牛某
最近我講緊康德會點回應當時嘅宗教，有興趣可以睇睇
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我以前讀哲學概論同西哲史
就會睇柏拉圖對話錄
可以睇入門果幾篇Apology, Meno, Crito, Phaedo, Symposium
或者再even睇左一D對話錄導讀先再睇原文
就會睇柏拉圖對話錄
可以睇入門果幾篇Apology, Meno, Crito, Phaedo, Symposium
或者再even睇左一D對話錄導讀先再睇原文
非台
其實要分返開 truth in a model 同 truth
truth in a model 只不過係一個理論入面嘅概念
呢個概念唔係 truth 本身
model 本身就係一種 idealized 嘅野，用嚟解釋 real language 嘅現象
real language (i.e., 自然語言) 語句嘅真假，唔係 relative to a model
而係純粹 true/not true
我諗你可能係撈埋咗幾個問題嚟問
如果你想問有冇 formula 係 true in all models
Assume 係 classical logic，咁答案係有
就係 classical validities (熟悉嘅例子有，e.g. Modus ponies, excluded middle, ...）
亦都冇咩值得迷思
如果你係問有冇 formula 係所有有可能出現嘅 formal logic 都係 true in all model
咁答案係冇
因為有 logic 冇 任何validities
你亦可能係想問real language入面有冇 necessary truth
或者 real language 入面，某啲似乎係 necessary truth 究竟係咪真係 necessary
咁呢個問題唔會直接搬套 logic 出嚟 settle 到
truth in a model 只不過係一個理論入面嘅概念
呢個概念唔係 truth 本身
model 本身就係一種 idealized 嘅野，用嚟解釋 real language 嘅現象
real language (i.e., 自然語言) 語句嘅真假，唔係 relative to a model
而係純粹 true/not true
我諗你可能係撈埋咗幾個問題嚟問
如果你想問有冇 formula 係 true in all models
Assume 係 classical logic，咁答案係有
就係 classical validities (熟悉嘅例子有，e.g. Modus ponies, excluded middle, ...）
亦都冇咩值得迷思
如果你係問有冇 formula 係所有有可能出現嘅 formal logic 都係 true in all model
咁答案係冇
因為有 logic 冇 任何validities
你亦可能係想問real language入面有冇 necessary truth
或者 real language 入面，某啲似乎係 necessary truth 究竟係咪真係 necessary
咁呢個問題唔會直接搬套 logic 出嚟 settle 到
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不過講開又講
「如果你想問有冇 formula 係 true in all models」我當呢個比較弱啲，而你亦答咗
呢種truth係with respect to model嘅true(varied by interpretation of domain)
咁我應該分開後兩個嚟問，
一、is there exist a formula that is constituted by any possible formal logic(like paraconsistent logic/classical logic) which is true in all model
二、for a real langauge itself, is there any necessary truth?
我不如集中問前者點解冇
「因為有 logic 冇任何validities 」，真係有？
呢種truth係with respect to model嘅true(varied by interpretation of domain)
咁我應該分開後兩個嚟問，
一、is there exist a formula that is constituted by any possible formal logic(like paraconsistent logic/classical logic) which is true in all model
二、for a real langauge itself, is there any necessary truth?
