Complications Cat Syllog

8/17/2019 Complications Cat Syllog

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Complications for CategoricalSyllogisms

PHIL 121: Methods of ReasoningFebruary 27, 2013

Instructor:Karin Howe

Binghamton University

http://creativecommons.org/licenses/by-nc-sa/3.0/


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Overall Plan

• First, I will present some problematic

propositions and explain how we can deal

with them (or not)

• Second, we will discuss some problematic

types of arguments and how we can deal

with them (or not)


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Problematic Propositions


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• Issue 1: Statements that don't contain a quantifier

– Example 1: Cats are mammals

• This is clearly making a universal claim (it's talking about

all cats, not just some of them)• Solution: Rewrite this statement as the following standard

form categorical proposition "All cats are mammals."• In general, when presented with a statement that seems

categorical but does not contain a quantifier, we will assume

it is making a universal claim unless we have good reasonsto think otherwise.

– Example 2: "Children are present"

• This example is clearly an exception to the general rule I just stated. Clearly, this is not stating that all the children are

present - it is merely saying that there are children here.• Solution: rewrite this statement as a standard form

categorical proposition as follows: "Some children are

people that are here." (or something like that)


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Issue 2: Compare the following two statements:– A bat is a mammal.

• This is clearly making a universal statement -- it's talking aboutALL bats, not about one particular bat.

• Solution: "A bat is a mammal" can be translated into thestandard form categorical proposition "All bats are

mammals."

– A bat flew in the window.• However, this sentence is not making a universal claim. Rather,

this is saying that there is some particular bat that flew into thewindow, and thus on the basis of this information we can makethe claim "Some bats are things that fly into windows," becausewe have at least one bat we can point to who did this.

• Solution: "A bat flew into the window" can be translated intothe standard form categorical proposition "Some bats are

things that fly into windows."


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– Likewise for sentences with "an" or "the" -- sometimesthese sentences make universal claims and sometimes

they make claims about particular individuals, and thusshould be treated as particular statements

– We have to use context clues and what we know aboutthe world in order to interpret them correctly

–How about this statement?• "The cat is a fine animal commonly mistaken for a

meatloaf."

– Does this statement make a universal claim, or is itmaking a particular claim?

• Kind of ambiguous …..

• Sometimes we're just going to have to make a judgment call


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• Issue 3: statements that are almost in standardform, but not quite

• Example 1: Racehorses are all thoroughbreds.– Clearly, this is making a universal statement. How

would we restate this as a standard form universalstatement?

– Careful! One temptation might be to take what follows"all" and make it the subject of the proposition, makingthe other term the predicate: "All thoroughbreds are

racehorses." (wrong!)

– Solution: rewrite this statement in standard form as"All racehorses are thoroughbreds."


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• Example 2: All roly-poly fishheads are not good dancers.

– Expressed symbolically: F = roly-poly fishheads, D =good dancers

• All F are not D

– Two possible solutions:

• Although we are make a distinction between "not D"and "non-D", this distinction is in a certain sense an

artificial and meaningless distinction. "not D" and"non-D" mean the same thing, so we could collapsethe distinction in this case and rewrite this statement as"All roly-poly fishheads are non-good-dancers."

• Another possible solution is to just think about what

this sentence is really saying. To say that all fishheadsare not good dancers is simply to say that "No roly-poly fisheads are good dancers" (note that thesesolutions are equivalent via obversion)

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• Issue 4: statements that are the negations of standard

form categorical propositions

– Not all birds can fly.

• Expressed symbolically: B = birds, F = things

that can fly

– Not all B are F

• Clearly, this is the negation of an A proposition.

Based on the square of opposition, this is clearly

equivalent to the corresponding O proposition.

• Solution: rewrite this statement as a standard

form categorical proposition as follows: "Some

birds are not things that can fly."


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• There are no penguins in the Arctic.

• This statement could naturally be read intwo different but logically equivalent ways:

– "It's false that some penguins are things that live

in the Arctic"– "No penguins are things that live in the Arctic."

