Thursday, 30 September 2010

Syllogism - Reasoning Questions and Answers

By on 04:20

Basic structure

A categorical syllogism consists of three parts: the major premise, the minor premise and the conclusion.
Each part is a categorical proposition, and each categorical position contains two categorical terms.[2] In Aristotle, each of the premises is in the form "All A are B," "Some A are B", "No A are B" or "Some A are not B", where "A" is one term and "B" is another. More modern logicians allow some variation. Each of the premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate of the conclusion); in a minor premise, it is the minor term (the subject) of the conclusion. For example:
Major premise: All men are mortal.
Minor premise: Socrates is a man.
Conclusion: Socrates is mortal.
Each of the three distinct terms represents a category. In this example, "men," "mortal," and "Socrates." "Mortal" is the major term; "Socrates", the minor term. The premises also have one term in common with each other, which is known as the middle term; in this example, "man." Here the major premise is universal and the minor particular, but this need not be so. For example:
Major premise: All mortals die.
Minor premise: All men are mortals.
Conclusion: All men die.
Here, the major term is "die", the minor term is "men," and the middle term is "mortals". Both of the premises are universal.
sorites is a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion. For example, if one argues that a given number of grains of sand does not make a heap and that an additional grain does not either, then to conclude that no additional amount of sand will make a heap is to construct a sorites argument.

Types of syllogism

Although there are infinitely many possible syllogisms, there are only a finite number of logically distinct types. We shall classify and enumerate them below. Note that the syllogism above has the abstract form:
Major premise: All M are P.
Minor premise: All S are M.
Conclusion: All S are P.
The premises and conclusion of a syllogism can be any of four types, which are labeled by letters[3] as follows. The meaning of the letters is given by the table:
aAllSarePuniversal affirmativesAll humans are mortal.
eNoSarePuniversal negativesNo humans are perfect.
iSomeSarePparticular affirmativesSome humans are healthy.
oSomeSare notPparticular negativesSome humans are not clever.
(See Square of opposition for a discussion of the logical relationships between these types of propositions.)
In Analytics, Aristotle mostly uses the letters A, B and C as term place holders, rather than giving concrete examples, an innovation at the time. It is traditional to use is rather than are as the copula, hence All A is B rather than All As are Bs It is traditional and convenient practice to use a,e,i,o as infix operators to enable the categorical statements to be written succinctly thus:
All A is BAaB
No A is BAeB
Some A is BAiB
Some A is not BAoB
This particular syllogistic form is dubbed BARBARA (see below) and can be written neatly as BaC,AaB -> AaC.
The letter S is the subject of the conclusion, P is the predicate of the conclusion, and M is the middle term. The major premise links M with P and the minor premise links M with S. However, the middle term can be either the subject or the predicate of each premise where it appears. The differing positions of the major, minor, and middle terms gives rise to another classification of syllogisms known as the figure. Given that in each case the conclusion is S-P, the four figures are:
Figure 1Figure 2Figure 3Figure 4
Major premise:M–PP–MM–PP–M
Minor premise:S–MS–MM–SM–S
Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, although this makes no difference logically). Each premise and the conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogism BARBARA above is AAA-1, or "A-A-A in the first figure".
The vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms. Even some of these are sometimes considered to commit theexistential fallacy, meaning they are invalid if they mention an empty category. These controversial patterns are marked in italics.
Figure 1Figure 2Figure 3Figure 4
The letters A, E, I, O have been used since the medieval Schools to form mnemonic names for the forms as follows: 'Barbara' stands for AAA, 'Celarent' for EAE, etc.
A sample syllogism of each type follows. Next to the name, the mood and figure of each syllogism appears (e.g., EIO-3 next to Ferison).
Next to each premise and conclusion is a shorthand description of the sentence. So in AAI-3, the premise "All fruit is nutritious" becomes "(MaP)"; the symbols mean that the first term ("fruit") is the middle term, the second term ("nutritious") is the predicate of the conclusion, and the relationship between the two terms is labeled "A" (All M are S).
Barbara (AAA-1)
All animals are mortal. (MaP)
All men are animals. (SaM)
All men are mortal. (SaP)
Celarent (EAE-1)
No reptiles have fur. (MeP)
All snakes are reptiles. (SaM)
No snakes have fur. (SeP)
Darii (AII-1)
All kittens are playful. (MaP)
Some pets are kittens. (SiM)
Some pets are playful. (SiP)
Ferio (EIO-1)
No homework is fun. (MeP)
Some reading is homework. (SiM)
Some reading is not fun. (SoP)
Cesare (EAE-2)
No healthy food is fattening. (PeM)
All cakes are fattening. (SaM)
No cakes are healthy food. (SeP)
Camestres (AEE-2)
All horses have hooves. (PaM)
No humans have hooves. (SeM)
No humans are horses. (SeP)
Festino (EIO-2)
No lazy students are students who pass exams. (PeM)
Some students are students who pass exams. (SiM)
Some students are not lazy students. (SoP)
Baroco (AOO-2)
All informative things are useful things. (PaM)
Some websites are not useful things. (SoM)
Some websites are not informative. (SoP)
Darapti (AAI-3)
All fruit is nutritious. (MaP)
All fruit is tasty. (MaS)
Some tasty things are nutritious. (SiP)
Disamis (IAI-3)
Some mugs are beautiful. (MiP)
All mugs are useful things. (MaS)
Some useful things are beautiful. (SiP)
Datisi (AII-3)
All the industrious boys in this school have red hair. (MaP)
Some of the industrious boys in this school are boarders. (MiS)
Some boarders in this school have red hair. (SiP)
Felapton (EAO-3)
No jug in this cupboard is new. (MeP)
All jugs in this cupboard are cracked. (MaS)
Some of the cracked items in this cupboard are not new. (SoP)
Bocardo (OAO-3)
Some cats have no tails. (MoP)
All cats are mammals. (MaS)
Some mammals have no tails. (SoP)
Ferison (EIO-3)
No tree is edible. (MeP)
Some trees are green things. (MiS)
Some green things are not edible. (SoP)
Bramantip (AAI-4)
All apples in my garden are wholesome fruit. (PaM)
All wholesome fruit is ripe fruit. (MaS)
Some ripe fruit are apples in my garden. (SiP)
Camenes (AEE-4)
All coloured flowers are scented flowers. (PaM)
No scented flowers are grown indoors. (MeS)
No flowers grown indoors are coloured flowers. (SeP)
Dimaris (IAI-4)
Some small birds are birds that live on honey. (PiM)
All birds that live on honey are colourful birds. (MaS)
Some colourful birds are small birds. (SiP)
Fesapo (EAO-4)
No humans are perfect creatures. (PeM)
All perfect creatures are mythical creatures. (MaS)
Some mythical creatures are not human. (SoP)
Fresison (EIO-4)
No competent people are people who always make mistakes. (PeM)
Some people who always make mistakes are people who work here. (MiS)
Some people who work here are not competent people. (SoP)
Forms can be converted to other forms, following certain rules.

