4.4 - Conversion, Obversion, and Contraposition
Obverse of an E proposition
All A are non-B
Converse of an A proposition (All A are B)
All B are A
Contrapositive of an A proposition
All non-B are non-A
What is the traditional square of opposition?
An arrangement of lines that illustrates logically necessary relationships among the 4 kinds of categorical propositions.
Term complement for one word term
Attach the prefix "non" to the term
What fallacy does this contrapositive inference form commit? No A are B Therefore, no non-B are non-A
Fallacy of illicit conversion
What fallacy does this contrapositive inference form commit? Some A are B Therefore, some non-B are non-A
Fallacy of illicit conversion
Is this contrapositive inference form valid or invalid? No A are B Therefore, no non-B are non-A
Invalid
Is this contrapositive inference form valid or invalid? Some A are B Therefore, some non-B are non-A
Invalid
Is this inference form valid or invalid? All A are B. Therefore, All B are A
Invalid
Is this inference form valid or invalid? Some A are not B. Therefore, Some B are not A.
Invalid
Fallacy of Illicit Contrary #1
It is false that all A are B No A are B
Fallacy of Illicit Contrary #2
It is false that no A are B All A are B
What does it mean when a proposition and it's converse have identical Venn diagrams?
It means that the proposition and it's converse are logically equivalent
Subalternation relation: If an A proposition is given as false, the corresponding I proposition is
Logically undetermined truth value. True truth value cannot be transmitted downward.
Subalternation relation: If an E proposition is given as false, then the corresponding O proposition is
Logically undetermined. False truth value cannot be transmitted downward.
Contrary relation: If an A proposition is given as false, the corresponding E will be
Logically undetermined. It could be either true or false (without violating the "at least 1 is false" rule).
Contrary relation: If an E proposition is given as false, the corresponding A proposition will be
Logically undetermined. It could be either true or false (without violating the "at least 1 is false" rule).
Subcontrary relation: If an I proposition is given as true, then the corresponding O proposition will be
Logically undetermined. It could be either true or false (without violating the "at least 1 is true" rule).
Subcontrary relation: If an O proposition is given as true, then the corresponding I proposition will be
Logically undetermined. It could be either true or false (without violating the "at least 1 is true" rule).
Subalternation relation: If an I proposition is given as true, the corresponding A proposition is
Logically undetermined. The true truth value cannot be transmitted upward.
Subalternation relation: If an O proposition is given as true then the corresponding E proposition is
Logically undetermined. Truth value cannot be transmitted upward.
Obverse of an A proposition
No A are non-B
Converse of an E proposition (No A are B)
No B are A
Contrapositive an E proposition
No non-B are non-A
Subalternation relation - upward
Only transmits falsity - falsity only flows upward. (Marked with the letter F (for "false"))
Subalternation relation - downward arrow
Only transmits truth - truth only flows down. (Marked with the letter T (for "truth")).
Obversion Step 2
Replace the predicate term with it's term complement
How do we represent the CONTRAPOSITIVE I proposition "Some non-B are non-A" on a Venn diagram
Since non-B is the area outside the B circle and non-A is the area outside the A circle, the place where these two areas overlap is the area outside both circles. So we put an X in this area.
How do we represent the CONTRAPOSITIVE E proposition "No non-B are non-A" on a Venn diagram?
Since non-B is the area outside the B circle and non-A is the area outside the A circle, the place where these two areas overlap is the area outside both circles. So we shade that area.
Obverse of an O proposition
Some A are non-B
Obverse of an I proposition
Some A are not non-B
Converse of an I proposition (Some A are B)
Some B are A
Converse of an O proposition (Some A are not B)
Some B are not A
Contrapositive an I proposition
Some non-B are non-A
Contrapositive an O proposition
Some non-B are not non-A
Steps in Testing Immediate Inferences
Step 1: Assume the premise is true. Step 2: Identify what type of proposition the premise is. Step 3: Identify what type of proposition the conclusion is. Step 4: Identify the relationship between the two propositions. 5. Given that the premise is a true certain type of proposition, what must the truth value of the conclusion be based on the relationship between the premise and conclusion's proposition types 6. Identify the truth value of the conclusion. Step 7: Based on the truth values of the premises, establish whether or not the argument is valid.
Steps in Contraposition
Step 1: Switch the subject term and the predicate term Step 2: Replace the subject term with its term complement Step 3: Replace the predicate term with it's term complement
Obversion
Step 1: change the quality of the proposition without changing the quantity Step 2: Replace the predicate term with it's term complement
True or False: Each categorically proposition is logically equivalent to it's obverse.
True
True or False: the Venn diagram for a E proposition is identical to the diagram of it's converse
True
True or False: the Venn diagram for a I proposition is identical to the diagram of it's converse
True
The relation of subalternation says truth does X and falsity does Y
Truth flows downward. Falsity flows upward.
