first order logic semantics - university of illinoisreason.cs.uiuc.edu/cs498ea/fol part 2.pdf ·...

First Order Logic Semantics

Codruta GırleaUniversity of Illinois at Urbana - Champaign

[email protected]

September 1, 2011

Introduction

Remember FOL signatures?

What’s a signature?

constant, predicate and function symbolsarityexample

constants: zeropredicates: greater(2), even?(1), isNat?(1)functions: succ(1), plus(2)

what’s the difference between predicate and function symbols?

succ(plus(x , succ(succ(zero))))succ(x , y)plus(zero, even?(x))

Introduction

constant, predicate and function symbols

arityexample

Introduction

constant, predicate and function symbolsarity

example

Introduction

constants: zero

predicates: greater(2), even?(1), isNat?(1)functions: succ(1), plus(2)

Introduction

constants: zeropredicates: greater(2), even?(1), isNat?(1)

functions: succ(1), plus(2)

Introduction

succ(plus(x , succ(succ(zero))))

succ(x , y)plus(zero, even?(x))

Introduction

succ(plus(x , succ(succ(zero))))succ(x , y)

plus(zero, even?(x))

Introduction

Remember WFF’s?

Base case: atoms

even?(succ(zero))¬plus(x , succ(y)) = zero

General WFF’s

∀x even?(zero)→ succ(succ(zero))∃x isNat?(x) ∧ ∀y isNat?(y)→ ¬x = succ(y)

is ∀y isNat?(y) → ¬x = succ(y) closed?

every even natural number except zero can be written as thesum of two natural numbers that are not even

Introduction

Remember WFF’s?

Base case: atoms

even?(succ(zero))

¬plus(x , succ(y)) = zero

General WFF’s

Introduction

Remember WFF’s?

Base case: atoms

General WFF’s

Introduction

Remember WFF’s?

Base case: atoms

General WFF’s

Introduction

Remember WFF’s?

Base case: atoms

General WFF’s

∀x even?(zero)→ succ(succ(zero))

∃x isNat?(x) ∧ ∀y isNat?(y)→ ¬x = succ(y)

Introduction

Remember WFF’s?

Base case: atoms

General WFF’s

Introduction

Remember WFF’s?

Base case: atoms

General WFF’s

Introduction

Remember WFF’s?

Base case: atoms

General WFF’s

Introduction

In this lecture - FOL semantics

We will assign meaning to our symbols and formulas

fix the domain of our problem - universe

interpret the symbols in this universe

assign meaning to variables

evaluate formulas according to the interpretation andassignment

relate formulas to each other according to how they evaluatein the semantics

Introduction

Lecture Outline

FOL models

What are FOL models?Isomorphisms of models

WFF satisfaction

AssignmentsInterpreting the truth of WFF’sConsequence and equivalence

Introduction

Lecture Outline

FOL models

WFF satisfaction

FOL Models

What are FOL Models?

FOL Model Example

Remember the signature at the begining of the lecture

One possible interpretation:

Suppose we are talking about natural numberszero is 0greater is >, so {(1, 0), (2, 0), (2, 1), ...}even? is {0, 2, 4, ...}isNat? is Nsucc is a function that adds one to every numberplus is the sum function

FOL Models

FOL Model Example

FOL Models

FOL Model Example

Suppose we are talking about natural numbers

zero is 0greater is >, so {(1, 0), (2, 0), (2, 1), ...}even? is {0, 2, 4, ...}isNat? is Nsucc is a function that adds one to every numberplus is the sum function

FOL Models

FOL Model Example

Suppose we are talking about natural numberszero is 0

greater is >, so {(1, 0), (2, 0), (2, 1), ...}even? is {0, 2, 4, ...}isNat? is Nsucc is a function that adds one to every numberplus is the sum function

FOL Models

FOL Model Example

Suppose we are talking about natural numberszero is 0greater is >, so {(1, 0), (2, 0), (2, 1), ...}

even? is {0, 2, 4, ...}isNat? is Nsucc is a function that adds one to every numberplus is the sum function

FOL Models

FOL Model Example

Suppose we are talking about natural numberszero is 0greater is >, so {(1, 0), (2, 0), (2, 1), ...}even? is {0, 2, 4, ...}

isNat? is Nsucc is a function that adds one to every numberplus is the sum function

