Inverse (statements)

Sep 2009
158
3
Is the inverse of the following compound statement correct?


Statement:

(For all fruit x, x is red) => [ (all houses are blue) or (some chickens are white) ]


Inverse:
(For some fruit x, x is not red) => [ (some houses are not blue) and (no chickens are not white) ]


My confusion comes from the hypothesis of the implication. It is of the form
[ (universal quantifier)(subject), (subject)(verb)(predicate) ]. I know a statement must be of the form [ (subject)(verb)(predicate) ] and also that the negation of [ (universal quantifier)(statement) ] is [ (existential quantifier)(negation of statement) ] but what is the negation of [
(universal quantifier)(subject), (subject)(verb)(predicate) ]?
 
Oct 2009
255
20
St. Louis Area
Is the inverse of the following compound statement correct?


Statement:

(For all fruit x, x is red) => [ (all houses are blue) or (some chickens are white) ]


Inverse:
(For some fruit x, x is not red) => [ (some houses are not blue) and (no chickens are not white) ]


My confusion comes from the hypothesis of the implication. It is of the form
[ (universal quantifier)(subject), (subject)(verb)(predicate) ]. I know a statement must be of the form [ (subject)(verb)(predicate) ] and also that the negation of [ (universal quantifier)(statement) ] is [ (existential quantifier)(negation of statement) ] but what is the negation of [
(universal quantifier)(subject), (subject)(verb)(predicate) ]?
\(\displaystyle \neg{\exists}P(x)\)=\(\displaystyle \forall{x} \neg{P(x)}\)

So it should be:

(For some fruit x, x is not red) => [ (some houses are not blue) and (all chickens are not white) ]
 
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Plato

MHF Helper
Aug 2006
22,461
8,633
You are incorrect on the 'chicken one'.
The negation of "some chickens are white" is "no chicken is white".
 
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Oct 2009
255
20
St. Louis Area
You are incorrect on the 'chicken one'.
The negation of "some chickens are white" is "no chicken is white".
Thank you, Plato. However, I don't see the difference between "no chicken is white" and "all chickens are not white".

Could you elaborate on this? Thanks.
 
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Plato

MHF Helper
Aug 2006
22,461
8,633
Actually they are saying the same thing.
However in this case it a matter of history and style.
Based on what is known as the square of opposition.
It has deep roots in the history of logic.

There are four types of statements.
Universal, positive statements are labeled A: All P is Q.
Universal, negative statements are labeled E: No P is Q.
Existential, positive statements are labeled I: Some P is Q.
Existential, negative statements are labeled O: Some P is not Q.
Statements A & O are negations of each other; as are E & I.

With that background in mind, we generally want students to use the classical form.
 
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