logic bram. what is logic? what is logic? logic is the study of valid reasoning. that is, logic...

180
Logic Bram

Post on 21-Dec-2015

215 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Logic

Bram

Page 2: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

What is Logic?

Page 3: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

What is logic?

• Logic is the study of valid reasoning.

• That is, logic tries to establish criteria to decide whether some piece of reasoning is valid or invalid.

• OK, so then what do we mean by ‘valid reasoning’?

Page 4: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Reasoning

• When we reason, we infer something (Y) from something else (X).

• That is, when we reason, we go like: “Well, if such-and-such-and-so (X) is the case, then this-and-that-and-the-other-thing (Y) must also be the case ”

• X and Y are thus things-that-can-be-the-case-or-not-be-the-case

X YReasoning Diagram:

Page 5: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Good vs Bad Reasoning

• What is the purpose of reasoning? Well, through reasoning, we try to gain new knowledge.

• ‘Good’ reasoning is a piece of reasoning that successfully fulfills this purpose, i.e. that indeed gives us a new piece of knowledge.

• ‘Bad’ reasoning is a piece of reasoning that, for some reason or other, is unsuccessful in this purpose.

• What can go wrong?– Reasoning is invalid– Reasoning is unsound– Reasoning is circular

Page 6: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Valid Reasoning

• While in every piece of reasoning something is believed to follow from something else, this may in fact not be so.

• Example: “If I win the lottery, then I’m happy. However, I did not win the lottery. Therefore, I am not happy.”

• A piece of reasoning in which Y is believed to follow from X is valid if Y does indeed follow from X. Otherwise, the reasoning is said to be invalid.

Page 7: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Sound Reasoning

• Not all valid reasoning is good reasoning.• Example: “If I win the lottery, then I’ll be poor.

So, since I did win the lottery, I am poor.”• This piece of reasoning is valid, but not very

good, since part of what it assumed is absurd (‘If I win the lottery, I’ll be poor.’ Huh??) (also, I did not win the lottery )

• A piece of reasoning where Y is believed to follow from X is sound if a) it is valid, and b) X is true (or at least acceptable/plausible).

Page 8: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth and Implication

• Logic studies the validity of reasoning.

• Logic does not study soundness.

• Therefore, logic alone cannot tell us whether an argument is good. Hence, logic alone is not a guide to truth.

• Instead, logic can tell us, assuming certain things to be true, what else will be true as well. Thus, logic is a guide to implication.

Page 9: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Arguments

• A piece of reasoning consists of a sequence of statements, some of which are claimed to follow from previous ones. That is, some are claimed to be inferred from others.

• Example: “Either the housemaid or the butler killed Mr. X. However, if the housemaid would have done it, the alarm would have gone off, and the alarm did not go off. Therefore, the butler did it.”

Page 10: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Arguments, Premises and Conclusion

• In logic, pieces of reasoning are analyzed using the notion of an argument

• An argument consists of any number of premises, and one conclusion

• Again, in logic, we are merely interested in whether the conclusion follows from the premises: we are not interested in whether those premises are true or acceptable.

• If you want to study all aspects of good reasoning, take my class Methods of Reasoning.

Page 11: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Deductive Validity vs Inductive Validity

• An argument is said to be deductively valid if, assuming the premises to be true, the conclusion must be true as well.

• An argument is said to be inductively valid if, assuming the premises to be true, the conclusion is likely to be true as well.

• For now, we will limit ourselves to deductive validity only!

• If you want to study non-deductive reasoning, take my Methods of Reasoning class.

Page 12: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Argument Forms

• “If I win the lottery, then I am poor. I win the lottery. Hence, I am poor.”

• This argument has the following abstract structure or form: “If P then Q. P. Hence, Q”

• Any argument of the above form is valid, including “If flubbers are gook, then trugs are brig. Flubbers are gook. Hence, trugs are brig.”!

• Hence, we can look at the abstract form of an argument, and tell whether it is valid without even knowing what the argument is about!!

Page 13: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Formal Logic

• Formal logic studies the validity of arguments by looking at the abstract form of arguments.

• Formal logic always works in 2 steps:– Step 1: Use certain symbols to express the

abstract form of premises and conclusion.– Step 2: Use a certain procedure to figure out

whether the conclusion follows from the premises based on their symbolized form alone.

Page 14: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Example Step 1: Symbolization

• Use symbols to represent simple propositions:– H: The housemaid did it– B: The butler did it– A: The alarm went off

• Use further symbols to represent complex claims:– H B: The housemaid or the butler did it– HA: If the housemaid did it, the alarm would go off– ~A: The alarm did not go off

Page 15: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Example Step 2: Evaluation

• One possible technique is to transform the symbolic representations using basic rules that reflect elementary valid inferences:

H B

HA

A

H

B

2, 3 MT

A.

A.

A.

5.

4.

3.

2.

1.

1, 4 DS

Since every step along the wayis an instance of an obviously valid inference, the conclusion does indeed follow from thepremises. So, valid argument!

Page 16: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Propositional Logic

• Propositional Logic studies validity at the level of simple and compound propositions.

• Simple proposition: An expression that has a truth value (a claim or a statement). E.g. “John is tall”

• Compound proposition: An expression that combines simple propositions using truth-functional connectives like ‘and’, ‘or’, ‘not’, and ‘if … then’. E.g. “John is tall and Mary is smart”

Page 17: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Predicate Logic

• Predicate Logic extends Propositional Logic by adding individuals, predicates, and quantifiers

• Individuals: ‘John’, ‘Mary’

• Predicates: ‘tall’, ‘smart’

• Quantifiers: ‘all’, ‘some’

Page 18: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Just to Put Things in Perspective

All arguments

All deductive arguments

All deductive arguments thatcan be analyzed using the formallogics we cover in class (And I’m probably

optimistic here!)

Page 19: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Uses of Formal Logic

• Evaluation/Checking: – Formal logic can be used to evaluate the validity of

arguments.

• Clarification/Specification: – Formal logic can be used to express things in a

precise and unambiguous way.

• Demonstration/Proof: – Formal logic can be used to figure out what follows

from a set of assumptions.

• Computation/Automated Reasoning: – Formal logic can be used for machine reasoning.

Page 20: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Logic, Computers, and AI

• Formal logic has many connections to computers:– Computation: Formal logic played a crucial role in the

development of the notion of ‘computation’ (See my class PHIL 4420 Computability and Logic)

– Circuit Design: Formal logic can be used for circuit design (See CSCI 2500 Computer Organization)

– Artificial Intelligence: Formal logic is central to many AI applications (See CSCI 4150 Artificial Intelligence)

Page 21: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Boolean Connectives

Page 22: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Propositional Logic

• Propositional Logic is the logic involving complex claims as constructed from atomic claims and connectives.

• Propositional Logic is not as powerful as Predicate Logic, but it has some powerful applications already.

Page 23: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-Functional Connectives and Boolean Connectives

• Connectives are usually called truth-functional connectives:– This is because the truth value of a complex claim

that has been constructed using a truth-functional connective is considered to be a function of the truth values of the claims that are being connected by that connective.

