# categorization of logic

(1). I was wondering about what are the relation and differences between formal and informal logic?

What topics does each of them have? For example, topics such as Meaning and Definition, Syllogistic Logic, Inductive Logic, Probabilistic/Statistical Reasoning, Deductive Logic

Is symbolic logic same as formal logic?

(2) what is the relation between informal/formal logic and deductive/non-deductive reasoning/inference?

must formal logic be only for deduction, not for non-deductive reasoning/inference?

(3) what are other usual/better ways to categorize various logic topics?

Thanks and regards!

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Informal logic is, well, informal. As such it is not unambiguously defined what the subject matter is, how different statements are related to each other, or (more ambitiously) what statements "mean". This type of activity is more popular in philosophy, linguistics and other areas that are not specifically mathematical. Modern interest in informal reasoning is often about how to formalize it, as in AI, natural language processing or data mining.

Formalized logic is as described in the other answers. Essentially, there are many types of formal logic, and each one is a game with its own well-defined rules (in principle, performable by a computer) for taking a set of statements -- informally thought of as "premises" -- and adding to it some additional statements, informally thought of as "conclusions". In spirit it is very close to computer programming and its study as a field resembles computer science. Some branches such as model theory or set theory are studied purely as mathematics in the sense that the explicitly combinatorial/linguistic aspect is in the background and the sentences are studied in a "semantic" approach that speaks in terms of the structures, objects and their properties (the material supposedly described by the logic) rather than the permissible syntactic operations in a formal logical system.

Mathematics is intermediate between these two extremes. It uses logic(s) that could be called potentially (or presumably, or in-principle) formalizable. Steps of the game are not codified precisely enough to be programmed on a computer, but they are standardized and sufficiently well-defined to correspond to known genuine moves in some actual formal systems, so there is a presumption that any correct non-formalized proof is robust enough to have many slightly different expansions into a very detailed formal proof in several of the formal systems suggested as a "foundation" for mathematics.

You mentioned probabilistic or statistical reasoning. This is a somewhat separate question because any deterministic setting can be enlarged a probabilistic one (e.g., instead of measuring attributes or their absence by $1$ or $0$, allow any numerical value between $1$ and $0$) and the logical rules modified accordingly in a known way. Similarly, any setting that uses ordinary or probabilistic logic on objects thought of as "data" can be placed in the context of statistics, where one considers how the data might have been generated in addition to the data itself, and the rules for reasoning and speaking about this richer picture can again be updated in a routine way. But these probabilistic/fuzzy/statistical enrichments are a kind of fixed recipe where the "logic" and "probabilistic" ingredients are cooked separately and mixed in an understood way. You could discuss the result as a new form of logic but really it is a repackaging of what you started with, not a fundamentally different mode of reasoning.

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Unfortunately I'm not versed in philosophical logic, so let's start with some "different categories of logic".

### Scope

The first relevant category of logic is scope, what is the logic talking about. Propositional logic talks about nonparametric propositions. A typical rule (modus ponens) allows you to derive a proposition B given that you know A and that A implies B.

Predicate logic talks about parametric propositions, for example "X is a poster". We can derive "There's X such that X is a poster" from "Tim is a poster", whereas in propositional logic "Tim is a poster" is monolithic and cannot be broken up to "Tim" and "X is a poster".

Predicate logic is also known as first-order logic. If we quantify over predicates then we get Second order logic. This allows us to use concepts such as "There's a predicate P such that P(X) iff X=Tim".

The scope is enlarged in a different direction by Modal logic, where we can talk about events (propositions) eventually happening, happening until some other event happens, and so on.

### Rules

The rules of logic people are usually taught constitute Classical logic. In classical logic you're allowed to use proof by contradiction. However, some mathematicians (well, logicians) dislike such rules since they're non-constructive (this is like set-theorists who consider what happens when you're not allowed to use AC, the axiom of choice).