FOL Models

FOL Model Example

Suppose we are talking about natural numberszero is 0greater is >, so {(1, 0), (2, 0), (2, 1), ...}even? is {0, 2, 4, ...}isNat? is N

succ is a function that adds one to every numberplus is the sum function

FOL Models

FOL Model Example

Suppose we are talking about natural numberszero is 0greater is >, so {(1, 0), (2, 0), (2, 1), ...}even? is {0, 2, 4, ...}isNat? is Nsucc is a function that adds one to every number

plus is the sum function

FOL Models

FOL Model Example

FOL Models

Another FOL Model Example

Same signature, another possible interpretation:

Talking about natural numbers again

isNat? is {1, 2, ..}zero is 0

even? is {0, 2, 4, ...}greater is >=, so {(0, 0), (1, 0), (1, 1), ...}succ is a function that adds two to every number

FOL Models

isNat? is {1, 2, ..}

zero is 0

FOL Models

even? is {0, 2, 4, ...}

greater is >=, so {(0, 0), (1, 0), (1, 1), ...}succ is a function that adds two to every number

FOL Models

even? is {0, 2, 4, ...}greater is >=, so {(0, 0), (1, 0), (1, 1), ...}

succ is a function that adds two to every number

FOL Models

This time we are talking about the cars in a parking lot

zero is the car in the center

greater is such that a car is greater than another if its size isbigger

even? is the set of red cars

isNat? is the set of minivans

succ is a function that returns, for each car, the nearest othercar, or itself is there is no other car

plus is a function that returns the bigger of the two cars

FOL Models

FOL Structure

Given signature (C , (F k)k≥0, (Rk)k≥0), a structure (interpretation,model) I consists of:

a set of objects |I | or u(I ) - the universe of discourse

an object I (c) or c I for each constant symbol c ∈ C -I (c) ∈ |I |a k-ary predicate I (ρ) or ρI for each predicate symbol ρ ∈ Rk

- I (ρ) ⊆ |I |k

a function of arity k I (f ) or f I for each function symbolf ∈ F k - I (f ) : |I |k → |I |

FOL Models

FOL Structure

- I (ρ) ⊆ |I |k

FOL Models

FOL Structure

an object I (c) or c I for each constant symbol c ∈ C -I (c) ∈ |I |

a k-ary predicate I (ρ) or ρI for each predicate symbol ρ ∈ Rk

- I (ρ) ⊆ |I |k

FOL Models

FOL Structure

- I (ρ) ⊆ |I |k

FOL Models

FOL Structure

- I (ρ) ⊆ |I |k

FOL Models

Lecture Outline

FOL models

WFF satisfaction

FOL Models

Isomorphisms of Models

An Example

Consider a signature with constant symbol zero and functionsymbol succ (of arity 1)

Let I be a structure where:

|I | is the set of natural numberszero I = 0succ I (x) = x + 1 for all natural numbers x

Let J be another structure with:

the same universe and interpretation for zerosuccJ(x) = x + 2 for all natural numbers x

What can you say about structures I and J?

Is there any way you can differentiate them with what youhave?

FOL Models

An Example

FOL Models

An Example

|I | is the set of natural numbers

zero I = 0succ I (x) = x + 1 for all natural numbers x

FOL Models

An Example

|I | is the set of natural numberszero I = 0

succ I (x) = x + 1 for all natural numbers x

FOL Models

An Example

FOL Models

An Example

FOL Models

An Example

the same universe and interpretation for zero

succJ(x) = x + 2 for all natural numbers x

FOL Models

An Example

FOL Models

An Example

FOL Models

An Example

FOL Models

Another Example

Consider a signature with constant symbol zero, functionsymbol succ (of arity 1) and predicate symbol even? (of arity1)

Let I , J be structures like I , J from the previous example,where additionally each interpret even? as the set of evennumbers

What can you say about I and J now?

FOL Models

Another Example

FOL Models

Another Example

FOL Models

Isomorphic Models

Given signature (C , (F k)k≥0, (Rk)k≥0) and two structuresI , J:

an isomorphism h : I → J is a function h : |I | → |J|, such that:for each c ∈ C , h(c I ) = cJ

for each k ≥ 0, ρ ∈ Rk and k objects o1, ..., ok ∈ |I |,ρI (o1, ..., ok) iff ρJ(h(o1), ..., h(ok))for each k ≥ 0, f ∈ F k and k objects o1, ..., ok ∈ |I |,h(f I (o1, ..., ok)) = f J(h(o1), ..., h(ok))

