Does bivalence assume truth values to preserve？

01-05

實在不好意思用英文寫問題，因為問題里「bivalence」，「preserve」和「assume」我都不太想得出合理的中文說法；還請見諒。
我是一所不知名大學的哲學系的二年級本科生，我最近在寫一篇論文。論文的要求是回應Michael Rea 2006年一篇論證「presentism, libertarianism和bivalence」不能共存的文章（「Presentism and Fatalism」, Australian Journal of Philosophy，Phil Paper鏈接在此： Michael C. Rea, Presentism and fatalism - PhilPapers）。我的motivation是希望有效地拒絕Rea的結論，希望在有presentism和bivalence的情況下拒絕fatalism。我之前跟教授談了這樣那樣的策略，教授也指出了這樣那樣的問題；最後，我在友人的幫助下，開始想有沒有可能target他對bivlanece的用法。

Rea提到的premise之一提到了這樣一個presmie「一個在t1為truth value為true的proposition，在t2時truth value也是true」*（Section 3."Fatalism"中的「(F2) Necessarily, if it was true at t* that Sally will stand up at t, then Sally stands up at t. 」）。我第一次寫這個問題時，印象里Rea說reject這條就是拒絕bivalence。但我在SEP上讀關於bivalence的東西時，只是提到Frege，說他在specifying 他的Begriffsschrift時，將「a sentence is (exclusively) either true or false」作為requirement。對比Rea說的和Frege定的，我不禁懷疑「一個proposition的truth value在時間中的真偽值不改變」是不是Rea自己的reading——還是classical logic本身就這樣規定過？
我上過我們學校開的Symbolic Logic，但是我們只講classical logic和如何進行文字和符號的翻譯，如何處理truth table和如何處理truth tree和如何做first order predicate logic的manipulation——並沒有具體講邏輯系統的分支或者它們意味著什麼。
我知道我有一個同學或許能解答這個問題，但她在備考GRE我實在不太敢打擾她，而且我真的很擔心她認為我之所以問她題就是因為她可以告訴我答案。我們學校教邏輯的教授也很神龍見首不見尾，我上次因為logistics找他，在他辦公室前程門立雪了大概兩個小時，也只在他上課前兩分鐘捉到了他；感覺他最近非常不願意host office hour。因為這樣的情況，我就只能來知乎問這個問題了。帶我這個論文的教授比較希望我reject presentism，因為她自己的研究比較傾向這個方向；她給我的論文也絕大多數都是reject presentism的。之所以說這些是因為，我感覺在知乎上提這個問題，很可能第一反應都是「你怎麼不問教授怎麼不自己看論文」；可能的確有我自己不夠努力的成分，但我現在的確比較疑惑應該往哪裡看。我知道這應該不是一個非常難的問題，但因沒有相關知識，非常希望有人可以回答一下這個問題。

再次感謝。
*我補充一下：友人向我普及過「Aristotle Sea Battle」，對於這個我不是很清楚如何回應，很大程度上正是因為我不知道classical logic是否真的assume了truth value會preserve。教授給我推薦了一篇她覺得pertinent的文章，但內容是「三個方法修改bivalence」。我不太知道是不是一定要modify bivalence才能carve out 關於preservation這一段。所以還是希望可以有人給我一些建議。謝謝。

In classical (propositional) logic, truth is preserved in the following sense: given a set of propositions ({p, q, r} for example), a "valuation", or a "truth assignment" is a function that takes each of the propositional variable p, q, r to a unique truth value (either "True" or "False") such that a propositional variable is assigned True if and only if its negation is assigned False. Note that the principle of bivalence is a semantic notion (notion about truth and falsehood in a given interpretation). A closely related syntactic (concerned with forms of sentences) notion is the law of excluded middle, which says that sentences of the form "p or not p" are always a valid inference from any set of premise.

Now back to your question. Classical logic "assumes" truth to preserve insofar as truth assignments are the only factor that determines the truth value of an atomic sentence (sentence with no logical connectives like "not", "and", "if", "or"). Note that classical logic is insensitive to tense (both the grammatical notion and the metaphysical notion of time) such that there"s no difference whatsoever between "he was alive", "he is alive", and "he will be alive" unless they are treated as atomic. The consequence of this (plus the principle of bivalence) is that statements about the future are either true or false, which, it seems to some people, gives classical logic a fatalist flavor. Indeed it seems counterintuitive to say "either it"s true that it"ll rain tomorrow or it"s false that it"ll rain tomorrow" instead of "it"s unknown whether it"ll rain tomorrow." A closely related consequence of ignoring tense is contrapositive inferences (from "if p the q" inferred "if not q the not p" and vice versa) of the following kind when it comes to counterfactuals: premise: if Kant didn"t die in 1804, then he still wouldn"t be alive today; conclusion: if Kant is alive today, then he would have died in 1804.

