Logic Notation Question: [[A]] vs. ⌈A⌉

Question

Let A stand for an arbitrary proposition. I've read some papers recently that use differing notation on expressing the notion that "A is true". The two I'm concerned about are as follows:

(1) [[A]] is true

(2) ^⌈A^⌉ is true

I've even seen these two different notations used in the same paper. What is the typical difference between the two, if any? I'm suspecting that [[A]] (often) refers, literally, to a set of worlds whereas ⌈A⌉ refers just to a proposition without making any commitments to whether it is a set of worlds or not, but I'm far from sure here.

Is anyone familiar with typical notation in the literature here? What about single brackets like [A]?

Answer 1

The first one (often called semantic brackets) is mostly found in formal semantics, and it's the name of the evaluation function, which maps expressions in a formal language to objects in the model of evaluation. Suppose A is the sentence "snow is white." Here's how semantic brackets are used:

[["A"]] is true ≡ snow is white

The second one (often called corner quotes) is mostly found in contexts where one's dealing with two languages: an object language and a metalanguage, and it's very important to keep the distinction in mind. It maps expressions in the metalanguage to expressions in the object language. Your example:

^⌈ A ^⌉ is true ≡ "snow is white" is true ≡ snow is white

The point of corner quotes here was to replace the metalinguistic A with the expression "snow is white", which unlike A is also found in the object language. Once the replacement is done, the corner quotes are replaced with ordinary quotation marks, and from there the usual Tarskian definition does its magic.

Lastly, I'm not familiar with any uses for [] in the context of semantics.

Answer 2

With regards to the bracket notation, you're basically correct. It's easiest to state the idea as follows. Let's say, for concreteness, we're dealing with a sentence "A or not-B". This is logically equivalent to "not-(not-A and B)" and "if B, then A" (at least in classical logic). But all three of these are distinct sentences (one's a disjunction, another's a negation of a conjunction, another a conditional). The notation "[A]" or sometimes "[[A]]" can be thought of as conveniently denoting the "propositional content" of A (which can be made mathematically more precise). For instance, in our original example, [A or not-B] = [not-(not-A and B)] = [if B, then A]. (I don't think there's a significant difference between single brackets and double brackets; as far as I know, it's merely notational preference.)

For instance, in the context of modal logic, [A] can denote the set of states in some model at which A is true. In the context of lattices or algebraic logic, [A] can be thought of as a point on a lattice which A (and all its logical equivalents) denotes.

Corner quotes, on the other hand, are more syntactic in nature. They're most often seen in the context of nonstandard models of PA, set theory, or whenever you're dealing with systems which are powerful enough to code sentences of their language (as you sometimes see, for instance, in debates about the Liar's paradox). Corner quotes placed around a proposition form a term in the language you're dealing with, and hence can appear and be referred to in formulae directly. Usually, though, corner quotes around logically equivalent propositions will not denote the same terms.

They're also sometimes used in order to properly account for the use-mention distinction with complex formulae, though this is more of a philosophical device/concern than a mathematical one. (See John MacFarlane's handout on Substitutional Quantifiers for an explanation.)