A Formal Proof is a derivation of a theorem that consists of a finite sequence of well-formed formulas. Every sentence in this sequence is either an axiom, an identified premise (a statement of "fact" that is not an axiom of the formal system), or follows from previous statements by the rules of inference of the system. The only "allowed moves" in a derivation are those "sanctioned" by the axiomatic system. If we accept the rules and axioms but nevertheless doubt the conclusions of a proof, the fault must lie in the validity of the premises.
An axiomatic system for sentential and predicate logic is somewhat arbitrary to set up. One scheme might take as an axiom or rule of inference what another scheme derives as a theorem from a slightly different set of axioms or rules of inference. In these notes we will designate as A an axiomatic system for sentential and predicate logic which has the following rules of inference. The statement preceded by the is the well-formed formula that follows from earlier well-formed formulas by the stated rule of inference. The ellipsis … is used to stand for possible steps in the derivation before or between the well-formed formulas required for the inference.
Conjunction: In a derivation if proposition p is a step and another step is proposition q, you may then conclude the conjunction of p and q.
Simplification: In a derivation if the conjunction of propositions p and q is a step, you may then conclude p.
Addition: In a derivation if proposition p is a step, you may then conclude the disjunction of p and any proposition q.
Disjunctive Syllogism: In a derivation if the disjunction of propositions p with q is a step and another step is the negation of p, you may then conclude q.
Excluded Middle Introduction (EMI): For any step in a derivation you may use the disjunction of any proposition p with the negation of p.
Universal Specification (US): If you have the assertion , you may then conclude for any s in the domain of the variable x .
Existential Specification (ES): If you have the assertion , you may then conclude where s represents a constant particular object in the domain of the variable x . This means that s is a new temporary constant symbol, not a variable!
Universal Generalization (UG): If you produce a derivation of , where x is a free variable representing any member of a certain domain, you may then conclude .
Existential Generalization (EG): If you produce a derivation of , where s represents a constant particular member of the domain of the variable x, you may then conclude .
Rules of Replacement: In any derivation you may generate a new step by replacing in a previous step an occurrence of any of the following propositions with its stated ( ) equivalent proposition. Note: all of these statements when considered as biconditionals are tautologies as can be verified with the appropriate truth table.
Title | Rule |
Commutative Property of OR | |
Commutative Property of AND | |
Associative Property of OR | |
Associative Property of AND | |
Distributive Property of OR | |
Distributive Property of AND | |
Double Negation | (The parentheses is added to avoid confusion.) |
Definition of Implication | |
Definition of Equivalence | |
DeMorgan's Rule: Negation of an OR | |
DeMorgan's Rule: Negation of an AND |
Rules of Replacement for Quantified Predicates:
Title | Rule |
Negation of a Universal Specification | |
Negation of an Existential Specification | |
Commutation of Universal Quantifiers | |
Commutation of Existential Quantifiers |
There are additional rules of inference which are valid for A. These rules can be derived from the initial set of rules already given for A , so in some sense they are redundant. Nevertheless, they are pretty intuitive and very useful.
Modus Ponendo Ponens (Modus Ponens for short)
Modus Tollendo Tollens (Modus Tollens for short)
Hypothetical Syllogism
Constructive Dilemma
Destructive Dilemma
Rules of Replacement:
Transposition
Exportation
As an example consider the proposition which is equivalent to the rule of inference Hypothetical Syllogism. A truth table establishing this statement as a tautology is shown below.
p | q | r | |||||
T | T | T | True | True | True | True | True |
T | T | F | True | False | False | False | True |
T | F | T | False | True | False | True | True |
T | F | F | False | True | False | False | True |
F | T | T | True | True | True | True | True |
F | T | F | True | False | False | True | True |
F | F | T | True | True | True | True | True |
F | F | F | True | True | True | True | True |
Suppose we wanted to prove just using the initial set of rules presented for A (i.e., we won't take Hypothetical Syllogism as a given rule of inference). Such a derivation is given in the following proof. The propositions are numbered from 1 starting at the the first step. After each step is the rule of inference used to generate that proposition.
Number Proposition Rule Used
1. EMI
2. Commutative Property of OR on 1.
3. Addition on 2.
4. Associative Property of OR on 3.
5. Associative Property of OR on 4.
6. Commutative Property of OR on 5.
7. Associative Property of OR on 6.
8. Associative Property of OR on 7.
9. EMI
10. Addition on 9.
11. Commutative Property of OR on 10.
12. EMI
13. Addition on 12.
14. Associative Property of OR on 13.
15. Commutative Property of OR on 14.
16. Associative Property of OR on 15.
17. Associative Property of OR on 16.
18. Addition on 12.
19. Commutative Property of OR on 18.
20. Associative Property of OR on 19.
21. Associative Property of OR on 20.
22. Commutative Property of OR on 21.
23. Commutative Property of OR on 22.
24. Associative Property of OR on 23.
25. Conjunction of 8 and 11.
26. Conjunction of 25 and 17.
27.