– This is also why propositional logic is also called truth-functional logic.

• For now, we will focus on three connectives: and, or, not; these are called the Boolean connectives.

Page 24: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Negation

• The claim “a is not to the right of b” is a complex claim. It consists of the atomic claim “a is to the right of b” and the truth-functional connective “not”.

• We will call the above statement a negation.• To express negations, we use the symbol ‘’• ‘’ should be put in front of what you want to be

negated.• If we symbolize the atomic claim “a is to the right

of b” as P, then the original claim will be symbolized as: P

Page 25: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-Table for Negation

• ‘’ is truth-functional, since the truth-value of a negation is the exact opposite of the truth-value of the statement it negates.

• We can express this using a truth table:

P P

TTF

F

Page 26: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Conjunction

• The claim “a is to the right of b, and a is in front of b” is called a conjunction.

• The two claims that are being conjuncted in a conjunction are called its conjuncts.

• To express conjunctions, we will use the symbol ‘’

• ‘’ should be put between the two claims.• Thus, the above statement can be

symbolized as: P Q

Page 27: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-Table for Conjunction

• ‘’ is truth-functional, since a conjunction is true when both conjuncts are true, and it is false otherwise.

• Again, we can show this using a truth table:

P P Q

T

TF

F

Q

T

FF

F

FFTT

Page 28: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Disjunction

• The claim “a is to the right of b, or a is in front of b” is called a disjunction.

• The two claims that are being disjuncted in a disjunction are called its disjuncts.

• To express disjunctions, we will use the symbol ‘’

• ‘’ should be put between the two claims.• Thus, the above statement can be

symbolized as: P Q

Page 29: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-Table for Disjunction

• ‘’ is truth-functional, since a disjunction is true when at least one of its disjuncts is true, and it is false otherwise.

• Again, we can show this using a truth table:

P P Q

T

TF

T

Q

T

FF

T

FFTT

Page 30: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Combining Complex Claims: Parentheses

• Using the truth-functional connectives, we can combine complex claims to make even more complex claims.

• We are going to use parentheses to indicate the exact order in which claims are being combined.

• Example: (P Q) (R S) is a conjunction of two disjunctions.

Page 31: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Parentheses and Ambiguity

• An ambiguous statements is a statement whose meaning is not clear due to its syntax. Example : ”P or Q and R”

• In formal systems, an expression like P Q R is simply not allowed and considered unsyntactical.

• Claims in our formal language are therefore never ambiguous.

• One important application of the use of formal languages is exactly this: to avoid ambiguities!

Page 32: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Exclusive Disjunction vs Inclusive Disjunction

• Notice that the disjunction as defined by ‘’ is considered to be true if both disjuncts are true. This is called an inclusive disjunction.

• However, when I say “a natural number is either even or odd”, I mean to make a claim that would be considered false if a number turned out to be both even and odd. Thus, I am trying to express an exclusive disjunction.

Page 33: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

How to express Exclusive Disjunctions

• We could define a separate symbol for exclusive disjunctions, but we are not going to do that.

• Fortunately, exclusive disjunctions can be expressed using the symbols we already have: (PQ) (PQ)

P (P Q) (PQ)

T

TF

T

Q

T

FF

T

FFTT

F

T

F

FT

F

T

TT

F

F

T

!

Page 34: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Conditionals

Page 35: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

The Material Conditional

• Let us define the binary truth-functional connective ‘’ according to the truth-table below.

• The expression P Q is called a conditional. In here, P is the antecedent, and Q the consequent.

P P Q

T

TF

T

Q

T

TF

F

FFTT

Page 36: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

‘If … then …’ Statements

• The conditional is used to capture ‘if … then …’ statements.• However, the match isn’t perfect. For example, we don’t want to

say that the claim “If grass is green then elephants are big” is true just because grass is green and elephants are big, nor that any ‘if … then’ statement is automatically true once the ‘if’ part is false or the ‘then part true. The problem is that most English ‘if…then’ expressions aren’t meant to make a claim that is truth-functional in nature.

• Still, any ‘if … then …’ statement will be false if the ‘if’ part is true, but the ‘then’ part false, and the conditional captures at least this important truth-functional aspect of any ‘if … then …’ statement.

• So, while we will from now on refer to the conditional as an ‘if … then’ statement, we must be careful about the use of this, just as care must be taken when applying Newtonian physics to some situation.

Page 37: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Necessary and Sufficient Conditions

• Conditionals can be used to express necessary and sufficient conditions:

• Sufficient Condition: Something (P) is a sufficient condition for something else (Q) iff P being the case guarantees Q being the case. Hence, if we know that P is true, we know that Q is true: P Q

• Necessary Condition: Something (P) is a necessary condition for something else (Q) iff P being the case is required for Q being the case. Thus, while P may be true without Q being true, we do know that if Q is true, P is true: Q P

Page 38: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

‘If’ vs ‘Only if’

• Sufficient conditions are expressed in English using ‘if’, while necessary conditions are expressed using ‘only if’.

• Thus:– ‘If P then Q’: P Q– ‘P if Q’: Q P– ‘P only if Q’: P Q– ‘Only if P, Q’: Q P

Page 39: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

‘If and only if’ and the Material Biconditional

• A statement of the form ‘P if and only if Q’ (or ‘P iff Q’) is short for ‘P if Q, and P only if Q’. Hence, we could translate this as (P Q) (Q P). However, since this is a common expression, we define a new connective ‘’:

P P Q

T

TF

F

Q

T

TF

F

FFTT

Page 40: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Logical Properties

Page 41: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth Tables

• Truth-tables can be used for:– defining the truth-conditions of truth-functional

connectives – evaluating the truth-conditions of any complex

statement

Page 42: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Tautologies

• A tautology is a statement that is necessarily true.

• Example: P P

P P

TF

P

TT T

TF

FF

T

Page 43: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Contradictions

• A contradiction is a statement that is necessarily false.

• Example: P P

P P

TF

P

FF T

TF

FF

T

Page 44: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Contingencies

• A contingency is a statement that can be true as well as false

• Example: P

P

TF

P

TF

Page 45: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Equivalences

• Two statements are equivalent if they have the exact same truth-conditions.

• Example: P and P

P

TF

P

TT

FF

FT

P

TF

Page 46: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Contradictories

• Two statements are contradictories if one of them is false whenever the other one is true and vice versa.

• Example: P and P

P

TF

P

TT

FF

FT

P

Page 47: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Implication

• One statement implies a second statement if it is impossible for the second statement to be false whenever the first statement is true.

• Example: P implies P Q

P P Q

T

TF

F

Q

T

FF

T

FFTT

P

FTTT

Page 48: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Consistency

• A set of statements is consistent if it is possible for all of them to be true at the same time.

• Example: {P, P Q, Q}

P P Q

T

TF

F

Q

T

FF

T

FFTT

P

FTTT

Q

TFTF

Page 49: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Consequence

• A statement is a consequence of a set of statements if it is impossible for the statement to be false while each statement in the set of statements is true.

