A question on logic.

7 + 5 ALWAYs equal to 12...that is a kind of "reasoned truth" that we can be certain of, not like other conceptual beliefs such as what Descarte say the "dream argument". We never know whether we're dreaming or not, and thus, we can never tell whether things in the external world exist or not because they can totally be illusions derived from brain impulses.

回應 簡單地說：你可想像一個宇宙的質能變換律是 E=mc十次方，但就不可想像一個宇宙有如此的數學公式：1+1=9。

A reply to a question from Benson Simply speaking, without invoking technicalities (like analyticity) and certain protracted philosophical debates, facts like "E=mc^2" not only depend on the meanings of terms which appear in the statements concerned, but also have recourse to the situations of our world. In science, for example, observations and experiments are usually needed to verify such statements. However, the truth of those statments which arise in logic and mathematics are independent of the situations of our world. Indeed their truth is mainly related to the definitions and meanings of the symbols and concepts laid down by us. As in your example, one needs not (and cannot) resort to, say, observations, to check whether 1+1=2 is true since the truth of this statement is only related to the meanings of "1","2","+" and "=". So, sometimes we say that statements of this kind are true in any possible universe, which is a concept borrowed from modal logic.

Re: the Justification of Truths What "Wor Jai" says in his reply to Benson is based on the alleged analytic/synthetic distinction,i.e., that there are certain statements (analytic statements) which are true in virtue of the meanings of the terms they contain whereas the truths of other statements (synthetic statements) depend on the verification obtained from facts. But in his famous paper, "Two Dogmas of Empiricism," W.V. Quine argues that this alleged analytic/synthetic distinction is only a dogma, at most, an exaggeration. I won't go into the details of his argument here, but anyone who is interested in the topic can take a look at his essay. Here I only want to offer a conjecture: ultimately, some of the "foundational" beliefs (including the so-called analytic truths) cannot be justified, or they are only justified by our "intuition" (please dont' misunderstand me; I'm not appealing to some kind of transcendental realm here). Certainly, a naturalistic approach towards this problem looks promising. But not until a detailed account of how such beliefs can be justified via a naturalistic mechanism, the justification of these beliefs is not as secure as what some people claim. Thus, I won't say that "logical truths are true in every possible world" outside the introductory logic class.

>Here I only want to offer a conjecture: ultimately, some of the "foundational" beliefs (including the so-called analytic truths) cannot be justified, or they are only justified by our "intuition" Sorry you lost me. Could you elaborate a little bit more (in layman terms perhaps) please?

Several remarks to Benson's question I agree with some points made by Faustus. My previous reply alluded to the so-called analytic/synthetic distinction which stirred up vehement debates in philosophy, Undoubtedly, Quine's paper "Two dogmas of empiricism" was influential in that it indirectly brought logical positivism to demise. But I think with respect to Benson's question it is unnecessary to let such a controversy come in. Instead, I would like to make several follow-up remarks. (1)Many people feel insecure when they have no watertight criteria to guide them in, say, making decisions. To me the crux of A/S debate is somehow related to the "fact" that impeccable criterion that distinguishes A and S statements seems hopeless. But can we say that these statements are just the same in every aspect under all circumstances? Is there really no ROUGH borderline between them, even in the case of statements like "1+1=2" and "The earth is round"? (2)It might be better if we try to use terms like "necessary" or "contingent" rather than "analytic" or "synthetic" in this context. (3)Though logic and mathematics are paradigmatic in the sense that they exemplify what absolutely sound knowledge should look like, they might not be as perfect as one thought. But I think it is inappropriate to elaborate this point here. (4)I want to end my reply with a toy example. If you were asked to prove "1+1=2", then what would you do? Mathematicians usually won't care about this "silly" question. But this doesn't mean that you can't prove it. A standard and rigorous proof usually invokes definitions of "2" as the succesor of "1" and addition by recursion as below: m+0:=m, m+n':=(m+n)' So "1+1=2" could be reduced to "1+1=1+1". I think myraids of people will be convinced at this step. But one could still say "Why is "1+1=1+1" true? I still can't see it!". At this stage a patient guy may try to explain the meaning of "=" to the "problem-maker". "Wow! Don't you see that "1+1=1+1" is a statement of the form "A is A", which can't be wrong!" "What? Why "A is A" can't be wrong?" Every student who has taken a normal elementary logic course must know that "A is A" is one of the three "Laws of Thought" first stated by Aristotle. "Hey! Prove it". Can we do it? I don't think so. Why? It's because any alleged proof MUST more or less presuppose "A is A". "Then, what should we do?" I think a possible way would be to show the consequences of denying "A is A" to justify our acceptance of it, as suggested by J.Hospers. At the end of the day, what we can rely on is our ABILITY TO REASON, which is the gist of the "APTITUDE APPROACH"(賦能進路) by Mr.Lee. Can you conceive of a world in which A is not A"?

