jeudi, février 16, 2006

Question of the Day


Necessitation, Collapse, Banalisation and other Issues

One of the curious properties of the Modal Semantics of Human Languages is that they do have a T principle and a Necessitation rule, and even Aristotle’s law, but lack other principles, like banalisation and collapse.
The T principle may be stated as follows(1):
T: A⊃A

In Logic the Necessitation Rule requires that if A is a thesis of a certain modal system S, then A is also a thesis of the S. In the study of Natural Languages one may, for instance, think of a variation of Necessitation by simply saying that if a string or sentence that expresses a proposition is a sentence of a language, then the sentence that expresses that the same proposition is necessary must also be a sentence of the same language. But that would be a very basic and elementary principle of natural language.
However, neither Necessitation nor T must be confused with the Banality and Collapse, which are quite different laws. Banality is a principle that claims that if a proposition is true than it is necessarily true:
(Ban) A⊃A

Collapse is a stronger law that establishes equivalence between necessity and truth:
(Coll) A≡A

Banality is a fallacy in Natural Languages. There are abundant examples that show this, such as the nonsense sentences below:
(1) #If Harper is the Prime Minister then he is necessarily the Prime Minister.
(2) #If it is raining outside, then it must be raining.
(3) #If it is true that your car is broken, then it is necessary that your car be broken.
(4) #If Governor General Michaela Jean came from Haiti, then she needs to have come from Haiti.

Of course, the same can be said about Collapse.
These facts indicate that the Modal Semantics of Natural Languages includes systems that, in the Logician's jargon, are not degenerated. Thus, the interesting and relevant question turns to be why is that so? There are several possible answers. One tentative manner to tackle this issue would be assuming that, inasmuch as the semantics of Natural Languages has a displacement property (the discourse is not limited to the here and now), they are modal languages, while banal systems are quasi-limits of modal systems. This kind of explication probably requires further elaboration.
The issue gets more complicated when we consider that instances of the P law below are odd statements:
P: A⊃◊A

Although it is very intuitive to assert that if A is true then A is possible, sentences with equivalent meaning have bizarre effects:
(5) #If Harper is the Prime Minister then he may be the Prime Minister.
(6) #If it is raining outside, then it might be raining.
(7) #If it is true that your car is broken, then it is possible that your car be broken.
(8) #If Governor General Michaela Jean came from Haiti, then she possibly has come from Haiti.
(9) #If today is Valentine's Day, then maybe it is Valentine's Day.

What is the problem with the sentences above? Does it have to do with truth-conditions? Or is it simply the case that these sentences are infelicitous?

(1) There are apparent exceptions though, in the cases where the lexical item that should correspond to the operator  have a deontic reading or actually corresponds to ◊. The (somewhat redundant) sentences below are instances where T works without doubt:
i. If it is necessarily true that oranges do not grow on apple trees, then it is true that they do not.
ii. If natural catastrophes necessarily happen sooner or later, then they (do/ will) happen sooner or later.
iii. Young males of necessity seek girls’ company and vice versa, which means that they seek each other.

jeudi, février 09, 2006

BRIEF THOUGHTS ON CRUCIAL MATTERS (one moment of your reflection, please)


To be one with the Universe and to exist in harmony with other creatures fall within human capacity though, humanity still seeks and walks throughout the perverse path of war and destruction.
Surely, in every conflict there is the stronger side, who happens to be the blinder one as well. Should they pursue the total annihilation of their weaker enemy, they imagine they will thereby prevail. But the truth is that one people cannot destroy another without destroying itself as well.
War was a deity in ancient mythology and so the acts of plunder and mass killing constituted forms of worship and devotion to the numina. And, for many centuries more after the advent monotheism, some wars have been deemed a holy and so the very acting of fighting and killing other humans have been considered instances of submission and obedience to the divine will.
In this aspect, Capitalism and the ascension of the bourgeois have rendered a great service to human culture, for they have completely de-sacralise and de-sanctified war in all aspects. Soldiers would no longer believe that they would die or kill to serve the cause of God, that they would be heroes, champions or martyrs in a ‘crusade’ or ‘jihad’. As time passed, the political conscience of the dominated classes grew and those in and out of the militias were aware that their fight was either a matter of collective survival (namely the survival of their nation) or simply the will of their politicians and generals.
The youth in the 1950’s and 1960’s have gradually and persistently worked in favour of the idea that war was something unholy and immoral. Young men and women in France and in the United States went even further on, denouncing the atrocities of the wars in Algeria and in Vietnam as continuations of a resilient colonialist mentality.
Nowadays nobody is naïve anymore. War is a business and it is impossible to ignore this basic truth. Industrial countries engage in invasive huge scale military operations either to conquer and annexe or to intervene in other countries for no primary reason other than the necessity of dynamising their economies. The process is basically twofold:
[1] On one hand the very warfare enterprise is a billionaire business that creates demands and it is up to the private sector to provide, i.e., to sell the materials and the services the armed forces will need.
[2] On the other hand, the country, which will be invaded, conquered or remain under intervention, will probably be deprived of its wealth (including its natural resources, lands, manpower, capitals, pieces of art and other assets) to the benefit of the private corporations behind the invasion forces.
Thus, the vastest majority of the citizens of the countries concerned, i.e., the taxpayers on both sides loose their money, paying for maintenance of their military personnel, the arsenals and the operations to attack and/or defend, and public treasury consequently incurs into deficit, so that some gigantic companies may have profit. The middle and lower classes send their children to die and pay all the war bills, so that the top wealthiest and safest classes increase their fortunes.
In the 21st century the war numina became faulty and imperfect humans who, nevertheless, still require all sorts of bloody sacrifices.

lundi, février 06, 2006

Propaedeutics: Counterfactuals

Some properties of counterfactuals


This note is intended to cover questions some readers have about some of my comments on works on counterfactual conditionals. In the following I repeat some of the things I have already said, and the examples I give are found in the literature.