Is this contrapositive inference form valid or invalid? All A are B Therefore, all non-B are non-A
Valid
Is this contrapositive inference form valid or invalid? Some A are not B Therefore, some non-B are not non-A
Valid
Is this inference form valid or invalid? No A are B. Therefore, No B are A
Valid
Is this inference form valid or invalid? Some A are B. Therefore, Some B are A
Valid
2 propositions have a *contrary* relation when
When at least 1 of the propositions is false because both cannot be true.
2 propositions have a *sub-contrary* relation when
When at least 1 of the propositions is true because both cannot be false.
Definition of Logical Equivalence
When two propositions necessarily have the same truth value.
What does the phrase "All A are non-B" assert in an OBVERTED E proposition
all members of A are outside B.
What does "all non-B are non-A" assert in a CONTRAPOSITIVE A proposition
all members of non-B are outside A, which means that no members of non-B are inside A.
How do we represent "Some A are not non-B" on an OBVERTED I propositions' Venn diagram
an X is placed in the area where A and B overlap.
How do we represent "Some A are non-B" on an OBVERTED O propositions' Venn diagram
an X is placed in the left-hand part of the A circle.
What does "Some non-B are not non-A" assert in a CONTRAPOSITIVE O proposition
asserts that at least 1 member of non-B is outside non-A. This means that at least 1 member of non-B is inside A.
What does "No A are non-B" assert in an OBVERTED A proposition
asserts that the area where A overlaps non-B is empty.
What does "No non-B are non-A" assert in a CONTRAPOSITIVE E proposition
asserts that the area where non-B overlaps non-A is empty.
What does the phrase "Some A are not non-B" assert in an OBVERTED I proposition
at least 1 member of A is not outside of B....which means that at least 1 member of A is inside B
What does the phrase "Some A are non-B" assert in an OBVERTED O proposition
at least 1 member of A is outside B
Obversion Step 1
change the quality of the proposition without changing the quantity
Which relation should you look at first when using the square of opposition to compute more than one truth value?
contradiction
What area does the term "non-A" designate on the Venn diagram of a CONTRAPOSITIVE A proposition?
designates the area outside
What does the term "non-B" designate in an OBVERTED A proposition
designates the area outside the B circle.
Example: the class complement of the class of dogs is what?
everything that is not a dog (cats, fish, trees, etc.)
A and E propositions cannot both be true in a contrary relation, but they can both be
false
Contradictory relation: If an A proposition is given as true, the corresponding O will be...
false
Contradictory relation: If an I proposition is given as true, the corresponding E will be...
false
Contrary relation: If an A proposition is given as true, the corresponding E will be
false
Contrary relation: If an E proposition is given as true, the corresponding A will be
false
I and O propositions can both be true in a SUBCONTRARY relation, but they cannot both be
false
Subalternation relation: If an I proposition is given as false, the corresponding A proposition is
false
Subalternation relation: If an O proposition is given as false, the corresponding E proposition is
false
obvert an A statement that is false, the resulting statement will be
false
obvert an E statement that is false, the resulting statement will be
false
obvert an I statement that is false, the resulting statement will be
false
obvert an O statement that is false, the resulting statement will be
false
An A proposition and it's converse are
logically unrelated with regards to truth value
An O proposition and it's converse are
logically unrelated with regards to truth value.
Example: Term complement of the term dog is what?
non-dog
How do we represent "No A are non-B" on an OBVERTED A propositions' Venn diagram
shading the left-hand part of the A circle.
What does "Some non-B are non-A" assert in a CONTRAPOSITIVE I proposition?
something exists in the area where non-B overlaps non-A.
Conversion
switching the subject term with the predicate term.
How do we represent "All A are non-B" on an OBVERTED E propositions' Venn diagram
the area where A overlaps B is shaded.
Class complement
the group consisting of everything outside the class.
Term Complement
the word or group of words that denotes the class complement.
2 propositions have a *contradictory* relation when
they necessarily have opposite truth values
Contradictory relation: If an A proposition is given as false, the corresponding O will be...
true
Contradictory relation: If an I proposition is given as false, the corresponding E will be...
true
Subalternation relation: If an A proposition is given as true, the corresponding I proposition is
true
Subalternation relation: If an E proposition is given as true, the corresponding O proposition is
true
Subcontrary relation: If a certain I proposition is given as false, the corresponding O proposition is
true
Subcontrary relation: If a certain O proposition is given as false, the corresponding I proposition is
true
obvert a true E statement, the resulting statement will be
true
obvert a true I statement, the resulting statement will be
true
obvert a true O statement, the resulting statement will be
true
obvert an true A statement, the resulting statement will be
true
How do we represent the CONTRAPOSITIVE O proposition "Some non-B are not non-A" on a Venn diagram
we place an X in the area where non-B overlaps A.
How do we represent a CONTRAPOSITIVE A proposition's Venn diagram
we shade the area where non-B overlaps A.
Contraposition involves how many steps
3
Which propositions are logically equivalent to their CONTRAPOSITIVES
A and O propositions
