# Socratic Paradox/Paradox 2

## Paradox

Socrates is supposed to have raised the question:

Is the following statement in italics true or false?

Suppose the statement in italics is true.

Then it is false.

That is, what it says is different from the way things actually are.

So, the sentence is not false.

Therefore it is true.

But it cannot be both.

We can more formally demonstrate this as follows:

Let $\map T P$ be a modal operator interpreted as "The proposition $P$ is true."

Assume $P \iff \map T P$, or in other words, that declaring the proposition $P$ is equivalent to declaring that $P$ is true.

Let $L$ be a proposition representing the liar sentence, and define $L$ as $\neg \map T L$.

We thus have by definition:

- $\neg \map T L \iff L$

and by assumption:

- $L \iff \map T L$

By Biconditional is Transitive, we have $\neg \map T L \iff \map T L$, a contradiction.

## Resolution

There are multiple proposed resolutions to this paradox, and there is no consensus as to which one to choose.

One resolution is due to Alfred Tarski's semantic theory of truth.

For any proposition $P$ in a language of truth, this theory requires that:

- $P$ is true if and only if $P$.

For instance:

- 'Snow is white' is true if and only if snow is white.

This is analogous to the principle stated above:

- $P \iff \map T P$

Tarski distinguishes object languages from metalanguages.

To talk about the truth of statements in language $A$, we must talk from the perspective of a higher language $B$.

The statement in language $A$ becomes an object of study, and not itself a vehicle with which to make truth assertions about statements in $A$.

Thus, the sentence "**Snow is white** is true if and only if snow is white" is composed of two languages:

- "
**Snow is white**" is an assertion in English

and:

- "... is true if and only if snow is white" is an assertion in some metalanguage.

Thus, the paradox's resolution is that the sentence:

- "this sentence is not true"

is meaningless, as the language within which the sentence is expressed serves as its own metalanguage.

In other words, the sentence makes an assertion of truth about itself, which is disallowed.

An issue with Tarski's resolution is that it imposes a severe limitation on our ability to coherently use the predicate "... is true".

For instance, consider the statement: "No language can talk about its own truth."

This statement expresses an assertion of truth about all languages, including itself.

This seemingly violates the very principle that Tarski uses to resolve the paradox.

## Also known as

This paradox is often seen referred to as the **liar paradox** or **liar's paradox**.

## Also see

## Source of Name

This entry was named for Socrates.

## Sources

- 1918: Bertrand Russell:
*The Philosophy of Logical Atomism*: 7. The Theory of Types and Symbolism: Classes - 1944: Eugene P. Northrop:
*Riddles in Mathematics*... (previous) ... (next): Chapter One: What is a Paradox? - 1964: Donald Kalish and Richard Montague:
*Logic: Techniques of Formal Reasoning*... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $(3)$ - 1979: Douglas R. Hofstadter:
*Gödel, Escher, Bach: an Eternal Golden Braid* - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**liar paradox** - 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*... (previous) ... (next): Chapter $1$: Introduction: $\S 1.1$: The origins of mathematical logic - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**liar paradox** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**liar paradox** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**liar paradox**