• Solution: always write statements of this

form ("there are no X that are Y") as E propositions ("no X are Y"), since that way

the statement will be in standard form


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• Issue 5: statements with non-standard quantifiers

such as "most," "many," "a few," "all but a few,"

"almost all," "not quite all," etc.

• Solution: although it loses some of the meaning

of these different quantifiers, we will translate all

of these quantifiers as simply "Some" (capturesthe minimal meaning of these quantifiers)

• Likewise, statements with the non-standard

quantifiers "every," "each," or "any" are bestunderstood as the standard quantifier "All."


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• Issue 6: statements involving "only"

• Example: Only mammals are marsupials

• Clearly, this is making some sort of universal claim.What universal claim is it making?

• Two options:

1. All mammals are marsupials.

2. All marsupials are mammals.• Another way to think about it: Can be seen to be making

the claim "If it's not a mammal, then it's not a marsupial"(because only mammals are marsupials). Well, this is thesame as saying "All nonmammals are nonmarsupials,"which is the contrapositive of option 2 above.

• Solution: rewrite all statements of the form "Only X are

Y" as "All Y are X"


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• Issue 7: exceptive propositions (propositions that say thingslike "All except employees are eligible")

• Clearly, this is making a universal claim, but what kind ofuniversal claim?

– In fact, is is asserting two universal claims. It is sayingboth that "all nonemployees are eligible" and that "noemployees are eligible."

• Solution: rewrite these kinds of statements as theconjunction of the two underlying standard formpropositions. Then, if we need to use these statements in anargument, we can just split them up into two premises,

right?…. ?• Okay, that works, sort of, but there are issues with this, aswe will come back to in the section about problematic

arguments


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• Issue 8: singular propositions (e.g., "Karin is a

kangaroo")

• What kind of claim is this making? Is it making a

universal claim, or is it making a particular claim?

• Standard solution:

– Symbolize the named individual as a unique unit class

(as a set containing only that individual); e.g. K =

{Karin}

– Symbolize the predicate class as normal; e.g. G =

kangaroos– Translate the statement as a universal statement; e.g.

All K are G


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• Copi says (in Ch 7) that it is "customary" to read these typesof sentences this way "automatically" -- in other words, we

simply interpret them this way naturally• Really???

• Another problem with this solution: translating the statement"Karin is a kangaroo" as "All K are G" ignores the existenceclaim that the original statement is making.

• Standard solution 2: Translate "Karin is a kangaroo" as theconjunction of the two statements "All K are G" and "SomeK are G"

• Both of these solutions are TERRIBLE. They are

non-intuitive (especially the part about the unitclass), and are in no way a match for the currentstandard logical treatment of these kinds of

statements.


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Summary: Problematic Propositions

• Issue 1: statements that seem categorical in nature,but which don't contain an explicit quantifier.

– Solution: we will generally treat these as making auniversal claim, unless given good reason to think

otherwise.• Issue 2: statements containing the words "a," "an,"

or "the" in place of a quantifier

– Solution: use context clues to determine whether the

statement is making a universal claim or a particular claim. Where this is ambiguous we will have to justmake a judgment call (I will try not to give you

statements where this is the case)


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• Issue 3: statements that are almost in standardform, but not quite

– Example 1: statements of the form "X are all Y" -Solution: rewrite these statements as "All X are Y"

– Example 2: statements of the form "All X are not Y" -Solution: rewrite these statements as "All X are non-

Y" or "No X are Y"• Issue 4: statements that are the negations of

standard form categorical propositions

– Solution: rewrite them as the corresponding

contradictory proposition (e.g., "Not all X are Y"

would be rewritten as "Some X are not Y")


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• Issue 5: statements with non-standard quantifiers

such as "most," "many," "a few," "all but a few,"

"almost all," "not quite all," "every," "each," "any"

etc.