• Example: P is a consequence of {PQ, Q}

P P Q

T

TF

F

Q

T

FF

T

FFTT

P

FTTT

Q

TFTF

Page 50: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Validity

• An argument is valid if it is impossible for the conclusion to be false whenever all of its premises are true.

• Example: P Q, Q P

P P Q

T

TF

F

Q

T

FF

T

FFTT

P

FTTT

Q

TFTF

Page 51: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Implication, Consequence, Validity

• The notions of implication, consequence, and validity are very closely related:

• A statement implies a statement if and only if is a consequence of the set of statements {}

• For implication and consequence we use the symbol ‘’: – If statement implies statement we write – If statement is a consequence of a set of statements {1,

…, n}, we write {1, …, n} • An argument consisting of premises 1, …, n and

conclusion is valid iff {1, …, n} • The terms implication, consequence and validity can

therefore be used interchangeably.

Page 52: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Summary• Logical properties of a single statement:

– Tautology: cannot be false– Contradiction: cannot be true– Contingent: can be true and can be false

• Logical properties of 2 statements:– Equivalent: always the same truth-value– Contradictory: always opposite truth-values– Contrary: cannot be both true– Subcontrary: cannot be both false– Implication: implied statement cannot be false while the implying statement is

true• Logical properties of a set of statements:

– Consistent: can be all true at the same time• Logical properties of a set of statements in relation to a single statement:

– Consequence: statement cannot be false if all of the statements of the set are true

• Logical properties of an argument:– Valid: conclusion cannot be false if all premises are true

Page 53: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

But Wait!

• Consider the statement ‘a=a’ • This statement is a tautology, since it

is always true.• However, since this statement does

not involve any truth-functions, propositional logic considers this an atomic statement, and symbolizes it as ‘P’. But ‘P’ is a contingency.

• What is going on??

Page 54: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-Functional Tautologies

• What is going on is that truth-tables only capture the truth-functional aspects of sentences.

• So, a statement may be a tautology for reasons other than truth-functional reasons. ‘a=a’ is an example.

• A statement that is a tautology because of truth-functional considerations is called a truth-functional tautology.

• Notice that while every truth-functional tautology is a tautology, not every tautology is a truth-functional tautology (again, ‘a=a’ is a tautology, but not a truth-functional tautology)

Page 55: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Normal Forms and Expressive Completeness

Page 56: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Logically Equivalent Statements

• To express that two statements P and Q are logically equivalent, we will write: PQ

• ‘’ is not a symbol of F!!

Page 57: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Some Important Equivalences

• Double Negation: – P P

• DeMorgan: (P Q) P Q (P Q) P Q

• Distribution:– P (Q R) (P Q) (P R)– P (Q R) (P Q) (P R)– (Q R) P (Q P) (R P)– (Q R) P (Q P) (R P)

Page 58: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

More Equivalences

• Commutation:– P Q Q P– P Q Q P

• Association:– P (Q R) (P Q) R– P (Q R) (P Q) R

• Idempotence:– P P P– P P P

• Subsumption:– P (P Q) P– P (P Q) P

Page 59: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Even More Equivalences

• Implication:– P Q P Q– (P Q) P Q

• Transposition:– P Q Q P

• Exportation:– P (Q R) (P Q) R

• Absorption:– P Q P (P Q)

• Equivalence:– P Q (P Q) (Q P)– P Q (P Q) (P Q)

Page 60: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Simplifying Statements I

• Using the principle of substitution of logical equivalents, and using the logical equivalences that we saw before (Double Negation, Association, Commutation, Idempotence, DeMorgan, Distribution, and Subsumption), we can often simplify statements.

• Example:

(A B) A (Commutation)

(B A) A (Association)

B (A A) (Idempotence)

B A

Page 61: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Generalized Conjunctions and Generalized Disjunctions

• Recall the Association equivalences: – P (Q R) (P Q) R– P (Q R) (P Q) R

• Because of this, we’ll allow to drop brackets:– P Q R– P Q R

• Thus we can generalize conjunctions and disjunctions– A generalized conjunction (disjunction) can have any

number of conjuncts (disjuncts)

Page 62: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Simplifying Statements II

• The conjuncts (disjuncts) of a generalized conjunction (disjunction) can be switched around in any way you want. This really helps with simplifying statements. Example:

C (A (B C)) (Distribution)

C (A B) (A C) (Subsumption)

C (A B)

Page 63: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

‘⊤’ and ‘⊥’

• A generalized conjunction is false if it has at least one false conjunct, otherwise it is true. – So, a generalized conjunction with 0 conjuncts

cannot have a false conjunct, and hence cannot be false. Therefore, it is a tautology! We will write this as ‘⊤’.

• A generalized disjunction is true if it has at least one true disjunct, otherwise it is false.– Hence, a generalized disjunction with 0

disjuncts can never be true, and is therefore a contradiction! We will write this as ‘⊥’.

Page 64: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Some equivalences involving ‘⊤’ and ‘⊥’

⊥ ⊤⊤ ⊥• P ⊥ ⊥ • P ⊤ ⊤ • P ⊤ P • P ⊥ P • P P ⊥ • P P ⊤

Page 65: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Simplifying Statements III

• Using ‘⊤’ and ‘⊥’, we can simplify statements even more. Example:

(A (B (A B)) (DeMorgan)

A (B (A B)) (Double Neg.)

A B (A B) (Distribution)

(A B A) (A B B) ⊥ ⊥

Page 66: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Negation Normal Form

• Literals: Atomic Sentences or negations thereof.• Negation Normal Form: An expression built up

with ‘’, ‘’, and literals.• Using repeated DeMorgan and Double Negation,

we can transform any truth-functional expression built up with ‘’, ‘’, and ‘’ into an expression that is in Negation Normal Form.

• Example:

((A B) C) (DeMorgan)(A B) C (Double Neg, DeM)(A B) C

Page 67: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Disjunctive Normal Form

• Disjunctive Normal Form: A disjunction of conjunctions of literals.

• Using repeated distribution of over , any statement in Negation Normal Form can be written in Disjunctive Normal Form.

• Example:

(AB) (CD) (Distribution)[(AB)C] [(AB)D] (Distribution (2x))(AC) (BC) (AD) (BD)

Page 68: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Conjunctive Normal Form

• Conjunctive Normal Form: A conjunction of disjunctions of literals.

• Using repeated distribution of over , any statement in Negation Normal Form can be written in Conjunctive Normal Form.

• Example:

(AB) (CD) (Distribution)[(AB) C] [(AB) D] (Distribution (2x))(AC) (BC) (AD) (BD)

Page 69: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-Functional Connectives

• So far, we have seen one unary truth-functional connective (‘’), and two binary truth-functional connectives (‘’, ‘’).

• Later, we will see two more binary connectives (‘’, ‘’)

• However, there are many more truth-functional connectives possible:– First of all, a connective can take any number of

arguments: 3 (ternary), 4, 5, etc.

