From a Complete Claim to Its Truth Value
In Predicates and Propositions, we distinguished a condition on inputs from a complete mathematical claim. For the integer-domain predicate \(P(x): x+2=5\), the instances \(P(3)\) and \(P(1)\) are both propositions, but they have different truth values.
Recognizing a proposition is only the first task. We must then ask what would justify calling it true or false. A particular equality may be settled by calculation. A claim about every integer requires an argument covering every integer, unless we can find an allowed input at which it fails.
A true statement correctly asserts what holds in that context; a false statement asserts something that does not hold. Neither status depends on whether the statement is useful, surprising, or easy to verify. In particular, a false proposition is still a proposition.
Evaluating Particular Statements
For a numerical equality, evaluate both sides using the meanings of the operations involved. The equality is true when both sides represent the same number and false when they represent different numbers. For an inequality, the comparison must also be checked: equality of the two sides does not satisfy a strict inequality.
| Statement | Truth value | Justification |
|---|---|---|
| \(3+2=5\) | True | The sum on the left is \(5\). |
| \(1+2=5\) | False | The sum on the left is \(3\), not \(5\). |
| \((-3)^2=9\) | True | \((-3)(-3)=9\). |
| \(4<4\) | False | A number is not strictly less than itself. |
| \(0\text{ is even}\) | True | \(0=2\cdot0\), with integer multiplier \(0\). |
The final row uses the definition of evenness given earlier: an integer is even when it equals \(2k\) for some integer \(k\). Definitions provide criteria for truth, not merely labels to memorize. The definition includes zero and negative even integers.
Worked Example: Checking Both Sides
Retain the real-domain predicate from the previous tutorial:
At \(x=-2\), the assertion is \((-2)^2=(-2)+2\). Its left side is \(4\), and its right side is \(0\). Therefore \(Q(-2)\) is false.
At \(x=-1\), the assertion is \((-1)^2=(-1)+2\). Both sides equal \(1\), so \(Q(-1)\) is true. These conclusions concern two specified instances. They do not assign a single truth value to the open sentence \(Q(x)\).
Read the Whole Statement
The previous tutorial introduced the ordinary mathematical wording “for every” and “there is.” These phrases change what is being asserted, and therefore change what must be checked.
For \(Q(x): x^2=x+2\), compare these complete statements:
- \(Q(2)\): true, since \(2^2=2+2=4\).
- There is a real number \(x\) for which \(Q(x)\) holds: true, because the allowed value \(2\) satisfies the condition.
- For every real number \(x\), \(Q(x)\) holds: false, because at the allowed value \(0\) the asserted equality becomes \(0=2\).
There is no conflict between the second and third conclusions. A condition can hold somewhere without holding everywhere. To evaluate a general statement, we must preserve its exact wording rather than replacing it with a statement about one convenient input.
Thus \(2\) is a witness for the existence claim above, and \(0\) is a counterexample to the claim about every real number. In each case, checking the input's eligibility is part of the justification.
Why One Counterexample Is Enough
A statement about every allowed input promises that none of those inputs fails. A counterexample directly violates that promise. It does not matter how many other inputs satisfy the condition.
Worked Example: A Pattern That Does Not Continue
Consider the statement:
For every positive integer \(n\), \(n^2+n+2\) is greater than \(2n^2\).
At \(n=1\), the comparison is \(4>2\), which is true. This is a successful instance, but it is not a proof of the general claim.
At \(n=2\), the two sides are
The claimed strict inequality is \(8>8\), which is false. Since \(2\) is a positive integer, it is a counterexample. The general statement is therefore false.
The value \(n=0\) would not be an eligible counterexample to this statement, regardless of the calculation, because its domain is the positive integers.
A false “every” statement need not fail everywhere. Disproving it establishes only that at least one allowed input fails. In this example, the successful instance \(n=1\) remains successful after the general claim has been disproved.
Establishing a True General Statement
To prove a claim about every integer, checking a list of integers is not enough: the list leaves other integers unexamined. A general argument instead starts with an arbitrary allowed integer and uses only properties that all allowed integers share.
Proof. Let \(n\) be an arbitrary even integer. By the definition of evenness, there is an integer \(k\) such that \(n=2k\). Squaring gives
Since \(k\) is an integer, \(2k^2\) is an integer. The displayed equation expresses \(n^2\) as twice an integer, so \(n^2\) is even by the definition of evenness. Because \(n\) was arbitrary among the even integers, the claim holds for every even integer.
