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The above is true because it is defined to be true. The meaning of the material conditional (i.e. "->") might itself produce some surprising results for anyone
by hetman 13y ago
The above is true because it is defined to be true. The meaning of the material conditional (i.e. "->") might itself produce some surprising results for anyone attempting to intuitively equate it with the English word "implies". Interestingly, there is still a lot of debate about how the material conditional should be used to model certain aspects of reality (or at least our statement about said reality).
So for example, if we're attempting to model a causal chain, we might apply the modus ponens inference rule to deduce the truthfulness of your above proposed statement. But the inference rule is itself an axiom and there's no reason why we can't define a system of logic where it does not hold.
Would that system of logic be useful in modeling reality? It very well may be for modelling some aspects of reality.
Is it possible to definitively say that there is a "Logic" that exists apart from logical systems. I'm not sure how you could define the former without the latter. As a result, this sounds somewhat akin to the classical Greek belief that the universe fundamentally consists of numbers and is probably more of a philosophical statement than anything else.
- Double_Cast 13y ago> But the inference rule is itself an axiom Is it? I've seen Boolean True explicitly derived from Modus Ponens. Apparently, p -> q = ~p v q which looks derivable from Aristotle's Third Law. http://matt.might.net/articles/logical-literacy/ http://matt.might.net/articles/logical-literacy/
- hetman 13y agoBut Modus Ponens is the axiom here. Without it, boolean logic is just rearranging arbitrary symbols on a page. Axioms like it are necessary to link those symbols to reality in some way.