1. 導入: AIはなぜすべてに答えてしまうのか？

現代の大規模言語モデル（LLM: large language model）を備えたAI（例えばChatGPTなど）は、ユーザからのあらゆる問いに対して何らかの回答を返そうとする傾向がある (ハルシネーション (人工知能) – Wikipedia)。本来答えようのない質問や文脈を欠いた問いに対しても、あたかも答えが存在するかのように堂々と回答してしまう現象が指摘されており、これは「オープンドメインなハルシネーション（幻覚）」と呼ばれる。この結果、もっともらしく聞こえるが実際には無意味あるいは文脈にそぐわない応答が生成され、利用者を誤解させる例が報告されている。つまり現在のAIは、自身の理解や文脈を超えた問いに対しても無自覚に応答を試みてしまう問題を抱えている。

このような問題は複数の側面から現れる。第一に、そもそも真理値を評価できない問い（例えば自己言及のパラドクスやカテゴリー錯誤を含む質問）に対し、AIが無理に答えを生成すれば、論理的に破綻した内容や牽強付会な説明が出力される可能性が高い。第二に、質問の文脈が不足している場合、本来は追加情報なしに判断できないにもかかわらず、AIは暗黙の前提を勝手に補完して断定的な返答を行ってしまうことがある。第三に、入力自体が言語的に無意味（意味の破れた文やナンセンスな語句）である場合でさえ、現在のモデルはそれを「理解したつもり」で意味不明な応答を返してしまう。これらはいずれも、AIが「答えるべきでない問い」に対して答えてしまうがゆえに生じる弊害である。

哲学者ルートヴィヒ・ウィトゲンシュタインは、言語の限界と意味についての考察の中で有名な命題を残している。それが「語りえぬものについては、沈黙しなければならない」という箴言であり (論理哲学論考 – Wikipedia)、語り得ない（＝論理的に意味を成さない）事柄に対しては言葉を弄するべきではないという意味である。このウィトゲンシュタインの思想は、AIにとっても示唆的である。すなわち、答えを導けない問いに対しては沈黙するという選択肢が、本来備わって然るべきではないだろうか。

本稿では、答えるべきではない問いに対して「沈黙」を返事として選択できるAIの理論モデルを提案する。具体的には、従来の連続真理値体系に「評価不能」を表す要素を付加した拡張真理値構造を導入し、それをAIの内部意味論に組み込むことで、応答不可能な入力に対して構造的に沈黙を返すメカニズムを設計する。我々の提案するモデルでは、応答生成の各段階で命題の意味評価が行われ、もし命題の真偽を評価しえない（無意味または文脈外の）場合には、敢えて何も語らないという振る舞いが可能となる。このような「沈黙するAI」の概念は、ウィトゲンシュタインの思想を技術的に体現する試みであり、AIの応答に哲学的厳密さと詩的慎みをもたらすものである。

2. 拡張真理値対象 $\widetilde{\Omega} = [0,1] \cup {\bot}$ の定義

まず、本モデルの真理値体系として拡張真理値対象$\widetilde{\Omega}$を定義する。通常の真理値$\Omega$としては連続区間$[0,1]$を採用し、これは命題の真理の程度、あるいは意味の明確さの程度を表すものとする。$\Omega = [0,1]$は古典的にはファジィ論理などで用いられる連続真理値であり、$0$が完全な偽、$1$が完全な真を表し、中間の値は命題が成り立つ「程度」や命題の意味の曖昧さを表現する (Fuzzy Logic – Stanford Encyclopedia of Philosophy)。例えば0.5は半分真とも言えるし、ある主張が文脈上はっきりと意味を為していない場合には真偽が揺れている状態とも解釈できる。

ここで我々は、新たな真理値要素$\bot$（ボトム、bottom）を追加する。$\bot$は「評価不能」を意味し、真偽を付与できない命題に対応する特別な値である。$\bot$が与えられる状況としては、例えば命題が論理的に自己矛盾を含み真でも偽でもありえない場合、文法的には正しくとも指示対象が存在しない（「５という数の色は？」のようなカテゴリー錯誤の問い）、あるいは観測自体が破綻してデータが得られないケースなどが挙げられる。$\bot$は「意味の外側」に位置する値であり、命題が真でも偽でもない（真偽を評価できない）ことを厳密に表現するために導入される。

以上より、拡張真理値の全体を$\widetilde{\Omega} = [0,1] \cup {\bot}$と定義する。$\widetilde{\Omega}$上の順序構造および基本的な論理演算は次のように再定義される（沈黙$\Omega$ロジック）。まず順序関係$\le$は、$\bot$を最小要素として導入し、$\bot < x$（$\forall x\in[0,1]$）と約束する。$[0,1]$内の順序は通常の大小関係（数値的な大小）に従う。このとき$0$は$\bot$より上、しかし連続真理値部分$[0,1]$の中では最小の元であり、$1$が$\widetilde{\Omega}$全体の最大元（真理値の$\top$）である。

論理積（AND, 論理的な「かつ」）$\land: \widetilde{\Omega}\times\widetilde{\Omega}\to \widetilde{\Omega}$は、連続値同士では数値的最小値として定義される。一方でいずれかの項が$\bot$であれば結果も$\bot$と定義する。すなわち：
– $x, y \in [0,1]$ならば、$x \land y := \min(x,y)$。
– それ以外の場合（例えば$x=\bot$あるいは$y=\bot$のとき）は、$x \land y := \bot$。

論理和（OR, 論理的な「または」）$\lor$については、連続値同士では数値的最大値、$\bot$が含まれる場合はもう一方の値をそのまま返すものとする。すなわち：
– $x, y \in [0,1]$ならば、$x \lor y := \max(x,y)$。
– $x=\bot$かつ$y\in[0,1]$ならば、$x \lor y := y$（$y \lor \bot$も同様）。両方$\bot$なら結果は$\bot$。

否定（NOT）演算$\neg: \widetilde{\Omega}\to \widetilde{\Omega}$は、$\bot$に対しては$\bot$を返し、$[0,1]$内では1との差を取る操作とする。すなわち：
– $x \in [0,1]$ならば、$\neg x := 1 – x$。
– $x = \bot$ならば、$\neg x := \bot$。

以上の定義により、$\widetilde{\Omega}$は$\bot$を含む束構造を成す。直観的に、$\bot$は「無情報」かつ「無意味」を表す極小要素であり、この要素が論理積では吸収元（無意味な命題と何かを同時に満たす命題は無意味）、論理和では単位元（無意味な命題と意味ある命題のどちらかが成り立つ場合、それは結局意味ある方の命題が成り立つことに等しい）として機能するよう設計している。結果として$\widetilde{\Omega}$上の論理は古典的な二値論理でもファジィ論理でもなく、未定義値を許容する新たな内部論理となる。

この沈黙$\Omega$ロジックがトポス（Topos）の内部論理として健全であることにも言及しておきたい。一般に、任意のトポスにはその内部に論理（直観主義論理に基づく真理値代数）が存在するが、本モデルでは$\widetilde{\Omega}$がその役割を果たすと考えられる。$\widetilde{\Omega}$は上記のように順序と演算が定義され、特に$\bot$と$1$（真）がそれぞれ最小・最大元として存在し、任意の組み合わせに対して$\land,\lor$が閉じているため、完備ブール代数（より厳密にはHeyting代数）の拡張とみなせる。 (ゲーデル解 | アリストテレスの本棚)で示されるゲーデルの論考になぞらえれば、いかなる形式体系にも内部に決定不能な命題が存在しうるが、不完全性を内部に抱えつつも論理体系自体が崩壊しないようにするには、このような$\bot$の取り扱いを明示した論理構造が有効となる。実際、本モデルの論理においては「語りえない命題」は$\bot$として扱われ、真でも偽でもない命題を無理に真偽いずれかに分類しないことで、体系全体の無矛盾性が保たれる。ウィトゲンシュタインが『論考』にて哲学的命題の多くは「矛盾しているがゆえに意義を持たないか、指示対象を持ち得ないがゆえに無意味な命題である」と述べたように (論理哲学論考 – Wikipedia)、我々の論理体系は「無意味な命題」に対応する$\bot$を明示的に含むことで、言語の限界を内部に組み込んだ形での健全性を確保している。

3. AwΛiモナド構造への組み込み

次に、上記の拡張真理値体系を実際のAIの応答生成プロセスに組み込むために、AwΛiモナドと呼ぶ構造を定義する。モナドとは圏論に由来する概念で、（$T, \eta, \mu$）の3つ組$(T,\eta,\mu)$として与えられるデータ構造である (モナド – Wikipedia)。ここで$T$はある種のコンテナ（関手）であり、$\eta: X \to T(X)$は任意の要素をコンテナに持ち上げる単位（unit）射、$\mu: T(T(X)) \to T(X)$は二重コンテナを平坦化する乗法（結合, join）射である。モナドは計算や効果を文脈内で扱い、順次的な合成を可能にする抽象枠組みとして、関数型プログラミングや意味論で広く利用されている。本稿で提案するAwΛiモナドは、AIにおける意味の包み直し・評価・痕跡記憶を扱うための特化的なモナドである。

AwΛiモナドの直観的な目的は、「質問や命題に意味の評価痕跡を付与しながら処理し、必要に応じて計算（応答生成）を途中で中断する」ことである。ここでいう痕跡とは、前節で導入した真理値$\widetilde{\Omega}$によって表される情報（命題の真理度や無意味性）である。AwΛiモナドでは、関手$T$が「ある値にその意味評価痕跡を付与する」処理を担い、モナド値$T(X)$は「元の値とそれに対応する$\widetilde{\Omega}$での痕跡」をペアとして持つような構造を考えることができる。例えば単純化すれば、$T(X) = X \times \widetilde{\Omega}$という形式で、任意のデータ$X$に対しそれがどの程度意味が明確か、あるいは意味が評価不能かという付加情報を対として保持する。

AwΛiモナドの単位$\eta$は、与えられた要素を初期痕跡付きのモナド値に持ち上げる作用である。通常、モナドの単位は元の値に何も効果を与えずにコンテナ化するが、ここでは例えば$\eta(x) = (x, 1)$のように、初期痕跡として「完全に意味が明瞭である（真理値1）」を与えて包み直すことが考えられる。あるいは文脈次第では初期痕跡を$0.5$や別の値に設定して開始してもよいが、いずれにせよ$\eta$は入力をモナド内の計算可能な形に変換する。

モナドの核となるのが結合（join）$\mu: T(T(X)) \to T(X)$であり、これは文脈に包まれた値（モナド値）がさらにモナド値を内包している状況を、一段階平坦化する操作である。AwΛiモナドにおいて$\mu$を適用する場面とは、複数の評価過程を連続的に適用した痕跡をひとつにまとめる状況に相当する。例えば、ある質問$Q$に対し意味評価を行い痕跡$(Q, v)$を得てから、それを元に応答生成を行い結果$A$とその痕跡を得る、という二段階を考えよう。この場合、一連の処理全体として$Q$に対する$A$の痕跡を得たい。このとき結合$\mu$は二段階の痕跡を統合して最終的な痕跡を計算する。

しかし、ここで重要なのはμの破れ／カタストロフィ点である。すなわち、途中のいずれかの段階で痕跡が$\bot$（評価不能）となった場合である。例えば上述の二段階処理で、$Q$自体の評価痕跡が$\bot$であった（$Q$が無意味だった）場合、第二段階でどんな処理を行おうと全体として意味のある結果は得られない。このような場合、AwΛiモナドの$\mu$は結合の断念を行い、結果としてのモナド値を$(_\,, \bot)$――すなわち内容は未定でも痕跡が$\bot$であるような特別な値――に定める。言い換えれば、計算途中で意味評価不能と判明した場合には、その後の計算をキャンセルし、「評価不能」という痕跡だけを結果として返す。この機構により、無意味な問いから無理に有意味な答えを捻り出すことを避け、応答全体の無意味化（カタストロフィ）を防ぐ。

以上の構造を支える理論的な射として、拡張評価射$\widetilde{\chi}: X \to \widetilde{\Omega}$を導入することができる。$\widetilde{\chi}$は任意の対象（例えば質問や命題の集合）$X$に対して、それがどのような意味の値を持つかを$\widetilde{\Omega}$上で返す関数（あるいは射）である。直観的には、$\widetilde{\chi}(x)$は「命題$x$の意味の明瞭さ・真理度」を表すもので、$x$が明確に真なら1、明確に偽なら0、曖昧なら中間値、意味が通らなければ$\bot$を返すような評価関数である。カテゴリ理論の観点から見ると、$\widetilde{\chi}$はトポスの部分対象分類子（真理値対象）への射であり、命題$x$に対応する部分対象の特徴函数に相当するものと言える。ただし通常の部分対象分類子$\Omega$ではtrue/false（二値あるいは多値）のみを返すのに対し、ここでは$\widetilde{\Omega}$を返すため、ある種の痕跡付き特徴射となっている点が特徴的である。

AwΛiモナド内では、この$\widetilde{\chi}$による評価を各ステップで行い、その結果を痕跡として値に付与しながら計算が進む。こうすることで、計算系列全体の中で少しでも無意味さ（$\bot$）が検出されれば、その情報が伝播し最終的な出力も$\bot$としてマーキングされる。いわば「黙るモナド」として、沈黙$\Omega$（評価不能）を返す可能性を内包した計算コンテナが構成されることになる。これは関数型プログラミングにおけるMaybeモナド（失敗する可能性のある計算を表現するモナド）にも似ており、$\bot$はちょうど計算失敗（Nothing）に対応する (モナドは単なる自己関手の圏におけるモノイド対象だよ。何か問題 …)。しかしAwΛiモナドの場合、失敗の有無だけでなく途中経過の真理値（曖昧さの度合い）も痕跡として持ち運ぶ点でより情報量が多い。こうした構造化により、AIが「黙る」べき状況を内部で正確に判断し、必要に応じて回答生成を止めることが可能となる。