我不如集中問前者點解冇
「因為有 logic 冇任何validities 」，真係有？
:^(
:^(
非台
//「因為有 logic 冇任何validities 」，真係有？//
Strong Kleene 3-valued logic 冇 validities
(一個經常被用嚟 invalidate excluded middle 嘅 logic)
Strong Kleene 3-valued logic 冇 validities
(一個經常被用嚟 invalidate excluded middle 嘅 logic)
非台
希望唔會講得太難明
1. 首先要講下 connectives 點 evaluate
[url]Strong Kleene truth-table[/url]，可以睇呢頁嘅Kleene (strong) K3 and Priest logic P3 個 part
https://en.wikipedia.org/wiki/Many-valued_logic
簡單嚟講，let's say 我地用 v(A) 表示 formula A 嘅 value
Classical logic 嘅 truth values，我地可以用 1, 0 表示平時嘅 T, F 二值
而家 3-valued logic, 就多咗 1/2 (先不理 1/2 代表乜)
Negation 嘅 truth table 其實等同： v(¬A) = 1 - v(A)
如果係 v(A) 係 classical value，咁 v(¬A) 都係同 classical logic 一樣情況
如果 v(A) ＝ 1/2, 咁 v(¬A) = 1/2
Conjunction 嘅 valuation scheme 就係: v(A ∧ B) = min(v(A), v(B))
如果 v(A), v(B) 係 classical value, 亦都同 classical logic 一樣,
let's say v(A) = 0， 咁已經知道 v(A ∧ B) = 0
多咗 1/2 其實 v(A ∧ B) 都係 depends on v(A), v(B) 之中最低 value果個
咁其他 connectives 都可以由 negation 同 conjunction 去界定
其實同 classical logic 嘅定義一樣
所以就唔特別講
A v B := ¬(¬A ∧ ¬B) [A v B即係唔會出現 ¬A 同 ¬B hold 嘅情況)
A → B := ¬(A ∧ ¬B）[A → B 即係唔會出現 antecedent holds ∧ consequent fails 嘅情況]
2. 咁其中一個重點嚟喇
let's say 我地係用 propositional language
係所有 atomic formula 都取 1/2 值嘅 model,
你可以 prove (by induction) 所有 formula 都係 1/2 值
3. 咁要知咩叫 validities 又要講 logical consequence
Let's say Γ 係 一個 formulas 嘅 set
Γ ⊨ A 意思: Γ logically entails A
⊨ A 意思係 A 係一個 validity，即唔理 Γ 有咩 formula，或者係 empty
A 都係 valid
咁 Strong Kleene 3-valued logic (K3) 嘅 logical consequence：
Γ ⊨ A iff 如果所有係Γ嘅 formula 都係取 1 值，A 都係取 1 值
4. 以下呢度同點解K3 冇 validities 冇關，可以先略過再睇返：
你諗多兩步，你可以想像到我地可以有 K3 consequence 以外嘅 consequence 定義，例如
Γ ⊨ A iff 如果所有係Γ嘅 formula 都係取 1, or 1/2 值，A 都係取 1 or 1/2 值
呢個就係另一套叫 logic of paradox LP 嘅 consequence
當然仲有其他組合，如果我地唔 require premise set Γ 同 conclusion 符合同一個標準，譬如你可以對 premise set 要求佢一定要取 1，但 conclusion 就可以鬆啲，取 1 or 1/2 都 ok, 咁又會得出另一套叫 strict-tolerant logic ST
仲有一套叫 tolerant-strict logic TS就係調返轉
(當然仲有其他可能比較武斷/離奇嘅可能性，但K3, LP, ST, TS 就係最 popular)
5. 因為有一個 model 係所有 formula 都係 1/2
咁呢個就係所有 formula 嘅 K3-counterexample
所以，K3 冇 validities
(但佢有 valid inferences, e.g modus ponens)
1. 首先要講下 connectives 點 evaluate
[url]Strong Kleene truth-table[/url]，可以睇呢頁嘅Kleene (strong) K3 and Priest logic P3 個 part
https://en.wikipedia.org/wiki/Many-valued_logic
簡單嚟講，let's say 我地用 v(A) 表示 formula A 嘅 value
Classical logic 嘅 truth values，我地可以用 1, 0 表示平時嘅 T, F 二值
而家 3-valued logic, 就多咗 1/2 (先不理 1/2 代表乜)
Negation 嘅 truth table 其實等同： v(¬A) = 1 - v(A)
如果係 v(A) 係 classical value，咁 v(¬A) 都係同 classical logic 一樣情況
如果 v(A) ＝ 1/2, 咁 v(¬A) = 1/2
Conjunction 嘅 valuation scheme 就係: v(A ∧ B) = min(v(A), v(B))
如果 v(A), v(B) 係 classical value, 亦都同 classical logic 一樣,
let's say v(A) = 0， 咁已經知道 v(A ∧ B) = 0
多咗 1/2 其實 v(A ∧ B) 都係 depends on v(A), v(B) 之中最低 value果個
