# Logic

In 1869, as women were not granted much access to laboratories and observatories, Christine Ladd-Franklin turned to mathematics and logic, which did not require any apparatus. Credit: Smithsonian Institution.
 Completion status: Been started, but most of the work is still to be done.

Logic is more than reasoning. Usually it is reasoning conducted or assessed according to strict principles of validity. Aristotelian logic is a particular system or codification of the principles of proof and inference.

 Educational level: this is a secondary education resource.

At a secondary level an introduction to logic may be helpful, where some of the more common operators are described. This introduction is a part of elementary logic at the undergraduate level. Here, there is at least one lesson available.

 Educational level: this is a tertiary (university) resource.

This learning resource is partly an article, in some subareas an essay, and mostly a lecture.

 Educational level: this is a research resource.

Logic is often considered a part of philosophy. And, most often is used in science to help create knowledge consisting of facts and truths. But, it finds needed applicability in law and the practice of law. A third popular field that confers a rigid structure on logic is mathematics.

 Resource type: this resource is an article.

Nearly all efforts, intellectual or otherwise, can be approached and have some understanding produced through the application of logic. This includes volition (e.g., emotion), affections, morality, and religion.

 Resource type: this resource contains a lecture or lecture notes.
 Subject classification: this is a law resource .
 Subject classification: this is a science resource .
 Subject classification: this is a mathematics resource .

## Notation

Notation: let the symbol Def. indicate that a definition is following.

Notation: let the symbols between [ and ] be replacement for that portion of a quoted text.

## Universals

To help with definitions, their meanings and intents, there is the learning resource theory of definition.

“[D]efinitions are always of symbols, for only symbols have meanings for definitions to explain.”[1] A term can be one or more of a set of symbols such as words, phrases, letter designations, or any already used symbol or new symbol.

In the theory of definition, “the symbol being defined is called the definiendum, and the symbol or set of symbols used to explain the meaning of the definiendum is called the definiens.”[1] “The definiens is not the meaning of the definiendum, but another symbol or group of symbols which, according to the definition, has the same meaning as the definiendum.”[1]

Def.

1.a(1): "a science that deals with the canons and criteria of validity of inference and demonstration : the science of the normative formal principles of reasoning"
(2): "a branch of semiotic; [especially: syntactics]"
(3): "the formal principles of a branch of knowledge"
b: "a particular mode of reasoning"
c: "interrelation or sequence of facts or events when seen as inevitable or predictable"

is called logic.[2]

Similar to the above dictionary, or lexical, definition is

Def. "[l]ogic is the study of correct argumentation."[3]

Def. "[a] method of human thought that involves thinking in a linear, step-by-step manner about how a problem can be solved"[4] is called logic.

Def. evidence that demonstrates that a concept is possible is called proof of concept.

The proof-of-concept structure consists of

1. background,
2. procedures,
3. findings, and
4. interpretation.[5]

The findings demonstrate a statistically systematic change from the status quo or the control group.

## Analogical reasoning

Def. "a representational mapping from a known "source" domain into a novel "target" domain"[6] is called analogy.

"In problem solving and learning, analogical reasoning promises to overcome the explosive search complexity of finding solutions to novel problems or inducing generalized knowledge from experience."[6]

Def. "familiar [mapped] elements or relations from the source into unfamiliar (or unknown) elements or relations in the target" are called analogical inferences.[6]

"Source, target, mapping, analogical inference, and confirmatory support [a broad spectrum of empirical evidence] are the basic materials of analogy."[6]

## Computer logic

Computer logic is a system of principles behind the arrangements of elements in a computer or electronic device for performing a specified task.

Def. "a system that provides algorithms for the symbolic manipulation of first-order formulas over some temporarily fixed language and theory"[7] is called a computer logic system.

"The aim of logic in computer science is to develop languages to model the situations [encountered], in such a way that we can reason about them formally. Reasoning about situations means constructing arguments about them; we want to do this formally, so that the arguments are valid and can be defended rigorously, or executed on a machine."[8]

## Deduction

Def. "[a] process of reasoning that moves from the general to the specific, in which a conclusion follows necessarily from the premises presented, so that the conclusion cannot be false if the premises are true"[9] is called deduction.

