Showing posts with label logic. Show all posts
Showing posts with label logic. Show all posts
Tuesday, April 16, 2013
The Problem of the Quick Detectives
Problem:
Three quick detectives require a second to think and a second to act. In a single second, they have grabbed their hats and coats and rushed out the door of Police Headquarters to join their chief, who is waiting for them beside his car. In their haste, all three detectives have put their hats on backwards.
"I observe that at least one of you gentlemen has his hat on backwards," said the chief. "I'm sure I don't have to tell you who you are."
Each detective looked at the other two, thought to himself, waited for a second, and then turned his hat around.
How did the three detectives know their hats were on backwards?
Solution:
This is a logical problem, but the solution requires imagination as well as logical thinking. The method can be seen more easily by first considering a simpler problem, where there are only two quick detectives, rather than three.
For Two Detectives: Name the detectives A and B, and consider what A does and thinks.
In the first second, A sees that B has his hat on backwards.
In the next second, A considers what B would think and do if A's hat were on forwards. Since at least one hat is on backwards, B would realize that his own hat must be on backwards, and in the third second he would turn his hat around.
But in the third second, B does not turn his hat around because A's hat is not on forwards.
In the fourth second, therefore, A knows that his hat is on backwards and he turns it around.
At the same time, B would be using the same reasoning, and he would also turn his hat around in the fourth second.
For Three Detectives: Name the detectives A, B, and C, and consider what A does and thinks.
In the first second, A sees that both B and C have their hats on backwards.
In the next second, A considers what B would think and do if A's hat were on forwards. Seeing that A's hat is on forwards, B considers what C would think and do if B's hat were also on forwards. Since at least one hat is on backwards, C would realize that his own hat must be on backwards, and in the third second he would turn his hat around.
If C does not turn his hat around in the third second, then B knows that his hat is on backwards, and in the third second he would turn his hat around.
But in the third second, B does not turn his hat around because A's hat is not on forwards.
In the fourth second, therefore, A knows that his hat is on backwards and he turns it around.
At the same time, B and C would be using the same reasoning, and they would also turn their hat around in the fourth second.
Eugene Paul
Tuesday, January 19, 2010
An Explanation of the Infamous Liar Paradox
The Liar Paradox is a statement that is expressed, variously, as "This statement is false", "This sentence is false", or "I am lying". We are told that the statement is paradoxical because:
* if it is false, as it claims, then it is true,
* if it is true, then it is false, as it claims.
Either way, we are told, the statement is both true and false at the same time. Logicians and philosophers profess their mystification.
Every student of logic has been introduced to logical propositions, which are statements that are either true or false, but never both true and false. If students of logic understood why propositions are never both true and false, they would understand why the statement of the Liar Paradox is not a proposition.
A true statement is an accurate statement about reality, whereas a false statement is an inaccurate statement about reality, e.g.,
* The Moon is a sphere, vs.
* The Moon is a cube.
These are statements of fact that can be verified by observing reality. But the Liar Paradox cannot be verified because it makes no statement about reality.
The Paradox is a non-propositional statement whose form imitates the form of an actual proposition. Consider three propositions:
1. Lead is denser than zinc.
2. Statement 1 is true.
3. Statement 1 is false.
The truth of statements 2 and 3 depends on the truth of statement 1. To say that statement 1 is true, is to restate statement 1, i.e.,
Statement 1 is true, i.e.,
It is true that lead is denser than zinc, i.e.,
Lead is denser than zinc.
Similarly, To say that statement 1 is false, is to restate statement 1 negatively, i.e.,
Statement 1 is false, i.e.,
It is false that lead is denser than zinc, i.e.,
Lead is not denser than zinc.
Thus, statements 2 and 3 are only valid propositions because they restate statement 1, which is a valid proposition. In contrast, the Liar Paradox is not a valid proposition because it does not restate any valid proposition. Consider two statements that resemble the Paradox:
1. Statement 0 is false.
2. Statement 2 is false.
We see that statement 1 is nonsense because it pretends to restate a statement that does not exist. Since it does not restate a statement of fact (a proposition), it is not a proposition.
Similarly, statement 2, which is the Liar Paradox and which pretends to refer to itself, is also nonsense because it pretends to restate a statement of fact that does not exist.
The Liar Paradox, then, is nothing more than a clever misuse of language.
Eugene Paul
* if it is false, as it claims, then it is true,
* if it is true, then it is false, as it claims.
Either way, we are told, the statement is both true and false at the same time. Logicians and philosophers profess their mystification.
Every student of logic has been introduced to logical propositions, which are statements that are either true or false, but never both true and false. If students of logic understood why propositions are never both true and false, they would understand why the statement of the Liar Paradox is not a proposition.
A true statement is an accurate statement about reality, whereas a false statement is an inaccurate statement about reality, e.g.,
* The Moon is a sphere, vs.
* The Moon is a cube.
These are statements of fact that can be verified by observing reality. But the Liar Paradox cannot be verified because it makes no statement about reality.
The Paradox is a non-propositional statement whose form imitates the form of an actual proposition. Consider three propositions:
1. Lead is denser than zinc.
2. Statement 1 is true.
3. Statement 1 is false.
The truth of statements 2 and 3 depends on the truth of statement 1. To say that statement 1 is true, is to restate statement 1, i.e.,
Statement 1 is true, i.e.,
It is true that lead is denser than zinc, i.e.,
Lead is denser than zinc.
Similarly, To say that statement 1 is false, is to restate statement 1 negatively, i.e.,
Statement 1 is false, i.e.,
It is false that lead is denser than zinc, i.e.,
Lead is not denser than zinc.
Thus, statements 2 and 3 are only valid propositions because they restate statement 1, which is a valid proposition. In contrast, the Liar Paradox is not a valid proposition because it does not restate any valid proposition. Consider two statements that resemble the Paradox:
1. Statement 0 is false.
2. Statement 2 is false.
We see that statement 1 is nonsense because it pretends to restate a statement that does not exist. Since it does not restate a statement of fact (a proposition), it is not a proposition.
Similarly, statement 2, which is the Liar Paradox and which pretends to refer to itself, is also nonsense because it pretends to restate a statement of fact that does not exist.
The Liar Paradox, then, is nothing more than a clever misuse of language.
Eugene Paul
Sunday, January 17, 2010
The Life Cycle and Natural Selection
We observe that all organisms tend to come into existence, mature, reproduce, and go out of existence. In humans this cycle is represented by birth, attainment of adulthood, giving birth, and dying.
We observe that human populations only survive because the old are replaced by the young. A population that did not reproduce would eventually die out and disappear. There thus appears to be a causal relationship between birth and death, and it appears that the purpose of birth is to compensate for death, which is widely considered to be a flaw in the scheme of things. But it sometimes happens that we confuse cause with effect, so for the sake of argument we might hold in reserve the possibility that the purpose of death is to compensate for birth, or even that the cycle of birth and death serves some unknown purpose.
We also observe that there are two kinds of reproduction, sexual and asexual. Asexual reproduction is a kind of self-cloning, in which new individuals are created using genes from a single parent. This is a much simpler and more reliable process than sexual reproduction, so we might wonder why humans and other complex organisms don't reproduce asexually, as many less complex organisms do.
The reason for sexual reproduction, I believe, has to do with the length of time it takes an organism to mature to the point where it can reproduce. A single-celled organism can reproduce asexually soon after it is created. Significantly, there are bacteria that begin reproducing after they are only 20 minutes old. Furthermore, there are strains of bacteria that are highly susceptible to mutations, i.e., errors in copying genes when they reproduce. Because of this short life-cycle, favorable mutations can propagate into the species rapidly.
