Truth values in the context of Platonic philosophy

Platonism is the philosophy of Plato and philosophical systems closely derived from it, though later and contemporary Platonists do not necessarily accept all of Plato's own doctrines. Platonism has had a profound effect on Western thought. At the most fundamental level, Platonism affirms the existence of abstract objects, which are asserted to exist in a third realm distinct from both the sensible external world and from the internal world of consciousness, and is the opposite of nominalism. This can apply to properties, types, propositions, meanings, numbers, sets, truth values, and so on (see abstract object theory). Philosophers who affirm the existence of abstract objects are sometimes called Platonists; those who deny their existence are sometimes called nominalists. The terms "Platonism" and "nominalism" also have established senses in the history of philosophy. They denote positions that have little to do with the modern notion of an abstract object.

In a narrower sense, the term might indicate the doctrine of Platonic realism, a form of mysticism. The central concept of Platonism, a distinction essential to the Theory of Forms, is the distinction between the reality which is perceptible but unintelligible, associated with the flux of Heraclitus and studied by the likes of physical science, and the reality which is imperceptible but intelligible, associated with the unchanging being of Parmenides and studied by the likes of mathematics. Geometry was the main motivation of Plato, and this also shows the influence of Pythagoras. The Forms are typically described in dialogues such as the Phaedo, Symposium and Republic as perfect archetypes of which objects in the everyday world are imperfect copies. Aristotle's Third Man Argument is its most famous criticism in antiquity.

Platonism is the philosophy of Plato and philosophical systems closely derived from it, considered the opposite of nominalism, or anti-realism. Platonism has had a profound influence on Western thought. Platonism or Platonic realism affirms the real existence of forms or abstract objects, originally to solve the problem of universals. Abstract objects are asserted to exist in a third realm distinct from both the sensible external world and from the internal world of consciousness. This can apply to properties, types, propositions, meanings, numbers, sets, truth values, and so on (see abstract object theory).

Plato's doctrine originally was an attempt to reconcile the reality which is perceptible but unintelligible, associated with the flux of Heraclitus and studied by the likes of physical science, and the reality which is imperceptible but intelligible, associated with the unchanging being of Parmenides and studied by the likes of mathematics. Geometry was the main motivation of Plato, and this also shows the influence of Pythagoras. The Forms are typically described in dialogues such as the Phaedo, Symposium and Republic as perfect archetypes of which objects in the everyday world are imperfect copies. Aristotle's Third Man Argument is its most famous criticism in antiquity.

Logical equality is a logical operator that compares two truth values, or more generally, two formulas, such that it gives the value True if both arguments have the same truth value, and False if they are different. In the case where formulas have free variables, we say two formulas are equal when their truth values are equal for all possible resolutions of free variables. It corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus.

It is customary practice in various applications, if not always technically precise, to indicate the operation of logical equality on the logical operands x and y by any of the following forms: