If they are part of the same logical argument, then they are contradictory.   Any argument that includes both of these statements will be an invalid argument.



Details:



The first statement says "either (H and J) or (N and J), or both, are true"

while the 2nd statement says "If (N or H) are true, then J is false."



Therefore, the two statements can not be simultaneously true.   



Below is a conditional logic proof to illustrate.



1. (H • J) v (N • J)      Premise

2. (N v H) --> ~J         Premise



3.:: H                    Conditional Proof (CP) assumption #1

4.:: J                    CP assumption #2

5.:: (H • J)              3,4, Conjunction (CONJ)

6.:: (H • J) v (N • J)    5, Addition (ADD)  

7.:: H v N                3, ADD

8.:: N v H                7, Commutation

9.:: ~J                   8,2, Modus Ponens (MP)

10.:: J • ~J              4,9 CONJ

11. J • ~J                3-10, CP



Line 11 shows "J and not J" follows from the two premises, and this is a contradiction.



EDIT:  1) Fixed line 10 to read '4,9 CONJ'  (was '4,10 CONJ')  and adding this

note:  2) Lines 5 & 6 are not strictly required, but included to show the CP assumptions lead to the first premise being true.

 

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