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Logic and rhetoric
A logical argument has a 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 absolutely guarantees the truth of the conclusion. Such an argument is valid. It is literally impossible for the premises of a valid argument to be true while the conclusion is false. No matter what other facts crop up, the premises imply the conclusion, thus a valid argument is a good deal more powerful than a merely strong one. What you're really after, though, is a sound argument: a sound argument combines validity with true premises. Since true premises guarantee a true conclusion in a valid argument, and the premises are true, the conclusion of a sound argument must be true as well.
(It's worth noting that although we call all sorts of things "valid" to mean they make good sense, in technical logical terms only a whole argument can be valid or invalid, not an individual statement. This makes sense because validity is a property of arguments and inferences, not of statements. On the other hand, we sometimes call arguments true or false. But, in logical terms, only individual statements, never whole arguments, are true or false. Truth and falsify come into play with respect to arguments when we consider the property of arguments called soundness. In short, an argument is sound if and only if it is (1) valid and (2) its premises are in fact true.)
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, many other things can be wrong with an argument, such as it having misleading premises or missing the point entirely. Such errors are informal. Valid deductive arguments can be constructed with entirely false premises. Such arguments have a solid logical structure and can make interesting hypothetical cases, or they can just be not even wrong.
Traditional (Aristolian) and propositional logic has as 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. 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 fuzzy logic, where a proposition has specific degrees of truth represented by real number values in [0,1]. However, fuzzy logic should not be confused with Bayesianism. Though fuzzy truth values and probability values are real numbers in [0,1] and both fuzzy logic and Bayesian reasoning are tools for inductive reasoning, fuzzy truth values are truth functional whereas probability values are not. By truth functional it is meant that the truth of compound logical statements like 'The ball is blue or it is orange' is determined by the truths of the atomic propositions 'The ball is blue' and 'The ball is orange' and the truth conditions of the logical operator (in this case the disjunction 'or'). To see the difference between fuzzy truth values and probability values, consider the following: Given a fair die, let A stand for 'you roll a 1, 2, or 3' and let B stand for 'you roll a 4, 5 or 6'. The Pr(A) = .5 and the Pr(A and A) = .5. However, while the Pr(A) = Pr(B) = .5, the Pr(A and B) = 0. On the other hand, in fuzzy logic, since it is truth functional, if A stands for 'The ball is blue' and B stands for 'The ball is orange' and the ball is exactly half blue and half orange, then the truth value of A = B = .5 and the truth value of 'A and B' = 1.
(These logical systems need not be in conflict, however. Fuzzy logic and Bayesian reasoning are tools for inductive reasoning, and the value they assign to a statement represents the confidence we should have in its truth, which is very different from its actual truth. Thus these systems assigning a partial value to a statement is compatible with the statement itself being simply true (or simply false) as traditional logic dictates.)
 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):
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 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[wp] "Schrödinger logics"
- 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)
- 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 intiutionism/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. (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.
 See also
- Logic, from Great Issues in Philosophy, by James Fieser (University of Tenessee at Martin)