Of Justification Benson: Basically, what I mean is that some of our basic beliefs (as Wor Jai suggests, e.g., 1 + 1 = 2) is not subject to proof. They can only be justified by our "intuition". Either we "perceive" the truth of it or we don't. It is almost like colour (certainly, the two cases are not exactly alike). Either you see that the apple is red or you don't. You can't "prove" to others that the apple IS red (even if you bring in the physical constition of the apple, the reflective properties of its surface, etc.). At the end of the day, as Wor Jai says, it's our ability to reason which does the job. Looking at Wor Jai's reply, I think that my position is not much different from his. But I do have some comments: 1) Yes, I agree that even if the analytic/synthetic distinction does not hold, there are still some differences between the two; namely that the beliefs (synthetic) which are at the periphery of our "web of beliefs" are more subject to revision than those in the core (analytic) of the web. But as Quine points out, both kinds of statements should be subject to "the tribunal of experience". That is, in principle, both of them are subject to revision in light of recalcitrant evidence. 2) Thus, even if I agree that some of our beliefs are justified by intuition (or our ability to reason), I regard this to be a "contingent" matter that we, as human beings, happen to reason that way. It is not difficult to imagine that some creatures do not share our intuition or ability to reason. In that case, I am hesitant to say that they are simply "wrong" in their reasoning. 3) Yes, there are certain "laws of thought" that most of us share. But as pointed out above, these laws are only contingent upon the physiological constitution of our body (or perhaps, our brain). They are not really "necessary". For example, some able philosophers and mathematicians do deny the law of contradiction. They think that a para-consistent logic should allow contradictions (for reference, see works by Chris Mortensen, Graham Priest, etc.). 4) As I said in my last reply, a naturalistic approach towards this issue seems promising. To take Wor Jai's example, how can we prove that "1 + 1 = 2"? Most mathematicians believe that numbers are sets. But are sets real? Are they independent of human thoughts? Can we "see" numbers (as sets) in the external world? If anyone's interested, I recommend that she should read "Naturalism in Mathematics" by Penelope Maddy. In that book, Maddy argues that sets are real entities inside the world. If that is the case, it seems that after all, "1 + 1 = 2" can have a naturalistic foundation. However, that naturalistic argument does depend on the way the world is and our perceptual mechanism. Since I can conceive a world which is vastly different from ours and inhabited by creatures who have different perceptual mechanisms, I still think that "logical truths are true in all possible worlds" is not well-founded.

