Logical Consistency and Truth in Philosophy
Logical Consistency and Truth
In philosophy, there are analytic propositions whose truth or falsity is established according to the principle of “no contradiction,” or mere logical consistency. For example: “We call any proposition or trial expression of the type ‘A is B’ or ‘A is not B’ knowledge. In philosophy, knowledge is expressed as judgment.”
Let A be “the whole is greater than the parts” and B be “the triangle has three angles.” Both seem true, but how do we know? We cannot physically grasp a whole and each of its parts, nor can we do the same with a triangle. Analytic propositions in physics are not relevant.
To ensure that they are true, we must understand the concept that functions as the subject, understand the concept that functions as the predicate, and check with thought if the second contradicts the first. If it does not contradict, it is true; if it does contradict, it is false.
In the case of propositions A and B, the predicates do not contradict the subjects. We can conclude that both statements are true. “Logical coherence” means non-contradiction, and propositions that express thoughts that we cannot think otherwise are necessary and are called analytic propositions.
Features of Analytic Propositions:
- They have no contradiction. The contradiction is needed to establish their truth or empirical falsity.
- Their truth or falsity is not established through experience in the natural world.
- They are necessary; we cannot think otherwise.
- They are universal; when you think, everyone thinks the same.
Analytic propositions express the knowledge of the so-called formal sciences of logic and mathematics. Their knowledge is axiomatic, according to the method.
Can We Be Sure of the Truth of Logical and Mathematical Propositions?
The truth of mathematics is not in a theorem as such. Consider our example, “the whole is greater than the parts.” This claim is deduced from these other two: (1) the parts alone are larger than the whole; (2) the parts are never equal to the whole. The first is true; the second is false.
Let N be the set of natural numbers and P the set of even natural numbers. P is a subset of N to the extent that P belongs to N, but not all of N belongs to P. We say that A and B have the same number of elements when a one-to-one correspondence (a bijection) can be established between them. Between P and N, we can establish a bijective relationship, meaning they have the same number of elements.
From this, it follows that it is *not* true because its opposite is unthinkable. Indeed, the whole *may* be equal to the parts. The relevant theorem to justify our negative response is called the Incompleteness Theorem.
Kurt Gödel and his Incompleteness Theorem show us that we cannot be certain that a set of axioms will not lead to contradictory statements. It is also obvious that, in mathematics, we must say goodbye to certainty.
Can We Be Sure that Analytic Propositions are Universal?
We ask two questions:
- Is the logic that we use to think the same in all cultures at all times?
- Consider the following: all human beings think; in thinking, we apply the rules of logic; analytic propositions are merely logical relationships. Why, then, do not all understand the analytic propositions of logic and mathematics?