Def. "inference in which the conclusion cannot be false given that the premises are true", or "Inference in which the conclusion is of no greater generality than the premises"[10] is called deductive reasoning.

"Deductive reasoning, also called deductive logic, is the process of reasoning from one or more general statements regarding what is known to reach a logically certain conclusion.[11]"[12]

"The theory of deduction is intended to explain the relationship between premisses and conclusion of a valid argument and to provide techniques for the appraisal of deductive arguments"[1].

## Dialectics

"Dialectic (also dialectics and the dialectical method) is a method of argument for resolving disagreement ... The dialectical method is dialogue between two or more people holding different points of view about a subject, who wish to establish the truth of the matter by dialogue, with reasoned arguments.[13] Dialectics is different from debate, wherein the debaters are committed to their points of view, and mean to win the debate, either by persuading the opponent, proving their argument correct, or proving the opponent's argument incorrect — thus, either a judge or a jury must decide who wins the debate. Dialectics is also different from rhetoric, wherein the speaker uses logos, pathos, or ethos to persuade listeners to take their side of the argument."[14]

## Induction

Def. "the derivation of general principles from specific instances" is called induction, from Wiktionary.

## Inference

Beginning the Wikipedia article on inference is "Inference is the act or process of deriving logical conclusions from premises known or assumed to be true.[15]"

## Logical calculus

"[A]n abstract logical calculus [consists of] "the vocabulary of logic, ... the primitive symbols ..., and the logical structure ... fixed by stating the axioms or postulates ... in terms of its primitive symbols."[16]

## Logic-based abduction

"In logic, explanation is done from a logical theory $T$ representing a domain and a set of observations $O$. Abduction is the process of deriving a set of explanations of $O$ according to $T$ and picking out one of those explanations."[17] "[T]o abduce $a$ [$a$$O$] from $b$ [$b$$T$] involves determining that $a$ is sufficient (or nearly sufficient), but not necessary, for $b$."[17]

"[T]o discover is simply to expedite an event that would occur sooner or later, if we had not troubled ourselves to make the discovery. Consequently, the art of discovery is purely a question of economics. The economics of research is, so far as logic is concerned, the leading doctrine with reference to the art of discovery. Consequently, the conduct of abduction, which is chiefly a question of heuretic and is the first question of heuretic, is to be governed by economical considerations."[18]

## Mathematical logic

In line with Boolean algebra which is a logical calculus is Boolean logic.

## Natural deduction

Per the article about natural deduction on Wikipedia: "In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning."

## Principle

"A principle is a law or rule that has to be, or usually is to be followed, or can be desirably followed, or is an inevitable consequence of something, such as the laws observed in nature or the way that a system is constructed. The principles of such a system are understood by its users as the essential characteristics of the system, or reflecting system's designed purpose, and the effective operation or use of which would be impossible if any one of the principles was to be ignored.[19]" from the Wikipedia article principle.

## Propositional logic

Propositional logic uses or may result in declarative sentences.

## Reasoning

Logic can also mean the quality of being justifiable by reason.

Def. "[t]he deduction of inferences or interpretations from premises" is called reasoning, from Wiktionary.

Another definition of reasoning may be

Def. "the drawing of inferences or conclusions through the use of" "statement[s] offered in explanation or justification" is called reasoning.[2]

## Sophistry

Def. "[a]n argument that seems plausible, but is fallacious or misleading, especially one devised deliberately to be so" is called sophistry, from Wiktionary.

## Symbolic logic

The systematic use of symbolic techniques to determine the forms of valid deductive argument may be deductive symbolic logic.

## Validity

Def. "the quality of state of" "having a conclusion correctly derived from premises" is called validity.[2]

A sequent, e.g. ϕ₁, ϕ₂, ϕ₃, … ⊢ Ψ, is valid when a proof for it can be found[20].

An argument is a formula of the kind PremicesConclusion and it is valid when for each interpretation under which the premises are all true, the conclusion is also true, or, in other words, when Premices ∧ ¬Conclusion = false.