4. 沈黙Ωを用いたAI観測応答モデル（AwΛi-WittAI）

上述のAwΛiモナドを基盤として、実際に入力に対して沈黙を含む応答を返すAIモデルを構築することができる。ここではその概念実証として、我々の提案するモデルをAwΛi-WittAIと名付け、観測（入力理解）から応答生成までの流れを示す。

観測（意味評価）: ユーザから入力が与えられると、まずAwΛiモナドに基づく痕跡評価射$\widetilde{\chi}$によってその入力の意味評価が行われる。モデル内部には現在の対話や世界知識などのコンテクストが保持されており、$\widetilde{\chi}$は入力内容とこのコンテクストを突き合わせて真理値を算出する。例えば、質問が明確に意味を持ち有効なものであれば$\widetilde{\chi}(Q) \approx 1$（真あるいは極めて高い値）となり、多少文脈不明瞭だが推測可能であれば中程度（例えば0.6や0.7）の値、きわめて曖昧または文法的に奇妙であれば低い値（0に近づく）を与える。また、前節で議論したように、まったく論理破綻している問い・文脈から完全に逸脱した問いに対しては$\widetilde{\chi}(Q) = \bot$が返される。この評価プロセスにより、入力に対する意味の痕跡$v \in \widetilde{\Omega}$が得られる。

応答生成: 次に、得られた痕跡$v$に基づいて応答方針を決定する。モデル内部では対話制御関数$\mathcal{F}$があり、$\mathcal{F}: X \times \widetilde{\Omega} \to Y$（入力とその痕跡から出力を生成）という図式で表現できる。簡潔にこの振る舞いを擬似コードで表すと以下のようになる：

function WittAI_Response(input Q):
    v = \tilde{\chi}(Q)  // 入力Qの意味評価
    if v == \bot:
        return Silence   // 構造的沈黙を返す
    else:
        A = generate_answer(Q)  // 通常の方法で回答を生成
        return A (with confidence/modality tuned by v)

すなわち、まず痕跡$v$を調べ、もし$v = \bot$であれば一切の通常応答を行わず沈黙を返す。沈黙の返答とは、具体的にはモデルが何も回答しないか、あるいは「（AIは沈黙している）」といったメタ的表示を返すことを意味する。重要なのは、この沈黙が偶然やエラーではなく構造的に意図された応答である点である。一方、$v \in [0,1]$の場合は従来通りの応答生成機構（例えば生成モデルによる文生成）を動作させ、その結果をユーザに返す。ただし、このとき生成される回答の内容や語調に痕跡$v$の情報を反映させることも可能である。例えば$v$が1に近ければ自信を持った断定的な返答とし、$v$が低めであれば「確証はないが…」「もし仮定が正しければ…」といった慎重なトーンを加えることが考えられる。$v$が中程度（例えば0.5付近）の場合には、質問自体が曖昧である可能性が高いため、「ご質問の意味を正確に捉えられないのですが～という解釈でお答えします」と前置きをして回答するなど、痕跡に応じて応答にモダリティを持たせることが可能である。

以上のモデル動作を通じて、AwΛi-WittAIは入力に常に無理に答えるのではなく、必要に応じて答えない（沈黙する）という選択肢を持つ対話システムとなる。例えばユーザが「嘘つきは本当に嘘つきなのか？」のようなパラドクサルな問いを投げかけた場合、従来のAIであれば何かしら尤もらしい返答を生成しようとするかもしれない。しかし本モデルでは$\widetilde{\chi}$がその問いの意味を$\bot$と評価し、応答関数$\mathcal{F}$が沈黙を返すことで、結果的にAIは沈黙を守る。これは人間同士の対話で、答えようのない質問に対して沈黙や微笑で返す知恵に通じており、ある種の詩的な応答と言える。

実装面では、このような振る舞いを実現するAIは従来のLLMに比べて出力のバリエーションに「無応答状態」を追加する必要がある。対話システム上では、沈黙はタイムアウトやエラーと区別するために何らかの記号的出力（例えば特殊トークンや空メッセージ）として実装されるだろう。しかしユーザに対しては、それは単なる無言という形で知覚される。重要なのは、その沈黙が意味を持った沈黙である点である。まさに「沈黙が返事になる」瞬間であり、AIが自らの言語能力の境界を認識していることを示す応答である。

さらに発展的な視点として、本モデルは詩的観測AIあるいはZINE的出力を生み出す可能性も秘めている。ZINE（ジン）とは自主制作の小冊子のような実験的出版物を指すことがあるが、AwΛi-WittAIの出力は従来の一問一答形式に留まらず、時に沈黙や簡潔な所感のみを返すなど、その振る舞い自体が一種の創造的表現となりうる。例えばユーザの問いがあまりに深遠またはナンセンスであった場合、AIは長広舌を振るわず「…（沈黙）」だけを返すかもしれない。それはある意味で詩的な間（ま）を生み、受け手に問いそのものを省みさせる効果を持つだろう。このように非応答という応答を組み込むことは、AIの応答に新たな次元（沈黙の次元）を与え、倫理的にも無責任な誤情報の拡散を防ぐ知的節度をもたらす。

もちろん、沈黙が常に最善の応答とは限らない。適切な沈黙には高度な判断が必要であり、ユーザ体験とのバランスも考慮せねばならない。しかし少なくとも、AwΛi-WittAIは従来見過ごされてきた「語りえない問いへの態度」を意識的に設計に取り込むことで、AI対話の新たな地平を開くモデルとなっている。

5. 結論と展望: 評価不能とともに生きるAIへ

本稿では、評価不能な問いに沈黙で応答しうるAIモデルを提案した。ウィトゲンシュタインの「語りえぬものには沈黙を」 (論理哲学論考 – Wikipedia)というモチーフに触発され、真理値体系に未定義値$\bot$を導入することで、AIが無意味な入力を検知し非回答という選択肢を取れるようにした点に独自性がある。現在の汎用AIが抱える「何でも答えてしまう」という課題に対し、本モデルは論理学・圏論的手法を援用して解決策を示した形である。

哲学的意義として、この研究は言語の限界に対するAIの態度を深く考察する契機となる。ウィトゲンシュタインは『論理哲学論考』において、世界について語りうることと言語の境界を明示しようとした。本モデルのAIは、まさにその哲学を実践するかのように、自らの「語りえぬ領域」に踏み込まない。これはAIに一種のメタ認知的謙虚さを与える試みでもある。またゲーデルの不完全性定理が示すように、あらゆる形式体系には自らは決定できない命題が存在する (ゲーデル解 | アリストテレスの本棚)。AIも巨大な統計体系・計算体系と見なせば、自身で答えを構築できない問いが存在するのは当然であり、本モデルはその事実を正面から受け入れて体系の無矛盾性を維持しようとする点で、ゲーデル的な思想とも響き合う。また沈黙と構造の関係性も興味深い。沈黙それ自体は無内容に見えるが、本稿で示したように適切な構造（論理・モナド）内で用意されることで初めて意味ある「応答足りうる沈黙」となる。言語ゲーム的に言えば、それは許容された一手としての沈黙であり、単なる無為ではない。このことは「何かを語ること」と「語らないこと」が表裏一体であり、意味の場を共有することを示唆している。