– Second, there are unary and binary connectives other than the ones listed above.

Page 70: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Unary Connectives

• What other unary connectives are there besides ‘’?• Thinking about this in terms of truth tables, we see

that there are 4 different unary connectives:

P *P

T

F

T

T

P *P

T

F

T

F

P *P

T

F

F

T

P *P

T

F

F

F

Page 71: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Binary Connectives

• The truth table below shows that there are 24 = 16 binary connectives:

P Q P*Q

T

T

T

T

F

FF

F

T/F

T/F

T/F

T/F

In general:n sentences

22n

different n-ary connectives!

2n truth value combinations(i.e. 2n rows in truth table)

Page 72: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Expressing other connectives using ‘and’, ‘or’, and ‘not’

• We saw that we can express the exclusive disjunction using ‘and’, ‘or’, and ‘not’.

• Q: Can we express all other connectives as well?

• A: Yes! We can generalize from this example:

P Q P*Q

T

T

T

T

F

FF

F

F

T

T

F

PQ

PQ (PQ) (PQ)

Step 1: Step 2:

Page 73: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-Functional Expressive Completeness

• Since I can express any truth function using ‘’, ‘’, and ‘’, we say that the set of operators {, , } is (truth-functionally) expressively complete.

• DeMorgan Laws: (P Q) P Q (P Q) P Q

• Hence, by the principle of substitution of logical equivalents, since {, , } is expressively complete, the sets {, } and {, } are expressively complete as well!

Page 74: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Applications: Computer Hardware and Software

• The results that we have seen on the previous slides have important applications in both computer hardware and software:– Digital Circuits– Machine Reasoning

Page 75: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Logic and Computer Circuitry

Page 76: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

1’s and 0’s

• All what modern digital computers do is transform strings of 1’s and 0’s, called bitstrings.

• Information is represented using 1’s and 0’s, and information is processed through the manipulation of those bitstrings.

• The 1’s and 0’s can be physically realized using any kind of physical dichotomy. We can therefore use pure mechanics (levers, pulleys, punchcards, etc.), electronics, optics, DNA, quantum physics, toilet paper and pennies, or just about anything else to physically implement the 1’s and 0’s.

Page 77: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Logic Gates

• To process information, bitstrings need to be manipulated.

• Thus, depending on whatever way the 1’s and 0’s are physically implemented, there needs to be a device to change those physical representations.

• But, we are not going to be interested in the physical nature of these devices, since this is just an issue of implementation.

• Rather, we are going to think of these devices as logic gates: thingamabobs that transform bitstrings into other bitstrings.

Page 78: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

‘And’, ‘Or’, and ‘Not’ Gates

OutIn1

In2

In1

In2

Out

In OutIn Out

1

1

0

0

In1Out

1

1

0

0

In2

1

00

0

0

0

1

1

In1Out

1

1

0

1

In2

1

00

1

0

0

1

1

Page 79: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Representing Numbers

• We normally represent numbers using the decimal system. That is, we take 10 as our base number to represent numbers.

• Example:

536277*100 = 7*1 = 72*101 = 2*10 = 206*102 = 6*100 = 6003*103 = 3*1000 = 30005*104 = 5*10000 = 50000

53627

Page 80: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Binary Numbers

• Binary numbers take 2 as their base.

• Example:

101100*20 = 0*1 = 01*21 = 1*2 = 21*22 = 1*4 = 40*23 = 0*8 = 01*24 = 1*16 = 16

22

Page 81: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Adding Binary Numbers

1101011010110011

110001001

1 1 1 1 1

Page 82: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Computing Binary Addition

• To compute the addition of two binary numbers, we need to implement the following architecture:

+

+

+

In10

In20

In11

In21

In12

In22

Out1

Out0

Out2

Carry1

Carry2

Page 83: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

2 Bit and Carry Adder

+In2n

In1n

Carryn

Outn

Carryn+1

In1n In2n Carryn Outn Carryn+1

0

0

0

1

0

1

1

1

0

0

0

0

1

1

1

1

0

0

1

1

0

0

1

1

0

1

0

1

0

1

0

1

0

1

1

0

1

0

0

1

Page 84: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Circuitry for the 2 Bit and Carry Adder (output bit)

In1n

In2n

Carryn

Outn

Page 85: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Circuitry for the 2 Bit and Carry Adder (carry bit)

In1n

In2n

Carryn

Carryn+1

Page 86: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Simplifying

• While the Disjunctive Normal Form provides us with a working circuitry (and thus guarantees us of one!), this circuitry may not be the most efficient one.

• Carryn+1 = (In1n In2n) (In1n Carryn) (In2n Carryn)

• Outn = (In1n XOR In2n) XOR Carryn

– where P XOR Q = (P Q) (P Q)

Page 87: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

An Interesting Trade-Off

• To keep production costs down:– Use as few gates as possible– Use as few different kinds of gates as

possible

• However, there is a trade-off between these two objectives: The fewer the number of kinds of gates one uses, the more gates of those kinds are needed.

Page 88: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Example of the Trade-Off

• The Disjunctive Normal Form tells us that we can build any circuit using only 3 kinds of gates.

• With more types of gates (e.g. the XOR), we could have saved on the total number of gates.

• On the other hand, because of the DeMorgan’s Law, we know that we can express any expression using only ‘and’ and ‘not’. Thus, we can also try and cut down on the number of types of gates, but this will mean an increase in the number of gates.

Page 89: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

The NAND

• Let us define the binary truth-functional connective ‘NAND’ according to the truth-table below.

• Obviously, P NAND Q (P Q) (hence the name!)

P P NAND Q

T

TF

T

Q

F

TF

T

FFTT

Page 90: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Expressive Completeness of the NAND

• The NAND has a very interesting property, in that it can express any truth-functional connective, i.e. {NAND} is expressively complete!

• Proof: We already know that we can express every truth-functional connective using only and . Furthermore:– P NAND P (P P) P– (P NAND P) NAND (Q NAND Q) ((P NAND P) (Q

NAND Q)) (P Q) P Q• In other words, we can build circuitry using only one

kind of logic gate!! Of course, the drawback is that we need many of those gates.

Page 91: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

The Miniac

• Behold! The world’s most powerful computer that fits in the palm of your hand: a penny!!

• Instructions: Ask any question with a yes or no answer. Flip the coin. Tails means ‘yes’ and heads means ‘no’. To see whether the Miniac’s answer is correct or incorrect, flip the coin a second time, asking: “Is your answer to this question just as correct as your answer to the previous question?”

• Question: How does this work?

Page 92: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Formal Proofs

Page 93: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Demonstrating Invalidity

• To demonstrate invalidity one has to show that it is possible for all premises to be true and the conclusion to be false all at the same time.

• One way to do this is to come up with a possible scenario (or possible world) in which all premises are true and the conclusion false (Tarski’s World). This is called a counterexample.

