I took a logics course this semester and I am still rather unsure about some basics.
Generally speaking, is many-valued logic more powerful than two-valued logic?
By powerful I mean: is there a many-valued logic expression that cannot be represented with two valued logic? (just like you can represent a decimal number in a binary system)
I am very sorry in advance if the question is trivial or obvious and hope you may help me.
greets, Daniel
On
It depends what you're talking about. In other words, it depends on how you evaluate 'power'.
In terms of formal expressions in propositional logic, in other words, well-formed formula of propositional logic ('wff', hereafter), every wff from any many-valued logic which is a tautology also qualifies as a tautology in two-valued logic. The converse does not hold. So, the set of tautologies of any many-valued logic is a subset of the tautologies of two-valued logic for propositional logic.
On the other hand, quantification in two-valued logic is limited to universal and existential quantification. There is no quantified such as 'very' or 'many' or 'quite a lot', which are possible in many-valued logic, especially when many-valued logic refers to a fuzzy, or in other words 'infinite-valued', logic.
So, many-valued logic has more expressive power in that it can represent more natural language expressions. But, two-valued logic is more powerful in that it has all of the wffs of many-valued logic, but many-valued logic does not have all the wffs of two-valued logic. Also, the notion of proof probably can more readily get analyzed in two-valued logic than in many-valued logic.
You might also want to consider intuitionistic logic, which is an infinite-valued logic. In some respects, it has more power than two-valued logic, and in some respects it has less power than two-valued logic.
From the semantic point of view, many valued logic (ML) is more expressive than bivalent logic (BL). ML is a semantic system (an interpreted formal language) in which any sentence can have more than two semantic values. It includes all BL type systems. Basically you can use it to describe a larger set of domains. However that doesn’t mean that a theory of type ML is more useful or more fecund than a theory of type BL. Having more values doesn’t imply that more things can be proved, if anything is the other way around (e.g. much of mathematics is based on the law of the excluded middle and it’s not available in the ML camp).