The ability of some viruses to quickly make themselves immune to our latest antibiotic medications is an example of natural selection at work. A strain of viruses survives because some mutations produce individuals that can survive better in an altered, hostile environment.
But in the case of complex multicellular organisms, the length of time needed to reach reproductive maturity becomes too great for asexual reproduction to be effective. New mutations would enter the gene pool very slowly. Sexual reproduction then becomes more effective, because mutant genes from many different individuals are shared when those individuals combine their genes during sexual reproduction. Plants tend to bear this idea out. Almost all plants can reproduce both sexually or asexually. Sexual reproduction is the preferred method, but since plants can't roam around looking for partners, they can resort to asexual reproduction when partners are not available. The species stagnates, but survives, during periods of asexual reproduction, and it thrives later by introducing more mutations into the gene pool during periods of sexual reproduction.
It should be noted that single-celled organisms do not experience natural death. They clone themselves into two new organisms when they reproduce. There doesn't seem to be anything about their cells that wears out after they have reproduced themselves a given number of times.
Since multicellular organisms are simply organized collections of single cells, it seems reasonable to assume that single-celled organisms existed first, and that multicellular organisms evolved from them later. But for some reason, the cells of multicellular organisms can only reproduce a given number of times before they, and the organisms, die.
When single-celled organisms reproduce asexually, the parts of the parent organisms are reused, and the species propagates itself most effectively. But when multicellular organisms reproduce sexually, the bodies of the parents cannot be reused during the process. In time, the parents will have contributed all of their potentially useful mutations to the gene pool, and they will have served their genetic purpose. It may be for this reason that they are designed to eventually self-destruct. Which is to suggest that the cycle of birth and death exists to serve the purpose of natural selection.
Eugene Paul
We observe that human populations only survive because the old are replaced by the young. A population that did not reproduce would eventually die out and disappear. There thus appears to be a causal relationship between birth and death, and it appears that the purpose of birth is to compensate for death, which is widely considered to be a flaw in the scheme of things. But it sometimes happens that we confuse cause with effect, so for the sake of argument we might hold in reserve the possibility that the purpose of death is to compensate for birth, or even that the cycle of birth and death serves some unknown purpose.
We also observe that there are two kinds of reproduction, sexual and asexual. Asexual reproduction is a kind of self-cloning, in which new individuals are created using genes from a single parent. This is a much simpler and more reliable process than sexual reproduction, so we might wonder why humans and other complex organisms don't reproduce asexually, as many less complex organisms do.
The reason for sexual reproduction, I believe, has to do with the length of time it takes an organism to mature to the point where it can reproduce. A single-celled organism can reproduce asexually soon after it is created. Significantly, there are bacteria that begin reproducing after they are only 20 minutes old. Furthermore, there are strains of bacteria that are highly susceptible to mutations, i.e., errors in copying genes when they reproduce. Because of this short life-cycle, favorable mutations can propagate into the species rapidly.
The ability of some viruses to quickly make themselves immune to our latest antibiotic medications is an example of natural selection at work. A strain of viruses survives because some mutations produce individuals that can survive better in an altered, hostile environment.
But in the case of complex multicellular organisms, the length of time needed to reach reproductive maturity becomes too great for asexual reproduction to be effective. New mutations would enter the gene pool very slowly. Sexual reproduction then becomes more effective, because mutant genes from many different individuals are shared when those individuals combine their genes during sexual reproduction. Plants tend to bear this idea out. Almost all plants can reproduce both sexually or asexually. Sexual reproduction is the preferred method, but since plants can't roam around looking for partners, they can resort to asexual reproduction when partners are not available. The species stagnates, but survives, during periods of asexual reproduction, and it thrives later by introducing more mutations into the gene pool during periods of sexual reproduction.
It should be noted that single-celled organisms do not experience natural death. They clone themselves into two new organisms when they reproduce. There doesn't seem to be anything about their cells that wears out after they have reproduced themselves a given number of times.