Terms in syllogism

We may, with Aristotle, distinguish singular terms such as Socrates and general terms such as Greeks. Aristotle further distinguished (a) terms that could be the subject of predication, and (b) terms that could be predicated of others by the use of the copula (is are). (Such a predication is known as a distributive as opposed to non-distributive as in Greeks are numerous. It is clear that Aristotle’s syllogism works only for distributive predication for we cannot reason All Greeks are Animals, Animals are numerous, therefore All Greeks are numerous.) In Aristotle’s view singular terms were of type (a) and general terms of type (b). Thus Men can be predicated of Socrates butSocrates cannot be predicated of anything. Therefore to enable a term to be interchangeable — that is to be either in the subject or predicate position of a proposition in a syllogism — the terms must be general terms, or categorical terms as they came to be called. Consequently the propositions of a syllogism should be categorical propositions (both terms general) and syllogism employing just categorical terms came to be called categorical syllogisms.
It is clear that nothing would prevent a singular term occurring in a syllogism — so long as it was always in the subject position — however such a syllogism, even if valid, would not be a categorical syllogism. An example of such would be Socrates is a man, All men are mortal, therefore Socrates is mortal. Intuitively this is as valid as All Greeks are men, all men are mortal therefore all Greeks are mortals. To argue that its validity can be explained by the theory of syllogism it would be necessary to show that Socrates is a man is the equivalent of a categorical proposition. It can be argued Socrates is a man is equivalent to All that are identical to Socrates are men, so our non-categorical syllogism can be justified by use of the equivalence above and then citing BARBARA.

Existential import

If a statement includes a term so that the statement is false if the term has no instances (is not instantiated) then the statement is said to entail existential import with respect to that term. In particular, a universal statement of the form All A is B has existential import with respect to A if All A is B is false if there are no As.
The following problems arise:
(a) In natural language and normal use, which statements of the forms All A is B, No A is B, Some A is B and Some A is not B have existential import and with respect to which terms?
(b) In the four forms of categorical statements used in syllogism, which statements of the form AaB, AeB, AiB and AoB have existential import and with respect to which terms?
(c) What existential imports must the forms AaB, AeB, AiB and AoB have for the square of opposition be valid?
(d) What existential imports must the forms AaB, AeB, AiB and AoB to preserve the validity of the traditionally valid forms of syllogisms?
(e) Are the existential imports required to satisfy (d) above such that the normal uses in natural languages of the forms All A is B, No A is B, Some A is B and Some A is not B are intuitively and fairly reflected by the categorical statements of forms Ahab, Abe, Ail and Alb?
For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:
"All flying horses are mythological" is false if there are not flying horses.
If "No men are fire-eating rabbits" is true, then "There are fire-eating dragons" is false.
and so on.
If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid (e.g. BaC,AaB->AiC).
These problems and paradoxes arise in both natural language statements and statements in syllogism form because of ambiguity, in particular ambiguity with respect to All. If "Fred claims all his books were Pulitzer Prize winners", is Fred claiming that he wrote any books? If not, then is what he claims true? Suppose Jane says none of her friends are poor; is that true if she has no friends? The first-order predicate calculus avoids the problems of such ambiguity by using formulae that carry no existential import with respect to universal statements; existential claims have to be explicitly stated. Thus natural language statements of the forms All A is B, No A is B, Some A is B and Some A is not B can be exactly represented in first order predicate calculus in which any existential import with respect to terms A and/or B is made explicitly or not made at all. Consequently the four forms AaB, AeB, AiB and AoB can be represented in first order predicate in every combination of existential import, so that it can establish which construal, if any, preserves the square of opposition and the validly of the traditionally valid syllogism. Strawson claims that such a construal is possible, but the results are such that, in his view, the answer to question (a) above is no.

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