EE1J2

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EE1J2 Discrete Maths Lecture 4

• Analysis of arguments
• Logical consequence
• Rules of deduction
• Rules of equivalence
• Formal proof of arguments
• See Anderson, Discrete Mathematics with
  Combinatorics (referred to in lecture 1)

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Logical Consequence

• Let ? be a set of formulae and f a formula
• f is a logical consequence of ? if for any
  assignment of truth values to atomic propositions
  for which all of the members of ? true, f is also
  true
• If f is a logical consequence of ?, write ??f
• Note this is consistent with ?f when f is a
  tautology
• This is important! It is the basis of
  formalisation of arguments

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Arguments

• An argument consists of
• A set ? of formulae, called the assumptions or
  hypotheses
• A formula f, called the conclusion
• If ??f then the argument is a valid argument

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Valid Arguments

• In other words
• An argument is valid if its conclusion is a
  logical consequence of its assumptions

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Notation

• An intuitive way to write an argument with a set
  of hypotheses ? and conclusion f is as follows

hypotheses
? --- ?f
conclusion
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Example 1

• Suppose ? p, p?q, q ?r, f p ? q ? r
• Then the corresponding argument can be written

p p?q q ?r ------------ ? p?q ?r
Assume that p is true, p?q is true, and q?r is
true. Then p and q and r are all true.
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Example 1 (continued)

• Test validity of argument using truth table

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Example 1 (continued)
This is the only row for which all hypotheses are
true. Conclusion is also true Therefore argument
is valid
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Example 2

• Suppose ? p?q, q ?r, r, f p
• Then the corresponding argument can be written

p?q q ?r r ------- ? p
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Example 2 (continued)

• Test validity of argument using truth table

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Example 2 (continued)
In this row all hypotheses are true. But the
conclusion is false Therefore argument is not
valid
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Proofs in Propositional Logic

• In mathematics, a proof is a sequence of steps,
  each of which is
• Self-evident (or axiomatic), or
• An explicitly stated assumption (or hypothesis),
  or
• Deduced from previous steps using rules of
  deduction

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Example traditional proof

• Show that if n is an integer and n is even then
  n2 is even.
• Proof
• n is even, so there exists an integer m such that
  n 2m (by definition of even)
• So, n2 (2m)2 4m2 2(2m2) (by rules of
  arithmetic)
• Hence n2 2l (where l2m2), so n2 is even.

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Rules of deduction

• The rules of logic which are used to derive new
  theorems from axioms, assumptions and existing
  theorems are called rules of deduction
• A rule of deduction is just a simple argument
  which is known to be valid.
• So, it consists of a set of hypotheses ? and a
  conclusion f such that ??f is true

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Rules of inference

• Rules of deduction are simple arguments which can
  be validated using truth tables
• For more complex arguments, involving many
  propositions, use of truth tables is not
  practical
• Complex arguments are validated via formal
  proofs, using simpler arguments which are known
  to be valid already

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Review of the last lecture

• Valid arguments
• An argument is valid if its conclusion is a
  logical consequence of its assumptions
• The simplest way to prove that an argument is
  valid is to use a truth table
• For any assignment of T and F to the elementary
  propositions in the assumptions and conclusion,
  if all assumptions are true then the conclusion
  must be true

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Review of last lecture

• Problems
• If the number of elementary propositions is large
  then the number of rows in the truth table will
  be very large
• If the formulae are complex, then the number of
  columns in the truth table will be large
• So, try to show that more complex arguments are
  valid using formal proofs

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Proofs in Propositional Logic

• In mathematics, a proof is a sequence of steps,
• Each step is
• Self-evident (or axiomatic), or
• An explicitly stated assumption, or
• Deduced from previous steps using rules of
  deduction

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Review of last lecture

• Rules of deduction are just simple arguments
  which have already been proved to be valid
• Because they are simple, can use truth tables to
  show that they are valid

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Example rules of deduction

• Law of Detachment
• Syllogism

p?q p ------- ? q
p?q q ?r ------- ? p ?r
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Proof using truth tables
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More rules of deduction

• Modus Tollens
• Addition

p?q ?q ------- ? ? p
p ------- ? p?q
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More rules of deduction

• Specialization
• Conjunction

p?q ------- ? p
p q ------- ? p?q
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More rules of deduction
p p?(r?s) r ?q s ?q ------- ? q

• Cases
• Case elimination

p?q p?(r??r) ------- ? q
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More rules of inference

• Reductio ad Absurdum
• Well see later that this is also known as Proof
  by Contradiction

?p?(r??r) --------------- ? p
Basically, this says that if q implies both r and
?r, then q must be false
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The symbol ?

