Logic

Logic (λόγος in Greek, logos, “thought”) is the most fundamental of all the Sciences and a major branch of Philosophy. Logic is the primary proof and method of what supports Physics, Mathematics, and Language, leading to Arithmetic, Geometry, Set Theory, and Computation, as well as Grammar, Philology, Linguistics, and Philosophy of Language, and indeed, all Argumentation in general. Logic is not only the study of systems of reasoning, guides for how intelligent systems make valid Inferences and of how Statements and Numbers are put together - it literally IS that reasoning. The earliest studies of Logic in relation to Philosophy and Mathematics can be found in virtually all Ancient Civilizations in one form or another, from the Americas to Africa, to Europe, to Asia. Anything which can be thought through rationally is done so using Logic.

Despite such a daunting introduction, Logic can be understood as simply what makes good sense within a given cultural context. In traditional contexts, Logic is “Wisdom”, the knowledge of the World and Reality passed down to each generation. In the “Western” Philosophy style, Logic is a highly developed system of determining which Arguments are good and which arguments are bad, that is, which are Valid and which are Fallacies. This means Logic is also a system of understanding what Truth is, at least Truth as established by good arguments. We can therefore explore Natural Language usages or increasingly Formal Logic descriptions of arguments which are used to tie concepts together.

In short, the fundamental nature of any example of reasoning can be described by Logic, and this is what it means for something to be “logical”. Logic is a wide-ranging, most fundamental of all fundamental Science, but it can be simplified by understanding that all of us use Logic every single day constantly, though we may not be aware of formal structures. Beyond this, we can describe Deduction, which is studying what if anything follows logically from a set of given Premises, and also Induction, which is studying how we can infer from some number of observed events a reliable generalization or projection. As the basis for all of Science, Logic is used to define the structure of Number, Proposition, and Argument, which are then used to build Language and Mathematics. Logic is very powerful, indeed.

All Ancient cultures developed intricate systems of Reasoning and Math, which is what Logic is, but calling the study “Logic” and focusing on explicit methods of analysis received sustained development in three regions: India in the Sixth Century BC, China in the Fifth Century BC, and Greece, between the Fourth and First Centuries BC. Our Modern, formal treatment of Logic descends from that early Greek tradition, and mainly from Aristotle. Unfortunately, Indian and Chinese Ancient Logic did not survive into the Modern era. The Qin Dynasty in China repressed the tradition of scholarly investigation into Logic, following the Legalism of Han Feizi. In India, innovations in the Scholastic School called “Nyaya”, continued into the early Eighteenth Century AD, but did not survive the British Colonial Period. In the Islamic World, the rise of the Asharite School suppressed original work in Logic, despite the Arabs having invented the numerals (0, 1, 2, 3, etc) still used today. During the Medieval Period, after it was shown that Aristotle's ideas were largely compatible with Faith, a greater emphasis was placed on Aristotle's Logic, and it became a central focus of Medieval Philosophy, when ideas were developed on how to engage in critical logical analyses of philosophical arguments.

Syllogistic Logic

The Organon was Aristotle's body of work on Logic, along with the Prior Analytics constituting the first explicit work in Formal Logic, introducing the “Syllogism”. Also known as “Term Logic”, Syllogisms express inferences as a major and minor premise and a conclusion:


Aristotle's work was regarded, even in classical times, but certainly throughout the Middle Ages in Europe and the Middle East, as the very picture of a fully worked out system. The only major rival was the Stoic system of Propositional Logic studied by medieval logicians. Today, Aristotle's system is sometimes seen as little more than a historical study, as the more recent “Sentential Logic”' and “Predicate Calculus” are more widely employed. Others use Aristotle in Argumentation Theory to help develop and critically question argumentation schemes used in areas such as Artificial Intelligence and Law. In many contexts, Syllogistic Logic can seem like sillygisms, because they often do not provide much insight.

Modal Logic

With Language, “Modality” means that parts of a sentence can have the Semantics modified by special verbs, or “modal particles”. For example, “We study Logic” can be modified to “We should study Logic”, “We can study Logic”, or “We will study Logic”. More abstractly, we might say modality affects the circumstances in which we take an assertion to be satisfied. The connection with Logic again dates back to Aristotle, who was concerned with the Alethic modalities of Necessity and Possibility, which he observed to be dualistic:


These are related to De Morgan's Laws:


While the study of Necessity and Possibility remained an important commentary to philosophers throughout the Middle Ages, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1910, who formulated a family of rival axiomatisations of the Alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modalities treated to include Deontic Logic and Epistemic Logic. The seminal work of Arthur Prior applied the same formal language to treat Temporal Logic and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of Frame Semantics which revolutionized the formal technology available to modal logicians and gave a new Graph Theory method of looking at modality which has driven many applications in Computational Linguistics and Computer Science, such as Dynamic Logic.

