Logic

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Logic

• Bram

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What is Logic?
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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?

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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-no
  t-be-the-case

X ? Y
Reasoning Diagram
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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

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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 Im 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.

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Sound Reasoning

• Not all valid reasoning is good reasoning.
• Example If I win the lottery, then Ill 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, Ill 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).

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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.

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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.

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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.

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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.

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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!!

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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.

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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
• H?A If the housemaid did it, the alarm would go
  off
• A The alarm did not go off

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Example Step 2 Evaluation

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

H ? B
A.
1.
H?A
A.
2.
Since every step along the way is an instance of
an obviously valid inference, the conclusion
does indeed follow from the premises. So, valid
argument!
?A
A.
3.
?H
2, 3 MT
4.
B
5.
1, 4 DS
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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

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Predicate Logic

• Predicate Logic extends Propositional Logic by
  adding individuals, predicates, and quantifiers
• Individuals John, Mary
• Predicates tall, smart
• Quantifiers all, some

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Just to Put Things in Perspective
All arguments
All deductive arguments
All deductive arguments that can be analyzed
using the formal logics we cover in class
(And Im probably optimistic here!)
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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.

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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)

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Boolean Connectives
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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.

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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.

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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

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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
T
F
T
F
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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

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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
Q
T
T
T
F
F
T
T
F
F
F
F
F
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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

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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
Q
T
T
T
F
T
T
T
T
F
F
F
F
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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.

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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!

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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.

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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
  (P?Q) ? ?(P?Q)

P
(P ? Q) ? ?(P?Q)
Q
T
T
T
T
F
F
F
T
T
F
T
T
T
T
F
F
T
T
F
F
F
F
T
F
!
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Conditionals
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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
Q
T
T
T
F
F
T
T
T
F
T
F
F
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If then Statements

• The conditional is used to capture if then
  statements.
• However, the match isnt perfect. For example, we
  dont 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
  ifthen expressions arent 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.

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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

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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

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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
Q
T
T
T
F
F
T
T
F
F
T
F
F
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Logical Properties
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Truth Tables

• Truth-tables can be used for
• defining the truth-conditions of truth-functional
  connectives
• evaluating the truth-conditions of any complex
  statement

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Tautologies

• A tautology is a statement that is necessarily
  true.
• Example P ? ?P

P ? ?P
P
T
T
T
F
T
F
T
T
F
F
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Contradictions

• A contradiction is a statement that is
  necessarily false.
• Example P ? ?P

P ? ?P
P
T
F
T
F
T
F
F
T
F
F
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Contingencies

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

P
P
T
T
F
F
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Equivalences

• Two statements are equivalent if they have the
  exact same truth-conditions.
• Example P and ??P

P
P
??P
T
T
F
T
T
F
T
F
F
F
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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
P
?P
T
T
F
T
F
T
F
F
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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
Q
P
T
T
T
T
F
T
T
T
T
F
F
T
F
F
F
F
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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
Q
P
?Q
T
T
T
T
F
F
T
T
T
T
T
F
F
T
F
F
F
F
F
T
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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 P?Q, ?Q

P
P ? Q
Q
P
?Q
T
T
T
T
F
F
T
T
T
T
T
F
F
T
F
F
F
F
F
T
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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
Q
P
?Q
T
T
T
T
F
F
T
T
T
T
T
F
F
T
F
F
F
F
F
T
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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.

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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

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But Wait!

• Consider the statement aa
• 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??

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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. aa 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, aa is a
  tautology, but not a truth-functional tautology)

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Normal Forms and Expressive Completeness
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Logically Equivalent Statements

• To express that two statements P and Q are
  logically equivalent, we will write P?Q
• ? is not a symbol of F!!

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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)

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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

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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)

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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
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Generalized Conjunctions and Generalized
Disjunctions

• Recall the Association equivalences
• P ? (Q ? R) ? (P ? Q) ? R
• P ? (Q ? R) ? (P ? Q) ? R
• Because of this, well 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)

62
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)
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? 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 ?.

64
Some equivalences involving ? and ?

• ?? ? ?
• ?? ? ?
• P ? ? ? ?
• P ? ? ? ?
• P ? ? ? P
• P ? ? ? P
• P ? ?P ? ?
• P ? ?P ? ?

65
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) ? ? ?
? ? ?
66
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
67
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

(A?B) ? (C?D) ? (Distribution) (A?B)?C ?
(A?B)?D ? (Distribution (2x)) (A?C) ? (B?C) ?
(A?D) ? (B?D)
68
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

(A?B) ? (C?D) ? (Distribution) (A?B) ? C ?
(A?B) ? D ? (Distribution (2x)) (A?C) ?
(B?C) ? (A?D) ? (B?D)
69
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.

70
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
P
P
P
P
P
P
T
T
T
T
T
F
T
F
F
T
F
F
F
T
F
F
71
Binary Connectives

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

P
Q
PQ
In general n sentences ?
T
T
T/F
T
F
T/F
2n truth value combinations (i.e. 2n rows in
truth table) ?
T
F
T/F
F
F
T/F
2n
2
different n-ary connectives!
72
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
PQ
Step 1
Step 2
T
T
F
? P??Q
T
F
T
? (P??Q) ? (?P?Q)
T
F
T
? ?P?Q
F
F
F
73
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!

