Converse Claim Definition

It will be useful to look at an example. The statement “The right triangle is equilateral” has the negation “The right triangle is not equilateral”. The negation of “10 is an even number” is the statement “10 is not an even number”. Of course, for this last example, we could use the definition of an odd number and say instead that “10 is an odd number”. We find that the truth of a statement is the opposite of that of negation. Let`s compare inverse and inverse statements to see if we can make judgments about them: in calculating the first-order predicate, all S are P can be represented as ∀ x. S ( x ) → P ( x ) {displaystyle forall x.S(x)to P(x)}. [8] It is therefore clear that categorical inversion is closely related to implicit inversion and that S and P cannot be exchanged for Any S are P. Be S a statement of form P implies Q (P → Q).

Then the inverse of S is the statement Q implies P (Q → P). In general, the truth of S says nothing about the truth of its overthrow,[2] unless the preceding P and the resulting Q are logically equivalent. Changing the hypothesis for the conclusion provides the opposite statement: in traditional logic, the process of exchanging the concept of subject with the term predicate is called conversion. For example, from “No S are P” to its inverse “No P are S”. In the words of Asa Mahan: For sentences A, the subject is distributed, while the predicate is not distributed, and therefore the conclusion of a statement A on its reversal is not valid. For example, for sentence A “All cats are mammals”, the opposite “All mammals are cats” is obviously false. However, the weaker statement “some mammals are cats” is true. Logicians define acciden conversion as the process of generating this weaker declaration. The inference of a statement about its reversal by accidens is generally valid.

However, as with syllogisms, this shift from universal to particular poses problems with empty categories: “All unicorns are mammals” is often considered true, while the reverse per accidens “Some mammals are unicorns” is clearly false. On the other hand, the opposite of a statement with mutually inclusive terms remains true, given the truth of the original statement. This is tantamount to saying that the opposite of a definition is true. Thus, the statement “If I am a triangle, then I am a three-sided polygon” is logically equivalent to “If I am a three-sided polygon, then I am a triangle” because the definition of “triangle” is “three-sided polygon”. A truth table makes it clear that S and the opposite of S are not logically equivalent unless the two terms involve each other: moving from a statement to its reversal is the mistake of confirming consistency. However, if the statement S and its inversion are equivalent (i.e. P is true, if and only if Q is also true), then the confirmation of the consequence is valid. What we see in this example (and what can be proven mathematically) is that a conditional statement has the same truth value as its counter-positive. We say that these two statements are logically equivalent.

We also see that a conditional statement is not logically equivalent to its inversion and its inverse. We now know these three facts about inverted and inverse statements: Before defining the inverse, counter-positive, and inverse statement of a conditional statement, we must examine the subject of negation. Every statement in logic is true or false. The negation of a statement simply involves inserting the word “no” in the correct part of the statement. The addition of the word “no” is done in such a way that it changes the truth status of the statement. If the statement is true, then the counter-positive is also logically true. If the opposite is true, then the opposite is logically true. In logic and mathematics, the inversion of a categorical or implicit statement is the result of the inversion of its two constituent statements. For the implication P → Q, the inverse is Q → P. For the categorical sentence All S are P, the inverse All P are S. In any case, the truth of the reversal is usually independent of that of the original statement. [1] Humans were not born to be logical.

Most people don`t start learning logic until they`re 10 years old. Logic is a learned mathematical skill, a method for detecting truth through certain formal steps and structures. Some of these structures of formal logic are inverted, inverse, counter-positive, and counter-example statements. If you find a substitute that tests the logical validity of the statement (but not its factual accuracy), you know that the statement is not always true and therefore not logically valid. The inversion of the implication P → Q can be written Q → P, P ← Q {displaystyle Pleftarrow Q}, but can also be denoted with P ⊂ Q {displaystyle Psubset Q} or “Bpq” (in Bocheński notation). [Citation needed] The inversion, which also appears in Euclid`s elements (Book I, Proposition 48), can be expressed as follows: In mathematics, the inversion of a theorem of the form P is → Q Q → P. The opposite may or may not be true, and even if it is true, proof can be difficult. For example, the four-vertex theorem was proved in 1912, but its opposite was not proven until 1997. [3] The logical result of all this work with reversed, inverse, counter-positive and counter-exemplary logical statements is that we learn that Jennifer is a living, breathable woman who eats.

In practice, when determining the inversion of a mathematical theorem, certain aspects of the precursor can be considered as determining the context. That is, the opposite of “Given P, if Q then R” will be “Given P, if R then Q”. For example, the Pythagorean theorem can be formulated as follows: “The original theorem is called Exposita; When it is converted, it is called in the other direction. The conversion is valid if and only if, conversely, nothing is claimed that is not confirmed or implicit in the Exposita. [5] Conditional instructions appear everywhere. In mathematics or elsewhere, it doesn`t take long to come across something in the form “If P then Q”. Conditional instructions are indeed important. It is also important to note statements that refer to the original conditional statement by changing the position of P, Q and the negation of a statement. Starting with an original statement, we get three new conditional statements called reverse, counterpositive, and inverse. Changing the conclusion of the hypothesis does not automatically prove the logical conditional statement, so the reverse statement may be true or false.

It turns out that although the inversion and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to each other. There is a simple explanation for this. We begin with the conditional statement “If Q then P”. The counter-positive of this statement is “If it is not P, then not Q”. Since the inverse is the counter-positive of inversion, the inverse and inverse are logically equivalent. If R {displaystyle R} is a binary relation with R ⊆ A × B , {displaystyle Rsubseteq Atimes B,}, then the inverse relation R T = { ( b , a ) : ( a , b ) ∈ R } {displaystyle R^{T}={(b,a):(a,b)in R}} is also called transposition. [4] If the opposite reverses a statement and the reversal denies it, could we do both? Could we reverse and deny that statement? Consider, for example, the true statement: “If I am a human being, then I am mortal.” The opposite of this statement is “If I am mortal, then I am a human being,” which is not necessarily true.