some comments on Faustus 1)和仔did not say that '1+1=2' cannot be proved. The proof of such proposition may not be done by u or me. But the study of mathematical logic is to provide a precise and adequate understanding of the notion of mathematical proof. Different mathematicians present different axiomatic theories to serve as a foundation for set theory. So, set theory are derived from axioms, but the problem is how to justify axioms?? The justifiability of axioms is based on what we called reason. But the use of the term 'intuition' is quite ambiguous both in perception .e.g perceiving colors and in reasoning .e.g the proof of '1+1=2'. The former case is perception and the latter case is reasoning, if we want to avoid the ambiguous term ' intuition' 2)We don't know the truth by perception, but we do know the world by perception, by our senses. We can perceive the property of a being. But truth is not a property of a being, say, 'some people are funny' then this statement is true if and only if some people are funny, this is the truth definitions by Tarski. 3)We have to agree that we, human beings, are the most rational beings amongs other beings in the world we known. If what u meant by 'wrong' is that we make empirically incorrect propositions, e.g. the sun goes round the earth, then both humans and other beings may commit the same kind of mistakes. But if it is fault in reasoning by the meaning of 'wrong', then i think humans should be more capable to judge what is wrong than other beings, say, if a monkey does a test on arithmetic and does it all wrong, then shall we say, ' since we are humans, happened to that way, and we are of different physical forms, we are hesitant to say that u are simply "wrong" in your reasoning.' 4)The so called 3 laws of thought are actually nothing but simple tautologies, if by laws of thought we meant tautologies, then we can easily write down some new 'law of thought' e.g (A-> (B-> (B->A))) which -> means imply My point is, there is no need to take the laws of thought to be so fundamental. 5)U mentioned a book 'Naturalism in Mathematics', i am very interested in that, do the author suggest that ALL set are REAL inside the world? if it is so, then is it REAL entities the set of all positive integers, or even null sets??

In reply to 翔 1) Certainly I agree that “和仔did not say that '1+1=2' cannot be proved.” As he said, “but this doesn't mean that you can't prove it. A standard and rigorous proof usually invokes definitions of "2" as the successor of "1" and addition by recursion as below: m+0:=m, m+n':=(m+n)'.” But when someone goes on to ask how we can justify the use of those terms in the first place and the relations among them, 和仔 does admit that “at the end of the day, what we can rely on is our ABILITY TO REASON.” Eventually, we are back to the ability of reason again. Either this argument is circular or the best we can do is by appealing to some kind of intuition which is indeed vague. But I don’t see how “ability to reason” is to a large extent clearer than “intuition”. 2) The concept of truth is quite puzzling and controversial. I agree (and I think no one will dispute) that truth is not an intrinsic property of things. But grounding truth entirely on the so-called “correspondence” (or matching, or verification) between propositions and facts does lead to many problems. There is a vast literature on this issue which is far beyond the scope of this discussion (I myself support the pragmatic theory of truth). 3) There is no doubt that we are the most rational beings among the known animals. What I am saying is that since the way how we reason is “contingent” upon the physiological constitution of our brains and it is possible that some other creatures (including some unknown ones) may have some other “ability to reason,” we do not have sufficient grounds to say that “logical truths (that is, truths according to us) are true in all possible worlds.” 4) Clearly, I am on your side about this. Laws of thought are not really fundamental. But the question is not just the laws (considered as expressions, literal or otherwise, of how we should reason) but also whether these expressions do correspond to some real regularities in the universe that we must follow. About that I am doubtful. 5) Yes, that’s a really interesting book. I do recommend reading it when you have time. Penelope Maddy. Naturalism in Mathematics. New York : Oxford University Press, 1997.

Thank you. Dear Faustus, 翔, 和仔: Thank you for all of your inspiring comments. As a graduate in physics, I know quite little about philosophy & logic. Though I still have my queries, I think I am in a better position now. It's such a great pleasure to meet you guys. Thanks again.

勁網友→勁網頁 這個網頁有如此質素如此水平的網友,稀有難得（本人只是偶然路經,不計在內）,尤其和仔、翔、Faustus三劍俠,勁甚,勁甚！

旁觀者兄过誉，此地臥虎藏龍﹐小弟不勝汗顏。

To Fauster 你說此處"卧虎藏龍",我同意,但你說我對你們的稱譽是"過譽"（該是漏了"譽"字？）,我就不同意了,試同其它許多網留言區比較一下,便知所言非虛。