Is there a logic of non-universal class characteristics?

Question

The proposition

(1) Dogs have four legs

is true, but if you tried to convert this proposition into predicate logic, the only reasonable candidate is the false

(2) for all x, x is a dog implies x has four legs

there are, after all, three-legged dogs.

It seems that predicate logic is missing something here, something that I would call a characteristic statement rather than a universal statement. It is a characteristic of the class of dogs that dogs have four legs, yet an individual can be a dog without having four legs.

Are there any logics that deal with this kind of proposition? I assume the logic would have to be intensional or have a peculiar form of modality.

ADDENDUM:

Several people have suggested default logics as an answer. While I appreciate the suggestion, this does not deal with the issue I am addressing. To clarify the question, let me add some philosophical background material.

I am talking about properties that are characteristic of class, properties that are natural for a class, not just properties that are common to a class. Here is one way of seeing the distinction: if you find a three-legged dog, then there is something unnatural about the dog; it has been maimed or has a congenital lack. By contrast, there is nothing unnatural about penguins not being able to fly or black swans being black. By saying something is "unnatural" I don't mean it's necessarily harmful. For example,

(3) Dogs can't talk.

If you found a talking dog it would not make (3) false because (3) is a statement about the nature of dogs. What it would mean that there is something unnatural about the particular dog you found.

Or here is another way to look at it. Mammals can be divided up into 0-legged (whales), 2-legged (humans, kangaroos), and four-legged. This is an informal conceptual category, not a scientific category, and there might be some fuzzy divisions (what are baboons?) but there aren't any 1-legged or 3-legged species of mammal. The dog species falls into the 4-legged category. So "all dogs have four legs" can be seen as a sort of claim about what category dogs fit into.

In other words, it is part of the very concept of being a dog that dogs have four legs. This doesn't mean that all dogs actually do have four legs because whether all dogs actually do have four legs is an empirical question, but whether it is part of the concept of being a dog that dogs have four legs is an analytic question.

Answer 1

Seems to me that you would enjoy Fines "the Logic of Essence".

Fine gives a formal language for talking about "truths in virtue of the nature of objects with certain properties". He does this by augmenting his logic with operators "square_F" for each predicate F. Informally, "square_F A" is "As truth is grounded in (objects with) the nature of F."

A couple of points off the top of my head:

(a): Fine holds particular views about essences (in particular non-modal). But this will be fine since

(b): your examples of a dog makes it seems like essences in the Finean sense are too strict. In particular, (going off a modal characterization of essence), a dog wouldn't contingently lose a characterstic essential to it.

(c): the strategy Fine adopts seems like it would fit the bill. Augment your language with operators that state that a property is "natural, likely, or expected". Then embed into a modal logic or defeasible logic to show how the operators interact with ones knowledge state.

Answer 2

If the goal is to model human reasoning, and that human reasoning involves making a presumption of existential quantification until an exception to the rule is found, than default logic serves the purpose of formalizing the mechanism from which by default existential quantification is applied; human reasoning is defeasible (SEP), and as such, logical formalisms have been devised to model it. Default logic is one such system. From WP:

Default logic can express facts like “by default, something is true”; by contrast, standard logic can only express that something is true or that something is false. This is a problem because reasoning often involves facts that are true in the majority of cases but not always. A classical example is: “birds typically fly”. This rule can be expressed in standard logic either by “all birds fly”, which is inconsistent with the fact that penguins do not fly, or by “all birds that are not penguins and not ostriches and ... fly”, which requires all exceptions to the rule to be specified. Default logic aims at formalizing inference rules like this one without explicitly mentioning all their exceptions.

Default logic is a type of non-monotonic logic, which a logic that does not obey monotonicity. From the Encyclopedia of Philosophy's entry Non-monotonic logic:

By contrast, in many instances of ordinary or ever-day reasoning, people arrive at conclusions only tentatively, based on partial or incomplete information, reserving the right to retract those conclusions should the learn new facts. Such reasoning is often called defeasible or non-monotonic, precisely because the set of accepted conclusions can become smaller when the set of premises is expanded.

One famous real-world example of this is the discovery that not all swans were white. Thus:

Swans are white.

Would be interpreted from natural language with universal quantification. Once a single instance of a black swan is discovered, then existential quantification would be the order of the day since the discovery of black swans defeats the original interpretation. In default logic, one uses default inference rules of the form A:B/C which is read as a A is known, and B is consistent with our knowledge, so we conclude C. Thus

Swans exist. : Swans are white. / All swans are white.
Dogs have legs : Dogs have four legs. / All dogs have four legs.

Of course, the 17th century discovery of black swans, or coming across a canine tripod as a child defeats the original reasoning.

Swans exist. : Swans are white and black. / Some swans are white.
Dogs have legs : Dogs have four or fewer legs. / Some dogs have four legs.

In this example, default logic is said to take a consistency-based approach to modeling reasoning because the second proposition, B, in A:B/C, is said to check our knowledge base for consistency. It should be noted that some approaches to defeasible reasoning attempt to minimize predicates, especially when dealing with a set of exceptions to a generalized premise. One such example is John McCarthy's circumscription logic.