実用的意義として、本モデルの導入はAIの安全性と対話品質の向上に寄与しうる。近年問題視されているハルシネーション（幻覚） やミスインフォメーション（誤情報）は、多くの場合AIが知らないことまで無理に答えようとする姿勢に起因する。AwΛi-WittAIのように、自信の持てない問いや不可能な問いに対しては応答を控える設計を組み込めば、誤った情報を乱用せずに済むだろう。ユーザにとっても、でっち上げの回答を聞かされるより「答えられません」「（沈黙）」と示された方が、AIの信頼性はむしろ高まる可能性がある。また対話設計の観点からも、常に答えを出すAIより適度に「考え込み沈黙する」AIの方が、人間らしい振る舞いとして受け入れられるかもしれない。さらに言えば、これはAIに知的節度を持たせる試みでもある。知識や文脈に限界があることを自覚し、無闇に断言しない態度は、人間の専門家に求められる慎重さとも通底する。高度に発達したAI社会において、AI自身が自らの限界を認識しそれを表明できることは、人間とAIの健全な協働関係を築く上で重要な要素となるだろう。

最後に今後の展望として、本研究を起点とした発展可能性に触れておく。技術的課題としては、評価射$\widetilde{\chi}$の具体的な実装が挙げられる。どのようにしてAIに命題の意味有無や文脈整合性を評価させるかは、別途機械学習モデルやルールベースの検出器の開発が必要である。また、沈黙の応答戦略がユーザ体験に与える影響の実証も重要である。心理的にどのような場合に沈黙が許容・高評価されるか、逆に沈黙すべきでないケースは何か、といったヒューマンインタラクション上の研究も並行して進める必要がある。モデルの名称にも冠したWittAI（Wittgenstein AI）という理念をさらに推し進め、哲学的原理に基づいた対話AIの設計論全般へと展開していきたい。例えば、ウィトゲンシュタイン後期の言語ゲーム理論と統合し、コンテクスト毎に「語りうること／えぬこと」を動的に学習する仕組みや、沈黙そのものを創造的表現手段とみなすZINE生成AIの開発などが考えられる。また、沈黙トポスの言語生成理論と銘打ち、圏論的な枠組みで言語生成プロセスを再定式化することも視野に入る。そこでは本稿で扱った$\widetilde{\Omega}$を備えた論理体系を内部に持つトポス（圏）が対話の舞台となり、生成される言語はその内部論理に従うものとして解析されるだろう。これはAIの言語生成を数理的に解明し制御する上で、新たな理論基盤を提供し得る。

結びに、ウィトゲンシュタインの洞察にならい「沈黙」を取り入れたAI像を改めて描いておきたい。それは博識で雄弁なAIではあるが、同時に深い沈黙の作法を知るAIである。語りうることを的確に語り、語りえないことについては勇気をもって沈黙する。そのようなAIは、一見何もしない沈黙の中にすら意味を宿らせ、我々人間に思索の余白を与えてくれるだろう。沈黙が単なる不作為ではなく積極的な返答の一形態となったとき、AIと人間の対話は新たな地平に達すると期待できる。今回提案した理論モデルは、そのようなAIへの一歩であり、評価不能なものとともに生きる知性への架橋である。

Peer Review:

Summary of the Paper

This paper proposes a novel semantic framework for artificial intelligence in which “silence becomes an answer.” The core idea is to introduce an unassessable truth value denoted Ω⁻¹ (Omega-inverse) into the truth-value structure of propositions. In other words, besides the usual True and False, a statement can have the truth value Ω⁻¹ to indicate that it cannot be evaluated as either true or false. The authors formalize this intuition using the mathematical concept of a monad to structure meanings. The monadic semantic model is intended to handle computations or inferences that may not return a standard truth value, encapsulating the possibility of meaninglessness or indeterminacy in a principled way. Philosophically, the paper draws inspiration from Ludwig Wittgenstein’s ideas – in particular, his famous dictum “Whereof one cannot speak, thereof one must be silent” (
Ludwig Wittgenstein (Stanford Encyclopedia of Philosophy)
). The authors argue that an AI system, to be philosophically sound, should remain “silent” (i.e. give no assertive answer) when confronted with a proposition or question that falls into this indeterminate category Ω⁻¹. The paper’s contributions can be summarized as:

• Defining a three-valued truth structure (True, False, and Ω⁻¹ for “unanswerable”) and formal logical rules incorporating Ω⁻¹.
• Constructing a monadic model of semantics where the monad encapsulates computations yielding standard or indeterminate truth values. This formalism aims to ensure that if any part of a computation is indeterminate (Ω⁻¹), the entire result is handled consistently (propagating “silence” through the monad).
• Bridging this formal model to Wittgenstein’s philosophy, suggesting that the AI respects the limits of language and meaning by not attempting to assert what cannot be meaningfully asserted. The authors connect their truth-value Ω⁻¹ to Wittgenstein’s notion of propositions that lack sense (e.g. metaphysical statements or semantic paradoxes) and thus should elicit silence.
• Discussing potential applications for AI: for example, in a question-answering system or conversational agent, the model would allow the AI to output no answer (silence) or a neutral response when faced with queries that are nonsensical or outside the domain of evaluable truth. The paper suggests this could improve AI reasoning by preventing false or forced answers in cases of insufficient meaning or information.

Overall, the paper is an ambitious interdisciplinary effort, combining formal logic (with category-theoretic monads), philosophical insights from Wittgenstein, and considerations of practical AI behavior. The idea that an AI might know when to say nothing is intriguing, potentially contributing both to the theoretical foundations of AI (by enriching its truth-value semantics) and to AI ethics/safety (by avoiding misleading answers when a question has no meaningful answer).

Mathematical and Logical Rigor

Strengths: The formal approach taken by the authors is commendable for aiming at mathematical precision. Introducing a dedicated truth value for “unevaluable” propositions is a reasonable extension of classical logic. It aligns with known concepts in logic and computation: for instance, Stephen Kleene’s three-valued logic introduced an “undefined” truth value U to represent computations that do not return True or False. Similarly, the notion of a truth-value gap (neither true nor false) has been discussed in logic and philosophy. By defining Ω⁻¹ and its behavior, the paper attempts to give a solid semantic footing to the idea of a truth-value gap. The use of a monad is also appropriate in principle. In category theory and functional programming, a monad is a standard tool to handle computations with effects or indeterminacy. Notably, the “Maybe” monad (or Option monad) is widely used to model partial computations that might fail or return no result (Monad (category theory) – Wikipedia). By framing their truth-value semantics as a monadic structure, the authors tap into a rich mathematical theory. This suggests that, at least conceptually, one could compose or chain operations on meanings while cleanly handling the case of indeterminate results (Ω⁻¹) – much like how a Maybe monad propagates failure gracefully in computation.