So citing time as grounds for rejecting bivalence appeals to the kind of consideration briefly sketched above. The argument is roughly this: if we want logic to capture our intuition that some propositions about the future are undetermined with regard to their truth value at the time of utterance, we must introduce a third truth value ("unknown" or "possible") so that "it"ll rain tomorrow" uttered on Jan. 1, 2017 has the truth value "unknown" on Jan. 1, 2017 but has the truth value "True" on Jan. 2, 2017 (supposing that it rained on that day).

Note that truth assignment is no longer the only factor that determines the truth value of an atomic sentence. What we need is a truth assignment relative to some time. This makes the resulting logic no longer classical. But truth is still "preserved" in the following sense: while a poposition can have different truth values at different times, for any given time a proposition has only one unique truth value. The kind of time-sensitive logic is often grouped under the umbrellla term "temporal logic". SEP has a nice article on that topic.

時隔這麼久，今天想起這件事挺有感觸的，我來update一下。其實我問這個問題的時候，論文已經交上去了。所以即使自己寫得非常錯誤，也沒有機會改了。但正是因為交上去了，才躍躍欲試地來知乎問大家。十分感謝各位的認真回答，我學到了很多很多。真的非常感謝。能問大家哲學問題實在是太好了。

今天的我仍然在好奇symbolic logic，下面這些話回復給當時提問的我自己

1."Classical logic assumes bivalence. According to bivalence, every sentence is either true or false. "

BUT

1.1"Bivalence does not even assume that we will know its truth value at some point in the future. Bivalence it is the claim that a sentence is either true or false, whether we know it or not."（從老師講課的PPT上抄的）

2.According to Burgess, "The only difference between sentences and "propositions" of real importance for our purpoes will be that sentences can change in truth value over time, where it is said that "propositions" can not (so that when a sentence changes truth value over time it is by expressing difference "propositions" at different times)". (從Princeton Foundations of Contemporary Philosophy的Philosophical Logic那本上的第3頁抄的)

3.Furthuer reading: Gaifman, Contextual Logic With Modalities for Time and Space https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/contextual-logic-with-modalities-for-time-and-space/2BAE1CB99A8D6B3EFAB9464B1E50D464

1. 先吐槽。題主寫論文前有沒有跟老師談過？題主作為二年級本科生，以及相關的知識儲備，至少我是不會建議你寫論文處理Rea的文章的。建議你去跟老師聊。

2. 「一個在t1為truth value為true的proposition，在t2時truth value也是true」--你這個具體是指哪個premise？Rea的文章我沒看過，剛才掃了一眼，沒找到這個。。。

3. 命題的真值會不會隨時間變化，這本身已經是個很複雜的問題了。不過討論Rea的文章可能不需要。我不確定你是不是在討論Rea文章里的（F1）。注意，（F1）里，那個命題是飽含著時間的，或者說時間是命題內容的一部分。這個命題如果真，那麼在任何時刻都真，如果假則在任何時刻都假。

題主，我認為你思考進了一條歧路。

這裡牽扯到了三種思想： presentism, fatalism和bivalence，問題中的那篇文章強調的是同時承認這三者會產生矛盾。

論文我讀過了，但這畢竟是你自己的工作，所以不能越俎代庖，只說說它的主要思路吧。

在fatalism那一節的論證：

通過bivalence得到了F1。
通過基本邏輯得到了從F1得到了F2，F2是最沒有爭議的。
F3說人類對於命題的真假無能為力。承認F3會導致fatalism。
b3在這個語境里被認為是真的。F4是前面b3和F3的直接推論，是fatalism。

由於b3和F2都是真的，那麼若想要讓F4的fatalism為假，意味著F1或F3得有一個是假的。

那麼現在問題就來了，文章論證了：

如果我們承認了presentism，在這個大前提下，承認了（滿足presentism的）bivalence，那麼我們就找不出合理的辯護來要求F3是假了。同時，如果還是處在presentism的大前提下，若否認fatalism，則找不出合理的辯護來要求bivalence是真。

結束。

這跟邏輯學基本沒關係。

具體的在留言吧。