This is also related with semantic entailment, e.g. ϕ₁, ϕ₂, ϕ₃, … ⊨ Ψ, which is a relation ⊨ that holds if Ψ evaluates to true whenever all formulas ϕ₁, ϕ₂, ϕ₃, … are evaluated to true.

Equivalently, a formula is defined as valid when it is true in every interpretation (is a tautology (logic)). To see this, it might be worth to rewrite ϕ₁, ϕ₂, ϕ₃, … ⊨ Ψ as its equivalent ⊨ ϕ₁∧ϕ₂∧ϕ₃∧… → Ψ.

A weaker concept, when formula can be true (but not necessary in all interpretations), is called satisfability. Valid formula is also satisfable but note vice-verse. However, negation relates the concepts more tightly: formula ϕ is satisfable iff ¬ϕ is not valid.

## References

1. Irving M. Copi (1955). Introduction to Logic. New York: The MacMillan Company. pp. 472.
2. Philip B. Gove, ed (1963). Webster's Seventh New Collegiate Dictionary. Springfield, Massachusetts: G. & C. Merriam Company. pp. 1221.
3. 72.174.74.68 (December 16, 2006). "Historical Introduction to Philosophy/Philosophical Method". Wikiversity: 1. Retrieved on 2011-11-29.
4. (July 7, 2012) "logic". Wiktionary. San Francisco, California: Wikimedia Foundation, Inc. Retrieved on 2012-07-30.
5. Ginger Lehrman and Ian B Hogue, Sarah Palmer, Cheryl Jennings, Celsa A Spina, Ann Wiegand, Alan L Landay, Robert W Coombs, Douglas D Richman, John W Mellors, John M Coffin, Ronald J Bosch, David M Margolis (August 13, 2005). "Depletion of latent HIV-1 infection in vivo: a proof-of-concept study". Lancet 366 (9485): 549-55. doi:10.1016/S0140-6736(05)67098-5. Retrieved on 2012-05-09.
6. Rogers P. Hall (May 1989). "Computational approaches to analogical reasoning: A comparative analysis". Artificial Intelligence 39 (1): 39-120. doi:10.1.1.94.7301. Retrieved on 2012-07-30.
7. Andreas Dolzmann, Thomas Sturm (June 1997). "Redlog: Computer algebra meets computer logic". ACM SIGSAM Bulletin 31 (2): 2-9. doi:10.1145/261320.261324. Retrieved on 2012-07-30.
8. Michael Huth and Mark Ryan (August 26, 2004). Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge, United Kingdom: Cambridge University Press. pp. 427. ISBN 0 521 54310 X. Retrieved 2012-07-30.
9. (April 11, 2012) "deduction". Wiktionary. San Francisco, California: Wikimedia Foundation, Inc. Retrieved on 2012-07-30.
10. (May 21, 2012) "deductive reasoning". Wiktionary. San Francisco, California: Wikimedia Foundation, Inc. Retrieved on 2012-07-30.
11. R. J. Sternberg (2009). Cognitive Psychology. Belmont, CA: Wadsworth. pp. 578. ISBN 978-0-495-50629-4.
12. (June 27, 2012) "Deductive reasoning". Wikipedia. San Francisco, California: Wikimedia Foundation, Inc. Retrieved on 2012-07-30.
13. The Republic (Plato), 348b
14. (July 15, 2012) "Dialectic". Wikipedia. San Francisco, California: Wikimedia Foundation, Inc. Retrieved on 2012-07-30.
15. http://www.thefreedictionary.com/inference
16. Patrick Suppes (1967). Sidney Morgenbesser. ed. What is a scientific theory? In: Philosophy of Science Today. New York: Basic Books, Inc.. pp. 55-67.
17. (July 28, 2012) "Abductive reasoning". Wikipedia. San Francisco, California: Wikimedia Foundation, Inc. Retrieved on 2012-07-30.
18. Peirce, C.S. (1902), application to the Carnegie Institution, see MS L75.329-330, from Draft D of Memoir 27
19. Alpa, Guido (1994). "General Principles of Law". Annual Survey of International & Comparative Law 1 (1, Article 2).
20. Logic in Computer Science: Modelling and Reasoning