Final Exam Review

Propositional Variable

Variable thats either True or False

Symbol for Negation in Propositional Logic

¬

Symbol for Conjunction in Propositional Logic

∧

Symbol for Disjunction in Propositional Logic

V

Symbol for Implication in Propositional Logic

→

Symbol for Biconditional in Propositional Logic

←→

Symbol for Truth in Propositional Logic

T

Symbol for False in Propositional Logic

⊥

Predicates

Describe properties of objects, take objects as arguments and evaluate to T or F

Functions

Map objects to one another

Quantifiers

allow you to reason about multiple objects

Constant symbols

unlike propositional variables, these refer to OBJECTS, not propositions

First order logic uses ____

Predicates

What is equality, what is its symbol, and what does it apply to?

(=), shows whether 2 objects are equal, only applies to OBJECTS

What is the biconditional, what is its symbol, and what does it apply to?

(←→), shows whether 2 objects are equal, only applies to PROPOSITIONS

Functions

Take objects as input and produce objects as output.

Is this proposition real? Why/Why not?
Venus → TheSun

No, you cannot apply connectives (V, ←→, →) to objects.

Is this proposition real? Why/Why not?

StarOf(IsRed(Sun) ^ IsGreen(Mars))

No, you cannot apply functions (IsBlue, IsWearing, etc.) to propositions

Connectives operate on ___ and produce ___

propositions, propositions

Predicates operate on ___ and produce ___

objects, propositions

Functions operate on ___ and produce ___

objects, objects

What is an existential quantifier? What is its symbol? Is it true in empty worlds?

a logical operator that indicates “At least one”

Its symbol is "∃"

it is false in empty worlds.

What is a universal quantifier? What is its symbol? Is it true in empty worlds?

A logical operator that indicates "For all".

Its symbol is "∀"

it is true in empty worlds.

Quantifiers have precedence above/below ~

below

Translate All A’s are B’s to First Order Logic

∀x (A(x) → B(x))

Translate Some A’s are B’s to First Order Logic

∃x (A(x) ∧ B(x))

Translate No A’s are B’s to First Order Logic

∀x (A(x) → ¬B(x))

Translate Some A’s aren’t B’s to First Order Logic

∃x (A(x) ∧ ¬B(x))

∀x P(x) is equivalent to..?

∃x ¬P(x)

∃x P(x) is equivalent to..?

∀x ¬P(x)

What is the Uniqueness quantifier, what is its symbol?

a logical operator that indicates “At least one object with a property exists and no other objects have that property”

Its symbol is "∃!"

What is Axiom 1’s equation in the Hilbert System?

⊢(A→(B→A))

What is Axiom 2’s equation in the Hilbert System?

⊢(A→(B→C))→((A→B)→(A→C))

What is Axiom 3’s equation in the Hilbert System?

⊢(~B→~A)→(A→B)

What is Axiom 4’s equation in the Hilbert System?

⊢∀xA(x)→A(a)

What is Axiom 5’s equation in the Hilbert System?

⊢∀x (A→B(x))→(A→∀xB(x)

What is the Modus Ponens Equation in the Hilbert System?

(⊢A→B⊢A)/(⊢B)

What is the Generalization Equation in the Hilbert System?

(⊢A(a))/(⊢∀xA(x))

How do you execute a query in SWI-Prolog?

you type the query into the prompt and end it with a period.

E.g. ? - likes(sam, X).

How do you stop a file in SWI-Prolog?

? - halt.

How do you write facts in SWI-Prolog?

State them in the form of a predicate, ending in a period

E.g.

food(pasta).

animal(dog).

What is the symbol used to write rules in a query?

:-