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Posts Tagged ‘Singular Statement’

Hume and The Impossibility of Falsification

Posted by allzermalmer on May 5, 2013

Hume’s logical problem of induction as Hume presents it and Popper presents it, deals with contingent statements. The affirmation or the negation of the same contingent statement is possible. Take the contingent statement that “All Swans are White”: It is both possible that “All Swans are White” and it is also possible that  not “All Swans are White”. Logic alone cannot decide if “All Swans are White” is either true or false. So it would be decided by some other way as to wither its affirmation or negation to be true. Hume, and Popper, say that experience cannot show the truth of the contingent statement “All Swans are White”.

“Hume’s argument does not establish that we may not draw any inference from observation to theory: it merely establishes that we may not draw verifying inferences from observations to theories, leaving open the possibility that we may draw falsifying inferences: an inference from the truth of an observation statement (‘This is a black swan’) to the falsity of a theory (‘All swans are white’) can be deductively perfectly valid.” Realism and The Aim of Science

(H) Hypothesis: All Swans are White
(E) Evidence: This is a Black Swan

Hume, as Popper takes him in his problem of induction, showed that we cannot show that (H) is true, no matter how many individual swans that are white we have observed. To show that (H) is true, we must verify every case of (H). (H) is a Universal statement, its scope is that of all times and all places. The universal statement is both omnipresent and omnitemporal in its scope. It makes no restriction on temporal location and spatial location. (E) makes a Singular statement, its scope is of a particular time and a particular place. It makes a restriction on temporal location and spatial location. Popper held that we can know (E) is true, ‘This is a Black Swan’. Thus, we cannot know (H) All Swans are White but we can know (E) This is a Black Swan.

Hume’s logical problem of induction, as Popper takes it, goes something like this:

(i) Science proposes and uses laws everywhere and all the time; (ii) Only observation and experiment may decide upon the truth or falsity of scientific statements; (iii) It is impossible to justify the truth of a law by observation or experiment.

Or

(i*) Science proposes and uses the universal statement “all swans are white”; (ii*) Only singular observational statements may decide upon the truth or falsity of ‘all swans are white’; (iii*) It is impossible to justify the truth of the universal statement ‘all swans are white’ by singular observational statements.

It is taken as a fact that (i) or (i*) is true. So there is no question about either (i) or (i*). So the conflict of Hume’s logical contradiction arises between (ii) and (iii) or (ii*) and (iii*). Popper accepts (iii) or (iii*). So the only way out of Hume’s logical problem of induction is to modify or reject (ii) or (ii*) to solve the contradiction.

Popper thus solves Hume’s logical problem of induction by rejecting (ii) or (ii*) and replacing it with a new premise. This new premise is (~ii).

(~ii) Only observation and experiment may decide upon the falsity of scientific statements
Or
(~ii*) Only singular observation statements may decide upon the falsity of ‘all swans are white’.

Popper rejects (ii) or (ii*), which basically said that only singular observation statements can show that either universal statements are true or false. Popper rejects this because of (iii), and says that Singular observation statements can only show that universal statements are false. Popper believes, as the quote at the beginning of the blog says, that Hume’s logical problem of induction doesn’t show that we can’t show that a universal statement is false by a singular observational statements. But is this what Hume showed to be true?

It does not appear that Hume’s logical problem of induction even allows Popper to escape with the modification of (ii) to (~ii). It appears that Hume’s logical problem of induction does not allow Popper to escape from “fully decidable” to “partially decidable”, i.e.  decide both truth or falsity to cannot decide truth but only falsity.

Take the singular observational statement that Popper gives in the quote, i.e. ‘This is a black swan’. It is a singular statement, but the statement contains a universal within it, it contains “swan”. “Swan” are defined by their law-like behavior, which are their dispositional characteristics, and is a universal concept. These dispositions are law-like, and thus universal in scope as well. And by (iii) we cannot determine if something is a “swan” because of that. The concept “swan” is in the same position as “all swans are white”. They are both universal, and because of (iii) cannot be shown to be true.

“Alcohol” has the law-like behavior, or disposition, or being flammable. So if we were to say that ‘This is alcohol’. We would have to check all the alcohol that existed in the past, present, future, and all places in the universe in which it was located. We would have to light them to see if they catch fire, and thus flammable. Only than could we say that “This is alcohol”, and know that it is alcohol. But to do so would be to verify a universal through singulars, which is impossible by (iii).

In fact, Hume even talks about dispositions and law-like behavior in his talks about the problem of induction. For example, Hume says that “we always presume, when we see like sensible qualities, that they have like secret powers, and expect that effects, similar to those which we have experienced, will follow from them.” Hume is specifically attacking dispositions as well, which means he is attacking universal concepts and universal statements.