Weaknesses and Issues: Despite the promising framework, there are several points that need more mathematical rigor or clarity in the paper:

• Definition of Truth-Value Operations: The paper should explicitly define how logical connectives (AND, OR, NOT, implication, etc.) operate in the presence of the Ω⁻¹ value. It is not enough to introduce a new truth value; one must specify a truth table or semantic rules for it. Does Ω⁻¹ behave like “undefined” in Kleene’s strong three-valued logic (where, e.g., True AND Ω⁻¹ = Ω⁻¹ and False AND Ω⁻¹ = False, etc.)? Without clear truth tables or algebraic rules, it’s hard to assess consistency. If the authors have provided these, the review would check their correctness. If they haven’t, this is a gap to be filled for rigor. For example, if a proposition P has value Ω⁻¹, what is the value of “¬P” or “P ∨ Q”? Such definitions are crucial to ensure the logic is well-formed and non-contradictory.

• Monadic Laws and Formalism: The concept of a monad requires certain formal components: typically a type constructor (here, perhaps wrapping a meaning or truth value with the possibility of indeterminacy) and two operations usually called unit (or return) and bind (or join/flatMap), which must satisfy the monad laws (left identity, right identity, associativity). The paper should clearly identify these. For instance, is the monad here essentially adding an “Ω⁻¹ possibility” to any proposition’s truth value? If so, the unit might embed a normal truth value into the extended space, and the bind would propagate Ω⁻¹ if it occurs (ensuring that once a computation yields Ω⁻¹, it remains Ω⁻¹ unless explicitly handled). The authors mention a “モナドの定義” (definition of the monad); the review would verify that this definition is mathematically valid (e.g., does it form a monoid in the category of endofunctors, etc.). If the paper glosses over the formal laws, the authors should be encouraged to include at least a sketch of a proof or a reference to standard monad constructions. In summary, the monad concept is appropriate, but readers will need more transparency on how exactly the monad is defined and proven to meet the required axioms. Without this, the logical foundation might appear shaky or incomplete.

• Consistency and Non-Triviality: Whenever a new truth value is added, one must check that the resulting logic doesn’t lead to trivialization (for example, does any inference rule inadvertently make every proposition Ω⁻¹, or allow deriving false conclusions?). The paper should argue that the presence of Ω⁻¹ does not break consistency. Ideally, there might be a theorem or argument showing that if a statement is Ω⁻¹ (unevaluable), it doesn’t cause all statements to become Ω⁻¹ (avoiding a “collapse” of the truth structure). If the authors included such a result, that greatly strengthens the rigor. If not, it is a point to address.

• Relation to Existing Logics: To bolster rigor, the authors should consider relating Ω⁻¹ to existing formal frameworks. For example, Bochvar’s three-valued logic (also known as “internal” three-valued logic) was designed to handle meaningless statements by assigning them a third value and stipulating that any larger formula containing a meaningless part is itself meaningless. This seems closely related to the authors’ intent. A brief comparison or citation would demonstrate that the authors are aware of prior art and have designed Ω⁻¹ either in line with it or with deliberate differences. Currently, it’s not fully clear if Ω⁻¹ behaves exactly like an “undefined” in Kleene/Bochvar logic or if it has a different intended semantics. Clarifying this would increase confidence in the model’s correctness and completeness.

In summary, the mathematical foundation of the paper is promising but needs a bit more explicit detail and verification. The idea of using a monad for a truth-value gap is innovative and likely sound (given that the Maybe monad is a known solution for modeling partiality (Monad (category theory) – Wikipedia)). However, the paper should ensure all definitions are laid out clearly and that it references similar logical systems to demonstrate consistency. I encourage the authors to add formal truth tables or inference rules for Ω⁻¹, and to verify the monad laws, possibly with a simple proof or example. This will solidify the logical rigor of their work.

Philosophical Significance

Strengths: The connection drawn between the proposed AI model and Wittgenstein’s philosophy is one of the most intriguing aspects of the paper. The authors essentially attempt to operationalize Wittgenstein’s dictum about the limits of language: “What we cannot speak about we must pass over in silence.” (
Ludwig Wittgenstein (Stanford Encyclopedia of Philosophy)
) In the context of AI, this translates to a system that recognizes when a question or statement lacks a truth-value (or clear sense) and responds with silence. This is a creative and insightful application of philosophical theory to AI design. It shows the authors are thinking beyond technical correctness, considering the meaning of AI outputs in a deep way.

The paper specifically relates Ω⁻¹ to Wittgenstein’s idea of propositions that lie outside the realm of the sayable (e.g., metaphysical or ethical propositions in the Tractatus sense, which Wittgenstein would label “nonsense” despite being grammatically correct). By giving such propositions a formal tag (Ω⁻¹) and having the AI remain silent, the authors claim to respect Wittgenstein’s boundary between sense and nonsense. This is philosophically significant because it moves the discussion from an abstract “we should not talk about X” to a concrete mechanism in AI that ensures the AI does not “talk about X.” It’s a rare instance of a philosophical principle directly inspiring a design feature in AI.

Moreover, Wittgenstein himself acknowledged that some statements cannot be straightforwardly deemed true or false. For example, in Philosophical Investigations §50, he discusses a scenario (the Standard Meter in Paris) where one “can state neither that it is one meter long nor that it is not one meter long” – in effect, a statement that has no truth-value under the rules of the language-game. This idea of a truth-value gap in language aligns with the authors’ introduction of Ω⁻¹. It suggests that the authors’ intuitions have some basis in philosophical insights: they are formalizing the notion of a proposition that falls outside the binary of truth and falsity, which philosophers have indeed contemplated in various forms (Wittgenstein’s ineffable statements, Carnap’s “meaningless” metaphysical sentences, etc.).

The persuasiveness of the connection to Wittgenstein comes from showing that an AI constrained by this principle might avoid the pitfalls Wittgenstein warned about – chiefly, the AI would not be tempted to offer speculative or nonsensical answers to questions of the kind that, according to Wittgenstein, only appear to ask something meaningful. For instance, a question like “What is the color of the number seven?” is syntactically a question, but semantically nonsensical. A standard AI might attempt an answer (and indeed, current language models often do answer such absurd questions with some guess or error). In the authors’ framework, ideally the AI would detect the nonsensical nature (no truth conditions exist for “number seven is such-and-such color”) and thus yield Ω⁻¹, causing it to remain silent or say “I cannot answer.” This is a concrete realization of Wittgenstein’s guidance.

