Truth is not fundamental

Traditionally, truth is considered a fundamental notion and logical consequence is considered less fundamental because it can be defined as truth-transmission. What is often forgotten is that things could also be presented the other way around: with logical consequence as fundamental and truth as derived from it. (This, I think is one of the important consequences of the work of Mike Dunn on algebraic methods for philosophical logic and David Lewis’ later work on the metaphysics of set theory).

Perhaps the main difference between logical consequence and truth is that the first one is a homogenous relationship, that is a relation between things of the same sort – propositions or sets of propositions, while truth is considered either not a relation or a heterogenous one – if one thinks of truth as truth-in-a-world, that is, a relation between things of different ontological kinds: a proposition and a world. So the first step in defining truth in terms of logical consequence is to level the land so to speak and consider a uniform ontological kind where propositions and worlds can live side by side. The simplest way is to conceive of possible worlds as sets of propositions, or vice versa, propositions as sets of possible worlds.

Now, we can easily define truth in terms of logical consequence: being true is being a logical consequence of the world. Thus a proposition is true (in the actual world) if it follows from the actual world, and in general, a proposition p is true in a world w iff p follows from w

Of course, one could complain that in order to get actual truth and not just truth-in-a-world, we still need actuality as a primitive and, well, you would be completely right to think so; thus, the most honest thing to say would be that truth is just actuality by other name and that truth-in-a-world is just another name for logical consequence.

Let W be a partially ordered set of (sets of) propositions/possible worlds (or, in general, evaluation points) with a top and a bottom. Since we are in an extensional framework, there are no distinct logically equivalent propositions/possible worlds. Now, my final point can be re-stated thus. Given two objects x and y in W, that x is less than or equal to y can be read either as 

  1. x is a logical consequence of y (if we want to take x and y as propositions) or 
  2. x is true in y (if we want to take x as as a proposition and y as a possible world) or 
  3. x is a subset of y (if we want x and y to be sets, either of propositions or of possible worlds) or 
  4. x is part of y (if we want x and y to be possible worlds and accept than not all possible worlds are total).


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