Two structures I , J are isomorphic iff there is an isomorphismh : I → J

FOL Models

Isomorphic Models

an isomorphism h : I → J is a function h : |I | → |J|, such that:

for each c ∈ C , h(c I ) = cJ

FOL Models

Isomorphic Models

FOL Models

Isomorphic Models

for each k ≥ 0, ρ ∈ Rk and k objects o1, ..., ok ∈ |I |,ρI (o1, ..., ok) iff ρJ(h(o1), ..., h(ok))

for each k ≥ 0, f ∈ F k and k objects o1, ..., ok ∈ |I |,h(f I (o1, ..., ok)) = f J(h(o1), ..., h(ok))

FOL Models

Isomorphic Models

FOL Models

Isomorphic Models

FOL Models

Examples and Discussion

What is the isomorphism in the case of the first example?

Can you think of a model that is isomorphic to I in the secondcase? What is the isomorphism?

How about this signature: constant first, binary relation next

Structure I with |I | = {0, 1, 2} interprets first as 0 and next as{(0, 1), (1, 2)}Structure J with |J| = {0, 1, 2, 3} interprets first as 0 and nextas {(0, 1), (1, 3)}Structure M with |M| = {0, 1, 2, 3} interprets first as 0 andnext as {(0, 1), (1, 2), (2, 3)}Which structures are isomorphic? What is the isomorphism?How many structures isomorphic to I can you think of?

How many isomorphic models are there?

FOL Models

Structure I with |I | = {0, 1, 2} interprets first as 0 and next as{(0, 1), (1, 2)}

Structure J with |J| = {0, 1, 2, 3} interprets first as 0 and nextas {(0, 1), (1, 3)}Structure M with |M| = {0, 1, 2, 3} interprets first as 0 andnext as {(0, 1), (1, 2), (2, 3)}Which structures are isomorphic? What is the isomorphism?How many structures isomorphic to I can you think of?

FOL Models

Structure I with |I | = {0, 1, 2} interprets first as 0 and next as{(0, 1), (1, 2)}Structure J with |J| = {0, 1, 2, 3} interprets first as 0 and nextas {(0, 1), (1, 3)}

Structure M with |M| = {0, 1, 2, 3} interprets first as 0 andnext as {(0, 1), (1, 2), (2, 3)}Which structures are isomorphic? What is the isomorphism?How many structures isomorphic to I can you think of?

FOL Models

Structure I with |I | = {0, 1, 2} interprets first as 0 and next as{(0, 1), (1, 2)}Structure J with |J| = {0, 1, 2, 3} interprets first as 0 and nextas {(0, 1), (1, 3)}Structure M with |M| = {0, 1, 2, 3} interprets first as 0 andnext as {(0, 1), (1, 2), (2, 3)}

Which structures are isomorphic? What is the isomorphism?How many structures isomorphic to I can you think of?

FOL Models

Structure I with |I | = {0, 1, 2} interprets first as 0 and next as{(0, 1), (1, 2)}Structure J with |J| = {0, 1, 2, 3} interprets first as 0 and nextas {(0, 1), (1, 3)}Structure M with |M| = {0, 1, 2, 3} interprets first as 0 andnext as {(0, 1), (1, 2), (2, 3)}Which structures are isomorphic? What is the isomorphism?

How many structures isomorphic to I can you think of?

FOL Models

Lecture Outline

FOL models

WFF satisfaction

WFF Satisfaction

Assignments

What about variables?

We will also assign them a meaning with respect to a certainstructure ⇒ assignment

Let I be a structure and V a set of variables

A (variable) assignment is a mapping α : V → |I |If x /∈ V is a variable, then we say that:

assignment α′ : V ∪ {x} → |I | extends assignment α : V → |I |ifα′(y) = α(y) for all y ∈ V

WFF Satisfaction

Assignments

WFF Satisfaction

Assignments

WFF Satisfaction

Assignments

A (variable) assignment is a mapping α : V → |I |

If x /∈ V is a variable, then we say that:

WFF Satisfaction

Assignments

WFF Satisfaction

Assignments

assignment α′ : V ∪ {x} → |I | extends assignment α : V → |I |if

α′(y) = α(y) for all y ∈ V

WFF Satisfaction

Assignments

WFF Satisfaction

Assignments

Terms in FOL Structures

Now we can interpret terms t with respect to a certainstructure I and a variable assignment α - t[α]

Variable assignments α : V → |I | are extended to termassignments:

if x ∈ V is a variable, then x [α] = α(x)if c ∈ C is a constant, then c[α] = c I

if f ∈ F k is a function symbol of arity k ≥ 0 and t1, ..., tk areterms,then f (t1, ..., tk)[α] = f I (t1[α], ..., tk [α])