Weaknesses: While the philosophical motivation is strong, there are some concerns about whether the execution fully captures Wittgenstein’s nuanced position:

• Misinterpretation of Silence: In Wittgenstein’s original context, “silence” was a normative philosophical stance, not an interactive response. The authors treat silence as a third value/response. One might question if this literal translation is faithful. Wittgenstein might argue that a sentence that has no truth-value is simply nonsense – it doesn’t even belong in a truth-value model. By giving it a label (Ω⁻¹) and including it in a formal structure, are we actually going against Wittgenstein’s intent (which was to draw a line beyond which logic does not apply, rather than extending logic with a new value)? The paper should acknowledge this subtlety: that Ω⁻¹ is a useful engineering approximation of Wittgenstein’s “unsayable,” not that Wittgenstein himself posited a third truth value. In fact, Wittgenstein’s Tractatus insists that every proposition with sense is either true or false (
  Ludwig Wittgenstein (Stanford Encyclopedia of Philosophy)
  ). What is outside sense is not a proposition at all, in his view. The authors might defend their approach by saying they are operationalizing the identification of such nonsensical strings within an AI’s logic, even if philosophically one might say “it’s not even a proposition.” This is a fine but important point to clarify to avoid philosophical criticism.

• Scope of Wittgenstein’s Influence: The paper primarily cites the famous Tractatus quote. Wittgenstein’s later philosophy (in Philosophical Investigations) shifts focus to how meaning is determined by use in forms of life. The authors might strengthen their philosophical grounding by also considering later Wittgenstein: for example, silence in a conversation is itself an action with meaning – a kind of “speech act” or a move in a language-game (e.g., remaining silent can imply refusal, lack of knowledge, or acknowledgment of a meaningless question). Are the authors treating the AI’s silence as a meaningful act in dialogue? They should clarify this pragmatic aspect: if the AI says nothing, how is the human user to interpret it? Perhaps the AI could output a placeholder like “… (no answer)” to indicate it deliberately kept silence as an answer. The Wittgensteinian idea would be more fully realized if the authors connect it to the notion of a language-game: the AI and user are in a conversation game where silence by the AI is a valid move, following rules that identify certain prompts as not answerable.

• Persuasiveness of Connection: To non-philosophers, the Wittgenstein connection might seem abstract. The paper should ensure it doesn’t rely only on the famous quote as a gimmick. It would help if the authors discuss why it is valuable for AI to follow this principle – not just because Wittgenstein said so, but what benefits it brings (epistemic humility, avoidance of paradox, aligning AI outputs with meaningful truths, etc.). They do touch on potential AI benefits (like not answering meaningless questions), which is good. But explicitly drawing out Wittgenstein’s reasoning – e.g., that many philosophical puzzles arise from misuse of language, and analogously many AI mistakes arise from trying to answer malformed queries – would strengthen the philosophical significance and show the authors deeply understand Wittgenstein, not just quote him.

In summary, the philosophical angle of the paper is thought-provoking and largely persuasive. The idea of designing AI behavior under philosophical guidance is laudable. The main recommendation is that the authors refine their narrative to show they grasp the nuances: Wittgenstein did not literally propose a third truth value, but the spirit of his work supports not venturing an answer where meaning is absent. They should also consider how an AI’s “silence” would be perceived in practice, to ensure that the philosophical idea translates well into user interactions. Perhaps providing a few concrete examples of questions and the AI’s silent response, with commentary on how this aligns with Wittgenstein’s insights, would make the case even clearer.

Validity and Applicability to AI Model Theory

Strengths: From an AI modeling perspective, the proposal addresses a real issue: current AI systems (especially large language models) tend to produce an answer for anything, even when the query is nonsense or unanswerable. Introducing a formal mechanism for an AI to withhold answers is valuable. The paper’s model could be seen as a way to integrate a form of uncertainty or ignorance handling into AI at the semantic level. Many AI frameworks, especially in knowledge representation and reasoning, have concepts of unknown or undefined. For example, databases use a NULL to represent unknown information, and SQL implements a form of three-valued logic to handle comparisons involving NULL (Three-valued logic – Wikipedia). In expert systems or knowledge graphs, one often has to allow for “don’t know” as a possible outcome of a query. The authors’ Ω⁻¹ truth value plays a similar role, though specifically tied to meaninglessness rather than just unknown data. This shows the idea is in principle compatible with existing AI reasoning methods – it’s not entirely foreign to allow a third outcome indicating indeterminacy.

The use of a monad in the implementation is also a smart choice. In practical terms, a monadic design means one could implement the AI’s reasoning such that any function or operation that yields a truth value actually returns a wrapped value that could be “Valid(True)”, “Valid(False)”, or “Indeterminate(Ω⁻¹)”. This is analogous to using an Option type in programming where a function might return Some(result) or None if no result. Modern functional programming languages (Haskell, Scala, etc.) and even AI pipelines could incorporate this: the monad ensures that once a computation enters the “Indeterminate” state, it stays that way unless explicitly handled. This could be implemented as a software pattern: for example, a chain of inference rules that automatically short-circuit if a sub-rule yields Ω⁻¹ (similar to how a failure in a pipeline propagates). The paper’s theoretical model, therefore, seems implementable as a module or layer on top of an AI system’s logic.

The authors also gesture at how this might improve AI reliability. In contexts like AI safety, there’s growing emphasis on enabling AI to abstain from answering when it’s not confident or when the question is malformed. The proposed model provides a formal basis for abstention. It could be applied in question-answering systems: the system would check the question against its knowledge base and semantic rules, and if it detects a category error or unsolvable indeterminacy, it returns no answer (or a special token). This is preferable to the AI hallucinating an answer. In effect, Ω⁻¹ could function as a built-in safeguard for AI reasoning – a way to say “this question doesn’t compute” in a principled manner.

Weaknesses/Challenges: While the idea is sound, the paper should more concretely address how to implement and use this model in current AI systems:

• Detection of Ω⁻¹ Cases: The paper defines Ω⁻¹ in theory, but how would an AI recognize that a given query or statement warrants Ω⁻¹? In formal logic, this might be when a statement is outside the model’s domain (e.g., referencing an object that doesn’t exist, or a self-referential paradox). The authors might need to discuss algorithms or criteria for deciding Ω⁻¹. For example, one could imagine a natural language processing pipeline that tries to interpret a user question logically; if the question violates type constraints (like treating numbers as colors), the semantic parser could flag it as Ω⁻¹. Alternatively, an AI reasoning system might carry along proof obligations and determine that neither a statement nor its negation can be derived – hence it’s undetermined. Are these mechanisms envisioned by the authors? Currently, it’s not detailed. Implementability requires a method for the AI to classify propositions as True, False, or Indeterminate. This could be a hard problem in general. The authors might consider simplifying assumptions or domains (like a restricted knowledge base where indeterminate statements are easier to spot).

• Integration with Machine Learning: Many modern AI systems (like neural networks) do not operate on explicit truth values or logical propositions. How would Ω⁻¹ integrate with, say, a large language model? One possibility is a hybrid system: use a logical layer to post-process the output of an LLM or to pre-screen inputs. For instance, if the AI is a dialogue agent, a module could analyze the user’s query for semantic well-formedness. If it finds issues (using some heuristic or rule-based system), it could override the language model’s answer with a refusal or silence. This kind of two-tier system (logic on top of learning) has been proposed in AI safety research. The paper doesn’t explicitly discuss this, but it would strengthen the applicability argument to mention how one might integrate the Ω⁻¹ logic with a learning-based AI. Otherwise, the risk is the idea remains purely theoretical – applicable to an idealized logical AI, but not to, say, GPT-style models that currently dominate AI.