Conjunction of 26 and 24.
28.
Associative Property of AND on 27.
29.
Distributive Property of OR on 28.
30.
Distributive Property of OR on 29.
31.
Distributive Property of OR on 30.
32.
Commutative Property of OR on 31.
33.
Commutative Property of OR on 32.
34. Distributive Property of OR on 33.
35. Distributive Property of OR on 34.
36. Commutative Property of OR on 35.
37. Commutative Property of OR on 36.
38. Distributive Property of OR on 37.
39. Double Negation on 38.
40. Double Negation on 39.
41. DeMorgan's Rule (~ of an OR) in 40.
42. DeMorgan's Rule (~ of an OR) in 41.
43. DeMorgan's Rule (~ of an AND) in 42.
44. Definition of Implication in 43.
45. Definition of Implication in 44.
46. Definition of Implication in 45.
47. Definition of Implication in 46.
Here the abbreviation QED at the bottom of the sequence of propositions is a signal that the derivation is complete. QED stands for Quod Erat Demonstrandum (Latin for "which was to be shown").
This proof, while correct according to the "rules of the game" of the axiomatic system A, is nearly impossible to understand! Even if you follow each and every step, it is quite easy to lose your way within the tangle of so many propositions. The overall strategy of the derivation is certainly not apparent. It would seem that so many trees have obscured the sight of the forest! More people would be convinced of the truth of this proposition by the eight-row truth table than by the formal proof. It must also be remembered that this is a proof of a rather elementary and self-evident result. Imagine the length and complexity of a difficult theorem! The situation is very analogous to programming a computer in its rather restricted machine language. Programs in machine language tend to be very long and hard to follow. To quote Roger Penrose in Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford University Press, 1994, page 72, "Rules can sometimes be a partial substitute for understanding, but they can never replace it entirely".
For all of these reasons strictly formal proofs are rarely used in practice. Instead a "higher level" more "human" language consisting of ordinary English (or any other "natural" language) and the rules of inference is employed. Similar steps are combined or skipped but noted. Some of the simpler details are left to the reader. This rather "loose" style of proof is called "informal proof" and is used throughout mathematics. A "good" informal proof outlines the main ideas or constructions of the corresponding formal proof but with less detail and more clarity. Ideally anyone who reads the informal proof could, if required, supply the missing steps and structure of the corresponding formal proof. As an example here is an informal version of the proof of the Hypothetical Syllogism proposition done formally above.
By EMI and addition the following four propositions are true.
Using the Associative and Commutative Properties of OR these can all be rearranged as follows.
.
Forming the conjunction of these four propositions and using the Distributive Property of OR to "factor out"
gives
, which by the Commutative Property of OR is equivalent to .
Again using the Distributive Property of OR this simplifies as follows:
By Double Negation and DeMorgan's Rule for the negation of a disjunction we have
Using DeMorgan's Rule for the negation of a conjunction yields
Finally, using the Definition of Implication we have .
While this proof could certainly be improved, it is considerably shorter and clearer than the corresponding formal proof!
Many if not most theorems are stated as conditionals of the form , where the conclusion q follows as an implication of the premise p. Thus, if we can establish that q follows from p, the conditional must be true. This observation is the basis for the method of Conditional Proof (CP). Any theorem proved by CP could also be proven by Formal Proof, so CP need not be assumed as an additional Rule of Inference. On the other hand, conditional proofs are often very intuitive and easy to understand. Furthermore they illustrate in a very direct way the logical connection between the premise p and the conclusion q. The form of CP is as follows:
The right arrow before p indicates that this statement is the initial premise of the CP. The initial premise and all statements up to and including the conclusion q are prefaced by a vertical line meaning these statements are "under the assumption" of p. The horizontal line "closes off" the steps under the assumption of p.
As an example consider the following conditional proof of the Hypothetical Syllogism proposition.
Note: This proof makes use of a Conditional Proof "nested" within a Conditional Proof! This is legitimate as long as each nested initial premise is closed off before any preceding initial premises are closed off. The structure of this proof makes a very convincing demonstration of the validity of the rule of Hypothetical Syllogism.
A special case of Conditional Proof is to assume p and then reach as a contradiction the conjunction of q and ~ q for some sentence q. This serves to establish that p was not true to begin with. Hence, we conclude ~ p. This method is attributed to Plato and often goes by the Latin name Reductio ad absurdum ("reduce to the absurd"). The method of Indirect Proof is related to the reasoning used in Hypotheses Testing in statistics (an application of Inductive Logic), where one assumes the Null Hypothesis and then tries to show that it can't be supported by the available empirical evidence. A formal justification of the method of Indirect Proof is presented below.
A famous example of Indirect Proof is Euclid's theorem that there are infinitely many prime numbers. A prime number is an integer larger than 1 that has a non-zero remainder when it's divided by any integer other than itself that's greater than 1.