Page 94: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Demonstrating Validity

• To demonstrate validity, we have to show that there is no possible way for all premises to be true and the conclusion false all at the same time.

• Showing a scenario in which all premises are true, and in which the conclusion is true as well, does not demonstrate validity, b/c there may still be a different scenario in which all premises are true and the conclusion false.

• Of course, we could try and generate all possible worlds, but this method is either impractical (b/c there are too many possible worlds), or simply impossible (b/c there are infinitely many possible worlds).

Page 95: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Proofs

• OK, so what do we do? Well, we can do what we do in everyday reasoning: we start with the premises, and we gradually work our way to the conclusion: “Either the housemaid or the butler killed Mr. X. Now, we know that if the housemaid would have done it, the alarm would have gone off. But, the alarm did not go off. Therefore, the housemaid did not do it. So, since it was either the housemaid or the butler, it must have been the butler.”

Page 96: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Intermediate Results

• The previous argument had 3 premises:– 1. Either the housemaid or the butler did it.– 2. If the housemaid did it, the alarm would have gone off.– 3. The alarm did not go off.

• The conclusion was: The butler did it.• We combined premises 2 and 3 to get an

intermediate result: The housemaid did not do it.• We then combined the intermediate result with

premise 1 to get the conclusion.• We use intermediate conclusions because without

them, the inference from the premises to the conclusion may not be obvious, but with them, each of the steps does become obvious.

Page 97: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

‘Obvious’

• In formal proofs, we try and formalize this step-by-step inference process, where each inference is obvious.

• OK, but ‘obvious’ is a bit of a vague term, as what is obvious to some, may not be obvious to others. So, what are going to count as ‘obvious’?

• We are going to play it safe: In formal proofs, we are only going to allow steps that are about as obvious as we can get. Thus, we are only going to allow ‘baby inferences’.

• So, in formal proofs, bigger inferences, which may still be obvious to many (if not all of us), will still have to be broken up into smaller ones!

Page 98: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Inference Rules

• Formal systems of logic come with a finite set of inference rules that reflect ‘baby inferences’.

• There are many formal systems of logic, each with their own set of inference rules:– The nature of the inference rules depends on

the symbols that the system uses to express statements.

– However, even if two systems use the same symbols, they may still have different inference rules.

Page 99: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

F: A ‘Fitch’-style Deductive System

• The formal system that our book uses is called F.• F has 2 inference rules for each connective:

– Introduction: A rule to infer a statement with that connective as its main connective

– Elimination: A rule to infer something from a statement with that connective as its main connective.

• Formal systems with these two types of inference rules are called ‘Fitch’-style systems.

• Warning: While Fitch-style systems are mathematically elegant, they are not always very user-friendly. In particular, it does not contain inference rules that reflect some ‘obviously’ valid inferences!

Page 100: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

The Structure of Proofs in F

• A formal proof in F will look like this:

P1

Pn

I1

Im

C

1

n

n+1

n + m

n + m + 1

Justification 1

Justification m

Justification m+1

Page 101: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Justification

• In a formal proof, you have to indicate from which premises or intermediate results you infer the new statement. Thus, each step needs to have its own justification.

• Inference rules may need any number of statements from which the new statement is inferred (though with too many statements, the rule may no longer be considered ‘obvious’).

• Most inference rules require one or two statements. • Some inference rules require no statements at all. This is

when the inferred statement is unconditionally true.• To help refer to previous statements, we are going to

number the statements.

Page 102: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Elim

• Conjunction Elimination ( Elim) allows one to infer any conjunct from a conjunction.

P1 P2 … Pn

Pi

Page 103: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Intro

• Conjunction Introduction ( Intro) allows one to conjunct any number of previously established statements in any order.

P1 P2 … Pn

P1

Pn

Page 104: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Intro

• Disjunction Introduction ( Intro) allows one to construct any disjunction using a previous result as one of its disjuncts.

P1 … Pi … Pn

Pi

Page 105: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Elim

• Negation Elimination ( Elim) allows one to infer P from P:

P

P

Page 106: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Do we Have Free Will?

• Either determinism is true or not. Now, if determinism is true, then my actions cannot be otherwise from what they are, i.e. I don’t any freedom to exert my will. On the other hand, if indeterminism is true, then my actions are partly determined by pure randomness, so there is no such thing as a will that is in total control of my actions. Either way, I don’t have free will.

Page 107: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Proof by Cases

• The proof we just saw follows a certain pattern: ‘Either P is the case or Q is the case. However, if P is the case, then S is the case, and if Q is the case, then S is the case as well. Either way, S is therefore the case. Hence, S is the case.’

• This pattern of reasoning is called Proof by Cases• Obviously, the above pattern can be generalized

to disjunctions with any number of disjuncts.• However, a very common form is to start with:

‘Either P is the case or P is not the case’.

Page 108: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Is Space Continuous?

• Suppose space is continuous. • Then between any two (different) points A and B

there exist infinitely many other points.• Thus, in order to move from any point A to any

other point B, you have to completely go through a sequence of infinitely many points. But, you can never reach the end of an infinite sequence. Hence, motion is impossible.

• But, things do move.• Contradiction!• So, space is not continuous. (thanks to Zeno!)

Page 109: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Proof by Contradiction

• The proof we just saw relied on the following pattern: ‘Assuming P to be the case, then I get some kind of impossibility or contradiction. Hence, contrary to my assumption, P cannot be the case.’

• This pattern of reasoning is called Proof by Contradiction (or Indirect Proof or Reductio ad Absurdum or simply Reductio).

Page 110: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Proof by Cases and Proofs by Contradiction

• Proof by cases and proof by contradiction are two important proof techniques that we would like to formalize.

• But, these proof techniques do not work by inferring some statement from some other statement(s).

• Rather, they work by pointing to the fact that I am able to infer something from something else.

Page 111: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Subproofs

• At any time during a proof, a subproof may be started by making an additional assumption which can then be used to draw further inferences.

• The subproof may be ended at any time. When it is ended, the individual statements from the subproof can no longer be used to infer others.

• Subproofs demonstrate that certain statements can be inferred when an additional assumption is made, and this result can be used in the proof itself. That is, the subproof as a whole can be used to infer other statements.

Page 112: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Formalizing Proof by Cases using Subproofs and ‘’

• Using subproofs, we can now formalize the Proof by Cases technique:– You have a disjunction P1 … Pi … Pn

• These are the possible cases

– You start a subproof for each of the possible disjuncts• This is going through each of the cases (what if P1 is the

case?; what if P2 is the case?, etc.)

– You infer the same statement (Q) in all subproofs• This show that in all cases, the same thing (‘Q’) can be

inferred

– You now point to the initial disjunction and all the relevant subproofs to conclude Q

Page 113: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Elim

• Disjunction Elimination ( Elim) is the formal counterpart of Proof by Cases:

P1 … Pi … Pn

P1

S

Pn

S

S

Page 114: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Subproofs and Scope

• An additional line is used to indicate the start and end of the subproof.