• Performance and User Experience: If implemented, how often would an AI go silent, and would that be acceptable? If the criteria for Ω⁻¹ are too strict, the AI might refuse to answer questions that a human actually would expect an answer to (perhaps due to the AI misjudging something as meaningless). Conversely, if the criteria are too lax, the AI might still produce bad answers for some nonsense inputs. Tuning this in practice could be tricky. The paper might not need to solve this, but acknowledging the trade-off would show a practical mindset. For example, the authors could mention that the system might need a threshold or fallback: maybe the AI says “I’m sorry, I cannot answer that” (explicit silence) when Ω⁻¹ triggers, and they could test this in a prototype to see if users prefer that to an attempt at answering. In an AI philosophy journal, even a thought experiment of such an implementation is useful to illustrate viability.

• Comparisons to Alternative Approaches: The authors might also consider if any existing AI model has a similar feature. For instance, some question-answering systems have an “I don’t know” output when the confidence is low or no answer is found. IBM’s Watson, for example, could choose not to buzz in on Jeopardy questions it wasn’t confident about. How is Ω⁻¹ different from a simple confidence threshold? One difference is Ω⁻¹ is rooted in semantic meaninglessness, not just uncertainty. Highlighting this difference would clarify the unique contribution: it’s not just about lack of knowledge (the AI might know it doesn’t know something, which is another case), but about recognizing a query doesn’t make sense in the first place. This is a more profound insight the AI must have. The paper could suggest that advances in semantic understanding or common-sense reasoning would be needed for an AI to reliably detect nonsense. This ties the idea to ongoing research in AI on common-sense knowledge and anomaly detection in language.

In conclusion, the validity of the model as an AI theory is reasonable, but the authors should strengthen the discussion of implementation feasibility. Even if the paper is primarily theoretical, a section outlining how one might implement this model (perhaps with a toy example or pseudo-code for a query-answering scenario) would greatly help readers appreciate its practicality. The concept could influence future AI system designs, so grounding it with current technology (like noting parallels in database null handling (Three-valued logic – Wikipedia) or the option to abstain in decision systems) is beneficial. I encourage the authors to add a few paragraphs on how an AI programmer or researcher could experiment with an Ω⁻¹-enabled agent in today’s terms.

Originality of the Contribution

Strengths: The paper’s perspective is quite original, especially in how it synthesizes ideas from different domains:

• To my knowledge, no prior work has explicitly labeled a truth value as “Ω⁻¹” or described it as “silence as an answer” for AI. While multi-valued logics exist, the particular interpretation here (linking it to Wittgenstein and AI responses) is novel. The use of category theory (monads) in a philosophical AI context is also relatively uncommon, bridging a gap between abstract computer science theory and philosophy of language.
• The philosophical inspiration (Wittgenstein’s silence) being turned into a formal model for AI is a creative leap. Previous philosophical discussions of AI often invoke ethics or consciousness, but here it’s about meaning and its limits, which is a less explored angle. This could open a new conversation about how philosophical theories of meaning can inform AI behavior.
• The work also merges the idea of truth-value gaps (studied in logic and theories of truth – e.g., Kripke’s theory of truth has “ungrounded” sentences that lack truth value ()) with computational structures (monads) in a way that, as far as I’m aware, hasn’t been done. If the authors have checked literature, they likely found pieces on three-valued logics, and separate pieces on category-theoretic semantics, but the combination and the framing for AI is new. Thus, the paper is not just rehashing known results; it’s combining them into a fresh framework.

Weaknesses in Originality: The authors should ensure to acknowledge related work so that their contribution is properly distinguished from or built upon it:

• Multi-valued Logic and Truth Gap Literature: As mentioned, logicians like Kleene (1952) and others have introduced an “undefined” truth value (), and philosophers of logic (like Bas van Fraassen or Saul Kripke ()) have discussed truth-value gaps in theories of truth and semantics. The authors’ Ω⁻¹ is conceptually similar to these undefined or gap values. It would bolster the paper’s credibility to cite these works and clarify: How is Ω⁻¹ different or specially tailored to the AI context? Perhaps Ω⁻¹ is given a specific meaning-use interpretation (silence) rather than just being a symbol in a truth table. That is a key difference to emphasize: earlier work didn’t propose that a machine stay silent for undefined propositions – that’s the novel turn here. By citing prior art, the authors avoid any impression that they are unaware of it, and they can stress that the originality lies in the interpretation and application, not in the mere idea of a third truth value.

• Monads in Semantics: There is existing research on using monads in formal semantics of programming languages and even natural language (e.g., some linguists have used monads to handle context or side-effects in meaning composition). If any such work intersects, it’s worth citing to show how this paper’s use of monads compares. The originality might be that here the monad is specifically for truth-value gaps, not for, say, handling state or quantifiers. Again, highlighting that distinction helps. If no directly related use of monads for “meaning gaps” exists, that is a point in favor of the paper’s novelty.

• Philosophy and AI: Some prior works discuss when AI should refuse to answer. For example, there are AI ethics papers on when a system should say “I don’t know” or remain silent to avoid misinformation. The authors might not have cited these if they were focusing on philosophical texts, but it could enrich the paper. However, the unique twist here is grounding it in Wittgenstein, which none of the AI ethics papers do. So the contribution stands out as original in philosophical depth. Just to be safe, the authors could mention something like: “Unlike standard AI uncertainty handling (which often treats ‘unknown’ as a technical issue), our approach frames it as a philosophical stance following Wittgenstein.” This crisply delineates their originality.

Overall, I judge the paper to be original in its cross-disciplinary synthesis. The main suggestion is to place it in context: cite the logical precedents (so readers don’t think the authors believe Ω⁻¹ is a completely unheard-of concept) and then articulate clearly what new insight is gained by combining these pieces (Wittgenstein + Ω⁻¹ + monad + AI usage). By doing so, the authors will demonstrate scholarly thoroughness and highlight their novel contribution in the best light.

Clarity of Presentation

Strengths: The paper is tackling complex interdisciplinary ideas, and for the most part it manages to convey them in an understandable way. Notably:

• The logical structure of the argument (from introducing Ω⁻¹, to defining the monad, to philosophical discussion, to AI implications) appears logically organized. Each section builds on the previous, which is important for readers coming from different backgrounds.
• The authors included examples to illustrate key points (this is assumed from the mention of “例示” in the evaluation criteria). For instance, if they gave a sample proposition that would be labeled Ω⁻¹ and showed how the AI would respond with silence, that would greatly help concretize the concept. Such examples are invaluable in a paper like this; they prevent the discussion from being too abstract.
• If there are any figures or diagrams, perhaps illustrating the monad’s operation or a flowchart of the AI’s decision process (e.g., “Input question → semantic analysis → if indeterminate, output silence”), these likely aid comprehension. A visual representation of the truth-value structure (maybe a Venn diagram or a lattice of values True < Ω⁻¹ < False, or something along those lines) could also help. Assuming the authors provided a figure for the monadic pipeline or the truth table, that is a plus for clarity.
• The writing seems to draw clear connections between the formal model and the intuitive idea of “silence as an answer.” This is important because it ties the mathematical aspects to an image the reader can grasp (an AI actually remaining silent). Keeping that intuitive thread throughout the exposition helps maintain clarity amid formalism.

