Identity (philosophy) in the context of "Mathematical object"

Relations are ways in which several entities stand to each other. They usually connect distinct entities but some associate an entity with itself. The adicity of a relation is the number of entities it connects. The direction of a relation is the order in which the elements are related to each other. The converse of a relation carries the same information and has the opposite direction, like the contrast between "two is less than five" and "five is greater than two". Both relations and properties express features in reality with a key difference being that relations apply to several entities while properties belong to a single entity.

Many types of relations are discussed in the academic literature. Internal relations, like resemblance, depend only on the monadic properties of the relata. They contrast with external relations, like spatial relations, which express characteristics that go beyond what their relata are like. Formal relations, like identity, involve abstract and topic-neutral ideas while material relations, like loving, have concrete and substantial contents. Logical relations are relations between propositions while causal relations connect concrete events. Symmetric, transitive, and reflexive relations are distinguished by their structural features.

Panentheism (/pæˈnɛnθiɪzəm/; "all in God", from the Greek πᾶν, pân, 'all', ἐν, en, 'in' and Θεός, Theós, 'God') is the belief that the divine intersects every part of the universe and also extends beyond space and time. The term was coined by the German philosopher Karl Krause in 1828 (after reviewing Hindu scripture) to distinguish the ideas of Georg Wilhelm Friedrich Hegel (1770–1831) and Friedrich Wilhelm Joseph Schelling (1775–1854) about the relation of God and the universe from the supposed pantheism of Baruch Spinoza. Unlike pantheism, which holds that the divine and the universe are identical, panentheism maintains an ontological distinction between the divine and the non-divine and the significance of both.

In panentheism, the universal spirit is present everywhere, which at the same time "transcends" all things created. Whilst pantheism asserts that "all is God", panentheism claims that God is greater than the universe. Some versions of panentheism suggest that the universe is nothing more than the manifestation of God. In addition, some forms indicate that the universe is contained within God, like in the Kabbalistic concept of tzimtzum. Much of Hindu thought is highly characterized by panentheism and pantheism.

Pride is a human secondary emotion characterized by a sense of security with one's identity, performance, or accomplishments. It is often considered the opposite of shame and, depending on context, may be viewed as either a virtue or a vice. Typically, pride arises from praise, independent self-reflection, and/or a fulfilled feeling of belonging.

The word pride may refer to group identity. Manifestations include one's ethnicity. It is notably known for Black Pride, which gained historical momentum during the U.S. Civil Rights Movement. Then, it became known for independence struggles—Feminist Pride, rooted in the women's rights movement with gender equality struggles and sexual identity (for example, Gay Pride or LGBT Pride, rising in visibility following the Stonewall riots). In this context of minority groups, the display of pride is in defiance of people outside of the minority in question trying to instill them with a sense of shame. There is also the sense of pride that can accompany national identity (patriotism), regional identity, or other affiliations (e.g. proud to be a university alumnus). In this context, the pride is more literal.

Essence (Latin: essentia) has various meanings and uses for different thinkers and in different contexts. It is used in philosophy and theology as a designation for the property or set of properties or attributes that make an entity the entity it is or, expressed negatively, without which it would lose its identity. Essence is contrasted with accident, which is a property or attribute the entity has accidentally or contingently, but upon which its identity does not depend.

Logic translation is the process of representing a text in the formal language of a logical system. If the original text is formulated in ordinary language then the term natural language formalization is often used. An example is the translation of the English sentence "some men are bald" into first-order logic as ${\displaystyle \exists x(M(x)\land B(x))}$. The purpose is to reveal the logical structure of arguments. This makes it possible to use the precise rules of formal logic to assess whether these arguments are correct. It can also guide reasoning by arriving at new conclusions.

Many of the difficulties of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in the sciences and humanities. It may play a key role for logic in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use of informal logic is an alternative to formalization since it analyzes the cogency of ordinary language arguments in their original form. Natural language formalization is distinguished from logic translations that convert formulas from one logical system into another, for example, from modal logic to first-order logic. This form of logic translation is specifically relevant for logic programming and metalogic.

An accident (Greek συμβεβηκός), in metaphysics and philosophy, is a property that the entity or substance has contingently, without which the substance can still retain its identity. An accident does not affect its essence, according to many philosophers. It does not mean an "accident" as used in common speech, a chance incident, normally harmful. Examples of accidents are color, taste, movement, and stagnation. Accident is contrasted with essence: a designation for the property or set of properties that make an entity or substance what it fundamentally is, and which it has by necessity, and without which it loses its identity.

Aristotle made a distinction between the essential and accidental properties of a thing. Thomas Aquinas and other Catholic theologians have employed the Aristotelian concepts of substance and accident in articulating the theology of the Eucharist, particularly the transubstantiation of bread and wine into body and blood.