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# Logic

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 “”Logic, my dear Zoe, merely enables one to be wrong with authority. —The Doctor, Doctor Who ("The Wheel in Space")

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.[1]

Logic is a useful guide to thinking, as it is neutral to properties of things and focuses only on their relationships and what that implies. It is easy when examining a matter to get distracted by what pleases you about it or by the positive social effects believing in a statement's truth might have. Logic abstracts from contents that would make one think that way and can therefore direct our thoughts and others in a more helpful direction.

A logical argument has а conclusion which follows from its premises. Arguments come in two types, deductive and inductive.

In a good inductive argument, the truth of the premises renders the conclusion likely, though not certain. Such an argument is described as strong. But further evidence could be added which would weaken an inductive argument so that even if the premises were true, the conclusion would no longer be likely.

In a good deductive argument, the truth of the premises guarantees the truth of the conclusion.[2] Such an argument is valid. It is impossible for the conclusion of a valid argument to be false if the premises are true. No matter what other facts crop up, the premises entail the conclusion; a valid argument is thus a good deal more powerful than a merely inductively strong one. What you're really after, though, is a sound argument: a valid argument with premises that actually are true. Since true premises guarantee a true conclusion in a valid argument, and the premises are true, the conclusion of a sound argument is true as well.[note 1]

The validity of an argument is determined by its structure. Where the argument structure breaks down is known as a formal logical fallacy. Of course, arguments can have other problems, like false or misleading premises. Within particular contexts, where some particular issue is under scrutiny, arguments might also fall short by missing the point. Such errors are informal, and are sometimes colloquially referred to as informal fallacies. Importantly, even arguments with no true premises can be deductively valid. Such arguments have a solid logical structure and can make interesting hypothetical cases, or they can just be not even wrong.

Traditional (Aristotelian) and classical logic share an operating assumption that all statements which aren't nonsense are either true or false. For instance, 2 + 2 = 4 is true, 3 - 7 = 84.6 is false. Various non-classical systems of logic include further possible truth values for statements. 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". Other extensions suggest that there are (technically) infinite states, as in fuzzy logic, where a proposition has specific degrees of truth represented by real number values between 0 and 1 (inclusive). However, fuzzy logic should not be confused with probability. Though fuzzy truth values and probability values are real numbers in [0,1], and both fuzzy logic and Bayesianism can be used as tools for inductive reasoning, fuzzy logic employs truth functional connectives, whereas probability does not. To say that a connective is truth functional means that the truth values of compound statements employing the connective, like 'or' in 'The ball is blue or it is orange' are functions of the truth values of their component propositions (here, 'The ball is blue' and 'The ball is orange'). For any given connective, the function is always the same. In contrast, the probability of a disjunction P(A or B) is not a straightforward function of the probabilities of its disjuncts P(A) and P(B); it also depends on whether A and B are independent.[note 2]

## 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, and 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 written like this (where p and q are variables ranging over propositions):

${\displaystyle p\rightarrow q}$
${\displaystyle p}$
${\displaystyle \therefore q}$

Formal logic is also known as symbolic logic or mathematical logic.

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 calculus, 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, strictly, concerned with the notions of necessity and possibility, though the term is often used in a broader sense.[3]
• 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 not true, 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:

• non-reflexive logic: allows for violations of or restrictions on the law of identity, such as Newton da Costa's "Schrödinger logics"[4]
• substructural logic: permits fewer 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)[5]
• linear logic: a system of logic based on the idea of constrained resources (a type of substructural logic)
• paracomplete logic: denies or restricts the law of the excluded middle (every statement must be either true or not true); the chief example is intuitionistic logic, which is inspired by the mathematical movements of intuitionism/constructivism
• many-valued logic: denies the principle of bivalence (every statement is either true or false); distinct from paracomplete logics, as many-valued logics can still validate the Law of the Excluded Middle
• paraconsistent logic: rejects the principle of explosion; permits valid reasoning from contradictory premises.[6] (All relevance logics are paraconsistent, but not all paraconsistent logics are relevant)
• infinitary logic: whereas classical logic only permits finite-length propositions and finite-length proofs, infinitary logic allows propositions and proofs of infinite length
• quantum logic: a system of logic used to reason about quantum mechanical systems

## The constitution of logic

The study of logic tries to relate formal logic to natural language argumentation. This has led to an old classification of the activities of the justification into parts, some of which are:

• semantics: The validity of an argument depends upon the meaning or semantics of the sentences that make it up
• inference: The account of how one moves from premises to conclusions in formal and natural language argumentation
• logical form: The identification of the kinds of inference used in argumentation and their representation in formal logic

These activities have remained part of logic since the times of Aristotle's Organon, although their nature has changed during the various revolutions that have happened in the subject.

## Reason and rhetoric

Outside the academy, arguments are rarely presented in a way that can be readily formalized. This is usually because a formalized rendition makes for poor natural language, and can require stating many things considered "obvious". While there are advantages to natural language, a well-crafted presentation can obscure weak points in an argument, including both dubious premises and questionable inferences. 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 presented with skillful rhetoric, fallacious inferences can appear reasonable or be disguised. Technical jargon, for instance, can be used to conceal weak reasoning from a non-expert audience, whose members likely cannot assess the accuracy of the usage. Lots of websites are guilty of this when it comes to science, donning the metaphorical white coat to dress up woo arguments as though they are grounded in sound scientific fact.

## Using what logic teaches

While it is often difficult to directly analyze arguments using formal techniques, it is worth the effort to try from time to time. This effort has the double reward of clarifying well and 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.