WFF Satisfaction

Assignments

WFF Satisfaction

Assignments

if x ∈ V is a variable, then x [α] = α(x)

if c ∈ C is a constant, then c[α] = c I

WFF Satisfaction

Assignments

WFF Satisfaction

Assignments

if f ∈ F k is a function symbol of arity k ≥ 0 and t1, ..., tk areterms,

then f (t1, ..., tk)[α] = f I (t1[α], ..., tk [α])

WFF Satisfaction

Assignments

WFF Satisfaction

Assignments

Example

Consider a signature with constant symbol zero and functionsymbols succ(1) and plus(2)

Consider interpretation I with natural numbers as the universeand

zero I = 0succ(x) = x + 1 for all natural numbers xplus(x , y) = x + y for all natural numbers x , y

Consider V = {x , y , z} assignment α with

α(x) = 1, α(y) = 5, α(z) = 9

What are the interpretations of the following terms,considering I and α?

succ(plus(succ(succ(zero)), succ(x + y)))succ(plus(succ(succ(y))))

WFF Satisfaction

Assignments

Example

α(x) = 1, α(y) = 5, α(z) = 9

WFF Satisfaction

Assignments

Example

zero I = 0

succ(x) = x + 1 for all natural numbers xplus(x , y) = x + y for all natural numbers x , y

α(x) = 1, α(y) = 5, α(z) = 9

WFF Satisfaction

Assignments

Example

zero I = 0succ(x) = x + 1 for all natural numbers x

plus(x , y) = x + y for all natural numbers x , y

α(x) = 1, α(y) = 5, α(z) = 9

WFF Satisfaction

Assignments

Example

α(x) = 1, α(y) = 5, α(z) = 9

WFF Satisfaction

Assignments

Example

α(x) = 1, α(y) = 5, α(z) = 9

WFF Satisfaction

Assignments

Example

α(x) = 1, α(y) = 5, α(z) = 9

WFF Satisfaction

Assignments

Example

α(x) = 1, α(y) = 5, α(z) = 9

WFF Satisfaction

Assignments

Example

α(x) = 1, α(y) = 5, α(z) = 9

succ(plus(succ(succ(zero)), succ(x + y)))

succ(plus(succ(succ(y))))

WFF Satisfaction

Assignments

Example

α(x) = 1, α(y) = 5, α(z) = 9

WFF Satisfaction

Assignments

Lecture Outline

FOL models

WFF satisfaction

WFF Satisfaction

Interpreting the Truth of WFF’s

The Satisfaction Relation

Now we can interpret the truth of a WFF φ with the respectto a model I and an assignment α : V → |I |

Satisfaction relation - we say that φ satisfies model I andassignment α and write: I , α |= φ, or I |= φ[α], or |=I φ[α]

If φ satisfies I for all variable assignments α, then φ satisfies Ior I is a model of φ

WFF Satisfaction

WFF Satisfaction

WFF Satisfaction

I |= ρ(t1, ..., tk)[α] iff (t1[α], ..., tk [α]) ∈ ρI

I |= t = t ′[α] iff t[α] = t ′[α]

I |= φ ∧ ψ[α] iff I |= φ[α] and I |= ψ[α]

I |= φ ∨ ψ[α] iff I |= φ[α] or I |= ψ[α]

I |= ¬φ[α] iff it is not the case that I |= φ[α]

I |= ∀x φ[α] iff I |= φ[α′] for all assignmentsα′ : V ] {x} → |I | that extend α

I |= ∃x φ[α] iff there is an assignment α′ : V ] {x} → |I | thatextends α such that I |= φ[α′]

WFF Satisfaction

I |= t = t ′[α] iff t[α] = t ′[α]

WFF Satisfaction

I |= t = t ′[α] iff t[α] = t ′[α]

WFF Satisfaction

I |= t = t ′[α] iff t[α] = t ′[α]

WFF Satisfaction

I |= t = t ′[α] iff t[α] = t ′[α]

WFF Satisfaction

I |= t = t ′[α] iff t[α] = t ′[α]

WFF Satisfaction

I |= t = t ′[α] iff t[α] = t ′[α]

WFF Satisfaction

Examples of WFF Satisfaction

Let I , J be the structures we talked about before, with|I | = |J| = N and:

zero I = zeroJ = 0even?I = even?J = {0, 2, 4, ...}succ I (x) = x + 1 and succJ(x) = x + 2 for x ∈ N

Which of the following WFF’s is satisfied by which structure?

