The Philosophy of Logic is a branch of Philosophy which deals with the scope and nature of Logic, and investigates philosophical problems raised by Logic, such as presuppositions often implicitly at work in theories and applications. It involves questions about how Logic is defined and how different logical systems are connected to each other, as well as to other disciplines. “Metalogic” is closely related to the Philosophy of Logic as the discipline investigating the properties of formal logical systems, like Consistency and Completeness. Also closely related are the Philosophy of Mathematics, as concerned with the nature of numbers and mathematical objects, and the Philosophy of Language, concerned with the meanings of words and relationship to Truth, Syntax, and Semantics. Of the three, Philosophy of Logic is the most primary or fundamental.

Many logical formal systems, or Logics, have been developed, mainly during the Twentieth Century, and Philosophy of Logic is used to classify them, to show how they are related to each other, and to address the problem of how there can be a manifold of Logics in contrast to one Logic. These Logics can be divided into Classical Logic, along with First-Order Logic, Modal and Temporal Logics, and Deviant Logics. There is even a “Metaphysics of Logic” speciality, so-named for investigating the metaphysical status of the Laws and Objects of Logic, but outside Academia, this is just part of Philosophy of Logic (as are many areas academics which to over-specialize in).

The “Meta” of Logic

As one might imagine, there is a great deal of commentary about what Logic is, what the relationships are between Logic, Language, and Mathematics, which of the sub-areas and sub-sub-areas are more important than the others, and so on. It is important to realize that Logic is not only a major branch of Philosophy, but a primary study period, so Metalogic, Philosophy of Mathematics, Mathematical Logic, Metaphysics of Logic, Philosophy of Language, Linguistics, and other specialized areas are all quite overlapped, sometimes coterminous as different names for the same thing, and the act of separating them out as if they were different fields is, well, “academic”, busywork, confusing, and unnecessary. Logic and Metalogic, or Logic and Philosophy of Logic, are perhaps the best overall terms to use for what is a large field of inquiry. But let's be clear: It's all Logic, and Philosophy.

Principally, Logic studies the Relation between a Premise and its Conclusion, because they represent not only the Event and Reality which we experience, but our thoughts about those events and realities. Whether based on valid inferences or logical truths, we are concerned with the Discipline and Investigation of the “Laws of Thought” (which is a potentially loaded term). Logic is not a strictly empirical or psychological discipline recording regularities found in actual thinking, but is one focused on the “Laws of Reasoning”, a science of valid argumentation, an objective expression of subjective reasoning. How normative a discipline Logic is depends upon how far we want to press the point that the Laws of Reasoning Logic investigates determine how we should think, such that violating them is irrational. A Softer stance is that Logic investigates the relations between propositions, and does not actually proscribe “correct reasoning”. Still, arguments that fail the Laws of Reasoning are fallacies. Formal fallacies are within the scope of Formal Logic whereas Informal Fallacies belong to Informal Reasoning.

Metalogicians sometimes hold that Completeness is a necessary requirement of a logical systems, and a formal system is complete if it is possible to derive from its axioms every theorem belonging to it. This would mean only those formal systems which are complete should be understood as constituting logical systems. One controversial argument for this approach is that incomplete systems cannot be fully formalized. So, First-Order Logic constitutes a complete logical system, but higher-order “Logics” are not, due to their incompleteness. Thus, it is the task of Logic to provide a general account of the difference between correct and incorrect inferences, or a set of premises together with a conclusion. An inference is valid if the conclusion follows from the premises, that is, if the truth of the premises ensures the truth of the conclusion.

For a very long time throughout History, Aristotelian syllogisms were considered the canon of Logic. Few substantial improvements arrived until two thousand years later with the works of George Boole, Bernard Bolzano, Franz Brentano, Gottlob Frege, and others. Their developments were often driven by a need to increase the expressive flexibility of Logic and to adapt it to specific areas of usage. The contemporary proliferation of logical systems encourages the question of how the systems are related to each other, and why deserve the title “Logic”. Should we have one Universal Logic, or a collection of multiple Logics? Is some “Grand Unification” needed, hopefully one which would avoid the failures of such attempts in Science? Perhaps a “Universal” or common Concept of Logic underlies and unifies different logical systems. One analogy is with our communicative languages. French, English, and Japanese are all very different languages, but few people fret over how to unify them. We simply accept that they express similar yet different mental concepts for their native speakers.

Conceptualism

Conceptualism is a theory that explains any universal nature of a Particular is a conceptualized framework situated within the thinking Mind. This is different from seeing universal mental acts corresponding with universal intentional objects, as well as the perspective that dismisses the existence of universals outside the mind. Consider the “Ship of Theseus” problem, which asks about Identity over a period of Time. If all parts of an Object are replaced, does the Object remain the same? With Conceptualism one claims the Identity is not an innate property, but a conceptual structure applied by the Mind.

Realism

Realism holds that logical entities exist independently of the human Mind. Thus humans do not invent Logic or Mathematics, but rather discover them, and any other intelligent beings in the Universe would presumably do the same. In this point of view, there is really one one sort of Logic and Mathematics that can be discovered: Triangles, for example, are real entities, not our creations.

Platonism

Platonism is the form of Realism suggesting logical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the naive view most people have of numbers. The term Platonism is used because such a view is seen to parallel Plato's belief in a “World of Ideas”, an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato's view partly came from Pythagoras, and his followers the Pythagoreans, who believed that the world was, quite literally, built up by the numbers.

Logicism

Logicism is the provable thesis that Mathematics is reducible to Logic, and hence a branch of Logic (Carnap 1931/1883, 41). Logicists hold that Mathematics can be known a priori, but suggest that our knowledge of Mathematics is just part of our knowledge of Logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, Logic is the proper foundation of Mathematics, and all mathematical statements are necessary logical truths.

Empiricism

Empiricism is a form of Realism that denies Logic and Mathematics can be known a priori at all. It says we discover mathematical facts by empirical research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th Century, but primarily arose in the middle of the century. This argues statements like “2 + 2 = 4” come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.

Formalism

Formalism holds that logical and mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the “game” of Euclidean Geometry (which is seen as consisting of some strings called “axioms”, and some “rules of inference” to generate new strings from given ones), one can prove that the Pythagorean Theorem holds (that is, you can generate the string corresponding to the Pythagorean Theorem). Mathematical truths are not about numbers and sets and triangles, in fact, they aren't “about” anything at all. Thus, Formalism need not mean that Logic or Mathematics are nothing more than meaningless symbolic games. It is usually hoped that there exists some interpretation in which the rules of the game hold.

Intuitionism

Intuitionism is a program of methodological reform whose motto is that “there are no non-experienced mathematical truths” (L.E.J. Brouwer). From this springboard, intuitionists seek to reconstruct what they consider to be the “corrigible” portion of Mathematics in accordance with concepts of constructibility, provability, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects. (CDP, 542)

Constructivism

Like Intuitionism, Constructivism involves the regulative principle that only logical entities which can be explicitly constructed in a certain sense should be admitted to discourse. In this view, Logic and Mathematics are an exercise of the Human Intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction.

Fictionalism

Fictionalism was introduced in 1980 when Hartry Field rejected and in fact reversed Quine's indispensability argument, that Mathematics was indispensable for our best scientific theories, and therefore should be accepted as true. Field suggested that Mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian Mechanics that didn't reference numbers or functions at all. He started with the “betweenness” axioms of Hilbert's Geometry to characterize Space without coordinatizing it, and then added extra relations between points to do the work formerly done with Vector Fields.