• The line can also be seen as the scope of the additional assumption made at the start of the subproof: every statement within that scope is inferred from the truth of that assumption and all previous assumptions.

• The line of the proof itself can be seen in exactly this way as well. Therefore, there is no real difference between subproofs and proofs.

Page 115: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Subproofs within Subproofs

• Within any subproof, another subproof can be started.

• Subproofs within subproofs must be ended before the original subproof is ended.

• The general rule is: one can use as justification all and only statements that is either one of the assumptions whose scope one is working in, or some statement inferred from those.

Page 116: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Formalizing Proof by Contradiction using subproofs and ‘’

• We can now formalize the Proof by Contradiction– Start a subproof, and assume P

• All proofs by contradiction start by assuming something, and this is the opposite of what you want to prove!

– In the subproof, derive • This shows that assuming P leads to a

contradiction

– Point to the subproof, and conclude P

Page 117: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Intro

• Negation Introduction ( Intro) is the formal counterpart of Proof by Contradiction:

P

P

Page 118: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Intro

Introduction ( Intro) allows one to infer from a pair of statements P and P:

P

P

Page 119: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

More on contradictions

• Theorem: For any statement P: P

• Proof: It is impossible for to be true, so it is impossible for to be true and P to be false, and hence for any P: P

• In other words: Anything is a logical consequence from a logical contradiction!

Page 120: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Elim

Elimination ( Elim) allows one to infer any statement P from :

P

Page 121: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Elim

• Conditional Elimination ( Elim) allows one to infer the consequent of a conditional, given the truth of its antecedent:

P

P Q

Q

This pattern is better known asModus Ponens

Page 122: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Conditional Proof

• We have seen two uses of subproofs: for Proofs by Contradiction, and for Proofs by Cases.

• A third use for subproofs is to do a Conditional Proof.

• A Conditional Proof infers some kind of conditional P Q from a given set of statements by making P an extra assumption, and trying to infer Q from the given statements and the additional assumption P.

Page 123: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Intro

• Conditional Introduction ( Intro) is the formal counterpart of Conditional Proof:

P

P Q

Q

Page 124: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Modus Tollens

1,3 Elim

1.

2.

3.

4.

Pattern: Proof:

5.

6.

2,4 Intro

3-5 Intro

Page 125: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Disjunctive Syllogism

1,3-5,6-6 Elim

2,3 Intro

1.

2.

3.

4.

Pattern: Proof:

5.

6.

7.

4 Elim

Page 126: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Fitch

• Fitch is the program that allows the user to construct formal proofs in F.

• Fitch has a number of additional features:– Checks whether a rule is applied correctly– Allows shortcuts that are not allowed in F– Provides CON rules

Page 127: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Metalogic

Page 128: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Logic and Metalogic

• Metalogic is the study of logic. That is, where logic has no specific subject matter (logic can be applied in any field where reasoning takes place), the subject matter of metalogic is logic itself.

• Metalogic makes claims about logical properties and relationships. For example: “A statement is a tautology if and only if its negation is a contradiction” is a metalogical claim.

• Of course, the paradox of metalogic is that it needs logic to support the claims it makes about logic! Hmmm…

Page 129: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

A Central Metalogical Result:Consequence as a Central Notion

• Many interesting logical properties can be expressed in terms of logical consequence. For example:– Tautology: A statement is a tautology iff {} – Contradiction: A statement is a contradiction iff

⊥– Equivalence: Two statements and are

equivalent iff and – Inconsistency: A set of statements {1, …, n} is

logically inconsistent iff {1, …, n}

Page 130: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Consequence and Formal Proof

• Since formal proofs can be used to demonstrate consequence, and since consequence can be used to demonstrate other logical properties, formal proofs can be used to demonstrate these other logical properties:– Tautology:

• To prove that something is a tautology, derive that statement from an empty set of premises.

– Contradiction: • To prove that a statement is a contradiction, derive from that statement as

a premise.– Equivalence:

• To prove that two statements P and Q are equivalent, do two proofs: – First, assume P as a premise, and derive Q. – Second, assume Q as a premise, and derive P.

– Inconsistency: • To prove that a set of statements is inconsistent, assume all those statements

as premises, and derive .

Page 131: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Talking about Proofs …

• How do we know if a formal proof does what it is supposed to do? That is, if I can derive a sentence from a set of sentences {1, …, n}, does that really mean that is a truth-functional consequence of {1, …, n}?

• Notice that this is a metalogical questions: it asks something about formal proofs in relation to a logical property. But of course, we want to settle this question through the use of a rigorous proof, i.e. we want to prove something about formal proofs!?!

Page 132: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Modus Bogus

• In order to demonstrate that the question on the previous slide is an interesting and meaningful question, consider the following rule:

P

P Q

Q

Obviously, a formal proof system that wouldcontain this rule would be able to ‘prove’things that just don’t follow!

Page 133: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

TF

• Recall: is a truth-functional consequence of {1, …, n} iff according to truth-functional properties it is impossible for to be false if each i is true.

• Let us use the symbol TF to indicate truth-functional consequence: TF iff is a truth-functional consequence of .

• Remember: If TF then , but not vice versa. E.g. LeftOf(a,b) RightOf(b,a), but not LeftOf(a,b) TF RightOf(b,a).

Page 134: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-functional Provability

• Let us define truth-functional provability with regard to some formal deductive logic system S (e.g F) as follows: Q is truth-functionally provable from a set of premises {P1, …, Pn} in the system S iff there exists a formal proof in S going from P1, …, Pn as premises and Q as the conclusion using the rules for ‘’, ‘’, ‘’, ‘’, ‘’, and ‘’ (or any other truth-functional connective defined by S).

Page 135: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

TF(S)

• Let us use the symbol TF(S) to indicate truth-functional provability in S: TF(S) iff is truth-functionally provable from in the system S.

• The subscript TF(S) indicates that we restrict our proofs to the truth-functional rules of S.

Page 136: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Two Very Important Properties

• For every deductive system of formal logic S we can define the following 2 properties:– 1. Truth-Functional Deductive Soundness: A

system S is truth-functionally deductively sound iff for any and :

• if TF(S) then TF – 2. Truth-Functional Deductive Completeness: A

system S is truth-functionally deductively complete iff for any and :

• if TF then TF(S)

Page 137: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

F is Sound

• Q: Is F truth-functionally deductively sound?

• A: Yes!

• If you want the full proof, take ‘Intermediate Logic’.

Page 138: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

A Trivially Sound System

• Let S be a logic system that has no inference rules. Then, trivially, all inference rules of S are sound. Hence S is deductively sound as well.

• In other words, it is trivial to make a deductively sound logic system: just don’t define any inference rules!

Page 139: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Deductive Completeness

• Q: Why is truth-functional deductive completeness important?

• A: If a logic system S is not complete, then for certain and , TF but not TF(S) . So, although is a truth-functional consequence of , the rules do not allow one to prove from !

• Q: How is that possible?• A: Easy. While P TF P, the system S will not be

able to prove this. Hence, S is not a truth-functionally deductively complete system!