By modifying (restricting) the rules, we arrive at Intuitionistic logic (or Constructive mathematics), in which every proposition that we prove has a constructive proof from the givens. In particular, you can't use a proof of the sort "if X, then Y; if not X, then also Y" unless you can decide whether X is in fact true. The resulting world is nice - for example all functions are continuous. On the other hand, not every theorem you can prove in classical logic is true in the constructive sense, although surprisingly many are (for example, the fundamental theorem of algebra is still provable).

In a different direction, in Infinitary logic the proof is infinite and there are logical rules with infinitely many premises. For example, you could have a rule deducing "P(n) for every natural n" from the (countably) many propositions P(0), P(1), ... In general these logics are not well-behaved, but if you choose the parameters correctly, they are (look for $L_{\omega_1,\omega}$).

### Goals

Different areas of logic have different goals. For example, in Structural Proof Theory one goal is to understand how simple a proof system can be made, and what are the consequences. Using this syntactic approach, you can show that certain statements are not provable in certain systems by converting them into a very simple form, bounding the resulting size of the simple proof, and show that it is too short to prove the given statement.

A complementary direction is Model Theory, which is all about what a given set of axioms describes. You can prove for example that a set of first-order axioms cannot characterize the size of the universe (if it's infinite). The model-theoretic approach to showing that some theorem is unprovable is to exhibit a model of the axioms in which the theorem isn't true.

In computer science, people are interested in Proof Complexity, which studies how long it takes to prove statements if you're only allowed to use basic means. The holy grail is proving a statement which is even stronger than $P \neq NP$. A different sub-field constructs systems in which every predicate proven to exist is efficiently computable.

### Categorical logic

Unfortunately I don't know anything about categorical logic. Apparently there are some kinds of categories (in the sense of category theory) which describe "universes of set theory" along with their associated logic, which is often constructive. These things are called Topoi (singular Topos).

### Philosophers' logic

Philosophers are interested in different aspects of logic. Unfortunately, I don't know much about these, so I'll only give some sporadic examples.

There is some interest in nice, succinct systems of logic whose axioms are "as natural as possible". In mathematical logic, by and large people aren't interested in the exact form of the axioms but in what they describe and what can be proven about the system using them.

Other interest is in what the allowable rules of logic should be. I described classical logic and intuitionistic logic. There are many systems in between (and even systems weaker than the latter), which might be natural given some philosophical stance. Most mathematicians don't even bother with intuitionistic logic, although some logicians (and even people from other branches) are intrigued by it.

There's also some interest in making sense of paradoxes - given a statement which is paradoxical (for example, the Liar paradox), what should its truth value be? The mathematical solution is to avoid self-reference in some systematic way, since mathematicians are more interested (in this case) in the well-being of the system rather than these "monstrous examples".

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@ Yuval: When you say FOL axioms "cannot characterize the size of the universe (if it's infinite)" is this to suggest that FOL cannot distinguish between one infinite cardinal and another (where the appropriate cardinal is the cardinal of the model, which comes to the cardinal of the universe of objects being interpreted)? And I'm guessing that this is a result of the upward l.s. Correct? – لويس العرب Jun 11 '13 at 1:51

Any formal system consists of:

1. a list of symbols (a vocabulary or alphabet),
2. a list of of typographical rules to connect these symbols to write statements (a grammar or syntax), and
3. a list typographical rules for deriving new statements from other statements (axioms or rules of inference).

By typographical rules, I mean rules that tell you only how to manipulate the symbols without reference to any meaning or interpretation that these symbols might have. Consider for example a rule of inference in propositional logic that I call the Join Rule:

If X and Y are true statements, then so is "(X) & (Y)".

Notice that I make no reference in the statement of this rule to any possible meaning of the "&" symbol. (You can probably guess that it is the AND-operator.)

You can play around with my own implementation of formal propositional logic using a freeware program I have developed, available at my website http://www.dcproof.com

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