“Our senses inform us of the colour, weight, and consistence of bread; but neither sense nor reason can ever inform us of those qualities which fit it for the nourishment and support of a human body…The bread, which I formerly eat, nourished me; that is, a body of such sensible qualities was, at that time, endued with such secret powers: but does it follow, that other bread must also nourish me at another time, and that like sensible qualities must always be attended with like secret powers?” Enquiry’s Concerning Human Knowledge

From Popper’s point of view, science can only show the falsity of a universal statement through the truth of a singular statement. The singular statement would have to contradict the universal statement and the singular statement would have to be true.

(h) If it rained then wet ground.
(e) Not a wet ground
(c)Thus, it didn’t rain.

If we assume that both (h) and (e) are true, then we accept a contradiction. Contradictions can’t possibly be true. So we know that at least one of these two must be false. But which one is false and which one is true, (h) or (e).

But how can we show the truth of a singular observational statement when it relies on a universal concept, and universal concepts fall for (iii) just as much as universal statements? Hume’s position of the logical invalidity of of induction, i.e. (iii), also holds not only with universal statements but also universal concepts, i.e. law-like behavior/ dispositional characteristics. How does Popper respond to this?

Popper accepts the invalidity of reaching universal statements through experience, but takes it that we accept singular observational statements based on conventions. We conventionally accept the singular observation statement as true.

Hume’s logical problem of induction shows this:

(H) All Swans are White
(E) This swan is black

Now we may either accept (H) as a convention or accept (E) as a convention, or both as conventions. Popper rejects accept (H) as a convention, because you cannot show that a convention is false. Showing something false is what (~ii) was used to solve the original problem of induction. He wants to show that (H) is false, which is consistent with (~ii), but the only way to do that is if (E) can be shown true. But (E) contains a universal concept and (iii) prevents us from experiencing dispositions or law-like behaviors, i.e. Swan or Alcohol. (iii) applies just as much to universal statements as it does to universal concepts. (E) is based on universal concepts and so has to be accepted as a convention, to escape (iii), in order to show that (H) is false and be consistent with (i) and (~ii). (H) has to have the ability to be shown false to be falsifiable, and not being a convention means it has the ability to be shown false.

Contrary to what Popper thinks, Hume’s logical problem of induction doesn’t even allow you to show a falsifying instance. Thus, following full implications of Hume’s logical problem of induction, we can neither show the truth of a universal statement or show the falsify of a universal statement.

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Did Popper Solve The Problem of Induction?

Posted by allzermalmer on October 3, 2012

Karl Popper said that he believed he had solved the “Problem of Induction”, or what he called “Hume’s Problem”. But did Karl Popper really solve the Problem of Induction or Hume’s Problem? Maybe we should (1) take a look at what Popper considered to be Hume’s problem, and (2) see what Popper says his solution to the problem is. (Whether or not Popper did correctly identify Hume’s problem, is of no concern here).

Before we do this, I think we should start out with something basic, or part of basic, logic.

(A) Universal Quantifier Affirmative (All S are P): For each x, if x is S, then x is P
(E) Universal Quantifier Negation (No S are P) : For each x, if x is S, then x is not P
(I) Existential Quantifier Affirmative (Some S are P): There exists at least one x, such that x is S and x is P
(O) Existential Quantifier Negation (Some S are not P): There exists at least one x, such that x is S and x is not P

“All of the categorical propositions illustrated above can be expressed by using either the universal quantifier alone or the existential quantifier alone. Actually, what this amounts to is the definition of the universal quantification of propositions in terms of existential quantification and the definition of existential propositions in terms of universal quantification.” p. 349 Formal Logic: An Introductory Textbook by John Arthur Mourant

Now this means that the Universal Quantifier (UQ) can be expressed in a logically equivalent form to an Existential Quantifier (EQ), and the Existential Quantifier can be expressed in a logically equivalent form to Universal Quantifier. For something to be logically equivalent means they mean the same thing in a logical sense. Logically equivalent statements have the exact same truth. One can’t be true and the other false, for this would mean they are both necessarily false.

Universal Quantifiers to Existential Quantifiers

A: For each x, if x is S, then x is P    There does not exist at least one x, such that x is S and x is not P
E: For each x, if x is S, then x is not P    There does not exist at least one x, such that x is S and x is P
I: Not for each x, if x is S, then x is not P    There exists at least one x, such that x is S and x is P
O: Not for each x, if x is S, then x is P   There exists at least one x, such that x is S and x is not P

A: For each x, if x is Crow, then x is Black  ↔  There does not exist at least one x, such that x is Crow and x is not Black
E: For each x, if x is Crow, then x is not Black  ↔  There does not exist at least one x, such that x is Crow and x is Black
I: Not for each x, if x is Crow, then x is not Black  ↔  There exists at least on x, such that x is Crow and x is Black
O: Not for each x, if x is Crow, then x is Black  ↔  There exists at least on x, such that x is Crow and x is not Black