Page 140: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Proving Deductive Completeness

• Proving completeness for some logic system S can be very difficult. This is not hard to understand: one needs to prove that for every and such that , there exists a proof in S going from to . But, there are an infinite number of pairs and such that , and the proofs don’t seem to follow any kind of systematic pattern.

• Note: If we already know a certain system S’ to be complete, we can try to prove S to be complete by demonstrating how S can prove anything that S’ is able to prove.

Page 141: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

F is Complete

• Q: Is F truth-functionally deductively complete?

• A: Yes, but the proof for this is rather complicated and outside the scope of this class. Again, if you’re interested, take ‘Intermediate Logic’.

Page 142: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

An Algorithm for Producing Formal Proofs

• Restricting ourselves to , , , and only:– Start a proof by contradiction (assume negation of conclusion).

Let be the set of statements you have. Then go through the following loop to obtain (keep adding results to ):

• If : stop

• If and : get by Intro

• If : get by Elim

• If (1 … n ) : get 1 … n by DeMorgan pattern

• If (1 … n ) : get 1 … n by DeMorgan pattern

• If 1 … n : get 1, … , n by Elim

• If 1 … n : set up subproof for each 1 and derive from {1} / {1 … n} with this same method. Then get with Elim

Page 143: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Hokus Ponens

• The logic system B (‘Bram’) contains only one inference rule, called Hokus Ponens:

Woohoo!

One line proofs!

This system is going to make me famous!

(Only one small problem …)

P

Page 144: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Automated Theorem Proving

Page 145: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Systematic Procedures

• A systematic procedure is a procedure that follows a certain step-by-step algorithm to perform a certain task.

• Examples are cookbook recipes and computer programs.

• A systematic procedure will either stop after a finite amount of time, or never stop (e.g. because it goes into an infinite loop).

• The truth-table method is a systematic procedure. • The method of formal proof is not a systematic

procedure (though it can be used to make one).

Page 146: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Positive Tests, Negative Tests, and Full Tests

• A positive test is a systematic procedure that tries to figure out whether certain things have a certain property.

• A negative test is a systematic procedure that tries to figure out whether certain things do not have a certain property.

• A full test is a systematic procedure that tries to figure out whether certain things do or do not have a certain property.

• The truth table method is a full test. It can answer that something is or is not a logical consequence of something else.

• The formal proof method, even if it were systematic, is not a full test, but only a positive test. It can answer that something is a logical consequence of something else, but it never concludes that it isn’t a logical consequence.

Page 147: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Soundness and Completeness for Tests and Decision Procedures

• A test (positive, negative, or full) is sound iff:– if the test claims that something has (not) a

certain property, then it has (not) that property.

• A test (positive, negative, or full) is complete iff:– If something has (not) a certain property, then

the test claims that it has (not) that property.

Page 148: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Decision Procedures

• A full test that is both sound and complete is called a decision procedure.

• The truth-table method is a decision procedure for truth-functional consequence.

• The Taut Con ‘rule’ (mechanism!) is also a decision procedure for truth-functional consequence.

• Questions: Is there a decision procedure for TF consequence that is more efficient than the truth-table method. In fact, how does Taut Con work?

Page 149: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth-Trees

Page 150: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Logical Possibility

• All logically interesting claims can be reduced to questions about logical possibility:– Logical Consistency: Is it possible for all statements to

be true?– Logical Validity: Is it possible for all premises to be true

and the conclusion false?– Logical Consequence: Is it possible for the implying

statements to be true and the implied statement to be false?

– Logical Equivalence: Is it possible for the two statements to have a different truth value?

– Logical Tautology: Is it possible for the statement to be false?

Page 151: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth Table Method

• The truth table method systematically exhausts all possible truth value combinations of the statements involved.

• In the truth-table we look for a row that reflects a certain possibility, and that will tell us the answer to whatever question we had (e.g. if there is no row where statement is false, then it is not possible for that statement to be false, and hence it is a tautology).

Page 152: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Drawback and Room for Solution

• A drawback of the truth table method is that the number of rows grows exponentially.

• Fortunately, there is room for a solution to this problem. Since all we are interested in, is the existence of a specific combination of truth values of the statements involved, all we need to find is one example of such a case. Once we have found such a case, there is no need to exhaust all other cases.

Page 153: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Simple Solution: Stopping Early

• One solution to the problem of big truth tables is therefore to simply stop once you have found a row that represents the combination of truth values you are interested in.

• Thus, rather than working out a truth table column by column, you may want to do it row by row, so that you can stop as soon as you have found a row of the kind you are looking for.

• A big drawback of this approach is that if no row of the kind you are looking for exists, then you have to complete the whole truth table after all.

Page 154: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

A More Focused Search

• Consider the following argument:

P (Q R)

R

R Q

• We are interested in whether all premises can be true and the conclusion false:– In order for the conclusion to be false, R must be false.

– In order for the second premise to be true while R is false, Q must be false.

– In order for the first premise to be true while Q and R are both false, P must be false.

Page 155: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

The Short Truth Table Method

• The Short Truth Table Method assigns truth values to the involved atomic and complex statements in order to try and obtain a certain combination of truth values:

P (Q R) RR QT FT FF T FF FF

/

• The Short Truth Table Method thus tries to generate one row of the truth table that has the combination of truth values you are interested in.

Page 156: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Short Truth Table Method and Indirect Proof

• As you assign truth values to certain statements, the truth values of other statements can be forced.

• If you are forced to make a statement both true and false, then you know that the combination of truth values you are looking for does not exist:

P (Q P)F FT FT

• The short truth table method is therefore a kind of indirect proof (proof by contradiction), except that you don’t always get a contradiction.

Contradiction, so the statement is a tautology!

Page 157: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Drawback of the Short Truth Table Method

• A drawback of the short truth table method is that you are not always forced to assign any further truth values:

• At this point, you can choose to assign certain truth values, but if your choice does not lead to the row you are looking for, then you need to try a different option, and the short truth table method has no tools to do go through all of your options in a systematic way.

R (Q P) R P(Q R) PTT T

QTT T

Page 158: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth Trees• The obvious solution to the drawback of the short

truth table method is to incorporate tools to systematically keep track of multiple options.

• One method that does so is the truth tree method:– The truth tree method tests for the consistency of a set

of statements and, as such, can be used to determine validity, tautologies, equivalence, etc.

– Like the short table method, it infers which other statements are forced to be true under this assumption.

– When nothing is forced, then the tree branches into the possible options.

Page 159: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth Tree Example

(((PQ)R) (P(QR)))

((PQ)R)

(P(QR))

(PQ)R

P(QR)

P(QR)

Q

R

(PQ) R

P Q× ×

×

PQR

PQ

P Q R

Q R××

×

All branches close the originalstatement cannot be false tautology!

Page 160: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Decomposition Rules for Truth Trees

PQ

PQ PQ

PQ

P

(PQ)

(PQ)

(PQ)

(PQ)

P

PP

P

P P

Q

Q

Q

Q

Q

P

P P P

P

Q QQ

Q

Q

Page 161: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Truth Trees as Decision Procedures

• The truth tree method can easily be made into a systematic procedure.

