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.
In symbolic logic, symbols can mean anything whatsoever. The only constraint is that the symbol must mean the same thing if used more than once under the same “existential quantifier”, such as ‘there is’ or ‘for all’. Y can be X, or it could be not-X, or a language sentence, proposition, a truth value, or a dog that neither exists nor does not exist, or nothing. Whatever y may be is not to be presumed, apart from the logical relations we seek to prove or disprove. The formula at 3 tells us only that x and y exist as terms of the same kind and can be related in some way, and 3 is formulated in a way that is allegedly logically equivalent to 2. The rationale is that, if we assume the law of identity (1) then 2 is necessarily follows, and if 3 (which implies the law of excluded middle, from what we know about the law of excluded middle) is logically equivalent to 2 then the law of identity implies the law of excluded middle. This article was entirely dedicated to explaining this equivalence, since it is not obvious.
The step 4 is implied by the assumption in 3 that there is no such y that violates the law of excluded middle.
The statements of the law of identity or excluded middle do not require the concept of truth values, and are not expressed in terms of truth values, although sometimes it is useful to introduce the concept to prove some higher order relation.
The schematic representation attempts to clarify the equivalence between Y being neither x nor not-x, and Y being identical to not-x and not-not-x, therefore the non-identity of x to itself.
Non-classical logics are either classical (but misrepresent their classical commitments), or are logically inconsistent (imply contradiction) in every case where they genuinely deny any of the ‘classical’ laws (but they invent spurious distinctions to conceal their non-sense). If the laws are logically translatable, equivalent, then any purported attempt to deny one law amounts to denying all of them, so if the ‘non-classical’ schools deny any of the laws but accept others, they are already contradicting themselves. The subject-matter in the present article is the equivalence of the law of identity and excluded middle, in order to show that the latter cannot be denied without also denying the former.
EDIT: my motivation for the visual explanation of the equivalence between ¬(y=x) and y=¬x was caused by the intuitive (but logically not equivalent) comparison with: ‘dog is not a cat’ does not imply that dog is identical to not-cat (something that is not-cat could be a tree). There is no analogy with the case of excluded middle because the existential domain of cats and dogs includes also trees (which are neither cats nor dogs), whereas excluded middle applies ‘between’ dog and not-dog, cat and not-cat etc. I needed this to show the equivalence of excluded middle and identity. The proof from non-contradiction was already done here: https://michaelkowalik.substack.com/p/proof-of-the-law-of-excluded-middle
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.
GPT conflates excluded middle with non-contradiction: “There's no third choice where it's both raining and not raining at the same time.” GPT is wrong, always.
The Law of Excluded Middle is a principle in logic stating that for any statement, either that statement is true, or its opposite is true—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 neither true nor false.
On the other hand, the Law of Non-Contradiction asserts that a statement and its opposite cannot both be true at the same time. So, "It is raining" and "It is not raining" cannot both be true simultaneously.
In summary, the Law of Excluded Middle ensures that every statement is either true or false, while the Law of Non-Contradiction ensures that a statement and its opposite cannot both be true at the same time.
Better. It would be appropriate to now ask what does “true” and “false” mean? The explanation that “it is raining” is true “if it is raining” is logically circular, analogous to ‘it is true if it is true’ leaving it unclear what “true” could mean.
I fail to understand - are you suggesting the actual, phenomenological impossibility of this four pronged statement about the dog (also called the catuscoti in eastern thought), merely its logical impossibility?
Nevertheless, you have the option of presenting the existential domain of the 3 cases you seem to refer to (I am not sure what the number 4 refers to; the challenge consist of 3 cases, since there are 3 laws).
Oh i actually misread the last sentence, i thought you had added a fourth term of “both a dog and a dog” and i guess my mind filled it in. What specifically do you mean when you say “existential domain”?
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.