The proof does not assume that \(n\) is positive. It includes \(n=0\) and negative even integers because the integer \(k\) may be zero or negative. Nor does it rely on observing that \(2^2\), \(4^2\), and \(6^2\) are even. Those calculations illustrate the result; the argument establishes its full scope.
The hypothesis matters. The different statement “The square of every integer is even” is false. The integer \(1\) is a counterexample: its square is \(1\), and \(1=2k\) would require \(k=1/2\), which is not an integer.
When Examples Settle a Claim
The role of an example depends on the scope of the assertion. For a “there is” statement, a single verified witness is a complete justification. For a “for every” statement, a single successful instance is generally only partial information.
| Kind of claim | To establish truth | To establish falsity |
|---|---|---|
| Every allowed input satisfies a condition. | Justify the condition for all allowed inputs. | Give an allowed input that fails. |
| There is an allowed input satisfying a condition. | Give a witness, or otherwise prove that one exists. | Show that no allowed input satisfies the condition. |
Worked Example: No Integer Can Work
Consider “There is an integer \(n\) such that \(n+2=n\).” Trying \(0\), \(1\), and \(-1\) produces three failures, but those failures alone do not rule out every integer.
To settle the claim, suppose an integer \(n\) satisfied \(n+2=n\). Subtracting \(n\) from both sides would give \(2=0\), which is false. Therefore no integer can satisfy the equation, and the existence claim is false.
Compare this with “There is an integer \(n\) such that \(n+2=0\).” The integer \(-2\) is a witness because \((-2)+2=0\). Here a single calculation settles the existence claim.
There is also an important case in which checking individual inputs proves an “every” statement: when the domain is finite and every input is checked. If the allowed values are exactly \(-1\), \(0\), and \(1\), then “Every allowed \(x\) satisfies \(x^2\leq1\)” is true because the three squares are \(1\), \(0\), and \(1\). This is exhaustive verification, not a sample from a larger domain.
False Is Not the Same as Undefined or Unknown
Before assigning a truth value, confirm that there is a meaningful, complete assertion. The open sentence \(x+2=5\), with \(x\) free, is not false merely because an input has not been specified. It is a predicate awaiting context.
Likewise, \(1/0=3\) is not an ordinary false equality of real numbers. Its left side is undefined in real arithmetic. As explained in the previous tutorial, a condition involving \(1/x\) requires an appropriate domain restriction before its instances can be evaluated.
A failed attempt at a proof does not disprove a statement. Similarly, an unsuccessful search for a counterexample does not prove it. We may remain uncertain about a complete, meaningful claim even though classical logic treats it as either true or false.
This distinction also applies to faulty arguments. An invalid argument can have a true conclusion. For example, checking \(2^2=4\) does not prove that every even integer has an even square, even though the theorem above shows that the conclusion is true.
A Method for Assessing Truth
Account for the variables and ensure that every expression is defined in the stated context.
Does the claim concern a specified input, every allowed input, or the existence of an allowed input?
Use a calculation, a witness, a counterexample, exhaustive verification, or a general argument as appropriate.
Do not turn several successful examples into a general proof, or a failed search into a conclusion of falsity.
Determining truth is not simply attaching a label. It is reading a claim precisely and supplying a justification that covers exactly what the claim says. That habit will remain essential when the statements in this course become more complex.
Check Your Understanding
Give a reason for each answer. When using a witness or counterexample, verify that it belongs to the stated domain.
- Determine the truth values of \((-4)^2=16\), \(0<0\), and “\(-6\) is even.” Justify the last answer using the definition of evenness.
- For \(Q(x): x^2=x+2\) on the real numbers, evaluate \(Q(3)\). Use earlier calculations to give a witness for “There is a real number satisfying \(Q\)” and a counterexample to “Every real number satisfies \(Q\).”
- A learner checks that \(n^2\) is even for \(n=2,4,6,8\) and concludes that the square of every integer is even. Identify both the gap in the argument and a counterexample to the conclusion.
- Determine whether there is an integer \(n\) satisfying \(n+5=n\). Give an argument that covers all integers rather than testing a list.
- Prove that the square of every even integer is a multiple of \(4\), where “a multiple of \(4\)” means \(4m\) for some integer \(m\). Explain why your proof includes zero and negative even integers.
- Distinguish a false proposition, an open sentence, an assertion involving division by zero, and a proposition you have not yet proved. Why should these not all receive the label “false”?