Existential Quantifiers to Universal Quantifiers

A: There does not exist at least one x, such that x is S and x is not P    For each x, if x is S, then x is P
E: There does not exist at least one x, such that x is S and x is P     For each x, if x is S, then x is not P
I: There exists at least one x, such that x is S and x is P   Not for each x, if x is S, then x is not P
O: There exists at least one x, such that x is S and x is not P    Not for each x, if x is S, then x is P

A: There does not exist at least one x, such that x is Crow and x is not Black  ↔  For each x, if x is Crow, then x is Black
E:
There does not exist at least one x, such that x is S and x is P  ↔  For each x, if x is Crow, then x is not Black 
I:
There exists at least one x, such that x is Crow and x is Black  ↔  Not for each x, if x is Crow, then x is not Black
O:
There exists at least one x, such that x is Crow and x is not Black  ↔  Not for each x, if x is Crow, then x is Black

It needs to be pointed out first that there are two types of statements.
(1)Necessary Truth: Statement whose denial is self-contradictory.
(2) Contingent Truth: One that logically (that is, without self-contradiction) could have been either true or false.

(1a) “All bachelors are unmarried males”
(2a) “Justin Bieber is an unmarried male”

A necessary truth is said to have no empirical content. A contingent truth is said to have empirical content.

Hume’s problem was that he found that he cannot justify induction by demonstrative argument, since he can always imagine a different conclusion.

What Popper takes to be “Hume’s Problem”

“It is usual to call an inference ‘inductive’ if it passes from singular statements (sometimes called ‘particular’ statements), such as accounts of the results of observations or experiments, to universal statements, such as hypotheses or theories. Now it is far from obvious, from a logical point of view, that we are justified in inferring universal statements from singular ones, no matter how numerous; for any conclusions drawn in this way may always turn out to be false: no matter how many instances of white swans we may have observed, this does not justify the conclusion that all swans are white. The question whether inductive inferences are justified, or under what conditions, is known as the problem of induction.” pg. 3-4 Logic of Scientific Discovery

“The root of this problem [of induction] is the apparent contradiction between what may be called ‘the fundamental thesis of empiricism’- the thesis that experience alone can decide upon the truth or falsity of scientific statements- and Hume’s realization of the inadmissibility of inductive arguments.” pg. 20 Logic of Scientific Discovery

Here’s an Inductive argument

Singular: (P1) There exists at least one x, such that x is Crow and x is Black
Singular: (P2) There exists at least one x, such that x is Crow and x is Black

Universal: (C) For each x, if x is Crow, then x is Black

Popper’s Solution to “Hume’s Problem”

“Consequently it is possible by means of purely deductive inferences (with the help of the modus tollens of classical logic) to argue from the truth of singular statements to the falsity of universal statements. Such an argument to the falsity of universal statements is the only strictly deductive kind of inference that proceeds, as it were, in the ‘inductive direction’ that is, from singular to universal statements.”pg. 21 Logic of Scientific Discovery

Here’s Popper’s solution

Universal: (P1) For each x, if x is Crow, then x is not Black
Singular: (P2) There exists at least one x, such that x is Crow and x is Black
Universal: (C) Not for each x, if x is Crow, then x is not Black

Singular statement leads to a universal statement. From there exists at least one x, such that x is Crow and x is Black, the conclusion is reached that not for each x, if x is Crow, then x is not Black.

Here’s Poppers understanding of Induction: “It…passes from singular statements…to universal statements…”

Here’s Poppers solution to the ‘Problem of Induction: “Such an argument to the falsity of universal statements is… from singular to universal statements.”

So going from singular statement to universal statement can be justified by  going from singular statements to universal statements. This falls for the problem of induction again, because this is a circular argument that is used to defend induction.

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Categorical Logic

Posted by allzermalmer on May 16, 2011

So what is Logic? Logic is “the study of methods for evaluating whether the premises of an argument adequately support its conclusion.” (The Power of Logic 4th edition). Now there are three words in such a definition that would need some answers as to what they themselves mean. Those words are argument, premises, and conclusion.
An argument  is a “set of statements where some of the statements, called the premises, are intended to support another, called the conclusion.” (The Power of Logic 4th edition).  So we can take away something from this. We have a set of statements that lead to another statement. Now we could wonder, what is a statement? A statement is “declarative statement that is either true or false” (The Power of Logic 4th edition. In some cases you can interchange between the words statement and proposition. In some texts they use the word statement, and in some other texts they use the word proposition.

Now that we know what statements are, and premises and conclusions are statements, we might wonder what is a premise and conclusion. A premise is “a statement we know, or strongly believe, to be true”. A conclusion is “a statement derived, or supported, by the premises”

Now sometimes it works that we come up with a statement, this would be our conclusion. Now we want to support this conclusions, this statement, and so we support this conclusion with a set of premises. We say, from these premises, we are lead to this conclusion.