∃x even?(x)even?(x)→ succ(x)even?(x)→ even?(succ(x))∀x even?(x)→ even?(succ(succ(x)))¬even?(succ(succ(zero)))

WFF Satisfaction

zero I = zeroJ = 0

even?I = even?J = {0, 2, 4, ...}succ I (x) = x + 1 and succJ(x) = x + 2 for x ∈ N

WFF Satisfaction

zero I = zeroJ = 0even?I = even?J = {0, 2, 4, ...}

succ I (x) = x + 1 and succJ(x) = x + 2 for x ∈ NWhich of the following WFF’s is satisfied by which structure?

WFF Satisfaction

∃x even?(x)

even?(x)→ succ(x)even?(x)→ even?(succ(x))∀x even?(x)→ even?(succ(succ(x)))¬even?(succ(succ(zero)))

WFF Satisfaction

∃x even?(x)even?(x)→ succ(x)

even?(x)→ even?(succ(x))∀x even?(x)→ even?(succ(succ(x)))¬even?(succ(succ(zero)))

WFF Satisfaction

∃x even?(x)even?(x)→ succ(x)even?(x)→ even?(succ(x))

∀x even?(x)→ even?(succ(succ(x)))¬even?(succ(succ(zero)))

WFF Satisfaction

∃x even?(x)even?(x)→ succ(x)even?(x)→ even?(succ(x))∀x even?(x)→ even?(succ(succ(x)))

¬even?(succ(succ(zero)))

WFF Satisfaction

Valid, Satisfiable, Unsatisfiable WFF’s

WFF φ is:

valid if I |= φ for all models Isatisfiable if there is a model I such that I |= φunsatisfiable if I |= φ for no model I

Note that all valid WFF’s are also satisfiable

Suppose that φ and ψ are two satisfiable formulas

What can you say about the satisfiability of φ ∧ ψ?

Say, φ is isPrime(Three) and ψ is ¬isPrime(Three)

How about φ ∨ ψ?

WFF Satisfaction

WFF φ is:

valid if I |= φ for all models I

satisfiable if there is a model I such that I |= φunsatisfiable if I |= φ for no model I

WFF Satisfaction

WFF φ is:

valid if I |= φ for all models Isatisfiable if there is a model I such that I |= φ

unsatisfiable if I |= φ for no model I

WFF Satisfaction

WFF φ is:

WFF Satisfaction

WFF φ is:

WFF Satisfaction

WFF φ is:

WFF Satisfaction

WFF φ is:

Suppose that φ and ψ are two satisfiable formulasWhat can you say about the satisfiability of φ ∧ ψ?

WFF Satisfaction

WFF φ is:

WFF Satisfaction

WFF φ is:

WFF Satisfaction

Examples of Valid, Satisfiable, Unsatisfiable WFF’s

Which of the following sentences are satisfiable? Which of themare valid?

∃x greater(x , y)

∃x greater(succ(succ(x)))

∀x x = zero ∨ greater(x , zero)

∀x ∀y (even?(x) ∧ greater(x , y))→ even?(x)

∀x ∃y (even?(x) ∨ greater(x , y)) ∧ ¬even?(x)

∀x ∀y greater(x , y)→ greater(y , x)

WFF Satisfaction

∃x greater(x , y)

WFF Satisfaction

∃x greater(x , y)

WFF Satisfaction

∃x greater(x , y)

WFF Satisfaction

∃x greater(x , y)

WFF Satisfaction

∃x greater(x , y)

WFF Satisfaction

∃x greater(x , y)

WFF Satisfaction

Lecture Outline

FOL models

WFF satisfaction

WFF Satisfaction

Consequence and Equivalence

Logical Consequence and Equivalence

Let φ and ψ be two WFF’s

ψ is a logical consequence of φ iff all models of φ are modelsof ψ

φ |= ψ - for all models I , if I |= φ, then I |= ψExamples:

even?(zero) |= ∃x even?(x)How about ∀x even?(x) |= even?(succ(zero)) ?How about ∃x φ |= ∀x φ ?How about ∀x φ |= ∃x φ ?

