The Liar's Paradox; Aristotle's Laws of Thought: Laws of Non-Contradiction (LNC), Excluded Middle (LEM); Plus, the Law of Bivalence (LBi)
Let: X := "This statement (X) is false".
QUESTIONS to Consider:
Q1. What is a proposition, i.e., what is the logical definition of a proposition? Is X a proposition?
Q2. Is it possible for a proposition to be both true and false?
Q3. Is it possible for a proposition to be neither true nor false, but some middle option between true and false or some otherwise third option besides true and false?
Q4. Can a proposition only bear one truth value, that (single) truth value being either true or false?
ANALYSIS OF THE LIAR'S PARADOX!
If it is true that 'X is false', then X is false, because X stands for: "This statement (X) is false". Therefore, if it is true that 'this statement (X) is false', then X is false.
If it is false that 'X is false', then X is true, because X states that 'this statement (X) is false'. Therefore, if it is false that 'this statement (X) is false', then X is true.
If it is both true and false that 'X is false', then two analyses exist:
If it is false 'that X is both true and false (together)', then X must be the other options besides being both 'true and false', namely that X is either true or false or neither true nor false.
On the other hand, if it is both true and false that 'X is false' then the following conjunction must hold: it is true that 'X is false' and it is false that 'X is false'. To the extent that X is true, it is true that 'X is false' implies 'X is false', and to the extent that X is false, it is false that 'X is false' implies 'X is true'. Therefore X can be both true and false, as was the very assumption that lead us to this.
Questions to consider:
- A1. Can X be both "neither true nor false" and "true"?
The joint denial, the neither-nor option, excludes the case that 'X is true' as well as the case that 'X is false', i.e. 'X is not true' and 'X is not false'. So, the conjunction 'X is true' and 'X is not true and X is not false' logically implies that 'X is not false', in which case 'X is true'. Here, another question arises: how can something be both true and not true (where not true is implied by the neither-nor option)?
- A2. Can X be "neither true nor false" and "false"?
The neither-nor option that states "X is neither true nor false" can be restated 'X is not true and X is not false'. Conjoining this statement with the statement that 'X is false' yields that X is both 'not false' and 'false' (a contradiction arises), which in its turn yields that 'X is not true' (i.e., X is false). Neither true nor false denies that X can be either true or false, therefore X cannot be false (a contradiction arises). How can a proposition X be both 'true' and 'not true' (i.e. false)? How can a proposition X be both 'false' and 'not false'? A contradiction arises yet again!
The Laws of Non-Contradiction, Excluded Middle, and Bivalence (i.e., the very definition of a proposition)
Questions to consider here: Is it logically permissible for a proposition to both be true and false? How about neither be true nor false? Let us examine both of these questions, making use of the rudimentary and foundational laws of classical logic. Let us analyze the definition of a proposition also.
There are four distinct options here for a given proposition X:
- [i]. X
- [ii]. ~X
- [iii]. Both X and ~X
- [iv]. Neither X nor ~X
which translates to...
- X is true
- X is false
- X is both true and false
- X is neither true nor false
where [i] = 1, [ii] = 2, [iii] = 3, [iv] = 4
Options 3 and [iii] are logically impermissible in classical bivalent (two-valued) logic.
The law of non-contradiction (henceforth, LNC) states: "Nothing can both be and not be," (where: "nothing" = no thing = not a thing). This can be restated as follows: "Something cannot both be and not be." That is, a proposition X and its negation ~X cannot both be true together: it cannot be the case that 'both X and ~X are true'. This is the Law of Non-Contradiction.
Therefore, the logical conjunction of a proposition (X) with its negation (~X) is excluded by logic i.e., made logically impermissible by the law of non-contradiction (LNC).
The law of non-contradiction also logically entails that any given proposition X cannot be both true and false (simultaneously, at the same time, in the same sense).
It is entailed by the syntax of the law of non-contradiction: ~[X & ~X], that states that it cannot be the case that the conjunction [X & ~X] is true (i.e., "contradictions cannot be"). The form ~[X & ~X] is a true formula for any X.