Since multicellular organisms are simply organized collections of single cells, it seems reasonable to assume that single-celled organisms existed first, and that multicellular organisms evolved from them later. But for some reason, the cells of multicellular organisms can only reproduce a given number of times before they, and the organisms, die.
When single-celled organisms reproduce asexually, the parts of the parent organisms are reused, and the species propagates itself most effectively. But when multicellular organisms reproduce sexually, the bodies of the parents cannot be reused during the process. In time, the parents will have contributed all of their potentially useful mutations to the gene pool, and they will have served their genetic purpose. It may be for this reason that they are designed to eventually self-destruct. Which is to suggest that the cycle of birth and death exists to serve the purpose of natural selection.
Eugene Paul
Saturday, January 9, 2010
Are You a Solar-Powered Parasite?
The seemingly outrageous suggestion that you might be a solar-powered parasite is but one of several easy deductions that can be made when we consider the implications of the fact that plants use photosynthesis to produce their own food whereas animals, yourself included, don't.
Virtually all plants make use of photosynthesis to capture energy contained in direct sunlight and use it to synthesize carbohydrates, fats, and proteins from the carbon dioxide available in the air and the water, nitrogen, and other materials available in the soil. But there is a very small class (logically speaking) of parasitic plants that exist by feeding on other plants rather than synthesizing their own food.
Animals do not produce their own food by means of photosynthesis. Herbivores are similar to parasitic plants in that they exist by feeding on plants instead of synthesizing their own food. And carnivores go a step farther by feeding on herbivores (and other carnivores) instead of synthesizing their own food. Finally, omnivores feed on both plants and animals instead of synthesizing their own food.
In other words, all animals feed on plants, either directly or indirectly, instead of synthesizing their own food, which is to say that all animals are parasites in the same way that parasitic plants are parasites. It follows from the foregoing, I regret to say, that you too are a parasite.
But while you are obviously a parasite, you may take some ecological consolation in the knowledge that you are a solar-powered parasite, however indirectly. The chemical energy that you derive by metabolizing the carbohydrates, fats, and proteins present in the tissues of your victims, both plant and animal, came originally from the radiant energy in sunlight, which the plants converted into chemical energy by means of photosynthesis.
Of the several easy deductions that can be made when we consider the implications of the fact that plants use photosynthesis to produce their own food whereas animals don't, the deduction that you are a solar-powered parasite is by far the least humiliating.
Eugene Paul
Virtually all plants make use of photosynthesis to capture energy contained in direct sunlight and use it to synthesize carbohydrates, fats, and proteins from the carbon dioxide available in the air and the water, nitrogen, and other materials available in the soil. But there is a very small class (logically speaking) of parasitic plants that exist by feeding on other plants rather than synthesizing their own food.
Animals do not produce their own food by means of photosynthesis. Herbivores are similar to parasitic plants in that they exist by feeding on plants instead of synthesizing their own food. And carnivores go a step farther by feeding on herbivores (and other carnivores) instead of synthesizing their own food. Finally, omnivores feed on both plants and animals instead of synthesizing their own food.
In other words, all animals feed on plants, either directly or indirectly, instead of synthesizing their own food, which is to say that all animals are parasites in the same way that parasitic plants are parasites. It follows from the foregoing, I regret to say, that you too are a parasite.
But while you are obviously a parasite, you may take some ecological consolation in the knowledge that you are a solar-powered parasite, however indirectly. The chemical energy that you derive by metabolizing the carbohydrates, fats, and proteins present in the tissues of your victims, both plant and animal, came originally from the radiant energy in sunlight, which the plants converted into chemical energy by means of photosynthesis.
Of the several easy deductions that can be made when we consider the implications of the fact that plants use photosynthesis to produce their own food whereas animals don't, the deduction that you are a solar-powered parasite is by far the least humiliating.