咁其他 connectives 都可以由 negation 同 conjunction 去界定
其實同 classical logic 嘅定義一樣
所以就唔特別講
A v B := ¬(¬A ∧ ¬B) [A v B即係唔會出現 ¬A 同 ¬B hold 嘅情況)
A → B := ¬(A ∧ ¬B）[A → B 即係唔會出現 antecedent holds ∧ consequent fails 嘅情況]
2. 咁其中一個重點嚟喇
let's say 我地係用 propositional language
係所有 atomic formula 都取 1/2 值嘅 model,
你可以 prove (by induction) 所有 formula 都係 1/2 值
3. 咁要知咩叫 validities 又要講 logical consequence
Let's say Γ 係 一個 formulas 嘅 set
Γ ⊨ A 意思: Γ logically entails A
⊨ A 意思係 A 係一個 validity，即唔理 Γ 有咩 formula，或者係 empty
A 都係 valid
咁 Strong Kleene 3-valued logic (K3) 嘅 logical consequence：
Γ ⊨ A iff 如果所有係Γ嘅 formula 都係取 1 值，A 都係取 1 值
4. 以下呢度同點解K3 冇 validities 冇關，可以先略過再睇返：
你諗多兩步，你可以想像到我地可以有 K3 consequence 以外嘅 consequence 定義，例如
Γ ⊨ A iff 如果所有係Γ嘅 formula 都係取 1, or 1/2 值，A 都係取 1 or 1/2 值
呢個就係另一套叫 logic of paradox LP 嘅 consequence
當然仲有其他組合，如果我地唔 require premise set Γ 同 conclusion 符合同一個標準，譬如你可以對 premise set 要求佢一定要取 1，但 conclusion 就可以鬆啲，取 1 or 1/2 都 ok, 咁又會得出另一套叫 strict-tolerant logic ST
仲有一套叫 tolerant-strict logic TS就係調返轉
(當然仲有其他可能比較武斷/離奇嘅可能性，但K3, LP, ST, TS 就係最 popular)
5. 因為有一個 model 係所有 formula 都係 1/2
咁呢個就係所有 formula 嘅 K3-counterexample
所以，K3 冇 validities
(但佢有 valid inferences, e.g modus ponens)
不過講開又講
1以前聽你講過，我亦都唔對1/2做解釋，當佢係syntactic嘅truth value算，而且佢都preserve咗classic logic嘅0和1嘅運用
2都明
3-4:
如果我冇估錯 我可以畫出一個表，
Let Gamma be {P}(premise set)
Let A be formula P or not P
喺classical logic, gamma |= A
喺Lp
If
v(P) = 0.5
Then
v(or(P,not P)) = 0.5
Antecedent
By iff, Gamma can logically entail A in Lp system
喺ST
If
v(P) = 0.5
Then
v(or(P,not P)) = 0.5
antecedent fail, conclusion true => true
Gamma can logically entail A in St system
However, in Strong Kleene 3 logic
If
v(P) = 0.5
Then
v(or(P, not P))= 0.5
By iff, Gamma can’t logically entail law of excluded middle
Not only in law of excluded middle, but also In general,
If all atomic formula is 0.5
Then all formula by induction is 0.5, 不論我取1- , min, max都一樣，就永遠唔會logically entail到任何validated嘅A，因為k3太強
上面有錯請指出
不過唔係幾明5，你係講緊 you can construct actual model係咁定係存在model係咁
2都明
3-4:
如果我冇估錯 我可以畫出一個表，
Let Gamma be {P}(premise set)
Let A be formula P or not P
喺classical logic, gamma |= A
喺Lp
If
v(P) = 0.5
Then
v(or(P,not P)) = 0.5
Antecedent
By iff, Gamma can logically entail A in Lp system
喺ST
If
v(P) = 0.5
Then
v(or(P,not P)) = 0.5
antecedent fail, conclusion true => true
Gamma can logically entail A in St system
However, in Strong Kleene 3 logic
If
v(P) = 0.5
Then
v(or(P, not P))= 0.5
By iff, Gamma can’t logically entail law of excluded middle
Not only in law of excluded middle, but also In general,
If all atomic formula is 0.5
Then all formula by induction is 0.5, 不論我取1- , min, max都一樣，就永遠唔會logically entail到任何validated嘅A，因為k3太強
上面有錯請指出
:^(
不過唔係幾明5，你係講緊 you can construct actual model係咁定係存在model係咁
:^(
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