Predicate Logic

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Predicate Logic
DAT911 - Foundations of Computer Science
Dario Garigliotti
March 9, 2016
Dario Garigliotti Predicate Logic

Predicate Logic
In Propositional Logic (P.L.) we could have:
r = It is snowing in Stavanger
s = It is snowing in Oslo
t = ...

Predicate Logic
In Propositional Logic (P.L.) we could have:
r = It is snowing in Stavanger
s = It is snowing in Oslo
t = ...

Predicate Logic
In Propositional Logic (P.L.) we could have:
r = It is snowing in Stavanger
s = It is snowing in Oslo
t = ...

Predicates and Logical Expressions
A predicate is a generalization of a propositional variable:
r(X) = It is snowing in X
...

Predicates and Logical Expressions
A predicate is a generalization of a propositional variable:
r(X) = It is snowing in X
...

Quantiﬁers
For our predicate
r(X) = It is snowing in X
if we want to express: It is snowing in at least one city
r(Berlin)...

Quantiﬁers
Recursive deﬁnition of logical expressions
Basis: every atomic formula is an expression
Induction: E, F logical...

Quantiﬁers
Recursive deﬁnition of logical expressions
Basis: every atomic formula is an expression
Induction: E, F logical...

Quantiﬁers: Bound and Free Variables
A quantiﬁer ”declares” a variable within a ”scope”
E.g. (∀X)r(X) ∨ (∀X)s(X)
Let’s thi...

Quantiﬁers: Bound and Free Occurrences of Variables
An occurrence of a variable X in an expression E is bound by a
quantiﬁ...

Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is deﬁned by:
Dario Garigliot...

Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is deﬁned by:
an interpretati...

Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is deﬁned by:
an interpretati...

Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is deﬁned by:
an interpretati...

Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is deﬁned by:
an interpretati...

Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is deﬁned by:
an interpretati...

Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is deﬁned by:
an interpretati...

Interpretations
We can give a deﬁnition of the interpretation of an expression E,
by induction in its expression tree
E ≡ ...

Tautologies - Substitutions and Equivalences
E is a tautology if its value is true for every interpretation of E
The previ...

Tautologies - Substitutions and Equivalences
E is a tautology if its value is true for every interpretation of E
The previ...

Tautologies - Substitutions and Equivalences
E is a tautology if its value is true for every interpretation of E
The previ...

Tautologies Involving Quantiﬁers
Let’s describe two techniques for tautologies with quantiﬁers
1 Rectifying expressions
Va...

Tautologies Involving Quantiﬁers
Let’s describe two techniques for tautologies with quantiﬁers
1 Rectifying expressions
Va...

Tautologies Involving Quantiﬁers
With those two steps, we can get an equivalent quantiﬁer-free
expression (or prenex-form ...

Tautologies Involving Quantiﬁers
With those two steps, we can get an equivalent quantiﬁer-free
expression (or prenex-form ...

Tautologies Involving Quantiﬁers
With those two steps, we can get an equivalent quantiﬁer-free
expression (or prenex-form ...

Tautologies Involving Quantiﬁers
With those two steps, we can get an equivalent quantiﬁer-free
expression (or prenex-form ...

Proofs in Predicate Logic: Basics
Given some hypotheses E1, E2, ..., Ek, a proof is a sequence of
expressions such that ea...

Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rule...

Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rule...

Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rule...

Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rule...

Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rule...

Provability and Truth
Idea of model: an interpretation which makes an expression
true
Similarly, a model of a collection o...

Predicate Logic
Thanks!
Questions?
Dario Garigliotti Predicate Logic

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