Eugene Paul
Friday, January 8, 2010
Sexual Utopianism vs. Biology
Proponents of same-sex marriage argue that male-male and female-female couples should have the same right to marry as male-female couples, but they ignore the biological nature of sex and the reason why there are two sexes.
We know from the study of human anatomy and physiology that male and female sex organs are designed to fit together for sexual intercourse. Male organs create genetic material (gametes) and transfer it to a female partner during intercourse. Female organs create comparable genetic material and, following intercourse, combine it with a male partner's genetic material to conceive a child and support its growth until it is born.
Sex hormones in both sexes create the desire to engage in sexual intercourse.
It is obvious from the preceding facts that pregnancy is the natural consequence of intercourse, but it may be less obvious that pregnancy is also the natural purpose of intercourse.
All members of any society are the result of pregnancy, and all (natural) pregnancy is the result of sexual intercourse between a male and a female. Therefore, all members of a society are the result of such intercourse. But if pregnancy is avoided totally, a society will have no new members and it will eventually die out and disappear. Therefore a society that habitually engages in intercourse without pregnancy cannot exist for long. In the long term, intercourse without pregnancy is impossible since there will be no one left to engage in intercourse. Therefore, the purpose of intercourse is pregnancy.
The argument made in favor of same-sex marriage, to the effect that male-male and female-female couples should have the same right to marry as male-female couples, is utopian because only a society of male-female couples can reproduce itself and survive.
In recent decades a same-sex counterculture has been created in opposition to the 'oppressive' male-female' culture. The builders of such a counterculture are social thinkers who have no idea why there are two sexes and why only a male-female society is possible.
A similar counterculture existed during the last century. The Shakers were a religious community that did not allow its members to engage in sexual intercourse. Consequently, this community did not reproduce itself and it was necessarily short-lived. It recruited new members from society at large, and it is said to have had a total of as many as 200,000 members, but never more than 6,000 at one time. Only four members remained near the end of the century. For the same biological reason the same fate must await the current same-sex counterculture.
Same-sex marriage, then, is a social institution of a society that cannot exist.
Eugene Paul
We know from the study of human anatomy and physiology that male and female sex organs are designed to fit together for sexual intercourse. Male organs create genetic material (gametes) and transfer it to a female partner during intercourse. Female organs create comparable genetic material and, following intercourse, combine it with a male partner's genetic material to conceive a child and support its growth until it is born.
Sex hormones in both sexes create the desire to engage in sexual intercourse.
It is obvious from the preceding facts that pregnancy is the natural consequence of intercourse, but it may be less obvious that pregnancy is also the natural purpose of intercourse.
All members of any society are the result of pregnancy, and all (natural) pregnancy is the result of sexual intercourse between a male and a female. Therefore, all members of a society are the result of such intercourse. But if pregnancy is avoided totally, a society will have no new members and it will eventually die out and disappear. Therefore a society that habitually engages in intercourse without pregnancy cannot exist for long. In the long term, intercourse without pregnancy is impossible since there will be no one left to engage in intercourse. Therefore, the purpose of intercourse is pregnancy.
The argument made in favor of same-sex marriage, to the effect that male-male and female-female couples should have the same right to marry as male-female couples, is utopian because only a society of male-female couples can reproduce itself and survive.
In recent decades a same-sex counterculture has been created in opposition to the 'oppressive' male-female' culture. The builders of such a counterculture are social thinkers who have no idea why there are two sexes and why only a male-female society is possible.
A similar counterculture existed during the last century. The Shakers were a religious community that did not allow its members to engage in sexual intercourse. Consequently, this community did not reproduce itself and it was necessarily short-lived. It recruited new members from society at large, and it is said to have had a total of as many as 200,000 members, but never more than 6,000 at one time. Only four members remained near the end of the century. For the same biological reason the same fate must await the current same-sex counterculture.
Same-sex marriage, then, is a social institution of a society that cannot exist.
Eugene Paul
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