Let me give an example: I say, “Socrates is mortal”. Now we want to know how did I arrive at this statement, which is the conclusion. So I would support this statement (conclusion), with another set of statements (premises). These premises would be, “All men are mortal” & “Socrates is a man”. Thus, our argument would look like this: (Premise 1) All men are mortal. (Premise 2) Socrates is a man. (Conclusion) Socrates is mortal. This forms an argument, and we find we have a conclusion that necessarily follows form its premises. Now this argument is valid.

Now the question would be? What does it mean for an argument to be valid? A valid argument is where “the premises succeed in guaranteeing the conclusion”. The other way of seeing this is as, the conclusion necessarily follows from the premises. Now the opposite of a valid argument is an invalid argument. So what is an invalid argument? An invalid argument is in which “the premises fail to guarantee the conclusion”. The other way of seeing this is as, the conclusion does not necessarily follow from the premises.

Valid and invalid are terms that are only used in deductive logic. These terms are used for deductive arguments. So the question becomes, what is a deductive argument? A Deductive argument is “an argument in which the premises are intended to guarantees the conclusion”. Deductive arguments are valid arguments, because the conclusion necessarily follows from the premises. The conclusion is guaranteed to follow from the premises.

Now one main point we can take is this: Logic is concerned about the form of the argument, which includes that the conclusion necessarily follows from the premises.

Now there are three axioms of logic. These are the law of identity, the law of excluded middle, and the law of non-contradiction.

The Law of Identity is “every individual thing is identical to itself”. It carries this symbolic form: A–>A or A=A. (the symbol of “–>” of “If…then” statements in propositional logic). An example would be like this: “If the sun is yellow in color, then the sun is yellow in color” or “the sun is yellow in color is the sun is yellow in color”

The Law of Excluded Middle is “every statement is either true or false”.  It carries this symbolic form: Av~A ( the symbol “v” is an “or” operator that we will come to in propositional logic). An example would be like this: “The sun is yellow in color or the sun is not yellow in color”. (The law of excluded middle is used when we get to truth-tables in propositional logic)

The Law of Non-Contradiction is “given any statement and its opposite, one is true and the other false”. It carries this symbolic form: ~(A&~A) (the symbol of “~” stands for “not“, and the symbol “&” stands for “and“). An example would be like this: “Not both the sun is yellow in color and the sun is not yellow in color”.

Now that we set up some of the basics, we can get into categorical syllogisms and propositional logic.

Categorical Syllogisms

Now categorical syllogisms deal with statements that tell us something about the relationship between categories. So we can give a definition to categorical statement. A Categorical Statement is “an assertion or denial that all or some members of the subject class are included in the predicate class”
( Logic: An Introduction 2nd edition) Now before we continue on with Categorical Syllogisms, now that we defined a Categorical statement, we should define what a syllogism is.

A Syllogism is “any deductive argument in which the conclusion is inferred from two premises” (Logic: An Introduction 2nd edition)

We have three types of statements in syllogisms. (1.) Universal statements, (2.) Particular statements, & (3.) Singular statements.

A Universal Statement is “a proposition that refers to all the members of a class”. A Particular Statement is “a proposition that refers to some but not all the members of a class. A Singular Statement is “a proposition that asserts that a particular individual has (or has not) some specified attribute”. (Introduction to Logic 13th edition)

Now I will try to break these three statements down some. All Universal and Particular statements involve general terms (categories). A General Term is “any word or phrase that describes or designates a general class of individuals” ( Logic: An Introduction 2nd edition). General terms are indefinite descriptions that follow the form of “a so-and-so”. They would be something like these:  “a cute baby”, “charming”, “drives a Buick”. General terms are symbolized with capital letters; So “a cute baby” can be symbolized with “C”. A cute baby=C.

All Singular statements have at least one singular term. A Singular Term is “a word or expression that truly can designate only one individual (e.g., a proper name or a definite description)”. (Logic: An Introduction 2nd edition). Singular terms are definite descriptions that follow the form of “the so-and-so”. They would be something like this: “the world’s cutest baby”, “this child”, “David”. Singular terms are symbolized with small letters; So “the world’s cutest baby” can be symbolized with “c”. The world’s cutest baby=c.

We have 8 forms of statements in syllogisms:
Four Categorical statements:
1. (A) Universal Affirmative: All A are B
2. (E) Universal Negative: No A are B
3. (I) Particular Affirmative: Some A are B
4. (O) Particular Negative: Some A are not B

The other four are singular statements, and two have a general term involved, and two have no general term:
5. Singular with one general term: a is B
6. Singular with one general term: a is not B
7. Singular with no general term: a is b
8. Singular with no general term: a is not b

Now I shall give an example of all 8 of these forms with actual english sentences.

