## Logical consequence

• 89
Logical consequence is said to be both necessary and formal.
Now, suppose that you are asked to resolve a modus ponens, and you reason in the following way, in a seemed way than simplifying equations.

(1) (P-> Q) & P->?
(2) (P-> Q) & P
(C) Q

The conclusion is both necessary and formal, but it seems not to be a "logical" consequence after all.
Thoughts?
• 314
I'm not sure I understand your question. Is your first statement, P implied Q, and P implies something unknown as well? Is your second statement, P implies Q, and not P implies something unknown? And then the third statement is simply Q?
• 2.6k
Well, it seems to me at first look that crossing out - if that's what you're doing - is more work that just reading and understanding the logic - because in order to cross out correctly you already have to have understood the logic.
• 89

The point is that the crossing out is other logic than "classic" modus ponens, but it is a formal way that leds to a valid conclusion (necessarily true if premises are true). But is seems not to be a logical consequence of premises.
• 2.6k
The question, then, is do you have an explicit "crossing out" rule that is not just a sleight-of-hand modus ponens? For example, (p=>q) = ~(p^~q). Can you do that with crossing out?
• 89

Obviously, the "crossing out" rule only works for certain premises and notations. My claim is that there are other "rules" with the same consequence for all logical system, which use always a linguistic notation (they are formal languages). For example, for (p=>q) = ~(p^~q) someone could interpret than "=>" means that we must cross out two "~" signs and one "^" sign, so pq=pq.
The point is that all notation system do not prevent the existence of heuristic rules that led both to formal and necessary conclusions. I try to show cases of logical luck.
• 24

Tautologies are logical truths.
In virtue of truth tables (p & q) -> q is a tautology.
and
((p -> q) & p) <-> (p & q), is a tautology.
therefore
((p -> q) & p) -> q is a tautology.
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