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⭐ Core Definition: Paradoxes

A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites".

In logic, many paradoxes exist that are invalid arguments, yet are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. Others, such as Curry's paradox, cannot be easily resolved by making foundational changes in a logical system.

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👉 Paradoxes in the context of Paradoxes of material implication

The paradoxes of material implication are a group of classically true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated with English words such as "implies" or "if ... then ...". They are sometimes phrased as arguments, since they are easily turned into arguments with modus ponens: if it is true that "if $P$ then $Q$ " ( $P\rightarrow Q$ ), then from that together with $P$ , one may argue for $Q$ . Among them are the following:

A material conditional formula $P\rightarrow Q$ is true unless $P$ is true and $Q$ is false; it is synonymous with "either P is false, or Q is true, or both". This gives rise to vacuous truths such as, "if 2+2=5, then this Wikipedia article is accurate", which is true regardless of the contents of this article, because the antecedent is false. Given that such problematic consequences follow from an extremely popular and widely accepted model of reasoning, namely the material implication in classical logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning.

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⭐ Core Definition: Paradoxes
👉 Paradoxes in the context of Paradoxes of material implication
Paradoxes in the context of Declarative sentence
Paradoxes in the context of Ineffability

Paradoxes in the context of Declarative sentence

Propositions are the meanings of declarative sentences, objects of beliefs, and bearers of truth values. They explain how different sentences, like the English "Snow is white" and the German "Schnee ist weiß", can have identical meaning by expressing the same proposition. Similarly, they ground the fact that different people can share a belief by being directed at the same content. True propositions describe the world as it is, while false ones fail to do so. Researchers distinguish types of propositions by their informational content and mode of assertion, such as the contrasts between affirmative and negative propositions, between universal and existential propositions, and between categorical and conditional propositions.

Many theories of the nature and roles of propositions have been proposed. Realists argue that propositions form part of reality, a view rejected by anti-realists. Non-reductive realists understand propositions as a unique kind of entity, whereas reductive realists analyze them in terms of other entities. One proposal sees them as sets of possible worlds, reflecting the idea that understanding a proposition involves grasping the circumstances under which it would be true. A different suggestion focuses on the individuals and concepts to which a proposition refers, defining propositions as structured entities composed of these constituents. Other accounts characterize propositions as specific kinds of properties, relations, or states of affairs. Philosophers also debate whether propositions are abstract objects outside space and time, psychological entities dependent on mental activity, or linguistic entities grounded in language. Paradoxes challenge the different theories of propositions, such as the liar's paradox. The study of propositions has its roots in ancient philosophy, with influential contributions from Aristotle and the Stoics, and later from William of Ockham, Gottlob Frege, and Bertrand Russell.

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Paradoxes in the context of Ineffability

Ineffability is the quality of something that surpasses the capacity of language to express it, often being in the form of a taboo or incomprehensible term. This property is commonly associated with philosophy, theology, aspects of existence, and similar concepts that are inherently "too great", complex or abstract to be communicated adequately. Illogical statements, principles, reasons and arguments may be considered intrinsically ineffable along with impossibilities, contradictions and paradoxes.

An object, event or concept is ineffable if it cannot adequately be expressed by the use of natural language.

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Paradoxes in the context of "Paradoxes of material implication"

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⭐ Core Definition: Paradoxes

👉 Paradoxes in the context of Paradoxes of material implication

Paradoxes in the context of Declarative sentence

Paradoxes in the context of Ineffability