Schematic analysis of excluded middle
: the absurdity of being between being and not-being
The relationship between the law of identity and the law of excluded middle can be formulated as follows:
1. ∀x(x=x)
2. → ¬∃x(¬x=x)
3. → ∀x¬∃y(¬(y=x).¬(y=¬x)) → ∀x¬∃y(y=¬x.y=x)
4. → ∀x(¬x∨x)
The law of identity (1-2) implies that, in the existential domain of x, there can be no third term that is neither x nor not-x (3), which is the sense of the law of excluded middle, since the third term implies the identity of x and not-x, therefore non-identity of x . The law of identity implies the law of excluded middle.
The following is an explanation of how, in the above formula, step 2 implies step 3, and therefore 4.
The existential domain at step 3 consists of x and y, that is, only x and y ‘may exist’. Since ‘identical’ (signified by the symbol =) means not-different, and ‘different’ means not-identical, the range of logical possibilities of how x and y could co-exist consists of either a) x and y being different, or b) x and y being identical.
By inspection of (a) in the illustration above, everything that is not y is identical to x, and everything that is not x is identical to y. The fragment ¬(y=x) from step 3 therefore implies ¬y=x and y=¬x. In other words, by excluding/negating a half of the domain (for example, y), only the other half remains (x), as there are no other logical possibilities remaining. Conversely, the fragment ¬(y=¬x) from step 3, which means that y is not identical to what is left when x is excluded, leaves only one possibility: the exclusion of x is logically equivalent to the exclusion of y, therefore y is identical to x, which is (b).
We thus obtain y=¬x from (a) and y=x from (b), and consolidating the identity of y over the two possibilities yields x=¬x, which is non-identity of x: a violation of the law of identity, therefore there is no such y.
Moreover, from the same fragments, y implies x and ¬x: a violation of the law of non-contradiction, therefore there is no such y.
The law of identity and the law of non-contradiction both imply the law of excluded middle.
Visual representation of the existential domain for a combination of terms allows us to distinguish the possibilities that are conceivable as reality from those that are inconceivable as reality, such as that there is a possibility of ‘being between being and not-being’, which is a string of words that does not make a meaningful whole, has no integrated sense, and is therefore what we call non-sense. One way of challenging the sense of the ‘non-classical’ schools of logic is to ask them to produce an image of the existential domain of a dog that is neither a dog nor not a dog, or a dog that is both a dog and not a dog, or a dog that is not a dog. The challenge can also be presented in a way that makes no reference to symbolic logic: draw a picture of a dog that neither exists nor does not exist (where the term ‘exist(s)’ refers to the ‘picture of a dog’.
Logical equivalence of the three laws can also be formulated as follows, which works in both directions:
∀x(¬x∨x) :excluded middle
→ ∀x¬∃y(x∨y∨¬x)
→ ∀x¬∃y(y→(¬x∧¬¬x))
→ ∀x¬(x∧¬x) :non-contradiction
→ ∀x¬(T(x)=T(¬x))
→ ∀x(T(x)=T(x))
→ ∀x(x=x) :identity
The function T(…) has two possible values: True, for the terms asserted in conjunction, and False, for the negation of True.
The argument needs shoring up, a little. Implicit assumption, leap from 3 to 4, and assumption of completeness. Let me try to explain. The argument—particularly in its transition from Step 3 to Step 4—reveals areas where its structure may benefit from further refinement. Kindly, but critically I share on the points where its reasoning could be strengthened.
Step 3 asserts: ∀x ¬∃y (¬(y = x) ∧ ¬(y = ¬x)). Here, the argument seeks to exclude the existence of any y that is neither x nor ¬x. This, prima facie, appears reasonable within a classical framework. Yet, a more careful inspection raises several questions.
Firstly, one wonders: what precisely is the nature of y? Is y intended to signify another entity, a potential state, or a truth value? The argument does not specify. If y is an entity distinct from x, the claim appears to exclude 'third things' in the existential domain of x. But if y is a truth value, the claim is more nuanced, requiring us to assume the domain of truth values is restricted to classical bivalence—true and false, with no intermediate values.
This ambiguity, though subtle, is significant. Without explicitly clarifying what y represents, the claim risks being interpreted differently by different audiences, particularly those inclined toward non-classical frameworks where y might inhabit a 'middle' or indeterminate state.
Step 4, the conclusion that: ∀x (¬x ∨ x), presents the Law of Excluded Middle, which assures us that every proposition is either true or false. While this principle is axiomatic in classical logic, its derivation from Step 3 is less self-evident.
Step 3 excludes the possibility of y being 'between' x and ¬x, yet this alone does not necessarily demonstrate that x is either true or false. Step 3 pertains to the existential domain—what y may or may not be. Step 4, however, *is a claim about propositional truth*—a subtle but critical distinction that may be behind the absurdity you mention. The derivation, as presented, seems to conflate existentialism (statement) and a proposition.
It would be most helpful if the argument explicitly bridged this conceptual divide. For example, one might introduce a principle asserting the exhaustiveness of x and ¬x in all possible scenarios. This additional premise would transform the argument from an implicit leap into a well-supported chain of reasoning.
We can also recognize that not all logic conforms to the classical mold. In other systems, such as paraconsistent or multi-valued logics, the claim of Step 4 is far from inevitable.
For instance: In paraconsistent logic, x and ¬x might both be true without trivializing the system. In multi-valued logics, truth values could inhabit a continuum, rendering x and ¬x neither mutually exclusive nor exhaustive. In constructive logic, the principle ¬x ∨ x holds only when one can constructively prove either x or ¬x.
The original argument would benefit from acknowledging these alternatives, not to adopt them, but to distinguish its conclusions as contingent upon the classical framework it employs.
To strengthen the argument while preserving its intent, three refinements may be considered: First, clearly define whether y represents an entity, a state, or a truth value. This removes ambiguity and ensures the argument is understood in its proper context. Second, provide an explicit justification for how excluding 'third terms' in the domain of y translates into the propositional truth of x ∨ ¬x. Introducing a principle of exhaustiveness or mutual exclusivity would solidify this connection. Third, briefly acknowledge that the argument’s conclusions rely on classical assumptions. I hope this helps!
PS There is also an implicit, unstated assumption about simultaneity of state - see Dialetheic logic... if it can oscillate between two states, we can say it is both, but we should be clear on the domain of logic we are using.
I asked GPT to explain it to me like I'm 12. (Is this correct?) Here's what it said:
The "Law of Excluded Middle" is a basic idea in logic that says every statement is either true or false—there's no middle option. For example, the statement "It is raining" is either true (if it's raining) or false (if it's not raining). There's no third choice where it's both raining and not raining at the same time.
This principle helps us think clearly and make decisions based on whether statements are true or false. However, some thinkers argue that not all situations fit neatly into this true-or-false framework. For instance, in complex scenarios or certain philosophical discussions, a statement might not be easily classified as entirely true or entirely false. In such cases, the Law of Excluded Middle might not apply perfectly.
Understanding this law is important because it forms the foundation of logical reasoning, helping us analyze arguments and solve problems by determining the truthfulness of statements.