Topic 2: Logic – CSc 245 v1.1 (McCann) – p. 1 Logic.

Topic 2:

Logic – CSc 245 v1.1 (McCann) – p. 1

Logic

What Is Logic?


Definition: Philosophical Logic

Definition: Mathematical Logic

What Is Logic?




The study of thought and reasoning, including

argu- ments and proof techniques. (This is the

classical no- tion of logic.)

What Is Logic?



The study of thought and reasoning, including

argu- ments and proof techniques. (This is the

classical no- tion of logic.)


The use of formal languages and grammars to represent

the syntax and semantics of computation.

Why Are We Studying Logic?


A few of the many reasons:

• Logic is the foundation for computer operation.





• Logical conditions are common in programs:






◦ Selection:

if (score <= max) { ... }






◦ Selection:


◦ Iteration:

while (i<limit && list[i]!=sentinel) ...






◦ Selection:


◦ Iteration:


• All manner of structures in computing have properties that need to

be proven (and proofs that need to be understood).

◦ Examples: Trees, Graphs, Recursive Algorithms, . . .






◦ Selection:


◦ Iteration:





• Programs can be proven correct.





◦ Selection:


◦ Iteration:





• Programs can be proven correct.

• Computational linguistics must represent and reason about human

language, and language represents thought (and thus also logic).

So we can relate to Mr. Spock

Simple Propositions (1 / 2)


Definition: Simple Proposition

Definition: Proposition

A proposition (a.k.a. statement) is a claim that is

either true or false with respect to an associated

context.



Definition: Proposition

A proposition (a.k.a. statement) is a claim that is

either true or false with respect to an associated

context.

Definition: Simple Proposition

A proposition containing no logical operators.


Example(s):

Some propositions:


Some non-propositions:


Example(s):

Some propositions:

Isotope 52Fe is radioactive


True



Example(s):

Some propositions:



Toronto is the capital of

Canada.

True

False (Ottawa)



Example(s):

Some propositions:




Canada.

5 ∗(1 + 4) = 5 ∗4 + 5

∗1

True

False (Ottawa)

True

(Distributivity)



Example(s):

Some propositions:




Canada.

5 ∗(1 + 4) = 5 ∗4 + 5

∗199.44 is irrational.

True

False (Ottawa)

True

(Distributivity)

False (rational)Some non-propositions:


Example(s):

Some propositions:




Canada.

5 ∗(1 + 4) = 5 ∗4 + 5


True

False (Ottawa)

True

(Distributivity)


log2x = 4 (what is x?)


Example(s):

Some propositions:




Canada.

5 ∗(1 + 4) = 5 ∗4 + 5


True

False (Ottawa)

True

(Distributivity)


log2x = 4

Is your name Ferb?

(what is x?)

(questions are not claims)


Example(s):

Some propositions:



Canada.

5 ∗(1 + 4) = 5 ∗4 + 5


True

False (Ottawa)

True

(Distributivity)


log2x = 4

Is your name Ferb?

Listen to me!

(what is x?)

(questions are not claims)

(a command, not a claim)


Proposition Labels

To save writing, it is traditional to label propositions with

lower-case letters called proposition labels or statement

letters.

Example(s):


Proposition Labels



letters.

Example(s):

t: Toronto is the capital of Canada.


Proposition Labels



letters. (also called propositional or statement variable)

Example(s):

t: Toronto is the capital of Canada.

Either:

• Select meaningful letters (like ‘t’, above), or

• Use letters in sequence starting with ‘p’

Note: Labeled propositions are still True or False.


In Other Words

A propositional or statement variable is just a variable that takes on the values of True or False.

A computer scientist might call them a Boolean variable or even a bit.

Compound Propositions

And with what do we combine them?


Definition: Compound Proposition

A statement that is a logical combination of multiple sim-

ple propositions.

Conjunctions

Conjunctions are:

• compound propositions formed in English with the word “and”,

• formed in logic with the caret symbol (“ ∧ ”), and

• true only when both participating propositions are true.

Example(s):


Conjunctions

Conjunctions are:




Example(s):


I bought gas.p:

q:I got a haircut.

Conjunctions

Conjunctions are:




Example(s):


I bought gas.

I got a haircut.

p:

q:

p ∧ q:

I bought gas and I got a haircut.

More Examples

My name is Eric and I am 6’10’ tall.

Is this a true or false proposition?What are the simple propositions that make it up?Are they true or false?

Disjunctions (1 / 3)


When two propositions p, q are joined by a disjunction, the compound proposition is true if at least one of them is true:

Disjunctions (1 / 3)


When two propositions p, q are joined by a disjunction, the compound proposition is true if at least one of them is true:

Example

n: My name is Erict: I’m 6’10” tall

n t: My name is Eric or I’m 6’10” tall.

Proposition is true.

Problems with English

While in mathematics, formal logic, and computer science “or” means disjunction (the prop is true if one or both of the joined props is true), in spoken English the meaning is not so clear.

Problems with English

While in mathematics, formal logic, and computer science “or” means disjunction (the prop is true if one or both of the joined props is true), in spoken English the meaning is not so clear.

For Example: If you drink the potion in the cup, you will live or you will die.

The speaker doesn’t mean you can both live and die. One or the other will happen. In other words, EXACTLY one has to be true. We call this an exclusive or.

Exclusive or

p

True if p is true or q is true but not both are true.

⊕

∨

is always called exclusive or

is sometimes called inclusive or to emphasize the difference

Negation

Negating a proposition simply flips its value.

