In logic, any sentence in the form “if x, then y” is called an implication. In practical life and in science, such a sentence can have several different meanings, all subsumed under the general term “implication”. Some of these multiple meanings are outlined below.

(a) Correlation. It is observed that in experience, observation, or deliberate experiment, when x occurs, y tends to occur also, with a stated probability; but the probability has to be reasonably high if we are to feel justified in using an implication type of statement to describe it. Example: For nations, a high rate of literacy is correlated with the degree of industrialization. Stated as an implication: If country P has a high degree of industrialization, it is likely also to have a high rate of literacy. No causality is proved or indicated by this statement; there may be direct causation, but we do not know in which direction it runs, or the two terms may be related through a third one, or the correlation may be a coincidence. (Tests for significance, i.e. non-coincidence, exist.)

(b) Cause and effect. This link exists if by the statement “if x, then y” we mean that x causes y. This simple statement, as it stands, means that x is a “sufficient” cause of y: every time that x happens, y is sure to follow. If we want to say that x is both sufficient and necessary for y, we would have to say “if and only if x, then y”. A statement of a “necessary” but not “sufficient” cause would be “only if x, then y”. There are several other ways of stating this, but these need not detain us. But we should note the irreversibility of the implication: “if x, then y” is a different statement from “if y, then x”. If we accept “if x, then y” as true, then “if not x” does not allow us to say anything about y. In causation this means that if x is a sufficient cause of y, we do not know whether y is also a (sufficient) cause of x or not. Example: If gravity acts on an object which is free to move, the object falls. Gravity is a sufficient cause to make the object fall, but if we see an object fall, we cannot conclude whether this is because of gravity or because it is an iron ball moving in a magnetic field. In other words, with a merely sufficient cause, there could also be any number of other causes which would produce the same antecedent. (An antecedent is the second part of an implication, the part preceded by the word “then”.)

(c) Prediction. If stated as an implication, this is not an absolute prediction (event z will happen), but a conditional prediction (event z will happen if x happens first, or possibly “if and only if” x happens first). (See the discussion above.) Example: “If you put you hand on the hot stove, it will get burned.” (Putting your hand there is sufficient, but not necessary, for getting burned, since you can also get burned in other ways.) Another example: “If the government raises taxes, it will lose the election.” This may turn out to be true or false – only future events will show – but someone can make this prediction and many will believe it. A third example: “If there is enough sun and rain next summer, the crops will be plentiful.”

(d) Warning, or a statement of actions and expected consequences. The first two examples given above under (c) can also be considered under this category However, the third example would not be a warning, since it refers to nature, not to actions by human agents.

(e) Threat. This takes the form “if you do this action which I don’t like, I will do something which you will not like.” This is different from a warning, where one merely anticipates what nature will do (e.g. burn your hand). because here one anticipates one’s own action in retaliation. Example “If you are unfaithful, then I will leave you”. Or, between nations, “if you invade my ally, I will destroy you” (extended deterrence). Threat is one instrument for exercising power over another person or group; another instrument of power (a kinder, gentler one, but power nevertheless) is given in the next paragraph.

(f) Promise. This takes the form “if you do this action which I want you to do but you seem reluctant, then I will give you something you will really like”. In politics, this is positive inducement, sometimes a bribe, perhaps patronage. Both threats and promises are used in bargaining, i.e. in waging conflict (not necessarily violent).

(g) Ends and means. Example: “If I want to go to work this morning, I have to get up on time.” Here the antecedent (“if’) is the end or goal that I intend to reach, the consequent (“then”) is the means that I have to use to reach that end. As with causation, the distinction between “sufficient” and “necessary” means must be made. In the above example, getting up is necessary for getting to work, but not sufficient — I must also catch the bus at the comer and remember to bring my briefcase. In Aristotle’s scheme of the four causes (material, formal, efficient, final), “ends and means” is the final cause, what we above called “cause and effect” is the efficient cause; so it would be expected that these two categories of implication would be related. A cause precedes its effect, both in time and in the sentence order of the implication. An end also precedes its means in the sentence, but occurs later in time than the means; however, in the formation of my intention (decision-making), the end still comes before the means.

(h) Observations and natural laws. Example: “If repeated observations show that the volume of a gas is inversely proportional to its pressure at a constant temperature, then there must be a natural law that this inverse proportionality will always hold in future observations too.” This is called induction and is the foundation of the scientific method. It is also widely used in daily life, not only in science. E.g. “If most men I know are taller than most women, then I conclude that generally men are taller than women.” The conclusion is not certain and never will be, no matter how many observations are made; but the approximation to certainty is good enough for most practical and even scientific purposes.

(i) Theory and phenomena. This is the other half of the scientific method. Once I form a theory on the basis of natural laws derived from observations by induction, I can deduce from the theory what other consequences would follow if the theory is true, and test them by experiment or observation. Example: “If the kinetic molecular theory of gases is true, Boyle’s law about the inverse proportionality of the volume and pressure of a gas would follow”, but also “If the kinetic molecular theory of gases is true, then equal volumes of gases at constant temperature and pressure must contain equal numbers of molecules.” The first statement is why the theory was first formulated (to explain the consequent), so of course it would be expected to hold. The second statement then derives a predicted new observation by deduction from theory, and we need to find out if it is true, to confirm or falsify the theory. As is well known, a scientific theory is never completely proved. This follows straight from the logical structure of the implication. In the statement “If theory x is true, then observation y should be such and such”, we assert only “if x then y” and it does not follow that “if y then x”. For the observation y might also be explainable by some alternative theory, perhaps one which has not yet been thought of.

(j) Assumptions (axioms) and theorems. This is deduction in mathematics. An example might be “If parallel lines never meet, then they could not form two sides of a triangle.” The conclusion is a certainty given the assumptions, but the assumptions can always be questioned, as this one was. On a spherical surface like the earth, two parallel lines (two meridians) do form a (spherical) triangle if extended to one of the poles. We then reformulate the assumption by specifying “parallel lines on a planar surface”, but may still run into trouble if space itself is curved.

In all these classes of implications, (and there could be more), we link two statements conditionally. Other logical structures (“and”, “or”, negation) are represented in computer hardware as “logic gates”, but I have not heard of an “implication gate”. It is probably represented somehow.

In the classification list presented, there are three types of inferences: logical (deductive) (b, i, j), empirical (inductive) (a, c, h), and normative (d, e, f, g). It is amazing that such widely differing fields of inquiry as mathematics, natural science, and ethics could be encapsulated in these simple “if…then…” statements.