Jerry,
The examples that you use from the Aristotelian Square of Opposition are
standard examples of contradictories, contraries, subcontraries, and
subalterns. The examples are not definitive of them, however. Every pair
of propositions (aside from self-referring propositions and that sort of
thing) is one of the following:
A pair of *contradictories* consists of two propositions such as 'John
is blue' and 'John is not blue', such that /each proposition is
equivalent to the other's negation/. That's to say, that they can't be
both of them true and they can't be both of them false.
A pair of *contraries* consists of two propositions such as 'John is
blue' and 'John is quiet and not blue', such that /each proposition
implies, without being implied by, the other's negation/. That's to say,
that they can't be both of them true, but they can be both of them
false. Another example is 'We have exactly five dogs' and 'We have
exactly four dogs'.
A pair of *subcontraries* consists of two propositions such as 'John is
not blue' and 'John is blue or not quiet', such that /each proposition
is implied by, without implying, the other's negation/. That's to say,
that they can both of them true, but they can't be both of them false.
A pair of *subalterns* consists of two propositions such as 'John is
blue' and 'John is blue or quiet', such that /each proposition neither
implies, nor is implied, by the other's negation/. That's to say, that
they can both of them be true, and they can both of them be false. Any
contingent proposition is subaltern with itself, that is, 'John is blue'
and 'John is blue' are equivalents and subalterns of each other.
Formally true propositions are equivalent and subcontrary to each other.
Formally false propositions are equivalent and contrary to each other.
No propositions are each the other's equivalent and contradictory.
In the Boolean Square of Opposition, A & E are each other's subalterns.
Likewise I & O.
Subalterns used to be distinguished from superalterns but that's in the
old terminology.
I won't provide references, look at 20th-Century logic text books.
Best, Ben
On 1/17/2015 3:10 PM, Jerry LR Chandler wrote:
List, Ben:
On Jan 17, 2015, at 12:16 PM, Benjamin Udell wrote:
Jerry,
But your 'S is P' & 'S is not P' are contradictories, not contraries;
they can't both be true and can't both be false.
'The dogs are four' and 'the dogs are five' are contraries: they
can't both be true but can both be false.
No idea about what your meaning is intended to confer, either to
pragmatism or logic.
First, let me make clear for I was using the term "contrary".
The distinction between contraries and contradictories are clearly and
distinctly presented in the Sanford Encyclopedia of Philosophy:
http://plato.stanford.edu/entries/square/
from which one reads:
*A* Every /S/ is /P/ Universal Affirmative
*E* No /S/ is /P/ Universal Negative
*I* Some /S/ is /P/ Particular Affirmative
*O* Some /S/ is not /P/ Particular Negative
and :
* ‘Every /S/ is /P/’ and ‘Some /S/ is not /P/’ are contradictories.
* ‘No /S/ is /P/’ and ‘Some /S/ is /P/’ are contradictories.
* ‘Every /S/ is /P/’ and ‘No /S/ is /P/’ are contraries.
* ‘Some /S/ is /P/’ and ‘Some /S/ is not /P/’ are subcontraries.
* ‘Some /S/ is /P/’ is a subaltern of ‘Every /S/ is /P/’.
* ‘Some /S/ is not /P/’ is a subaltern of ‘No /S/ is /P/’.
and which shows a clear diagram illustrating the difference between
contraries and contradictories.
'The dogs are four' and 'the dogs are five' are contraries: they
can't both be true but can both be false.
This sentence, from either a logical or mathematical sense, does not
mean to me.
Are you certain you intended to use the word "the" in this context?
The two phrases as in quotation marks which suggests that you may
intend to independent concepts.
Your two phrases are contrary if and only if the phrases refer to the
same sign for the set of dogs you have in mind.
Your response illustrates very nicely the point of my original post.
:-)
Cheers
Jerry
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