predicate logic:
Everything is black or white
A dog is a quadruped
Everybody loves somebody
Someone is loved by everyone
There is someone whom everyone loves
No one loves himself unless it’s John
If you love a woman, kiss her or lose her
People who live in New York love it