Formal/Symbolic Logic

Formal Logic, or Symbolic Logic, is concerned primarily with the structure of Reasoning using Valid forms of Argument. Symbolic Logic deals with the relationships between concepts and provides a way to compose proofs of statements. Concepts are rigorously defined, and sentences are translated into a precise, compact, and unambiguous symbolic notation. For example, a statement which defines p as a true mathematical formula implies a given reverse formula holds as “q”:


Propositions can be combined using Logical Conjunction, Disjunction, or the Conditional as shown above. It essentially says “if p then q”. These are called Binary Operators, and they combine propositions into Compound Propositions. These proposition can be true or false, valid or invalid, when used to construct arguments by means of specific forms.

Predicate Logic

Gottlob Frege, in his Begriffsschrift, discovered a way to rearrange many sentences to make their logical form clear, to show how sentences relate to one another in certain respects. Frege's work started contemporary Symbolic Logic. Prior to Frege, Symbolic Logic had not been successful beyond the level of Sentential Logic. It could represent the structure of sentences composed of other sentences using such words as “and”, “or”, and “not,” but it could not break sentences down into smaller parts. Without “predicates”, Symbolic Logic could not show how that “cows are animals” entails “parts of cows are parts of animals.”

Frege expanded Sentential Logic to include words such as “all”, “some”, and “none”, and showed how we can introduce Variables and Quantifiers to arrange sentences:



Frege treated simple sentences without subject nouns as predicates, and applied them to “dummy objects” (x). The logical structure in discourse about Objects can then be operated on according to the rules of Sentential Logic, with some additional details for adding and removing Quantifiers. So, Frege added (1) the vocabulary of Quantifiers (∀, ∃) and Variables (x, y, etc), (2) a Semantics to explain that Variables denote individual Objects and Quantifiers have the force of “all” or “some” in Sets relating to those Objects, and (3) methods for using these in Language. To introduce '∀', you assume an arbitrary variable, x, prove something that must hold true of it, and then prove that it didn't matter which Instantiations you chose, for the proposition would have held true for all of them. An “All” quantifier can be removed by applying the sentence to any particular Object at all. '∃' can be added to a sentence true of any Object, or removed, in favor of a term about which you are not already presupposing any information. In other words, we can claim properties are true of all Objects in our given set, or show one or more exceptions. We can expand this to introduce “Propositional Functions” with “Domains of Discourse”:


To form a statement, one uses Universal or Existential Quantifiers to denote Objects in a Domain and Functions to be applied. In this way, Propositional Logic is extended into Set Theory, and Venn Diagrams are also useful to visualize Sets and Domains with their member Elements.

Philosophy of Logic

Philosophical Logic, or the Philosophy of Logic, is concerned with the elucidation of ideas such as Reference, Predication, Identity, Truth, Quantification, Existence, and others, and it has an equal concern with the connection between Natural Language and Logic. Such close connections are similarly found with Mentation, or Reason, studied in the Philosophy of Mind, Cognitive Psychology, and Neurological Science.

The bulk of “normal” reasoning in which we engage can be captured by Logic, if one can only find the right method (and enough time) to translate Ordinary Language into Logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard “Logics”, such as Free Logic, Tense Logic, and others, as well as extensions of Classical Logic including Modal Logic into non-standard Semantics for such Logics. For example, Saul Kripke developed a technique of Supervaluation in the Semantics of Logic.^[1]

Consistency, Completeness, Soundness

There are three valuable properties Formal Systems can have:

• Consistency, meaning the theorems of a system do not contradict.
• Completeness, meaning there are no true sentences in the system that cannot, at least in principle, be proved using the derivation rules (and axioms, if any) of the system.
• Soundness, meaning the system's rules of derivation will never let you infer anything false, so long as you start with only true premises.

So if a system is sound (and its axioms, if any, are true), then the theorems of a sound formal system are the Truths. All of the theorems of a system which has no axioms are its Truths, sometimes called “Logical Truths”. A Corollary: A system that is not consistent cannot be sound. Not all systems, whether thinking machines, natural languages, or other constructs, achieve all three virtues. It has been proven by Kurt Gödel that a system with enough axioms and/or rules of derivation to derive the principles of arithmetic, cannot be both consistent and complete at the same time.