All A are B: ‘All men are a cute person’
No A are B: ‘No men are a cute person’
Some A are B: ‘Some men are a cute person”
Some A are not B: ‘Some men are not a cute person’
a is B: ‘Gondoliere is a cute person’
a is not B: ‘Gondoliere is not a cute person’
a is b: ‘Gonoliere is the cutest person’
a is not b: ‘Gondoliere is not the cutest person’

Now with our four categorical statements of A, E, I, and O, they each have their own logical equivalents. Being Logically equivalent  means that the statements “have the same meaning, and may therefore replace one another whenever they occur”. For example, in english Bachelor means unmarried male. So any time we say bachelor, we are saying they are an unmarried male. And when we say someone is an unmarried male, we are also saying that they are a bachelor. They are equivalent to each other. So we can make an immediate inference. An Immediate inference is “an inference that is drawn directly from one premise without the mediation of any other premise”. (Introduction to Logic 13th edition)

We have the immediate inferences of conversion, obversion, and contraposition.

Conversion is “the result of interchanging the subject and predicate terms in a categorical statement”. (The Power of Logic 4th edition). The statement that we immediately derive in conversion is known as the converse. The Converse is “the result of interchanging the subject and predicate terms in a categorical statement”. We can do conversion with E and I categorical statements to obtain their converse. So this is how it would look:

(E) No A is B= No B is A
No plants are animals= No animals are plants
(I) Some A is B= Some B is A
Some plants are trees= Some trees are plants

Now before I talk about obversion, I have to set something up so we can better understand obversion. Each class/category, has its complement. A Complement is “the complement of class/category X is the class containing all things that are not a member of class/category X”. (The Power of Logic 4th edition). For example, the complement of the class/category of tree is the class/category containing all nontrees, which is, everything that is not a tree (horse, hawks, humans, hamburgers, and son).

Now each term has a term-complement. A Term-Complement is “a word or phrase that denotes a class/category complement” (The Power of Logic 4th edition). So the term-complement of “dogs” is “nondogs”, which denotes the class of everything that is not a dog. Now a term-complement should not be confused with a contrary term. So, the term “winner” should not be confused with “loser”, but is “nonwinner”, which includes nonwinners like players who tie, nonplayers, and losers.

Now when we get to term-complement with more than one word, like “wild dogs”, the term complement would not be “non wild-dogs”, since it contains a class that includes everything outside of the class connotated by the term wild dogs. So the term-complement of “wild dogs” is “things that are not wild dogs”, which includes a class that includes both tame dogs and nondogs. So this would include, in nondogs, as wild geese, since they are nondogs.

So Obversion is “a process of immediate inference by which a logically equivalent statement is formed from a categorical statement by changing its quality and negating the predicate term” (Logic: An Introduction 2nd edition). Now we change the statement to its obverse, and the Obverse is ” the result of changing the quality of a categorical statement and replacing the predicate terms with its term-complement”. (The Power of Logic 4th edition)

So here is how obversion works with all four categorical terms:
(A) All A are B= No A are not-P
All trees are plants= No trees are nonplants
(E)
No A are B= All A are not-B
No cats are trees= All cats are nontrees
(I)
Some A are B= Some A are not not-B
Some trees are oaks= Some trees are not nonOaks (this involves double negation)
(O) Some A are not B= Some A are non-B
Some trees are not oaks= Some trees are nonoaks

Contraposition is “a process of immediate inference by which the subject and predicate of a categorical statement trade places, and each is negated. (Logic: An Introduction 2nd edition).

(A) All A are B= All non-B are non-A
All cats are mammals= All nonmammals are noncats
(O) Some A are not B= Some non-B are non-A
Some plants are not weeds= Some nonweeds are not nonplants

Now the four categorical statement forms form what we call the square of opposition. The Square of opposition is “A diagram displaying the logical relations between categorical statements having the same subject and predicate terms”. (Logic: An Introduction 2nd edition)

There are two diagrams of the square of opposition, and here is the Traditional Square of Opposition:

Now here is the Modern Square of Opposition:


Now, there is a vital difference between the traditional square of opposition, and the modern square of opposition. But before we go over those differences, we should start out with how the Traditional Square of Opposition works.

The traditional square of opposition has the terms contradictories, contraries, subcontraries, and subalternation. Now we should go over what some of these terms mean, and how they work.

Contradictories are “two propositions so related that one is the denial or negation of the other”. (Introduction to Logic 13th edition). This means that contradictory statements cannot both be true, or false, at the same time. This means that both statements, which are contradictory, have the same subject and predicate terms, but they are different to each other in both quantity and quality. So the categorical statements of (A) and (O) are contradictories, as are the categorical statements of (E) and (I). So All A are B is contradicted by Some A is not B, and No A is B is contradicted by Some A are B.

