By KP Mohanan and Tara Mohanan
When we are given certain premises, we draw conclusions from them, often without thinking twice. For example, if we are told (a) and (b), we conclude (c):
1 a. Martians have three eyes.
b. Simson is a Martian.
c. Simson has three eyes.
There are two questions that we may ask of (1).
A) Is it legitimate to draw conclusion (1c) from (1a) and (1b)?
B) Is statement (1c) true?
Question (A) is about the VALIDITY of reasoning in (1).
Question (B) is about the SOUNDNESS of the rational justification provided in (1).
Our judgments on the legitimacy of our conclusions are based on certain patterns in the FORM of our argument, quite independently of the meanings of the relevant statements, and the truth of the statements. Thus, our reasoning in concluding (c) above takes the following form:
2 If it is true that: a. Martians have three eyes, and
b. Simson is a Martian,
Then it is true that: c. Simson has three eyes.
Even if we cannot tell if the premises and conclusion are true, given the FORM of the argument, we can draw the conclusion that IF the premises are true, THEN the conclusion must be true.
But that doesn’t mean that the conclusion is true. For instance, if (2a) is false, then (2c) need not be true even if the reasoning is valid.
Let us define validity and soundness as follows:
VALIDITY: An argument is valid if and only if the conclusion follows logically from the given premises.
SOUNDNESS: An argument is sound if and only if (i) its reasoning is valid; and
(ii) all the premises are true.
Notice that an argument is valid as long as there is no flaw in the reasoning, regardless of whether or not the premises are true.
For an argument to be sound, the reasoning has to be valid, but in addition, the premises have to be true. This means that a valid argument may be unsound, but an invalid argument can never be sound. To put it differently, we can have valid but unsound arguments, but we can never have an argument that is invalid and sound.
Let us consider an example:
3 a. Every insect has six legs.
b. Wasps have six legs.
c. Therefore wasps are insects.
In this example, all the three statements can be taken to be ‘true’. However, the conclusion (c) doesn’t follow from the premises. (1a) says that every insect has six legs. It doesn’t say that ONLY insects have six legs. Therefore, though by (1b), wasps have six legs, (1a) doesn’t tell us anything about wasps. The conclusion doesn’t follow logically from the premises. Hence the reasoning is invalid. As a result, the argument is unsound, even though we judge the conclusion in (3c) to be true.
Take another example:
4 a. The area of a right-angled triangle is the product of the length of the non-hypotenuse sides.
b. The length of the hypotenese of a right-angled triangle ABC is 5cms, and those of the other two sides are 3 cms and 4 cms.
c. Therefore, the area of ABC is 6 sq. cms.
Given (4a) and (4b), the area of ABC should be 12 sq. cm. So the conclusion doesn’t follow from the premises. Hence the reasoning is invalid, making the argument unsound.
Here is a final example:
5 a. The area of a right-angled triangle is the product of the length of the non-hypotenuse sides.
b. The length of the hypotenese of a right-angled triangle ABC is 5cms,
and those of the other two sides are 3 cms and 4 cms.
c. Therefore, the area of ABC is 12 sq. cms.
In this case, the conclusion (5c) follows from (5a, b). Hence the reasoning is valid. However, given the condition that for the argument to be sound, the premises have to be true, the argument in (5) is unsound. This is because, within Euclidean geometry — the kind of geometry we have all studied in school — the area of a triangle is not base x height, but ½ base x height. [This example requires subject knowledge, but only of the level of Grade 8.]
Let us go back to questions we began with, in a more general form:
A) Is it legitimate to draw conclusion X from premises Y-Z?
B) Is statement X true?
We have distinguished between the concepts of validity (relevant for (A)) and soundness (relevant for (B)), but there is a further distinction that we need to make about (B). To understand this, consider the following scenario:
You are looking at a creature, and someone says “That creature has four legs.”
On the basis of what you see, you say, “Yes, that is true.” or “No, that is not true.”
Validity and soundness are not relevant for your judgment on the truth or falsity of the statement, “That creature has four legs.” Here is why:
Your conclusion that it is true/false is not based on the rational justification in support of or against the statement. It is based on your experience, without the mediation of conscious reasoning. The concepts of validity and soundness are relevant only in the context of rational justification.
The distinction here is between a “true statement” and “sound justification for a statement.” This distinction becomes clearer when we examine the nature of truth in mathematics. We say that a mathematical conjecture is true in a given theory if and only if the conjecture follows logically from the given set of axioms and definitions of the theory. For instance, take the following statement:
“Given any two distinct points, one and only one straight line can be drawn through both.”
We say that the statement is true in Euclidean geometry, because it can be derived from Euclidean axioms and definitions. But it is not true in spherical geometry, because the axioms and definitions of spherical geometry yield the following statement:
“There exist pairs of distinct points through which infinitely many straight lines can be drawn.”
That raises a question:
Are mathematical proofs sound?
Given our above definitions of validity and soundness, we must say that mathematical proofs can be valid, but, given that the premises (axioms and definitions) are neither true nor false, mathematical proofs cannot be said to be sound. Thus, even when a conjecture is accepted as true in mathematics, the proof that justifies that conclusion does not qualify as a sound proof.
In short, soundness is not a relevant consideration for mathematical proofs.
Photo by Giammarco Boscaro on Unsplash