Multi-Valued Logic(s)

The Logics discussed above are all “Bivalent” or “two-valued” Logics. The Semantics for each of them will assign to every sentence either the value “True” or the value “False.” Systems which do not always rely on this distinction are known as Multi-Valued Logics, or sometimes, “non-Aristotelian” Logics. In the early 20th century, Jan Lukasiewicz investigated the extension of the traditional true/false values to include a third value, “possible”. Logics, such as Fuzzy Logic, have since been devised with an infinite number of “degrees of truth”. Fuzzy values are represented by a Real Number between Zero and One. Bayesian Probability, for example, can be interpreted as a system of Logic where Probability itself is the Subjective Truth Value.

Mathematical Logic

Mathematical Logic is the use of Symbolic Logic to study mathematical reasoning. During the early Twentieth Century, philosophical logicians, starting with Gottlob Frege and Bertrand Russell, proved that Mathematics was “reducible” to Logic - in other words, that Logic was the “parent” of Math, that Logic was the more primary Language. This created such a firestorm among mathematicians and logicians that even now mathematicians of course deny it.^[2] At any rate, Logic is now accepted as an accurate and fundamental way to describe an Reasoning of any kind, including mathematical. Further, all Number Theory behind Mathematics requires Logic for its proof.^[3]

“Logicism”, also pioneered by Frege and Russell, was an outcome of this Reductibility, and showed that mathematical theories were logical tautologies. The road was rocky though, with Frege's Grundgesetze challenged by Russell's Paradox, and Hilbert's Program by Gödel's Incompleteness Theorem. What was required was the establishment of a new area of Philosophy of Logic in the application of Mathematics to Logic (and vice versa) using of Proof Theory. Despite the negative effect of the incompleteness theorems, the resulting Proof Theory and Model Theory, another new application, showed that every rigorously defined mathematical theory can be exactly captured by a first-order logical theory. Frege's Proof Calculus described the whole of Mathematics, though is not equivalent to it. Logic is more primary to Mathematics, though not its substitute, and through these developments, Mathematical Logic (or Philosophy of Logic) showed how complementary mathematical and logical areas of reasoning have actually been.

Proof Theory and Model Theory have not been the whole story, though, as Set Theory originated in the study of Infinite Sets by Georg Cantor. This has been the source Cantor's Theorem, the Axiom of Choice and the question of the independence of the Continuum Hypothesis, as well as the Modern debate on Large Cardinal Axioms. Further, Recursion Theory captured the idea of Computation in logical and arithmetical terms. The classical achievements here were the Undecidability of the Computability Problem by Alan Turing, and his presentation of the Church-Turing Thesis. Today, Recursion Theory is also concerned with Complexity Classes, in studying when a problem is Efficiently Solvable. This involves the Turing Degrees of Unsolvability.

Computation and Computability

Alan Turing's work on the Computability Problem followed from Kurt Gödel's work on the Incompleteness Theorems, and the notion of general purpose computers which came from all of this was of fundamental importance to the designers of early computer machinery in the 1940s. Throughout the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using Logic and mathematical notation, it would be possible to create a reasoning machine, an Artificial Intelligence. In Logic Programming, a program consists of a set of axioms and rules. Logic Programming systems compute the consequences of axioms and rules in order to answer a query. True Artificial Intelligence, though, has turned out to be a bit more difficult to achieve than expected, due to the more recently discovered complexities of human reasoning. Put simply, Computer Science as a discipline would be unthinkable without Logic.

Scholarship by M.R.M. Parrott

Synthetic A Priori: Philosophical Interviews Interviews, Discussion ©1998-1999 M.R.M. Parrott First Published: 99,00,02,08,11 Published by rimric press 0-9662635-6-1 | 978-0-9662635-6-5 232 Pages, Paperback & eBook, 2025 2025 Edition Extras: Both Prefaces, Notes on the Text and Cover Art Amazon Paperback (author) Barnes & Noble Paperback (author) Waterstones Paperback (author)

The Generation of 'X': Philosophical Essays 1991-1995 Academic Papers ©1991-1995 M.R.M. Parrott First Published: Oct 2002 Published by rimric press 0-9662635-0-2 | 978-0-9662635-0-3 160 Pages, Paperback & eBook, 2025 2025 Edition Extras: Afterword Amazon Paperback (author) Barnes & Noble Paperback (author) Waterstones Paperback (author)

References

1. Philosophy of Language is closely related, and has to do with the study of how our Language engages and interacts with our thinking, or vice versa..
2. This is partially evidenced by the Pseudopedia version of this page, strongly slanted toward the pro-mathematical view..
3. 1 + 1 = 2 is a mathematical proposition which requires logical proof, despite the fact that few would question it on face value..