In symbolic logic, symbols can mean anything whatsoever. The only constraint is that the symbol must mean the same thing if used more than once under the same “existential quantifier”, such as ‘there is’ or ‘for all’. Y can be X, or it could be not-X, or a language sentence, proposition, a truth value, or a dog that neither exists nor does not exist, or nothing. Whatever y may be is not to be presumed, apart from the logical relations we seek to prove or disprove. The formula at 3 tells us only that x and y exist as terms of the same kind and can be related in some way, and 3 is formulated in a way that is allegedly logically equivalent to 2. The rationale is that, if we assume the law of identity (1) then 2 is necessarily follows, and if 3 (which implies the law of excluded middle, from what we know about the law of excluded middle) is logically equivalent to 2 then the law of identity implies the law of excluded middle. This article was entirely dedicated to explaining this equivalence, since it is not obvious.
The step 4 is implied by the assumption in 3 that there is no such y that violates the law of excluded middle.
The statements of the law of identity or excluded middle do not require the concept of truth values, and are not expressed in terms of truth values, although sometimes it is useful to introduce the concept to prove some higher order relation.
The schematic representation attempts to clarify the equivalence between Y being neither x nor not-x, and Y being identical to not-x and not-not-x, therefore the non-identity of x to itself.
Non-classical logics are either classical (but misrepresent their classical commitments), or are logically inconsistent (imply contradiction) in every case where they genuinely deny any of the ‘classical’ laws (but they invent spurious distinctions to conceal their non-sense). If the laws are logically translatable, equivalent, then any purported attempt to deny one law amounts to denying all of them, so if the ‘non-classical’ schools deny any of the laws but accept others, they are already contradicting themselves. The subject-matter in the present article is the equivalence of the law of identity and excluded middle, in order to show that the latter cannot be denied without also denying the former.
EDIT: my motivation for the visual explanation of the equivalence between ¬(y=x) and y=¬x was caused by the intuitive (but logically not equivalent) comparison with: ‘dog is not a cat’ does not imply that dog is identical to not-cat (something that is not-cat could be a tree). There is no analogy with the case of excluded middle because the existential domain of cats and dogs includes also trees (which are neither cats nor dogs), whereas excluded middle applies ‘between’ dog and not-dog, cat and not-cat etc. I needed this to show the equivalence of excluded middle and identity. The proof from non-contradiction was already done here: https://michaelkowalik.substack.com/p/proof-of-the-law-of-excluded-middle
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.
GPT conflates excluded middle with non-contradiction: “There's no third choice where it's both raining and not raining at the same time.” GPT is wrong, always.
Is it still wrong?
The Law of Excluded Middle is a principle in logic stating that for any statement, either that statement is true, or its opposite is true—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 neither true nor false.
On the other hand, the Law of Non-Contradiction asserts that a statement and its opposite cannot both be true at the same time. So, "It is raining" and "It is not raining" cannot both be true simultaneously.
In summary, the Law of Excluded Middle ensures that every statement is either true or false, while the Law of Non-Contradiction ensures that a statement and its opposite cannot both be true at the same time.
Better. It would be appropriate to now ask what does “true” and “false” mean? The explanation that “it is raining” is true “if it is raining” is logically circular, analogous to ‘it is true if it is true’ leaving it unclear what “true” could mean.
I'll give a whirl... GPT is a tool GIGO
I fail to understand - are you suggesting the actual, phenomenological impossibility of this four pronged statement about the dog (also called the catuscoti in eastern thought), merely its logical impossibility?
Nevertheless, you have the option of presenting the existential domain of the 3 cases you seem to refer to (I am not sure what the number 4 refers to; the challenge consist of 3 cases, since there are 3 laws).
Oh i actually misread the last sentence, i thought you had added a fourth term of “both a dog and a dog” and i guess my mind filled it in. What specifically do you mean when you say “existential domain”?
But if this is not clear then i can rephrase the challenge: draw a picture of a dog that neither exists nor exists.
This is what the article explains, specifically, with pictures.
Yes