• Recall that the symbol ? means logical
  equivalence
• Recall that if f and g are both formulae
  involving the elementary propositions p1,,pN,
  then f ? g if and only if f and g have the same
  truth table.
• So, in principle the simplest way to show that
  two formulae are logically equivalent is to
  construct their truth tables and show that they
  are the same

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Standard equivalences

• Using this method we can establish a set of
  standard equivalences which we can use later in
  proofs
• Suppose p, q and r are elementary propositions
• Commutative Laws
• p ? q ? q ? p
• p ? q ? q ? p

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More standard equivalences

• Associative Laws
• (p ? q) ? r ? p ? (q ? r)
• (p ? q) ? r ? p ? (q ? r)
• Idempotent Laws
• p ? p ? p
• p ? p ? p

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More standard equivalences

• Distributive Laws
• p ? (q ? r) ? (p ? q) ? (p ? r)
• p ? (q ? r) ? (p ? q) ? (p ? r)
• De Morgans Laws
• ?(p ? q) ? (?p) ? (?q)
• ?(p ? q) ? (?p) ? (?q)

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Example proof
p ? (q ? r) ? (p ? q) ? (p ? r)
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More standard equivalences

• Law of double negation
• ??p ? p
• Equivalence of contrapositive
• p ? q ? ?q ? ?p
• Another useful equivalence
• p ? q ? ?p ? q
• p ? ?p is a tautology
• p ? ?p is a contradiction

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Definition of proof

• A proof that a formula f is a logical consequence
  of a set of assumptions ? is a sequence of
  statements, ending with f, each of which is
• True by assumption (i.e. in ?), or
• An axiom or definition, or
• A previous theorem, or
• A statement implied by previous statements by a
  rule of deduction, or
• Logically equivalent to a previous statement

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Example proof 1

• Show that
• is a valid argument

p ?q ?r??q ?r ------- ? ?p
Intuitively, if we assume ?r and ?r??q are both
true, then ?q must be true. If, in addition, p
?q is true, then ?q ? ?p is true. Combining ?q
and ?q ? ?p, ?p must be true.
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Example proof 1 (continued)

1. p ? q (given)
2. ?r ? ?q (given)
3. ?r (given)
4. ?q (from 2, 3 and Law of Detachment)
5. ?q ? ?p (from 1 by Equivalence)
6. ?p (from 4, 5 and Law of Detachment)

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Example proof 2

• Show that
• is a valid argument

p ? q q ? r p ? s ------- ? r ? s
Intuitively, if p ? q is assumed true, then
either p or q must be true (or both). If p is
true and p ? s is true, then s must be true. If q
is true and q ? r is true, then r must be true.
Hence r or s must be true (or both)
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Example proof 2

• Where do we start?
• The assumptions combine the ? and ? connectives
• These are related by p ? q ? ?p ? q
• Also ??p ? p, so ?p ? q ? p ? q
• So, conclusion can be written ?r ? s ? r ? s
• So, can we get ?r ? ?q ? p ? s ?

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Example proof 2 (continued)

1. p ? q (given)
2. q ? r (given)
3. p ? s (given)
4. ?r ? ?q (2 and Equiv. of contrapositive)
5. q ? p (1 and commutativity)
6. ? ?q ? p (5 and law of double negation)
7. ?q ? p (6 and standard equivalence)
8. ?r ? s (4, 7 3 and syllogism)
9. r ? s (8 and standard equivalence)

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Example 3

• Lets try to prove

• ?( p ? q )
• ? r ?q
• -------
• ? p ?r

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The Completeness Theorem

• In a more rigorous course on logic, a distinction
  would be made between the statements
• f is a logical consequence of ? (written ??f ),
  and
• f is proveable from ? (written ??f )
• The completeness theorem says that these two
  notions are the same.

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Completeness Theorem for Propositional Logic

• If ? is a set of formulae in Propositional Logic,
  and f is a formula in Propositional Logic, then
  ??f if and only if ??f
• f is a logical consequence of ?
• if and only if
• f is provable from ?

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Summary

• Formal proof in propositional logic
• Rules of inference
• Rules of equivalence
• Example proofs