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The definition of a proposition is as follows

A proposition is a declarative statement that is either true or false but not both

Let us categorize declarative statement in to four types

Type 1: True declarative statements

Ex: 2 + 3 = 5

Type 2: False declarative statement

Ex: 2 * 3 = 5

Type 3: Dual declarative statement (both true and false)

Ex: x + 2 = 5

Type 4: Declarative statement that is neither true nor false

My doubt is whether type 4 exists or not? Does type 4 comes under paradox? Are both(type 4 and paradox) same? Are the following paradoxes come under type 4?

This statement is false.

I know that i know nothing

If not, provide an example for type 4?

hanugm
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    Type 3 does not exist; see your def: " a declarative statement that is either true or false but not both". – Mauro ALLEGRANZA Jan 26 '18 at 07:33
  • True, In question, the example I wrote under type 3 has to be under type 4. – hanugm Jan 26 '18 at 07:48
  • I suggest replacing "a declarative statement that is either true or false but not both" with "a well-formed formula in a logical system with no free variables"... – Derek Elkins left SE Jan 26 '18 at 09:00
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    It seems like you're saying a statement is true when it's true for some assignment of free variables, and false when it's false for some assignment of free variables. Using this definition, a statement in 1st-order logic must be either true or false or both. – Ibrahim Tencer Jan 26 '18 at 16:32

1 Answers1

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There are many examples of such statements. Generally, ill-defined statements fall under the category of: 'neither true nor false'.

For example, consider this statement: $$\text{This ice-cream is tasty} $$ This cannot be true or false without an appropriate definition of "tasty" as applied to "the icecream." And yes, your first example statement also falls under this category.

Also, see here, and these previous MSE questions:

  • How "This ice-cream is tasty" doesn't come under type 3? x+2 = 5 and its truth value depends on x. Here taste can be considered as x, which can vary in a set of people? – hanugm Jan 26 '18 at 07:32
  • See @Mauro's comment, sir. Those type of statements are not propositions. –  Jan 26 '18 at 07:39