Symbols representing negation include:

¬x x

Example(s):


Negation



x ′

¬x x∼ x

Example(s):

We will not use these forms.


Negation



¬x x

Example(s):


h: Homework 1 is easy

Negation



¬x x

Example(s):

h: Homework 1 is easy

h: Homework 1 is not easy


Conditional Propositions (1 / 3)


Definition: Conditional Proposition

A proposition expressible in the form “if p, then q” (p →

q, read “p implies q”), where p and q are propositions, is

a conditional proposition.

Example:



Definition: Conditional Proposition

A proposition expressible in the form “if p, then q” (p →

q, read “p implies q”), where p and q are propositions, is

a conditional proposition.

Example:

If Sauron gets the ring, then all Middle-earth will be destroyed.


In “if p, then q”, p and qare known by various names:

Common forms of “if p, then q” (Rosen 7/e, p. 6):

—p q





—p q

antecedent — consequent





—p q

antecedent

hypothesis

— consequent— conclusion





—p q

antecedent

hypothesis

sufficient

— consequent— conclusion— necessary





—p q

antecedent

hypothesis

sufficient

— consequent— conclusion— necessary


⊲ if p, then q ⊲ q if p⊲ if p, q ⊲ q when p⊲ p implies q ⊲ q whenever p⊲ p only if q ⊲ q follows from p⊲ p is sufficient for q ⊲ q is necessary for p⊲⊲

a necessary condition for p is q

q unless ¬p

⊲⊲

a sufficient condition for q is pand I’m sure you could find more!


Example(s):

You will get a citation if you part in the loading zone.

≡ If you park in the loading zone, then you will get a citation.


The crops will grow well only if they are adequately fertilized.

≡ If the crops grow well, then they are adequately fertilized.

[Switching p and q gives an incorrect statement; crops need more than

just fertilizer.]

Inverse, Converse, and Contrapositive


Definition: Inverse

Definition: Converse

Definition: Contrapositive

r s

T T

T F

F T

F F

(Inverse)

¬r → ¬s

(Converse)

s → r

Thus, Original ≡ Contrapositive Inverse ≡ Converse

(Orig.)(Contrapos.)

¬r ¬s r → s ¬s → ¬r

F F

F T

T F

T T



(Original) (Contrapos.)

If UA is in session, beer sales increase.If beer sales don't increase, UA isn't in session.

(Converse) (Inverse)

If beer sales increase, UA is in session.If UA isn't in session, beer sales do not increase.



(Orig.) (Contrapos.) (Inverse) (Converse)

r s ¬r ¬s r → s ¬s →

¬r

¬r → ¬s s → r

T T F F T T T T

T F F T F F T T

F T T F T T F F

F F T T T T T T

Contraposition


An implication p→q gives you interesting information in two scenarios:

- If p is TRUE, then q must be TRUE- If q is FALSE, then p must be FALSE

.

Biconditional Propositions and if (1 / 2)


What is the meaning of:

A triangle is equilateral if and only if all three angles are equal.

Biconditional Propositions and if (1 / 2)What is the meaning of:


t: A triangle is equilateral a : All three angles are equal

IF AND ONLY IF





IF AND ONLY IF


∧




IF AND ONLY IF


t if a

if a , then t

( a →

t) ∧




IF AND ONLY IF


t if a

if a , then t

t only if a

if t, then a

( a →

t) ∧ ( t →

a )

Biconditional Propositions and if (2 / 2)Definition: Biconditional Proposition

The expression “p if and only if q” (“p iff q”) is called

the biconditional proposition (a.k.a. biimplication)

and is represented by the symbol ↔ (as in: p ↔ q).


Biconditional Propositions and if (2 / 2)

r s

T T

T F

F T

F F

Definition: Biconditional Proposition




r → s s → r ( r → s) ∧ ( s → r )

r ↔ s


Biconditional Propositions and if (2 / 2)Definition: Biconditional Proposition





r s r → s s → r (r→ s) ∧ (s → r )

r— s

T T T T T T

T F F T F F

F T T F F F

F F T T T T

Precedence of Logical Operators


Total agreement is hard to come by:



Rosen 7/ePrecedence p. 11

Highest

Lowest

¬

∧

∨

→

↔

(Note: We’ll cover → and ↔ soon.)






Precedence

Rosen 7/e

p. 11

Gersting 5/e

p. 6

Highest ¬

∧

’

∧,∨

∨ →

Lowest

→

↔

↔





Precedence

Rosen 7/e

p. 11

Gersting 5/e

p. 6

Hein 2/e

p. 351

Highest ¬

∧

’

∧,∨

¬

∧

∨ → ∨

Lowest

→

↔

↔ →





Precedence

Rosen 7/e

p. 11

Gersting 5/e

p. 6

Hein 2/e

p. 351

Epp 1/e

p. 24

Highest ¬

∧

’

∧,∨

¬

∧

∼

∧,∨

Lowest

∨

→

↔

→

↔

∨

→

→ ,↔




In this class: Use parentheses to avoid confusion.


Precedence

Rosen 7/e

p. 11

Gersting 5/e

p. 6

Hein 2/e

p. 351

Epp 1/e

p. 24

Highest ¬

∧

’

∧,∨

¬

∧

∼

∧,∨

Lowest

∨

→

↔

→

↔

∨

→

→ ,↔

Topic 2: Logic – CSc 245 v1.1 (McCann) – p. 1 Logic.

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Transcript of Topic 2: Logic – CSc 245 v1.1 (McCann) – p. 1 Logic.