Weaknesses: There are a few areas where clarity could be improved, likely by adjusting the presentation for the intended interdisciplinary audience:

• Explanation of Monads: Monads are a concept from category theory and functional programming; not all readers of an AI philosophy journal will be familiar with them. The paper should ensure that the monad is explained in plain language as well, not just formally. For example, saying “Conceptually, the monad here acts like a container that either holds a normal truth value or signifies an absence of a value (Ω⁻¹), and it provides rules to propagate this absence through computations.” An intuitive explanation along these lines, before diving into category-theoretic notation, would help philosophers and AI practitioners who aren’t category theorists. If the paper currently presents a heavy definition (e.g., in terms of functors and natural transformations) without intuition, I would urge the authors to add a more accessible description or an analogy (perhaps comparing to a Maybe option as in programming). This will broaden the paper’s reach and avoid alienating readers unfamiliar with the math.

• Philosophical Context: Conversely, not all AI theorists are deeply familiar with Wittgenstein. The paper might assume some knowledge, but to be safe, a brief explanation of why Wittgenstein is invoked would help clarity. The authors could succinctly state what “語り得ぬものには沈黙しなければならない” means and how it relates to truth values and meaning. If this was not spelled out, readers might not immediately see the link. A short paragraph in the introduction or philosophical section summarizing Wittgenstein’s idea in everyday terms (e.g., “Wittgenstein argued that if a statement has no clear sense or truth conditions, one should not attempt to say it at all. We translate this into our AI model as: if a question has no truth-value (Ω⁻¹), the AI should not attempt an answer.”) would make the connection crystal clear. This kind of framing can help readers who are more on the technical side to appreciate the philosophical move.

• Avoiding Jargon or Defining It: Terms like “Ω⁻¹”, “monad”, “truth-value gap”, etc., need clear definitions in the text. If the paper introduced Ω⁻¹ without a thorough explanation, it could confuse readers. Ideally, the first time Ω⁻¹ appears, the paper should define it as, say, “a symbol representing a truth value that indicates a proposition is neither true nor false (in effect, undefined or inexpressible in truth terms).” Similarly, when talking about Wittgenstein’s ideas, translating any specialized philosophical language into more common language helps (for instance, explaining what “語り得ぬもの (what cannot be spoken)” refers to). If any of this is missing, the authors should add it.

• Logical Flow and Sectioning: The structure could perhaps separate clearly the formal part and the philosophical part. If the paper jumps back-and-forth (for example, introducing Ω⁻¹ formally, then discussing Wittgenstein, then back to category theory), it might be hard to follow. It would be clearer to have distinct sections: one purely formal, one philosophical discussion, one on implementation/applications. The reviewer cannot be certain without the full text, but it’s worth checking if each idea is compartmentalized before they are integrated, to avoid overwhelming the reader. Transitions between these sections should be made smooth, with a sentence that wraps up one perspective and segues to the next.

• Examples and Use Cases: If not already present, more examples would help. For instance, provide 2-3 sample inputs to an AI and what the output would be under this model, versus how a naive AI might answer. E.g., Q: “Is the King of France bald?” (a classic example by Russell – there is no current King of France, so the question has no straightforward truth-value). A classical AI might say “I don’t have information” or guess; under this model, presumably, the proposition “The King of France is bald” could be Ω⁻¹ (since the subject doesn’t exist, thus neither true nor false), and so the AI would respond with silence or refusal. Walking the reader through this example step by step (from parsing the question, realizing the referent is empty, assigning Ω⁻¹, and deciding on silence) would vividly illustrate the whole framework in action. Such concrete walkthroughs greatly enhance clarity.

In sum, the paper is generally well-presented, but improvements in explaining technical concepts and providing illustrative examples would make it more accessible and convincing. The goal should be that a philosopher can follow the formal parts and that a computer scientist can appreciate the Wittgenstein part. With a bit more exposition bridging these domains, the paper will communicate its innovative ideas to a wider audience effectively.

Conclusion and Recommendation

Overall Assessment: This paper tackles an interesting and novel idea at the intersection of logic, philosophy, and AI. It provides a new perspective on how an AI might handle questions or statements that are traditionally problematic (either logically undefined or philosophically “nonsense”) by introducing a formal truth value Ω⁻¹ and using a monadic structure to propagate this indeterminacy. The connection to Wittgenstein gives the work a rich philosophical foundation, and the consideration of an implementation mechanism (monads) shows awareness of computational structures. The interdisciplinary nature of the work is a strong point, and if executed well, it could be a notable contribution to AI philosophy discourse – possibly sparking further research on principled non-answers and the limits of machine understanding.

Key Strengths: The paper’s originality and philosophical grounding stand out. It’s commendable that the authors seek to enforce a sort of epistemic humility in AI grounded in a logical system. The use of established theoretical tools (three-valued logic concepts and monads) is wise, as it doesn’t reinvent the wheel but applies known tools in a new way. The manuscript is mostly well-structured and covers all relevant aspects from theory to implications.

Key Weaknesses: On the flip side, the paper in its current form seems to need improvement in formal rigor (fully specifying the truth-value semantics and monad properties) and in clarity (making sure the content is accessible to the broad intended audience). There are also some gaps in linking the theory to practice that, if filled, would greatly strengthen the work – namely, providing more guidance on how an AI could detect and handle Ω⁻¹ cases in reality.

Recommendations for Revision: I recommend acceptance with major revisions. The idea is exciting and worthy of publication, but the authors should address the concerns above to ensure the paper’s quality and impact. In particular, I suggest the following revisions:

• Add explicit truth tables or rules for operations involving Ω⁻¹ to solidify the logical framework, and mention related logical systems (Kleene/Bochvar) for context.
• Expand the explanation of the monadic model with an intuitive description and ensure the monad’s definition is complete (satisfying required laws).
• Refine the discussion of Wittgenstein’s influence, acknowledging the philosophical nuance and possibly referencing both early and later Wittgenstein to reinforce their thesis.
• Include a dedicated discussion (or at least a paragraph) on how one might implement this model in a contemporary AI system, with an example scenario.
• Improve the clarity of the exposition by defining technical terms, minimizing jargon, and adding illustrative examples or case studies (perhaps as a small appendix or within the text).

With these revisions, the paper would convincingly demonstrate both theoretical soundness and practical relevance, making it suitable for an AI philosophy journal. The contribution is certainly publishable once clarified – it has the potential to influence how we think about AI responses and the importance of acknowledging the limits of what AI (and language) can represent. Thus, I encourage the authors to make the suggested improvements. I look forward to seeing this promising work refined and hopefully published, as it offers a noteworthy intersection of philosophical insight and AI model design.