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logical relations between mathematics and propositional forms metamathematics

In propositional logic are logical relationships between propositional forms. Typically

relationships that can exist between a pair of propositional forms are contradictoriness, annoyance or inconsistency, implication and logical equivalence subcontrariedad. Two

propositional forms are contradictory when it is impossible to be true at the same time in any case of replacement and, moreover, can not possibly be false at the same time.
Technically, for any substitution of the truth table of the same case that both are true and it happens that both are false.
Another definition: given two propositional formulas A and B can not simultaneously be true or may be false simultaneously.
Another definition: two propositional forms are contradictory if and only if when one is true the other is false.

Two propositional forms are contrary (or incompatible) when they can not simultaneously be true in any interpretation (or substitution) but false.
Another definition, are inconsistent when they are both true in any case sutitución table true but false in some cases.
This definition is different from the misguided "two propositional forms are inconsistent when they can not be simultaneously true in any case of replacement," because this is equivalent (under the assumption that logic is bivalent, that is the principle of non contradiction) "two propositional forms are inconsistent when they are simultaneously true in any case of replacement."
But if you take two propositional forms A and-A (not A), second definition is satisfied, so these would also be contrary contradictory.
But contradictoriness means that both can not be simultaneously either simultaneously true or false, so if being contrary are also contradictory, then there may be both false, and there is no difference between disappointment and contradictoriness.

A propositional form involves another if it can happen that the former is true and the second is false. That is, if there is no substitute in any way that the former is true and the second false.

Two forms are logically equivalent propositional when they engage each other. Consequently, there will be no substitution for which they are both true, nor any case they are both false.

Two propositional forms are subcontrarias when they can not simultaneously be both false but true. In other words, there are false simultaneously in any case of replacement, but there is at least one case they are both true.