量子力學x輪迥x外星人 v0.0.1

甚至係現在, 同一個時間可以有同一個靈魂喺兩個唔同嘅人度

亦難怪難咁多

政治便是一條大賓周，每個人都要幫手含撚

所有ring都有subring of any other ring that is isomorphic to integer ，因為map完兩個identity，就會有延伸出去f(1+1), … ，咁integer其實可以叫做building block

雖然都冇人用中文做科學學術研究

夏日時空啲plot twist同故仔完整性好勁 歌好聽

其實幾鍾意啲風格 啲人係引導人慢慢諗出嚟

3D 螺旋
https://youtu.be/PVPk_HxgDeE?si=FHpF3uM3CBRB7yG7&t=80

多謝兩位老師的教導，古人遠比今人進步許多，我哋都係失傳不合格的後代
所有嘢都係好似torus咁由外到內到外，我哋就以為係人生哲學曲解咗佢，其實係物理學，同我講嘅輪迴即係量子糾纒論有異曲同工之妙
所有野都要有其各自position，所有能量都會一出一入

我如今都鍛鍊緊自己嘅肛門
為了迎接源源不絕的生命能量做好準備

等我幫你注入生命能量

True: if there exist at least one structure and it is true in every assignment function of that structure

Valid: true under every structure and every assignment fucntion

Tautology: true under every possible truth assignment of all variable in propositional logic

Free variable: the truth is indeterminate, hence it is determined by structure

Substituable: the variable of that term being substituted is either not free(which means the scope restricts what substitution can assign to), or otherwise, substitution variable itself does not coincide with variable with quantifier, I would not mention any other case of formulas)

Valid and tautology may seem like they are pretty the same, but tautology can be interpreted as propositional reduct of valid formulas(which has 5 kinds of formulas), therefore if that is tautology, it must be valid, but if it is valid, it doesn’t have to be tautology, however, if it is not valid, it is not tautology

v(conclusion) = T, then conclusion in formula form(consider 5 different case）is true in every structure
For that particular structure, formula is false in some assignment function, hence after being propositional reduct(consider 5 different case), it is possibly be false, therefore it is not tautology

Logically entail: if the antecedents is valid, and the conclusion is valid, then antecedent deduces conclusion under every structure, since empty set is always true, ie. it can never be falsified by any counterexample, then it always true, so I can say empty premise set |= conclusion

Propositional consequence and tautology:
WTS: if we eliminate the situation that v(conclusion) can never be false when all premises are true, then it must be tautology
by def, propositional consequence, means every truth assignment makes each premises true also make conclusion true
First, we express every formula in premise set in propositional reduct form, eg. v(Ai) = T for each i, and express conclusion as B, by def above, v(B) = T, hence it eliminate the situation that conclusion is false, hence propositional consequence->tautology
WTS: if it is tautology, then it is propositional consequence
By def of tautology, conclusion must never fail
It can be proven by tautology->valid, express conclusion in 5 cases of formula, and prove each case by using the satisfaction definition in every structure with every assignment function, and each formula in premise set is true is one of the situation that it is true, hence tautology -> propositional consequence

Soundness: If there is a way to prove it, thats true, since non-logical axiom is indeed true, logical axiom is always valid(identity, universal/existential quantifier axiom), rule inference is true if thats true in each formula(universal/existential quantifier rule and structure satisfies conclusion(logically entailed) if every structure satisfies antecedent(valid) and antecedents satisfies conclusion(propo->tauto->valid) ）

睇到2.7，唔知上述有冇錯

仆街，讀咗兩年phy先發現中伏，好似揾唔𠹌到工
GPA高好似都無咩用，讀到phd應該都爭唔到教席
係咪應該讀定啲CS course先，第時讀master轉行

我唔係金主
如果唔係我請你補phy
前途=CS
興趣=Phy

突然發現自己只是凡人

痴線 你兩個都咁夜都唔訓

免費幫你補
諗返起以前中學，有個教bio嘅miss係hku phd，佢話自己研究無funding所以要走嚟教書