In this proof we take as given the Fundamental Theorem of Arithmetic: every integer greater than 1 can be expressed uniquely (apart from a reordering of factors) as a product of its prime factors. If the integer is itself prime, it is the only factor in the product. For example, , and the only way 3560 can be written as a product of prime factors is to have exactly three factors of 2, one factor of 5, one factor of 89, and no other factors.
Assume there are a finite number, n, of prime numbers. Call them . Now consider the number . This number is larger than all of the existing prime numbers and is therefore not a prime. All of the n prime numbers when divided into L leave a remainder of 1. Since L is not itself prime, it has no prime factors in contradiction to the Fundamental Theorem of Arithmetic. Therefore, our assumption of a finite number of primes is false. There must be an infinitely many prime numbers.
Suppose you want to establish . By the Rule of Replacement for the Negation of a Universal Specification this is equivalent to showing . We need only find a particular instance in the domain of x for which P(x) is false. This is often a good strategy when confronted with a universal statement whose validity you suspect. Maybe it's not true! All you have to do is find a case that doesn't work. For example, consider the assertion that for all integers n, n > 1, there are no positive integer solutions for x, y, and z to the equation . Since for n = 2, x = 3, y = 4, and z = 5 is a counter example, the assertion must be false. For n > 2, however, the statement is true and is known as Fermat's Last Theorem. This famous conjecture remained unresolved until Andrew Wiles finally proved it in 1994.
The principle of Mathematical Induction, despite the word "induction", is a method of Deductive Logic. It is used to prove sentences of the form .
Here the domain of n is the set of Integers ( ) , b is a specific integer, and P(n) is a predicate. The principle can be stated as follows:
The idea behind Mathematical Induction is that we establish two facts.
1. Base Case is true. This is usually established by substitution of b into the predicate P(x).
2. Induction Step . This is usually established by Conditional Proof. We assume P(n) (this assumption is called the Induction Hypothesis) and show by valid steps that P(n+1) follows. Note: Since we are trying to establish a Universal Generalization, it is crucial that n be a free variable representing any integer.
Once we have these two facts we can construct the following "argument".
This proof never stops. It is an infinite string of implications. It seems reasonable to make the assertion that we have established P(n) for all integers greater than or equal to b. Mathematical Induction is not a rule of inference of predicate logic. Its validity must be assumed in addition to the rules of logic already presented. Mathematical Induction is sometimes considered as a rule of "metalogic". This means that it's a rule we can use to prove properties about axiomatic systems rather than just proving statements within axiomatic systems. Sometimes Mathematical Induction is pictured as a line of dominoes set up so that each domino falling over causes the next to fall (this is the Induction Step). Thus, knocking over the first domino (the Base Case) causes all of the dominoes to fall. The only problem with this visualization is that in Mathematical Induction we've got infinitely many dominoes which is beyond our experience!
As an example of Mathematical Induction we can prove that the Rule of Replacement known as the Distributive Property of OR can be extended to a disjunction with a conjunction of n sentences.
Base Case: n = 2 is just the accepted Distributive Property of OR.
Induction Step: Assume is true. Now consider . By the Associative Property of OR this is equivalent to . Using the Distributive Property of OR this is equivalent to . Using the equivalence stated in the Induction Hypothesis gives the equivalent expression.
. Finally, by the Associative Property of AND this last expression is equivalent to
. This completes the Induction Step.
As a second example consider the formula for the sum of the cubes of the first n natural numbers. Natural numbers are the positive integers {1, 2, 3, 4, 5, 6, … } .
Here is the Greek letter upper case sigma and stands for summation. The variable j is a dummy variable indexing the terms in the sum which in this case start at j = 1 (the meaning of the expression below the ) and end at j = n (the meaning of the expression above the ). S(n) is a function (more about these in Section V) whose input is n and whose output is the stated formula.
This formula can be verified by computation for any specific value of n. For example,
.
To verify this formula for any natural number requires Mathematical Induction.
Base Case: n = 1
Induction Step: Assume
is valid.
Consider , by the Induction Hypothesis this is equal to
This establishes the Induction Step. Hence the formula is valid for any natural number n.
Mathematical Induction is an indispensable tool of mathematics, but it generally only validates results that we arrive at by other means. In the above example, Mathematical Induction verifies the formula for S(n) but does not come up with the formula to begin with!
In some problems a modified version of Mathematical Induction called "Strong" or "Complete" Mathematical Induction is used. The version presented above is then called "Weak" or "Ordinary" Mathematical Induction. Complete Mathematical Induction can be formulated as follows:
In fact, Complete Mathematical Induction can be derived from Ordinary Mathematical Induction by defining Q(n) to be the predicate . Certainly , so Ordinary Induction on Q(n) establishes Complete Induction on P(n). For some theorems the use of Complete Mathematical Induction makes the argument easier to follow.
IV. Methods of Proof <----------> Table of Contents <----------> V. Naïve Set Theory