Contradictories:
(A) All A is B :: (O)  Some A is not B
All cats are animals :: Some cats are not animals
(E) No A is B :: (I) Some A is B
No cats are animals :: Some cats are animals

Contraries  are “two propositions so related that they cannot both be true, although both may be false” (Introduction to Logic 13th edition). They cannot both be true, but they can both be false. Take this example of two statements: ‘The Ravens will win the game against the Steelers”‘ or “The Steelers will win against the Ravens”. If these two propositions refer to the same game, then one has to be true and the other false. Now they are not contradictory since the game could be a draw, and so both would be false. So they cannot both be true, but they could both be false. So (A) propositions are contrary to (E) propositions.

Contraries:
(A) All A are B :: (O) No A are B
All poets are dreamers :: No poets are dreamers
* Now they cannot both be true, but they could both be false since there might not be any poets at all.

Subcontrariers are “two propositions so related that they cannot both both be false, although they may both be true”. (Introduction to Logic 13th edition).  Both particular statements have the same subject and predicate terms, but they are differing in quality (one affirming the other denying). So “some diamonds are precious stone” and “some diamonds are not precious stone” could both be true, but they could not both be false. (I) and (O) statements are subcontrariers.

Subcontrariers:

(I) Some A are B :: (O) Some A are not B
Some diamonds are precious stones :: Some diamonds are not precious stones

Subalternation are “are the relation on the square of opposition between a universal proposition and its corresponding particular proposition.” (Introduction to Logic 13th edition). Two statements have the same subject and predicate terms, and agree on quality (affirming/denying), but differ in quantity (one universal and one particular). So the (I) statement is the subalternation of (A) statement, and the (O) statement is the subalternation of the (E) statement.

Subalternation:
(A) All A are B :: (I) Some A are B
All spiders are 8 legged animals :: Some spiders are 8 legged animals
(E) No A are B :: (O) Some A are not B
No whales are fish :: Some whales are not fish

Now that we have gone over traditional square of opposition, it is time to speak of the problems with the traditional square of opposition. It is because of these problems that a modern square of opposition was developed. Once we go over the problems of the traditional square of opposition, we will go over the modern square of opposition and see why we use the modern square of opposition now.

Problems of traditional square of opposition:

This problem revolves around existential import. Existential import is “an attribute of those propositions which normally assert the existence of objects of some specified kind.”(Introduction to Logic 13th edition). (I) and (O) statements carry existential import, and traditional interpretation of categorical statements say that (A) and (E) statements carry existential import. However, the modern interpretation differs from traditional interpretation on these issues.

So the (I) statement of “Some soldiers are heroes” says that there exists at least one soldier who is a hero. The (O) statement of “Some dogs are not companions” says that there exists at least one dog that is not a companion. So (I) and (O) statements do assert that the class designated by their subject terms are not empty, and each has at least one member in it. So by subalteration we find that (I) and (O) statements validly follow from (A) and (E) statements. So (A) and (E) statements must also have existential import, because a statement with existential import cannot be derived validly from another that does not have such import.

We know that (A) and (O) statements are contradictories on the traditional square of oppositions. So “All Danes speak English” is contradicted by “Some Danes do not speak English”. Contradictories cannot both be true, and both cannot both be false, since one must be true. Now if both (A) (Universal statement) and (O)  (particular statement) have existential import, then both contradictories could both be false. Let us take another example: “All inhabitants of Mars are blond” and “Some inhabitants of Mars are not Blond”. They are contradictories and would, under traditional assumption, have existential import. So we assume that both statements are asserting that there are inhabitants of Mars, then both propositions are false if Mars has no inhabitants. So if Mars has no inhabitants, then the class of its inhabitants are empty. Thus, both of the propositions would be false. Now if they are both false, then they cannot be contradictories.

Now if the traditional square of opposition is correct about existential import, and it states that (A) is the contradictory of (O) and (E) is the contradictory of (I), then the square is not correct that they are contradictory based on the idea of existential import. We are clearly left with a contradiction, which violates the Law of Non-contradiction ~(A&~A). Not both the traditional square of opposition has contradictories and the traditional square of opposition does not have contradictories.

Modern solution to Traditional Square of Opposition:

We can solve this problem to the traditional square of opposition, but the price can seem high in doing so. In order to solve this problem, we presuppose that the categories are not empty. For example, take the question “Have you stopped beating your wive?”. This question presupposes that you have beat your wife, so it presuppose that the category is not empty. So if we can answer this question “yes” or “no” that I have stopped, or not, of beating my wife. Now if we we don’t presuppose that I have beat my wife, then I don’t have to answer “yes” or “no”. The category is empty when we don’t make that presupposition, and the category is not empty when we do make the presupposition.

So to rescue the square of opposition, then we insist that all statements, like (A), (E), (I), and (O), that the categories that they refer do have members  (so they are not empty). So the truth and falsehood of the statements, and logical relation among them, can be answered (under this interpretation) if we never presuppose that they are empty. Under this interpretation of presupposition; (A) and (E) will remain contraries; (I) and (O) will remain contraries; (A) and (O) will remain contradictories, as well (E) and (I).