• As such, the truth tree method becomes a decision procedure for truth-functional consequence that is, on average, quite a bit more efficient than the truth-table method.

Page 162: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Resolution

Page 163: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Running Example

• As a running example, let us consider the following argument with premises:– Horned(a)– Horned(a) Elusive(a)– Dangerous(a) Horned(a)– (Elusive(a) Mythical(a)) Rare(a)– Dangerous(a) (Elusive(a) Rare(a))

• And conclusion:– Rare(a)

Page 164: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Horn Clauses

• A Horn clause is a disjunction of literals with at most one positive literal (i.e. atomic statement).

• We can differentiate between 3 types of Horn Clauses:– Q P1 … Pn

P1 … Pn Q

– (Q and each of Pi is atomic)

Page 165: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Efficient Check of Consistency of a Set of Horn Clauses

• Horn Clauses form an important class of statements, since the consistency of a set of Horn clauses can be checked in a systematic and efficient manner. E.g. the short truth table method will always work:– 1. Write ‘T’ under all clauses.– 2. Make all forced moves until:

• A. You get a contradiction: The set is inconsistent.• B. Nothing is forced anymore: The set is consistent.

Page 166: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Example

Horned(a) H

Putting into Horn Clauses:

Horned(a) Elusive(a) H E

Dangerous(a) Horned(a) D H

(Elusive(a) Mythical(a)) Rare(a) E M R

Dangerous(a) (Elusive(a) Rare(a)) D E R

Negating Conclusion: Rare(a) R

H H E D H E M R D E R RShort Truth Table:

T F T T T T T FT F T T T F T F F T T F T T F T F

Page 167: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Horn Clauses in Conditional Form

• Horn clauses can be written in conditional form:– Q

– (P1 … Pn)

– (P1 … Pn) Q

Page 168: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Algorithm for Horn Clauses in Conditional Form

• To check the consistency of a set S of Horn Clauses in conditional form:– 1. Make a new set of statements T, starting with

the set of all atomic statements Q.

– 2. If there is a statement (P1 … Pn) in S, and each Pi is in T, then you can stop: S is inconsistent. Otherwise, go to 3.

– 3. If there is a statement (P1 … Pn) Q in S, and each Pi is in T, then add Q to T, and go to 2. Otherwise stop: S is consistent.

Page 169: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Example

H H

Putting into conditional form Horn:

H E H E

D H H D

E M R (E M) R

D E R (D E) R

R R

{H} {H,E} {H,E,D} {H,E,D,R}

Page 170: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Prolog

• The programming language Prolog is based on Horn clauses.

• A Prolog program consists of 2 types of lines:– Facts: These are Horn clauses of type Q. In Prolog: P.

– Rules: These are Horn clauses of the form (P1 … Pn)

Q. In Prolog: P :- P1 , … , Pn.

• A Prolog program is run by asking whether some atomic statement Q follows from the facts and rules. In Prolog: Q?

• The Prolog program will answer ‘Yes’ or ‘No’.

Page 171: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

The Prolog Algorithm

• Prolog checks whether Q follows from the facts or rules as follows:– 1. Make a set of goals G, starting with Q.– 2. If G is empty, stop with answer ‘Yes’.– 3. If a statement P is in G that is a fact, remove P from

G.

– 4. If P is in G and there is a rule P :- P1 , … , Pn, then remove P from G, and add each Pi to G.

– 5. If you get stuck, try a different rule P :- P1 , … , Pn.

– 6. If all options fail, stop with answer ‘No’.

Page 172: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Prolog Example

H H

Putting into Prolog:

H E E :- H.

H D D :- H.

(E M) R R :- E, M.

(D E) R R :- D, E.

Query: R?

{R}

{E, M}

{H, M}

{M}

{D, E}

{H, E}

{E}

{H}

{} ‘Yes’!

Page 173: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Resolution

• While the consistency of a set of Horn Clauses can be checked systematically and efficiently, not all statements are Horn Clauses, nor can they always be transformed into Horn Clauses.

• However, all sentences can be put into Conjunctive Normal Form (CNF).

• Horn Clauses are a special case of claims in CNF.• The algorithms for Horn Clauses are a special

case of the more general method of resolution as defined over any set of statements in CNF.

Page 174: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

The Rule of Resolution

• The Rule of Resolution is defined over disjunctions of one or more literals:

P1 … Pi-1 X Pi+1 … Pm

P1 … Pi-1 Pi+1 … Pm Q1 … Qi-1 Qi+1 … Qn

Q1 … Qi-1 X Qi+1 … Qn

(each of Pi and Qi are literals; X is atomic)

Page 175: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Clauses

• A clause is a set of literals.• Assuming a clause to represent a disjunction of

all literals that are in that clause, we can resolve two clauses as follows:

{P1 , … , Pi-1 , X , Pi+1 , … , Pm}

{P1 , … , Pi-1 , Pi+1 , … , Pm , Q1 , … , Qi-1 , Qi+1 , … , Qn}

{Q1 , … , Qi-1 , X , Qi+1 , … , Qn}

(each of Pi and Qi are literals; X is atomic)

Page 176: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

The Method of Resolution

• The method of resolution checks whether some set of statements S is consistent. It does this as follows:– 1. Make a set T of clauses representing all

conjuncts of the CNF of each statement in S.– 2. Resolve any two clauses from T that can be

resolved, and add the result to T.– 3. If two clauses resolve to the empty set, stop:

the original set of statements was inconsistent.

Page 177: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Resolution Example

(Elusive(a) Dangerous(a)) (Elusive(a) Rare(a))

Horned(a) Magical(a)

Magical(a)

(Rare(a) Dangerous(a)) Horned(a)

(E D) (E R)

(R D) H

H M

M

(E D) (E R)

(R D) H (R D) H

(R H) (D H)H M

Negate Conclusionand put into CNF

Page 178: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Resolution Example (Cont’d)

(E D) (E R) (R H) (D H) H M M

{E, D} {E, R} {R, H} {D, H} {H, M} {M}

{H}

{}

{E, H} {E, H}

{H}From CNF toclauses and resolve

Inconsistent, so valid!

Page 179: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Soundness and Completeness of Resolution

• The method of Resolution is sound and complete with regard to truth-functional consistency in the sense that:– If the method finds a set of statements to be

inconsistent, then that set of statements is indeed inconsistent (soundness).

– If a set of statements is inconsistent, then the method can find that set of statements to be inconsistent by deriving the empty clause (completeness).

Page 180: Logic Bram. What is Logic? What is logic? Logic is the study of valid reasoning. That is, logic tries to establish criteria to decide whether some piece

Algorithms for Resolution and ATP’s

• Algorithms for resolution will differ in the order in which clauses get resolved.

• Many ATP’s are based on some such algorithm:– Snark– Otter– Taut Con in Fitch