The law of non-contradiction thus makes contradictions logically impermissible, where contradiction arises where X and ~X are both true together, and not when X and ~X are both false together! A contradiction is the the joint affirmation of X and ~X, where both X and ~X are taken to be true together. In other words, the option that states X and ~X are both true is excluded by the Law of Non-Contradiction, which makes contradictions logically impermissible.
Options (iv) and 4 are logically impermissible in classical bivalent (T,F) logic!
The law of excluded middle (LEM): X V ~X, where V = inclusive disjunction = "or", as opposed to an exclusive disjunction (+) = "xor", which excludes the joint affirmation of X and ~X, whereas the "or" operator includes the contradiction (X & ~X), because an inclusive disjunction takes either of the disjuncts or the conjunction of the disjuncts.
The negation of the inclusive-either-or option is the joint denial of X and ~X, which states neither X is true nor ~X is true; that is, the law of excluded middle logically excludes the option in which both X and ~X are both false together!
Either a proposition X is true or its negation ~X is true or both X and ~X are true together.
The logical complement of the inclusive-or ("or") is the neither-nor option (nor)
LEM excludes the joint denial (the "neither-nor" option). Therefore, LEM can be reformulated as stating that "X and ~X cannot both be false together!" Either X is true or ~X is true, and it cannot be neither, but it can be both! That is, by LEM, at least one of X and ~X has to be true, including the option that X and ~X are true together, but excludes the option where X and ~X are both false together!
LEM permits the contradiction: ('X is true' AND '~X is true'); this contradiction is only restricted by non-contradiction, not by excluded middle!
The law of bivalence logically excludes both the joint affirmation and the joint denial
A proposition is defined to obey the law of bivalence:
Proposition X can bear only one truth value, that truth value being true or false; where "or" is to be understood as an exclusive disjunction. Therefore the law of bivalence can be reformulated as stating "A proposition X is either true or false; where "or" is to be understood as an exclusive disjunction.
The law of bivalence is the logical conjunction of excluded middle (LEM) and non-contradiction (LNC). The law of bivalence states EXACTLY ONE of (X, ~X) is true, and the other false. The law of bivalence logically includes both the option that 'X is true and its negation ~X is false' as well as the option that 'X is false, and its negation ~X true'. The law of bivalence logically excludes both the option that 'both X and ~X are true' as well as the option that 'neither X nor ~X is true'.
Therefore, bivalence (LBi) states: [X (+) ~X] = [X xor ~X], where (+) = xor = exclusive disjunction = exclusive-either-or; whereas, LEM states [X V ~X] = [X or ~X]. Not a trivial distinction.
Therefore LBi states that: "something is not neither or both what it is and what it is not".
BY the law of bivalence: (X xor ~X) yields: either X or ~X is true, and it cannot be both, and it cannot be neither!
Please Note the Important Differences Between the Laws of Excluded Middle and Bivalence!
- Note that the law of excluded middle (LEM) uses the operator "or"
(inclusive disjunction), while the law of bivalence (LBi) uses "xor"
- Further note that LEM (excluded middle) contains a negation ("not" =
"~") in its formula; whereas, LBi (bivalence) does not include a
negation in its expression!
- Moreover note that LEM is only expressed in terms of the truth value
of 'true' and does not include 'false'! Bivalence, on the other hand,
is mathematically expressed in terms of 'true' and 'false' and does
not include ("not" = "~") as a connective.
- In LEM, the negation operator ("not" = "~") serves as a logical
connective, not as a truth function! Whereas, LBi (bivalence) is a
principle about negation ("not" = "~") as a truth function!
- Therefore, LEM is a syntactical principle of logic, while LBi is a
semantical principle of logic.
Bivalence states that a truth variable X, i.e., a proposition ("truth-bearer"), can only carry one truth value at a time, that (single) truth value being either "true" or "false"; where or is to be understood as an exclusive disjunction, which logically excludes the conjunction of the contradictory disjuncts X and its negation ~X.
The Law of Bivalence is the Conjunction of the Laws of Excluded Middle and Non-Contradiction!
LEM makes the joint denial (the "neither-nor" option) logically impermissible.
LNC makes the joint affirmation (the "both-and" option) logically impermissible.
Therefore, LBi - being the conjunction of LEM and LNC - makes both the joint denial and the joint affirmation impermissible!