– Solution: translate non-standard quantifiers like

"most," "many," "a few," "all but a few," "almost all,"

"not quite all," as "Some," and translate non-standard

quantifiers like "every," "each" and "any" as "All"• Issue 6: statements involving "only" (e.g. "Only

mammals are marsupials")– Solution: translate all statements of the form "Only X

are Y" as "All Y are X"


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• Issue 7: exceptive propositions (e.g., "All

except employees are eligible")– Solution: rewrite these kinds of statements as theconjunction of the two underlying standard formpropositions. Then, if we need to use these statementsin an argument, we can just split them up into two

premises, right?…. ?• Issue 8: singular propositions (e.g., "Karin

is a kangaroo")

– We cannot deal with these types ofstatements in syllogistic logic. Any attempts

to do so are highly bizarre.


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Problematic Arguments


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Issue 1: Sorites

• Pronounced so-ri-teas

• In this context, a sorites is a categorical

argument that contains three or more

categorical propositions as premises

• Like categorical syllogisms, categorical

sorites can also be said to have a standardform


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Standard Form Categorical Sorites

1. All statements are standard form categoricalpropositions

2. Each term appears exactly twice in the sorites.

3. Propositions are arranged in such a way that

every proposition has one term in common withthe proposition that follows it, except for the lastproposition.

4. A line is drawn under the last proposition in thesorites.

5. The conclusion of the sorites appears under theline.


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Example

1. All babies are illogical persons.

2. All illogical persons are despised persons.

3. No persons who can manage crocodiles are despised persons.__

Therefore no babies are persons who can manage crocodiles.

In theory, we can use the Venn diagram

technique to diagram this argument, with

a little adjustment!


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• Step 1: symbolize the argument1. All babies are illogical persons.

2. All illogical persons are despised persons.

3. No persons who can manage crocodiles are despised persons.__

Therefore no babies are persons who can manage crocodiles.

B = babies, I = illogical persons, M = people

who can manage crocodiles, D = despisedpersons

1. All B are I

2. All I are D3. No M are D

Therefore No B are M


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1. All B are I

2. All I are D

3. No M are D

No B are M

• Basically, what's going with a sorites is that

there is a suppressed subconclusion that makes itall hang together - we need to bring out thatsubconclusion in order to split the argument intotwo standard form categorical syllogisms!

1. All I are D 1. No M are D2. All B are I 2. All B are D

All B are D No B are M


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• We can then diagram these sub-arguments using twoseparate Venn diagrams.

• If both the diagrams show that the individual sub-argumentsare valid, the the argument as a whole is valid! (if eitherdiagram shows invalidity, then the argument is invalid)

1. All I are D 1. No M are D

2. All B are I 2. All B are D

All B are D No B are M


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Issue 2: Arguments with

Inconsistent Premises Explain why it is impossible for a standard

form categorical syllogism to have

inconsistent premises, making use of whatyou know about inconsistent premises and

the relationships between standard form

categorical propositions, as well as thedefinition of a standard form categorical

syllogism.


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Walking it through…

• What is the definition of a set of inconsistentpremises?

– A set of premises is inconsistent if and only if it isimpossible

for them all to be true at the same time

• What is the only way two categorical propositions canbe inconsistent? (Hint: think about the Square ofOpposition)

– Answer: they have to be contradictory statements!

(note: this is not universally true of a set ofinconsistent statements, but is true in any set of twostatements which are inconsistent)


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• Okay, so pick any two pairs of contradictory

statements and make them the premises of yourargument:

1. All X are Y

2. Some X are not Y

Therefore ….• What do you notice about this argument?

• It is not in standard form, because there is nomiddle term!!

• And that my children is why it is impossible tohave a standard form categorical syllogism withinconsistent premises.


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• But wait … didn't we say earlier that all

arguments with inconsistent premises arevalid?

• Does that mean that there are some validcategorical syllogisms that we can't analyze

using our Venn Diagram technique??

• Well, maybe …. but maybe we can …. isthere a way that we can use the Venn

Diagram technique to show that anargument with inconsistent premises isvalid?