ψ and φ are semantically equivalent iff they have exactly thesame models

ψ and φ are equivalent iff φ |= ψ and ψ |= φExample: ∀x even?(x) and even?(x)

WFF Satisfaction

φ |= ψ - for all models I , if I |= φ, then I |= ψ

Examples:

WFF Satisfaction

even?(zero) |= ∃x even?(x)

How about ∀x even?(x) |= even?(succ(zero)) ?How about ∃x φ |= ∀x φ ?How about ∀x φ |= ∃x φ ?

WFF Satisfaction

even?(zero) |= ∃x even?(x)How about ∀x even?(x) |= even?(succ(zero)) ?

How about ∃x φ |= ∀x φ ?How about ∀x φ |= ∃x φ ?

WFF Satisfaction

even?(zero) |= ∃x even?(x)How about ∀x even?(x) |= even?(succ(zero)) ?How about ∃x φ |= ∀x φ ?

How about ∀x φ |= ∃x φ ?

WFF Satisfaction

ψ and φ are equivalent iff φ |= ψ and ψ |= φ

Example: ∀x even?(x) and even?(x)

WFF Satisfaction

Some More Examples of Equivalent WFF’s

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

What can you say about the following?

¬¬φ and φ(∃x φ ∧ (∃x ψ)) and (∃x φ) ∧ (∃x ψ)(∃x φ ∧ (∃y ψ)) and (∃x φ) ∧ (∃y ψ)∃x φ ∧ ∃x ψ and ∃x φ ∧ ψ¬∀x φ and ∃x ¬φ¬∃x φ and ∀x ¬φ∃x ∀y φ and ∀y ∃x φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ

¬(φ ∨ ψ) and ¬φ ∧ ¬φWhat can you say about the following?

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

¬¬φ and φ

(∃x φ ∧ (∃x ψ)) and (∃x φ) ∧ (∃x ψ)(∃x φ ∧ (∃y ψ)) and (∃x φ) ∧ (∃y ψ)∃x φ ∧ ∃x ψ and ∃x φ ∧ ψ¬∀x φ and ∃x ¬φ¬∃x φ and ∀x ¬φ∃x ∀y φ and ∀y ∃x φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

¬¬φ and φ(∃x φ ∧ (∃x ψ)) and (∃x φ) ∧ (∃x ψ)

(∃x φ ∧ (∃y ψ)) and (∃x φ) ∧ (∃y ψ)∃x φ ∧ ∃x ψ and ∃x φ ∧ ψ¬∀x φ and ∃x ¬φ¬∃x φ and ∀x ¬φ∃x ∀y φ and ∀y ∃x φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

¬¬φ and φ(∃x φ ∧ (∃x ψ)) and (∃x φ) ∧ (∃x ψ)(∃x φ ∧ (∃y ψ)) and (∃x φ) ∧ (∃y ψ)

∃x φ ∧ ∃x ψ and ∃x φ ∧ ψ¬∀x φ and ∃x ¬φ¬∃x φ and ∀x ¬φ∃x ∀y φ and ∀y ∃x φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

¬¬φ and φ(∃x φ ∧ (∃x ψ)) and (∃x φ) ∧ (∃x ψ)(∃x φ ∧ (∃y ψ)) and (∃x φ) ∧ (∃y ψ)∃x φ ∧ ∃x ψ and ∃x φ ∧ ψ

¬∀x φ and ∃x ¬φ¬∃x φ and ∀x ¬φ∃x ∀y φ and ∀y ∃x φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

¬¬φ and φ(∃x φ ∧ (∃x ψ)) and (∃x φ) ∧ (∃x ψ)(∃x φ ∧ (∃y ψ)) and (∃x φ) ∧ (∃y ψ)∃x φ ∧ ∃x ψ and ∃x φ ∧ ψ¬∀x φ and ∃x ¬φ

¬∃x φ and ∀x ¬φ∃x ∀y φ and ∀y ∃x φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

¬¬φ and φ(∃x φ ∧ (∃x ψ)) and (∃x φ) ∧ (∃x ψ)(∃x φ ∧ (∃y ψ)) and (∃x φ) ∧ (∃y ψ)∃x φ ∧ ∃x ψ and ∃x φ ∧ ψ¬∀x φ and ∃x ¬φ¬∃x φ and ∀x ¬φ

∃x ∀y φ and ∀y ∃x φ

WFF Satisfaction

De Morgan’s laws:

¬(φ ∧ ψ) and ¬φ ∨ ¬φ¬(φ ∨ ψ) and ¬φ ∧ ¬φ

first order logic semantics - university of illinoisreason.cs.uiuc.edu/cs498ea/fol part 2.pdf ·...

Documents