Never Stop Questioning


“The important thing is not to stop questioning. Curiosity has its own reason for existing. One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of the marvellous structure of reality. It is enough if one tries merely to comprehend a little of this mystery every day. Never lose a holy curiosity. Besides; it is a miracle that curiosity survives formal education.”

– Albert Einstein

Necessary and Sufficient Condition


In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.

Lightning

– A necessary condition of a statement must be satisfied for the statement to be true. Formally, a statement P is a necessary condition of a statement Q if Q implies P.
For example, the ability to breathe is necessary to a human’s survival. Likewise, for the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime.

– A sufficient condition is one that, if satisfied, assures the statement’s truth. Formally, a statement P is a sufficient condition of a statement Q if P implies Q.
Thus, jumping is sufficient to leave the ground, since an intrinsic element of the concept jumping is leaving the ground. A number’s being divisible by 4 is sufficient (but not necessary) for its being even, but being divisible by 2 is both sufficient and necessary.

A condition can be either necessary or sufficient without being the other. For instance, being a mammal (P) is necessary but not sufficient to being human (Q), and that a number q is rational (P) is sufficient but not necessary to q‘s being a real number (Q) (since there are real numbers which are not rational). A condition can be both necessary and sufficient.
For example, at present, “today is the Fourth of July” is a necessary and sufficient condition for “today is Independence Day in the United States.” Similarly, a necessary and sufficient condition for invertibility of a matrix M is that M have a nonzero determinant.

Necessary Condition
The assertion that P is necessary of Q is colloquially equivalent to “Q cannot be true unless P is true,” or “if P is false then Q is false.” By contraposition, this is the same thing as “whenever Q is true, so is P”. The logical relation between them is expressed as “If Q then P” and denoted “Q [Rightarrow] P” (Q implies P), and may also be expressed as any of “P, if Q,” “P whenever Q,” and “P when Q.” One often finds, in mathematical prose for instance, several necessary conditions which, taken together, constitute a sufficient condition, as shown in Example 3.

Example 1: Consider thunder, technically the acoustic quality demonstrated by the shock wave that inevitably results from any lightning bolt in the atmosphere. It may fairly be said that thunder is necessary for lightning, since lightning cannot occur without thunder, too, occurring. That is, if lightning does occur, then there is thunder.

Example 2: Being at least 30 years old is necessary of serving in the U.S. Senate. If you are under 30 years old then it is impossible for you to be a senator. That is, if you are a senator, it follows that you are at least 30 years old.

Example 3: In algebra, in order for some set S together with an operation star to form a group, it is necessary that star be associative. It is also necessary that S include a special element e such that for every x in S it is the case that e star x and x star e both equal x. It is also necessary that for every x in S there exist a corresponding element x” such that both x star x” and x” star x equal the special element e. None of these three necessary conditions by itself is sufficient, but the conjunction of the three is.

Sufficient Condition
To say that P is sufficient for Q is to say that in and of itself, knowing P to be true is adequate grounds to conclude that Q is true. The logical relation is expressed as “If P then Q” or “P [Rightarrow] Q,” and may also be expressed as “P implies Q.” Several sufficient conditions may, taken together, constitute a single necessary condition, as illustrated in example 3.

Example 1: An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense that hearing thunder, and unambiguously recognizing it as such, justifies concluding that there has been a lightning bolt.

Example 2: A U.S. president’s signing a bill that Congress passed is sufficient to make the bill law, regardless of the fact that even in the event of a presidential veto it still could have become law through a congressional override.

Example 3: That the centre of a playing card should be marked with a single large spade (♠) is sufficient for the card to be an ace. Three other sufficient conditions are that the centre of the card be marked with a diamond (♦), heart (♥), or club (♣), respectively. None of these conditions is necessary to the card’s being an ace, but their disjunction is, since no card can be an ace without fulfilling at least (in fact, exactly) one of the conditions.

Linguistic Sign


There are many models of the linguistic sign. According to a linguistic classic model, language is made up of signs and every sign has two sides:

– The signifier, the ‘shape’ of a word, its phonic component. For instance; the sequence of letters or phonemes.

– The signified, the ideational component, the concept or object that appears in our minds when we hear or read the signifier. For instance; a small domesticated feline – the signified is not to be confused with the ‘referent’. The former is a ‘mental concept’, the latter the ‘actual object’ in the world.

Furthermore, speech acts (la parole) are separated from the system of a language (la langue). Parole was the free will of the individual, whereas langue was regulated by the group, albeit unknowingly.

The theory also postulated that once the convention is established, it is very difficult to change, which enables languages to remain both static, through a set vocabulary determined by conventions, and to grow, as new terms are needed to deal with situations and technologies not covered by the old.

According to the theory, the relation between the signifier and the signified is ‘arbitrary’, meaning; there is no direct connection between the shape and the concept.
For instance, there is no reason why the letters ‘C-A-T’ – or the sound of these phonemes – produce exactly the image of the small, domesticated animal with fur, four legs and a tail in our minds. It is a result of ‘convention’: speakers of the same language group have agreed and learned that these letters or sounds evoke a certain image.

Compare an aerial drawing of London (a field of potential signifies) with a grid (a field of signifiers) placed on it. The grid is arbitrary. Its structure – however motivated – divides the drawing into areas, which can then be referred to. The division of the drawing is arbitrary.
A square ‘EC1’ is an inseparable fusion of grid and area of drawing. This is a sign – just like two sides of the same sheet of paper – which refers to ‘real’ land. EC1 does not have to refer to the particular part of London it does. Drawing + grid = map = language.

Two concepts are often cited to disprove the theory’s claim, however, it provides reasons as to why these concepts are irrelevant. They are:

– Onomatopoeia, which applies only in a very limited number of cases, and stems from phonetic approximation of sounds, which can themselves evolve into a more standard linguistic sign.

– Interjections, which fall much to the same logic as onomatopoeia, as is demonstrated by comparisons of the same expression in two languages. For instance; the French aïe and the English ouch.

Likewise, the figures made in writing are arbitrary, and not connected to the sounds which they inspire. The only requirement is the ability to differentiate between separate figures, such as t, l and f, and that the difference in the symbols is understood by the collective consciousness. Meaning for instance; ‘i’ is recognized as ‘i’ by all members of the community, no matter what word it is placed in.