# Logic

(Redirected from Formal logic)
 Part of the series on Logic Key articles General logic Bad logic v - t - e

Logic is the formal study, and use, of the interrelationship between statements in order to determine whether arguments yield useful, coherent and correct results, or bullshit. It is achieved through the ritual of kolinar and most often used by long-lived, prosperous individuals.

A logical argument is one that follows from its premises. A correctly constructed logical argument should be self-evidently true because each logical step is unarguable, and therefore the conclusion follows from the premise. In certain circumstances, this is considered more reliable than mere demonstration of a conclusion. For example, the mathematical sum 23,005 x 233,290 can be calculated by constructing a grid that is 23,005 rows by 233,290 columns and counting the cells formed or it can be derived from various rules of multiplication, derived from far simpler axioms such as 1 x 0 = 0 and 1 + 0 = 1.[1]

Where the argument structure breaks down is known as a formal logical fallacy. Where conclusions can be incorrect due to faulty premises or other argumentation flaws, the fallacy is said to be informal. Where a correct logical conclusion is made from premises known to be false, it's termed not even wrong, as the argument fails the logical test that would describe it as either true or false.

Traditional (Aristolian) logic poses that statements can have one of two states: true, or false. For instance, 2 + 2 = 4 is true, 3-7 = 84.6 is false. Extensions to logic include further possible values for a statement. Extending it like this isn't entirely as ludicrous as it sounds (contrast with paraconsistent logic); for example, three-valued logic poses three states of "true", "false" and "unknown". Further extensions suggest that there are (technically) infinite states, such as observed in probability theory and fuzzy logic, where a proposition has a specific likelihood of being true. This is seen frequently in Bayesian rationality.

## Formal logic

In formal logic, any natural language used in an argument is reduced to abstract symbolism, with the results looking pretty much like equations in algebra or set theory. At its core, logic is the process of boiling down statements into pieces so that each individual step is unobjectionable. Indeed, looking at a single logical step, one might be forgiven for thinking logic is nothing more than stating the obvious, so has no practical use! Yet on another level, that is exactly what it is - each step is unobjectionable, but when placed together we can derive far more complicated ideas and know that they're right because each little jump is "obvious". This abstraction allows the clear and concise analysis of the content of the argument - i.e., not getting bogged down in things like "well it depends on what the definition of 'is' is".

A simple example would be modus ponens, which at a formal level is read like this (where p and q are variables ranging over propositions):

$p \rightarrow q$
$p$
$\therefore q$

Formal logic is also known as symbolic logic or mathematical logic. It forms part of mathematics, and is often considered the foundational discipline upon which the rest of mathematics can be built.

Formal logic is not a single system, but rather many, with competing and contrary principles; the discipline concerns itself with studying the properties of these different logical systems, both as an end-in-itself (pure mathematics), but also to try to find which formal system best reflects our pre-existing intuitive ideas of what is "logical".

Logical systems can be distinguished on the basis of which types of statements they concern themselves with:

• propositional calculus is concerned with the relationships between propositions, but not the internal structure of those propositions
• predicate calculus breaks propositions down into subject and predicate, and provides quantifiers (all, some). It is broken up into first-order predicate calculus, which can assert that entities have properties, but cannot talk about those assertions or properties themselves; and higher-order predicate caclulus, which enables assertions to be made about propositions and predicates.
• type theory extends predicate calculus with the notion that entities belong to certain types; restrictions are imposed on what can be said about entities of different types, to avoid paradoxes such as Russell's paradox
• modal logic is concerned with the notions of necessity and possibility.
• temporal logic formalizes temporal statements, and provides past, present and future tense (and aspect also)

There is one particular approach to logic which is known as classical, since it is the most popular approach, and the one which is generally presented first in textbooks. This approach is based on certain assumptions, such as the law of the excluded middle (everything is either true or false, but not neither) and the law of non-contradiction (nothing can be both true and false simultaneously). Non-classical logics question some of the assumptions of classical logic:

• substructural logic: permits less rules of inference than those permitted in classical propositional calculus
• relevance logic: attempts to better model our informal ideas of implication, by insisting the premise must be relevant to the conclusion (a type of substructural logic)
• linear logic: a system of logic based on the idea of constrained resources (a type of substructural logic)
• intuitionistic logic: denies the law of the excluded middle (everything must be true or false); inspired by the mathematical movements of intiutionism/constructivism
• paraconsistent logic: rejects the law of non-contradiction; permits valid reasoning from contradictory premises. (All relevance logics are paraconsistent, but not all paraconistent logics are relevant)
• infinitary logic: whereas classical logic only permits finite-length propositions and finite-length proofs, inifinitary logic allows propositions and proofs of infinite length
• quantum logic: a system of logic used to reason about quantum mechanical systems

## Reason and rhetoric

Rarely are arguments outside of formal logic classes presented in a way that can be readily abstracted. This is usually because a formalized rendition makes for poor natural language, and often would require stating many things considered "obvious". The dangers come when logical fallacies sneak their way in, disguised by the way natural languages are conjugated and expressed, and when the "obvious" assumptions that are only implied or taken for granted are themselves false, or at least debatable. The study of logic without formalisms is known as informal logic.

When good arguments are assembled into high-quality rhetorical speech, they form robust and even brilliant presentations. When poor arguments are disguised by translation into rhetoric, they can lead many people astray.

## Using what logic teaches

While it is often difficult to directly analyze arguments using formal techniques, it is worth the effort to at least try from time to time. This effort has the double reward of clarifying or refuting well or poorly constructed arguments, and reminding one how to construct a good argument oneself. A high quality argument could literally be footnoted or deconstructed in an appendix, expressing every element it contains at a formal level.

Si vous voulez cet article en français, il peut être trouvé à Logique.