# Formal versions of exotic logical connectives in natural language

Formal logic tends to be concerned with minimal or at least almost-minimal sets of logical connectives. The standard logical connectives are and, or, implies, iff, neg (I couldn't use Latex for their formal symbols) and for quantification forall, exists. In fact, even and, or, neg, forall, exists is a minimal set.

In natural language however, we have all kinds of logical constructions that are syntactically different and allow you to structure sentences in a different ordering than if one had to write them out in the basic logical connectives. For the purpose of this question I'll call this "Extended propositional logic": For example we can say things like:

"Extended propositional logic": A, except when B in which case C. Propositional logic: not B implies A, and B implies C and not A

However, I've never seen such more "exotic" connectives be formalized. In fact, the "except-connective" is not a binary connective, but can be arbitrarily complex:

"Extended propositional logic": A, except when B in which case C, or when D in which case E, or when F in which case G. Propositional logic: not (B or D or F) implies A, and B implies (C and not A), and D implies (E and not A), and F implies (G and not A).

There are certainly others (e.g. even the simple ternary connective "if A then B, else C" isn't used in formal logic, and instead written as A implies B, and notA implies C). It seems like some propositions can be stated more succinctly using appropriate natural language connectives than in terms of the standard formal connectives.

Question: Is there a literature that attempts to formalize of such more "exotic" connectives that people use in natural language, and in particular define symbols for them or at least formal grammars/languages with precise semantics, and with a procedure for turning those "exotic formal symbols" into formulas using only standard connectives? I can imagine that there are also quantifiers that are more exotic than forall, exists but can be restated in terms of those two.

Note: I'm not asking for things like modal logics, which have an entirely different semantics.

Edit: Here is a quantifier-example based on the word "also":

"Extended predicate logic": Only A(a), except when B in which case also A(b) Predicate logic: notB implies (A(a) and forall x not equal to a, notA(x)), and B implies (A(a) and A(b) and forall x not equal to a and not equal to b, notA(x))

As you say, there are many connectives in natural languages such as English, while formal logic attempts to express these in just a few. To some extent this is an inevitable tension. We want our formal logic to be simple, practical and easy to produce proofs. Natural languages on the other hand are messy: they express all kinds of subtleties and nuances that are bound up with the pragmatic purposes that we put language to. Sometimes in logic we just have to make do with an approximation to the meaning of English. This is particularly true if we want our logical connectives to be truth functions.

There is a table of English connectives on the Wikipedia page on logical connectives. Many of these are only approximately correct. For example, 'implies' is only very roughly approximated by material implication. John Corcoran in his paper "Meanings of Implication" (Dialogos 25, 1973, pp.59-76) distinguishes more than twelve different meanings of 'implies' in English.

Another example from David Sanford: 'if A then B' is usually taken to be equivalent to 'A only if B'. But there is clearly a difference between:

``````If you learn to play the cello, I’ll buy you a cello.
You’ll learn to play the cello only if I buy you a cello.
``````

With conditionals, the antecedent is typically causally, and/or temporally, and/or epistemologically prior to the main clause. This is a feature that is not captured by using a simple truth function.

Another example: 'A unless B' is usually understood to be the same as 'A if not B'. If we take 'if' here to be a truth function then it follows that 'A unless B' has the same truth conditions as 'A or B' (inclusive or). But in ordinary English 'or' is not synonymous with 'unless'. For one thing, 'or' is commutative but 'unless' is usually not. There is a difference between:

``````Tomorrow, I will go to the beach, unless it rains.
Tomorrow, it will rain, unless I go to the beach.
``````

With quantifiers, again there are many in English, such as 'most', 'a lot', 'many', 'few', 'hardly any', etc. Many of these express proportions and cannot be represented in simple first order predicate logic without using arithmetic, so they are not reducible to just 'forall' and 'exists'. There is a considerable literature on what are called generalized quantifiers. The SEP article gives a good overview.

I'm not aware of any particular work that treats the subject you delineate in-depth, but linguists and philosophers of language who don't use exclusively truth conditional notions of semantics often appeal to pragmatics to understand how various linguistical aspects such as implicature, concepts which entail connotation, and conceptual metaphors shade the semantics of logical connectives. For example, note the difference in meaning:

S1 I went to the store, but I bought a book.
S2 I went to the store, and I bought a book.

S1 seems to communicate something different than S2 such as the purchase wasn't the intended activity, that the agent was thwarted in action, or perhaps it carries an implicit statement or instance of indexicality of sorts. S1 and S2 both have a compound predicate, and in propositional logic they'd likely be translated the same, but the 'but' seems to be a word of contrast. Again:

S'1 I went to the store, but I bought a book (because they didn't have pencils).
S'2 I went to the store, and I bought a book (because I need it to draw).

This extra-logical meaning is confirmed by Merriam-Webster:

Definition of but

(Entry 1 of 5)
1a : except for the fact
// would have protested but that he was afraid
...
c : without the concomitant
// that it never rains but it pours

Since the complexity given the phonological, syntactical, semantic, and pragmatic aspects permeate natural language, you might want to post the same question on Linguistics SE.

• Yes even though it wasn't what I asked for this is interesting. – user56834 Oct 28 '20 at 13:22
• (The reason that it wasn't what I asked for is that in my "extended propositional logic", there is no additional semantics, there is just a more succinct way of representing the same semantics. – user56834 Oct 28 '20 at 13:29
• @user56834 That's an interesting pursuit. If you attempt to reduce the complexity of propositional logic with additional definitions to shorten the syntax and efface the logical primitives, you certainly would have some sort of "compressed" formal system for logical expression. It would be like a syntax for macros in a logical proof system. It doesn't seem that Bumble's article about the generalization quantifiers qualifies since it uses a more expansive syntax to enrich the semantics of logic... – J D Oct 28 '20 at 16:18
• I did read your request more along the lines of using connectives to add conceptual information to the logical operations which isn't what you want, and yet the use of mathematical notation to reduce logical operations like in conditioned disjunction also misses the mark because it's syntax is artificial... – J D Oct 28 '20 at 16:27
• Maybe a philosophical objection to methodology explains the lack of theory? The Analytical tradition strives for reduction to comprehensive formal primitives? Isn't one of the chief complaints of the ordinary language movement that such linguistic edifices are essentially the pursuit of obscurum per obscurious? – J D Oct 28 '20 at 16:43