# Can classical logic be interpreted modally?

a. is it correct to say that Modal logic can be interpreted classically by using Kripke possible world semantics, i.e. that in each world classical logic holds?

This is then a possible argument for the primacy of classical logic.

b. To argue then for logical pluralism, one may suppose there is a way of interpreting classical logic modally - but is this possible?

• It is not clear what it means for a logic to 'hold' or have primacy over others. Which logic, if any, has primacy, depends on what you are trying to do with the logic. But why must there be one single logic which has absolute 'primacy'? Jul 23 '13 at 16:38
• I don't subscribe to logical monism, I tend to pluralism. But in the context of this question, I'm asking if logic L1 is interpretable in logic L2, that is there is someway of expressing in L2 every sentence of L1 (it may not be a very natural or simple one) then you may as well say that L2>L1. If this holds for every logic Li, then L2>Li then you may as well say logic L2 is best because it dominates all the others. Jul 23 '13 at 19:02
• In the context of logic, interpretation is assigning meanings to sentences to give truth values. But the rules of the logic cannot be interpreted in this sense or else they become propositions with truth values, not logical rules. If you could express L1 as L2, this would just mean you can fully create transformation rules between the two systems. This simply cannot be done if L1 has modal operators and L2 does not. You either have modal operators and a modal logic, or you have neither. Jul 24 '13 at 1:05
• Unless you change the meaning of 'modal', but that's cheating! Jul 24 '13 at 1:06
• @adrianos: Ok, transformation rules is roughly the technical term I'm looking for.( I was using interpretation in its ordinary language sense: to express by other means - I'm aware that in formal logic its used to model truth). So, do you disagree with what I've said above - that in each Kripke world classical logic holds? How about, say I attach a set theory to classical logic, and internally develop the notion of axiomatic systems - and then choose any formalisation of modal logic? Jul 24 '13 at 2:07

Let's assume that classical logic refers mainly to the law of excluded middle, more precisely a bivalent logic. Let's assume that Modal logic refers mainly to the situation were there is a (loosely specified) universe with worlds and a reachability relation between these worlds, and that we are in one specific (well specified) world of this universe and talk about propositions in our own world and the (relatively well specified) worlds that are reachable from our own world (and perhaps also the worlds reachable from worlds reachable from our own world, and finite iterations of this construction).

Let's focus on propositional logic to simplify things. In classical logic, some natural language sentences correspond to propositions, and every proposition is either true or false. In classical logic, you better resist the temptation to assign a proposition to every natural language sentence, because the negation of an ambiguous sentence might still be ambiguous, and hence the functoriality of logic might fail.

One of the problems with a classical logic account of Modal logic is that there can be natural language sentences which correspond to propositions in one world, but fail to correspond to propositions in another world. One idea to remedy this situation is to only consider natural language sentences which correspond to propositions in every world of the universe. But how much will you restrict the expressive power and the useful applications of Modal logic by doing so?

• "One of the problems with a classical logic account of Modal logic is that there can be natural language sentences which correspond to propositions in one world, but fail to correspond to propositions in another world" : Can you give an example of such propositions? Is modal logic done naturally not susceptible such problems? Jul 23 '13 at 19:14
• @MoziburUllah I guess you heard that example/sentence before: "The present King of France is bald." I'm still learning modal logic, so I'm not yet an expert on how modal logic is done naturally. But modal logic is closer to the logic of normal everyday language than classical logic, and normal everyday language deals quite well with such problems. In classical first-order logic, syntactic rules are enough to separate the sentences which correspond to propositions from those which do not. This no longer works for the sentence given above. Jul 23 '13 at 21:14
• It is not that 'the present king of France is bald' necessarily fails to refer to proposition but that it contains a non-referring expression. The problem of vague sentences in classical logic is quite different. A better example is something like "john is still a child" or "we should get going." MOST sentences are vague in one way or another. Sentences 'referring to propositions' is a metaphysical reading of logical systems as facts when they are better understood as encoding inferential rules. Jul 24 '13 at 0:50
• @klimpel: Yes, now that you remind me, I have. "syntactic rules are enough to separate the sentences which correspond to propositions from those which do not". You mean as propositions those that are quantified? Natural language does have more resources though, I could for example suppose you're relating a narrative - a possible world - is this not covered by modal logic? Or that you're being deliberately misleading! But this I assume means some notion of intentionality of a sentence has to be described, and that I suppose is not something that modal logic can provide. Jul 24 '13 at 5:08
• @adrianos: yes, most sentences are vague, as are most definitions in ordinary language. This is why when Socrates is taking pot-shots at Euthyphro for failing to provide precise definitions of piety, I'm more sympathetic towards his point of view, rather than Socrates. Jul 24 '13 at 5:15

Modal logics is "classical" logics plus the operators for necessity and possibility (however interpreted).

The interpretation of these operators, then, relies on access to possible worlds.

But within the possible world, classical logics holds. There is no "p and (not p)" world.

• What does it mean for classical logic to 'hold'? If the answer to the question 'does Jones like Jazz' is yes and no (P & not P), or neither yes nor no, does this mean classical logic doesn't 'hold'? Logical calculus are devices for formalizing inferential relations but how they 'hold' depends on the context of use. Jul 23 '13 at 16:41