Meanings and Overlap of Double-arrow Conditionals

[bethlynnjean1395]

I've been experiencing some confusion regarding double-arrows and their related meanings. Please let me know if these are correct and whether I am missing any important inferences! This is in response to the Rule Identification Drill in the LG workbook that gave me lots of difficulties. If there is a place I can go where this information is all summarized and diagrammed, please let me know!


1. NOT A NOT B
Rule: Either A or B must be included (allows for both but does not allow for neither)
Equivalent to: NOT A B (contrapositive: NOT B A) Should there be another conditional associated with this because it is a double arrow (both terms are sufficient and necessary)??

2. A NOT B
Rule: Either A or B must be present but not both (does not allow for neither)
Equivalent to: A NOT B (contrapositive: B NOT A) and NOT B A (NOT A B)

3. A B
Rule: Either A or B must be present, both not both (allows for neither) (somewhat of an overlap of #2??)
Equivalent: A NOT B (contrapositive: B NOT A) Should there be another conditional associated with this because it is a double arrow (both terms are sufficient and necessary)??

4. A B
Rule: If A then B and if B then A
Equivalent: A B (contrapositive: NOT B A) and B A (contrapositive: NOT A B)


Thank you in advance for your help!!

[James Finch]

Hi Beth,

1-3 look good! The most important thing is to be able to quickly identify the possibilities and certainties based on knowledge of one condition, so write out out all the conditionals until you feel comfortable with what the biconditionals mean.

4 looks like the contrapositives were diagrammed incorrectly. A B means that if you have one variable included, you must include both. This also means that if one isn't present, then the other cannot be present either. So:

A B
B A
B A
A B

Hope this helps!

[bethlynnjean1395]

Thank you so much!

Just to clarify, with a double-not-arrow it does not entail two different conditional statements (just one and then its contrapositive), correct?

[Adam Tyson]

Correct, bethlynnjean1395! A double-not-arrow arises when you have a sufficient condition that is positive ("if X is selected") and a necessary condition that is negative ("then Z is not selected"). That diagrams as:

X Z

and it can be read as "X and Z cannot be selected together". You can have just one, or just the other, or neither, but you cannot have both.

Good work!