74
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

75
Logic and Computer Circuitry
76
1s and 0s

• All what modern digital computers do is transform
  strings of 1s and 0s, called bitstrings.
• Information is represented using 1s and 0s, and
  information is processed through the manipulation
  of those bitstrings.
• The 1s and 0s 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 1s and 0s.

77
Logic Gates

• To process information, bitstrings need to be
  manipulated.
• Thus, depending on whatever way the 1s and 0s
  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.

78
And, Or, and Not Gates
Out
In1
In2
1
1
1
In1
Out
0
0
1
In2
1
0
0
0
0
0
Out
In1
In2
In1
1
1
1
Out
0
1
1
In2
1
1
0
0
0
0
In
Out
In
Out
1
0
1
0
79
Representing Numbers

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

53627
7100 71 7
2101 210 20
6102 6100 600
3103 31000 3000
5104 510000 50000
53627
80
Binary Numbers

• Binary numbers take 2 as their base.
• Example

10110
020 01 0
121 12 2
122 14 4
023 08 0
124 116 16
22
81
Adding Binary Numbers
1
1
1
1
1
11010110
10110011
110001001
82
Computing Binary Addition

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

In10
Out0

In20
Carry1
In11
Out1

In21
Carry2
In12
Out2

In22
?
83
2 Bit and Carry Adder
Carryn
In1n
In2n
Carryn
Outn
Carryn1
In1n
Outn
1
1
1
1
1

1
1
1
0
0
1
1
0
1
0
Carryn1
In2n
0
1
0
0
1
1
0
1
1
0
0
0
1
0
1
0
0
0
1
1
0
0
0
0
0
84
Circuitry for the 2 Bit and Carry Adder (output
bit)
In1n
In2n
Outn
Carryn
85
Circuitry for the 2 Bit and Carry Adder (carry
bit)
In1n
In2n
Carryn1
Carryn
86
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.
• Carryn1 (In1n ? In2n) ? (In1n ? Carryn) ?
  (In2n ? Carryn)
• Outn (In1n XOR In2n) XOR Carryn
• where P XOR Q (P ? ?Q) ? (?P ? Q)

87
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.

88
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 DeMorgans 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.

89
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
Q
T
F
T
F
T
T
T
T
F
T
F
F
90
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.

91
The Miniac

• Behold! The worlds 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 Miniacs
  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?

92
Formal Proofs
93
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
  (Tarskis World). This is called a counterexample.

94
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).

95
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.

96
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.

97
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!

98
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.

99
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!

100
The Structure of Proofs in F

• A formal proof in F will look like this

P1
1
?
Pn
n
I1
Justification 1
n1
?
Im
n m
Justification m
C
n m 1
Justification m1
101
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.

102
? Elim

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

P1 ? P2 ? ? Pn
?
Pi
103
? Intro

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

P1
?
Pn
P1 ? P2 ? ? Pn
104
? Intro

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

Pi
?
P1 ? ? Pi ? ? Pn
105
? Elim

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

??P
?
P
106
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 dont 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 dont have free will.

107
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.

108
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!)

109
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).

110
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.

111
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.

112
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

113
? Elim

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

P1 ? ? Pi ? ? Pn
P1
?
S
?
Pn
?
S
S
114
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.

115
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.

116
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

117
? Intro

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

P
?
?
?P
118
? Intro

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

P
?
?P
?
?
119
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!

120
? Elim

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

?
?
P
121
? Elim

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

P ? Q
This pattern is better known as Modus Ponens
?
P
?
Q
122
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.

123
? Intro

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

P
?
Q
P ? Q
124
Modus Tollens
Pattern
Proof
? ? ?
? ? ?
1.
??
??
2.
?
3.
??
?
1,3 ? Elim
4.
?
5.
2,4 ? Intro
6.
??
3-5 ? Intro
125
Disjunctive Syllogism
Pattern
Proof
? ? ?
? ? ?
1.
??
??
2.
?
3.
?
?
2,3 ? Intro
4.
?
5.
4 ? Elim
?
6.
1,3-5,6-6 ? Elim
?
7.
126
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

127
Metalogic
128
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

129
A Central Metalogical ResultConsequence 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 ? ?

130
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 ?.

131
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!?!

132
Modus Bogus

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

P ? Q
?
Obviously, a formal proof system that
would contain this rule would be able to
prove things that just dont follow!
?P
?
?Q
133
?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).

134
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).

135
?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.

136
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) ?

137
F is Sound

• Q Is F truth-functionally deductively sound?
• A Yes!
• If you want the full proof, take Intermediate
  Logic.

138
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 dont define
  any inference rules!

139
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!

140
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 dont 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.

141
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 youre interested, take Intermediate
  Logic.

142
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

143
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
144
Automated Theorem Proving
145
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).

146
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 isnt a logical
  consequence.

147
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.

148
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?

149
Truth-Trees
150
Logical Possibility

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