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)

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

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

Valid Arguments In other words: An argument is valid if its conclusion is a logical consequence of its assumptions

Notation An intuitive way to write an argument with a set of hypotheses  and conclusion f is as follows:  ---  f hypotheses conclusion

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.

Example 1 (continued) Test validity of argument using truth table: pqrp pqpq qrqr pqrpqr TTTTTTT TTFTTFF TFTTFTF TFFTFTF FTTFTTF FTFFTFF FFTFTTF FFFFTTF

Example 1 (continued) pqrp pqpq qrqr pqrpqr TTTTTTT TTFTTFF TFTTFTF TFFTFTF FTTFTTF FTFFTFF FFTFTTF FFFFTTF This is the only row for which all hypotheses are true. Conclusion is also true Therefore argument is valid

Example 2 Suppose  = {p  q, q  r, r}, f = p Then the corresponding argument can be written: p  q q  r r  p

Example 2 (continued) Test validity of argument using truth table: pqr pqpq qrqr rp TTTTTTT TTFTFFT TFTFTTT TFFFTFT FTTTTTF FTFTFFF FFTTTTF FFFTTFF

Example 2 (continued) pqr pqpq qrqr rp TTTTTTT TTFTFFT TFTFTTT TFFFTFT FTTTTTF FTFTFFF FFTTTTF FFFTTFF In this row all hypotheses are true. But the conclusion is false Therefore argument is not valid

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

Example ‘traditional’ proof Show that if n is an integer and n is even then n 2 is even. Proof 1. n is even, so there exists an integer m such that n = 2m (by definition of ‘even’) 2. So, n 2 = (2m) 2 = 4m 2 = 2(2m 2 ) (by rules of arithmetic) 3. Hence n 2 = 2l (where l=2m 2 ), so n 2 is even.

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

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

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

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

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

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

Example rules of deduction Law of Detachment: Syllogism: p  q p  q p  q q  r  p  r

Proof using truth tables pq pqpq pq TTTTT TFFTF FTTFT FFTFF pqr pqpqqrqrprpr T T TTTT TTFTFF TFTFTT TFFFTF FTTTTT FTFTFT FFTTTT FFFTTT

More rules of deduction Modus Tollens: Addition: p  q  q   p p  p  q

More rules of deduction Specialization: Conjunction: p  q  p p q  p  q

More rules of deduction Cases: Case elimination: p p  (r  s) r  q s  q  q p  q p  (r  r)  q

More rules of inference Reductio ad Absurdum: We’ll 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

The symbol  Recall that the symbol  means logical equivalence Recall that if f and g are both formulae involving the elementary propositions p 1,…,p N, 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

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

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

More standard equivalences Distributive Laws: p  (q  r)  (p  q)  (p  r) p  (q  r)  (p  q)  (p  r) De Morgan’s Laws:  (p  q)  (  p)  (  q)  (p  q)  (  p)  (  q)

Example proof p  (q  r)  (p  q)  (p  r) pqr(p  (q  r))((p  q)  (p  r)) TT T TTT T TTTT T TTT TTFTTTFFTTTTTTF TFTTTFFTTTFTTTT TFFTTFFFTTFTTTF FTTFTTTTFTTTFTT FTFFFTFFFTTFFFF FFTFFFFTFFFFFTT FFFFFFFFFFFFFTF

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

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

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.

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) Assumptions Conclusion

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)

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 ?

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)

Example 3  ( p  q )  r  q  p  r Let’s try to prove:

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.

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 

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