Logical consequence
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) ([s]P[/s]-> Q) & [s]P[/s]
(C) Q
The conclusion is both necessary and formal, but it seems not to be a "logical" consequence after all.
Thoughts?
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) ([s]P[/s]-> Q) & [s]P[/s]
(C) Q
The conclusion is both necessary and formal, but it seems not to be a "logical" consequence after all.
Thoughts?
Comments (6)
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.
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.
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.