Logic in Mathematics
This is a guide on logic in mathematics.
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Assumptions, Proofs, and the Discipline of Clear Thinking
In mathematics, one of the most important habits of mind is to keep a clear and constant distinction between what we assume and what we prove. This distinction is not merely a matter of stylistic preference; it is the very foundation upon which mathematical reasoning rests. When we write a proof or construct an argument, we must always be aware of which statements we are taking as given—whether by definition, by hypothesis, or by prior established result—and which statements we are deriving through logical steps. Confusing the two leads to circular reasoning, unstated assumptions, and arguments that appear valid but are not.
The rules of deduction that govern mathematical reasoning are, at their core, essentially rules of common sense. They are not mysterious or arbitrary conventions; they are formalized versions of the patterns of reasoning that people use in everyday life when they reason carefully. Among the most fundamental of these patterns is the "if...then" sentence, which we use whenever one statement implies another. For example, we might say: "If 2x = 5, then x = 5/2." This is a true statement, and it follows the general sentence structure: if A, then B. Here, A is the statement "2x = 5" and B is the statement "x = 5/2." The statement asserts that whenever A holds, B must also hold.
Converse Statements and the "If and Only If" Construction
Given any statement of the form "if A, then B," we may form what is called its converse, which is the statement "if B, then A." The converse of our example is: "If x = 5/2, then 2x = 5." In this particular case, the converse is also true. When both a statement and its converse are true, we can save ourselves space and express the relationship more compactly by saying: "2x = 5 if and only if x = 5/2." This single phrase captures both directions of implication at once.
It is worth pausing to consider the phrase "only if," because it has a logic of its own that is easily misunderstood. The statement "A only if B" means precisely "If A, then B." In other words, A can occur only when B occurs; B is a necessary condition for A. However, using "only if" by itself, outside the context of the full phrase "if and only if," tends to sound awkward in English. Because of the structure of the English language, there is a natural tendency to interpret "A only if B" as meaning "if B, then A"—that is, to mistake it for the converse. To avoid this confusion entirely, one sensible policy is never to use the phrase "only if" by itself, but only as part of the complete phrase "A if and only if B," which is unambiguous.
Examples Illustrating Converse and Biconditional Statements
Several examples help to clarify these distinctions. Consider the assertion: "If x = -3, then x² = 9." This is a true statement, since squaring -3 indeed yields 9. Its converse, however—"If x² = 9, then x = -3"—is a false statement, because x may equally well be equal to 3. Since the converse fails, the biconditional statement "x² = 9 if and only if x = -3" is also false. The failure of the converse is enough to invalidate the "if and only if" claim.
A second example comes from geometry. The statement "If two lines are perpendicular, then they have a point in common" is true. Its converse—"If two lines have a point in common, then they are perpendicular"—is false, because two lines may intersect at any angle, not necessarily a right angle. Once again, the biconditional fails.
A third example shows a case where the biconditional succeeds. The statement "Two circles are congruent if and only if they have the same radius" is true. Here both directions of implication hold: congruent circles must have equal radii, and circles with equal radii must be congruent. This is the sort of clean, two-directional relationship that the phrase "if and only if" is designed to express.
Proof by Contradiction
Mathematicians often establish the truth of a statement through a technique known as proof by contradiction. The method works as follows. Suppose we wish to prove that a certain statement A is true. We begin by supposing, temporarily, that A is false. Then, by logical reasoning starting from the supposition that A is false, we arrive at an absurdity—that is, at a contradiction of a statement we know to be true. Since our reasoning was valid, the only possible source of the contradiction is our initial supposition. We therefore conclude that the supposition "A is false" cannot hold, and hence that A must be true.
A classic example of this method occurred in the proof that √2 is not a rational number. The proof proceeded by assuming that √2 is rational, which means it can be expressed as a fraction in lowest terms. By reasoning from this assumption, one shows that both the numerator and the denominator of this fraction must be even. But a fraction in lowest terms cannot have both numerator and denominator even, since they would share a common factor of 2. This contradicts the hypothesis that √2 could be expressed as a fraction in lowest form. We therefore conclude that √2 is not a rational number. The power of the method lies in its indirectness: rather than proving the statement directly, we show that its negation is impossible.
Meaningless Assertions and the Need for Qualification
Not every string of mathematical symbols constitutes a meaningful statement. Some assertions are true, some are false, and some are simply meaningless. Often a set of symbols is meaningless because certain letters, such as x or a, appear without being properly qualified—that is, without any indication of what kind of object they are meant to denote or what range of values they may take. When we write an equation such as 2x = 5, the context is ordinarily supposed to make it clear that x denotes a number. However, if there is any chance of doubt, this should always be specified explicitly. A more adequate formulation would be: "If x is a real number and 2x = 5, then x = 5/2." Similarly, one might write: "Let x be a real number. Then 2x = 5 if and only if x = 5/2."
The symbols "2x = 5" by themselves are called an equation. As it stands, the equation merely indicates a possible relationship; to give it definite meaning, we must say something more about x. For instance, we might assert any of the following:
There exists a number x such that 2x = 5. (This is true.)
For all numbers x, we have 2x = 5. (This is false.)
There is no number x such that 2x = 5. (This is false.)
If x is a real number and 2x = 5, then x < 7. (This is true.)
Each of these sentences gives the symbols a definite meaning by quantifying over x or by placing the equation within a conditional statement. The bare equation, by contrast, does not assert anything on its own.
Ambiguity in Problem Statements
The symbols 2x = 5 can also appear in a context such as: "Determine all numbers x such that 2x = 5." This sentence is actually somewhat ambiguous, because of the word "determine." In one sense, the equation itself already determines such numbers x; the equation specifies a condition that x must satisfy. To avoid such ambiguities, one might prefer a more explicit formulation: "Express all numbers x such that 2x = 5 in the form m/n, where m and n are integers and n is not equal to zero." This is what one should understand when faced with the earlier sentence, or with a similar instruction such as "Solve for x in the equation 2x = 5."
The Importance of Complete Sentences
In writing mathematics, it is essential that complete sentences be used. Many mistakes occur because one allows incomplete symbol strings, such as "2x = 5," to occur without the proper qualifications. An equation written in isolation is not a statement; it is merely a fragment. To function as a mathematical assertion, it must be embedded in a sentence that makes clear what is being claimed—whether an existence claim, a universal claim, a conditional, or something else. Careful writers of mathematics therefore take pains to ensure that every assertion is complete, qualified, and unambiguous.
A Final Example
Consider the symbols "x² = 2" by themselves. These symbols are merely an equation; they do not assert anything. The sentence "There exists a rational number x such that x² = 2" is false, because no rational number squares to 2. By contrast, the sentence "There exists a real number x such that x² = 2" is true, because √2 is such a real number. The same equation, placed in different contexts, yields statements with different truth values—a vivid reminder of why qualification and completeness matter so much in mathematical writing.