Can counterfactuals be explained temporally, via concepts like time-line/ history? To answer this question, one has firstly to define what one means by counterfactuals and which properties characterise them.
Counterfactuals ought not to be confused with contrapositions. Contraposition is a kind of transposition in traditional logic. Thus, although in the literature it is common to find sentences like (3) and (4) as examples of counterfactuals, they are actually contrapositions:
(1) If Jerry is a cat then he eats fish.

(2) If you must to go to Lisbon, you take train A.


(3) If Jerry does not eat fish, then he is not a cat.

(4) If you do not take train A, you cannot go to Lisbon.

Counterfactuals are conditionals of a different nature. First of all they state things that are contrary to the facts. If a conditional states something that is in accordance with the facts, then the statement is not counterfactual. Counterfactual statements serve as kind of explanations and, perhaps, are closely related to reduction to the absurd demonstrations. Furthermore, Counterfactuals display some key properties of counterfactuals, whereby some modes of inference do not apply:
Failure of the Hypothetical Syllogism (Transitivity)

(5) a. If Hoover had been a Communist, he would have been a traitor.

b. If Hoover had been born in Russia, he would have been a Communist.

*⇒ c. If Hoover had been born in Russia, he would have been a traitor.

Failure of Contraposition

(6) (Even) if Goethe hadn’t died in 1832, he would still be dead now.

*⇒ b. If Goethe was alive now, he would have died in 1832.

Failure of Strengthening the Antecedent

(7) a. If kangaroos had no tails, they would topple over.

*⇒ b. If kangaroos had no tails but used crutches, they would topple over.

Failure of Inversion with equal values (T&T or F&F)

(8) a. If kangaroos had no tails, they would topple over. (F→F)

*⇒ b. If kangaroos had tails, they would not topple over. (nonsense) (T→T)

(9) a. If Goethe did not die in 1960, he would still be dead now. (T→T)

*⇒ b. If Goethe died in 1960, he would still be alive now. (nonsense) (F→F)

Failure of Modus Ponens

(10) If high interest rates were a remedy to control inflation, piglets could fly.

*⇒ a. High interest rates are a remedy to control inflation.
b. Piglets can fly.

These properties can be explained in terms of different histories, i.e., alternate time-lines. (A particular history h can in principle be identified with a possible world w, but we shall set aside the discussion on whether we have really distinct concepts or just different terminologies to old ideas.) First assume a set of all possible events E. If we impose an ordering upon the elements of E, then we derive a set of possible histories or time-lines. Similarly, if we take a finite subset of E, let us say E1= {e1…en} and we impose an ordering upon the elements of E´, then each different order of events will be a different possible time-line or history.
Now assume that we are in a particular time-line h*, which is history as it has been (our time-line). We know or imagine that History could have been otherwise, the several imaginable possibilities being the alternate time-lines. In our time-line the allies have won World War II, China has annexed Tibet, Armstrong was the first man to walk on the Moon and the United States lost Vietnam War. Each of these points in our time-line is crucial. History would have been very different if, for instance, the Nazi-Fascists regimes had won World War II, the Tibet had annexed China, Gagarin was the first man to walk on the Moon and the United States won the Vietnam War. We can pick up each of these crucial points in our time-line and relate it to alternate time-lines via branching nodes [1]. Accordingly, reasoning on abstract grounds, there was previously a point in the time-lime where one of the possible outcomes of World War II would be inserted. If we entertain two possibilities, i.e., either the Allies would win or the Nazi-Fascists regime would win, then we have a point where the line splits into two: we are in the branch of the time-line where the first possibility was the event that succeeded the preceding historical events. In the alternate time-line, the other possible took place. Thus, considering the other paring of events, i.e., the annexation of China by Tibet versus the annexation of Tibet by China, etc, we get more branches of the time-line (or time-tree, if you like).
In this sense, a counterfactual conditional is a statement about sequences of events that could have happened, but did not. This is precisely why they state things contrary to the facts and cannot be subject to modus ponens.
In other words, counterfactuals may be characterised as statements about time branching, about the alternate time-lines. If someone says one of the sentences below, he will be talking about different sequences of events:
(11) If you stroke these matches, they would have lit.

(12) If you stroke these matches and soaked them in the water, they would not have lit.

By changing the events that are included in the line or by changing their orders, the results affect the ways we interpret the sentences and so make certain modes of inference unavailable.
Let us examine the examples above. In (11) the conditional construction accesses a line with two ordered events:
Event A: the matches are striken;

Event B: the matches lit.

Event B is the outcome. In (12) another event C is interposed between events A and B, but then B is no longer accessible: event C causes the line to split and creates a binary branching, leading to another outcome ¬B. So the addition of the clause and soaked them in the water does not strength the antecedent, it changes the time-line that is accessed, it creates a space-time branching.
Examples like those in (5) seem trickier, but they are not: although (5a) and (5b) share one common event, the one in which Hoover would have been a Communist, they access two different alternate time-lines, with two different orderings. (5a) branches from some point in our actual history after the event of Hoover being born in the US. While (5b) departs from a branching exactly at instant of Hoover’s birth: in the actual time-line Hoover’s place of birth is the US, in the alternate time-line his place of birth is Russia and the counterfactual accesses the latter.

To be continued…