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1. All S are P

2. Some S are not P__

No S are M

• What happens if wetry to diagram thisargument?

– The premisesbasically cancel eachother out

• Problem: conclusion

is not diagrammed• So, not really a fit for

the technique


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Issue 3: arguments with

exceptive premises• Consider the following statement: "All

except students are wealthy."

• We could express this symbolically as theconjunction of these two statements: AllnonS are W and No S are W (S = students,W = wealthy people)

• Okay, what sort of conclusion could followfrom this?


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1. All nonS are W

2. No S are W____

Therefore ….?

One option: W could be the middle term, in which case

nonS would be the major term and S would

be the minor term.

However, that's not really right -- that's seeing

three terms where there are only two


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1. All nonS are W

2. No S are W____

Therefore ….?

Easiest fix: No S are W => No W are S (conversion) =>

All W are nonS (obversion)1. All nonS are W

2. All W are nonS____

Therefore ….? Ooops! Not a standard form categorical

syllogism!


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• Could this problem be fixed by adding another premise?In other words, suppose this was really a sorites, in

disguise• Example: All except students are wealthy. All wealthy

people are happy. Therefore all nonstudents are happy.

1. All nonS are W

2. All W are nonS3. All W are H______

All nonS are H

Yes, this would fix it -- all we would have to do is

ignore the extraneous premise (P2). (note however that itisn't really a sorites, though, because we still only have

three terms)…. ?


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Issue 4: Can the existential

fallacy be fixed?• Consider the following argument:

1. All cats are cute things.

2. All cute things are soft things.

Therefore some cats are soft things.

The problem with this argument is that it

depends on cats existing, which isn't givenby the premises but which we know to infact be the case.


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• How about this as a solution?– Recall what Copi said in Chapter 3 -- it isn't really a

problem that universals don't imply existence in theBoolean interpretation, because where existence isnecessary for an argument, we can always add it in bywriting the relevant universal statement both as auniversal and a particular statement.

1. All cats are cute things.2. Some cats are cute things.

3. All cute things are soft things.


Again, like in the previous example, it fixes theissue, but does so by ignoring one of the originalpremises (P1)…. ?


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1. All cats are cute things.

2. Some cats are cute things.3. All cute things are soft things.


Again, like in the previous example, itfixes the issue, but does so by ignoringone of the original premises (P1)

Again, this isn't really a problem (we canand should ignore irrelevant premises),but it's weird and non-standard.


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Issue 5: Arguments containing

singular statements1. All kangaroos can fly.

2. Karin is a kangaroo.__

Karin can fly. Since we can't analyze "Karin is a kangaroo"

as a standard form categorical proposition,

we are not able to analyze this argument

using the Venn Diagram technique, even

though it is clearly valid.


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Summary: Problematic Arguments

• Issue 1: Sorites• Things you should know for the exam:

– In this context, a sorites as a categorical argument that

contains three or more categorical propositions

– Like standard form categorical syllogisms, there is sucha thing as a standard form sorites.

– In theory, we can use our Venn Diagram technique to

deal with these types of arguments

– You should understand how to do this in theory, butyou will not be asked to actually do it on the exam.


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• Issue 2: arguments with inconsistent premises

– The Venn Diagramming technique cannot be adapted

to analyze these arguments

• Issue 3: arguments with exceptive premises– These sometimes work, but sometimes don't -- it

depends on the number of premises in the argument

and the form of all of those premises …. ?• Issue 4: fixing the existential fallacy

– If an argument commits the existential fallacy, but inthis case the existence of the relevant class(es) is not inquestion, then we can simply add in the necessaryparticular statements.…. ?

– Important caveat: Only works if we know the content of

the argument -- can't be done if all we have is the form.


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• Issue 5: arguments containing singular

propositions

– Since we cannot translate singular

statements easily into standard form

categorical propositions, this means that wecan't deal with any arguments involving

these types of statements using the Venn

Diagram technique.


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