The first problem with these presuppositions is this: If we presuppose that the class designated has members, we can never be able to formulate a statement that denies that it has members, since we presuppose that they do have members. The second problem with these presuppositions is this: Sometimes what we say doesn’t presuppose that there are any members of the category that we are talking about.

Now the modern square of opposition does not make any presupposition. Instead, it works as follows:

(1.) (I) and (O) statements continue to have existential import. So the statement “Some A are B” is false if the category of A is empty, and same for “Some A are not B”.

(2.) Universal statements, (A) and (E), are contradictories of particular statements, (O) and (I).

(3.) Universal statements are interpreted as having no existential import. So if category A is empty, the statement “All A are B” can be true, and so can the statement “No A are B”. Take this example: “All unicorns have horns” and “no unicorns have wings” may both be true, even if there are no unicorns. However, if there are no unicorns, the (I) statement “Some unicorns have horns” is false, and so is the (O) statement “Some unicorns do not have wings”.

(4.) We can utter universal statements which we intend to assert existence. However, in doing so requires two statements, one existential in force but particular, and the other universal but not existential in force.

(5.) Corresponding (A) and (E) statements can both be true and are thus not contraries. They would carry this propositional logic form, which we get to later on. (A) All unicorns have wings= If there is a unicorn, then it has wings; (E) No unicorns have wings= If there is a unicorn, then it does not have wings. And both of these “if…then” statements can be true even if there are no unicorns.

(6.) Corresponding (I) and (O) statements are not subcontraries. They do have existential import, and can both be false if the subject class is empty.

(7.) Subalternation, inferring an (I) statement from an (A) statement or an (O) statement from an (E) statement, is generally not valid. This is because one may not validly infer a statement that has existential import from one that does not.

(8.) Now for this point, I have given conversion, contraposition, and obversion, all based on the modern interpretation. So this point is not necessary to explain what it was. However, with the traditional square of opposition, there were more conversions and contrapositions, but they are not really allowed with the modern interpretation.

(9.) Relations along the sides of the square are undone, but the diagonal, contradictory relations remain in force.

The existential presupposition is rejected by modern logicians. So it is a mistake to assume that a class has members if it is not asserted explicitly that it does. So if an argument relies on assumption of existential import, without explicitly stating it, relies on the existential fallacy.
Finding out if a syllogism is valid:

Here is a syllogism, and one that is valid:

Premise 1: No heroes are cowards (No A are B) (E)
Premise 2: Some soldiers are cowards (Some A are B) (I)
Conclusion: Some soldiers are not heroes (Some A are not B) (O)

The proposition above has three terms in it. It carries the terms heroes, cowards, and soldiers. Our conclusion is important in figuring out how to create a valid syllogism. Remember how I said that we usually come up with the conclusion first, and then support it with some premises? Well, this is where that comes back into play. In the conclusion, we find that the last sentence has a subject and predicate.The subject is soldiers and the predicate is heroes.

There are three terms involved in a valid syllogisms. They are major term, middle term, and minor term. The Major Term is “the term of the categorical syllogism that appears as the predicate term of the conclusion”, which would be “heroes” in the foregoing argument. (Logic: An Introduction 2nd edition). The Minor Term is “the term in the categorical syllogism that occurs as the subject term of the conclusion”, which would be “soldiers” in the foregoing argument. (Logic: An Introduction 2nd edition). The Middle Term is “the term in a categorical syllogism that appears in both premises but not in the conclusion”, which would be “cowards” in the foregoing argument. (Logic: An Introduction 2nd edition)

The premises of a syllogism also have terms to go along with them, besides the terms that show up in them. There are three premises in a syllogism. They are major premise, minor premise, and conclusion. We have already gone over conclusion, so now it is time to deal with major premise and minor premise.

A Major premise is “the premise of a categorical syllogism containing the major term”, which would be the first premise of the foregoing argument (Logic: An Introduction 2nd edition). A Minor premise is “the premise of a categorical syllogism containing the minor term”, which would be the second premise of the foregoing argument. (Logic: An Introduction 2nd edition).

So let us go back over the syllogism that I started out with, but list what they are and what is going on.

(E) No heroes are cowards:: It is the major premise, and has “heroes” as the major term and “cowards” as the middle term.
(I) Some soldiers are cowards:: It is the minor premise, and has “soldiers” as the minor term and “cowards” as the middle term.
(O) Some soldiers are not heroes:: It is the conclusion, and has “soldiers” as the minor term (subject) and “heroes” as the major term (predicate)

Each categorical syllogism has a mood. A Mood is “the specification of the categorical forms of a syllogism arranged in standard order (major premise, minor premise, and conclusion)”. (Logic: An Introduction 2nd edition). For example, each premise of a categorical syllogism is made up (A), (E), (I), and (O) premises. These types of statements make up the mood of an argument. For example, the argument that I presented carries the mood of (E), (I), and (O).

Now categorical syllogisms also have a figure. A figure is “the arrangement of the middle term in the premise of a categorical syllogism”. (Logic: An Introduction 2nd edition). There are four figures to a categorical syllogism. Here are the four figures of a categorical syllogism.

That is the four figures of a categorical syllogism, and here are the rules to it.

(1.) Middle term may be the subject term of the major premise and predicate term of the minor premise.
(2.) Middle term may be the predicate term of both premises.
(3.)  Middle term may be the subject term of both premises.
(4) Middle term may be the predicate term of the major premise and the subject term of the minor premise.

Now there are 256 possible syllogisms that one can come up with (4×64=256). However, there are only 15 syllogisms that are valid. Let us go over the 15 valid forms of a categorical syllogism.

There are four valid syllogisms following figure 1. They all carry a certain mood along with them, as they follow a certain figure. So these syllogisms all follow figure 1.
1. AAA-1:: barabara
All M are P
All S are M
All S are P
2. EAE-1:: celarent
No M are P
All S are M
No S are P
3. AII-1:: darii
All M are P
Some S are M
Some S are P
4. EIO-1:: ferio
No M are P
Some S are M
Some S are not P

There are four valid syllogisms following figure 2. They all carry a certain mood along with them, as they follow a certain figure. So these syllogisms all follow figure 2.
1. AEE-2:: camestres
All P are M
No S are M
No S are P
2. EAE-2:: cesare
No P are M
All S are M
No S are P
3. AOO-2:: baroko
All P are M
Some S are not M
Some S are not P
4. EIO-2:: festino
No P are M
Some S are M
Some S are not M

There are four valid syllogisms following figure 3. They all carry a certain mood along with them, as they follow a certain figure. So these syllogisms all follow figure 3.
1. AII-3:: datisi
All M are P
Some M are S
Some S are P
2. IAI-3:: disamis
Some M are P
All M are S
Some S are P
3. EIO-3:: ferison
No M are P
Some M are S
Some S are not P
4. OAO-3:: bokardo
Some M are not P
All M are S
Some S are not P

There are three valid syllogisms following figure 4. They all carry a certain mood along with them, as they follow a certain figure. So these syllogisms all follow figure 4.
1. AEE-4:: camenes
All P are M
No M are S
No S are P
2. IAI-4:: dimaris
Some P are M
All M are S
Some S are P
3. EIO-4:: fresison
No P are M
Some M are S
Some S are not P

There are six rules to make a valid syllogism. I shall list those 6 rules, but it should be known that if you follow the figure and mood that I gave above, then you have a valid syllogism.

Rule 1. Standard-form syllogism must contain exactly three terms, each of which is used in the same sense throughout the argument. Violation: Fallacy of four terms

Rule 2. In a valid standard-form categorical syllogism, the middle term must be distributed in at least one premise. Violation: Fallacy of Undistributed Middle

Rule 3. In a valid standard-form categorical syllogism, if either term is distributed in the conclusion, then it must be distributed in the premises. Violation: Fallacy of illicit major, or fallacy of illicit minor

Rule 4. No standard-form categorical syllogism having two negative premises is valid. Violation: Fallacy of exclusive premises

Rule 5. If either premise of a valid standard-form categorical syllogism is negative, the conclusion must be negative. Violation: Fallacy of drawing an affirmative conclusion from a negative premise

Rule 6. No valid standard-form categorical syllogism with a particular conclusion can have two universal premises. Violation: Existential fallacy

However, Harry Gensler in Introduction to Logic, gives us a very simple way to find out if a syllogism is valid or not. It is called the Star Test.

So let us return to the 8 statements of syllogisms:
1. All A are B
2. No A are B
3. Some A are B
4. Some A are not B
5. a are B
6. a are not B
7. a are b
8. a are not b

Pay attention to the ones that are in bold and italicized, since that is very important to the Star Test.

A letter is distributed in a valid syllogism if it occurs just after “All” or anywhere after “No” or “not”.

So by definition, (1.) The first letter after “All” is distributed, but not the second. (2.) Both letters after “No” are distributed. (3.) The Letter after “not” is distributed.

So the Start Test for a syllogism goes as follow:  Star the distributed letters in the premises and undistributed letters in the conclusion. Then the syllogism is VALID if and only if every capital letter is starred exactly once and there is exactly one star on the right-handed side.

So a valid argument must satisfy two conditions: (1.) Each capital letter is starred in one and only one occurrence. (Small letters can be starred any number of times.) (2.) Exactly one right-handed letter (letter after “are” or “are not”) is starred.

Let me give you an example of a valid argument with the Star Test.

Example 1:
All A* are B
Some C are A
Some C* are B*

Now we notice that the argument is in figure 1, and has the mood of AII. So it is AII-1, which is a Darii syllogism.

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