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Logical


How to define Agnosticism?



A:     
Both a believer and an atheist are believers. Each of them accepts certain axiomatic truths as his/her own faith and the rest follow from those truths. That's why I am an agnostic. Ali's posting is a good example: you believe certain info at the outset (for example, that everything in Quran is true — beyond questions), the rest become so evident then.
B:
What worries me is your implicit definitions of theism and atheism. My understanding is that you want to say that the theists accept certain axioms regarding God or something similar. Perhaps 99.99% theists are like that. But I would like to call them God-believers, not theists. I think being a theist or an atheist (or even an agnostic) should not be a matter of axioms, rather it should be a matter of conclusion. Take Σ to be an axiom set (for an individual) and G is the statement "There is a God". According to your definition atheism means not-G is in Σ. I am not clear how you would define agnosticism then. Very likely you would say that in case of agnosticism neither G nor not-G is in Σ. But by that count Euclid's Geometry will be a case of agnosticism too. If you think in that way then you cannot define an -ism by what is included or not included in Σ. I am an atheist but there is no not-G in my Σ . By the same count an enlightened theist may claim that there is no G in her/his Σ. I am an atheist because I think I can draw not-G as a conclusion from my Σ. If I were a theist then I could draw G as a conclusion from my Σ. How can you fit your agnosticism here? You may say then the agnostic can draw neither G nor not-G as conclusion from Σ. Again, Euclidean Geometry will be Agnosticism by that definition. You cannot define agnosticism in either way: (1) Neither G nor not-G is contained in Σ (2) Neither G nor not-G can be drawn as a conclusion from Σ. Now apparently there seems to be a way out, if you like to define agnosticism by 3-valued logic. By 3-valued logic a sentence can have three truth-status, T (True), F (False) and ? (something neither true nor false). By that count agnosticism is a position if you can draw ?-G (which intuitively means that G is neither true nor false) as a conclusion. But to do that you have to find out a proof-theory suitable for three-valued logic.
A:
That's interesting! I never thought that way. But here is something to counter you (and I don't think that three-valued logic is a good option for an agnostic). That Euclidean geometry is not agnosticism because that set, I mean the very axiom set pertaining to that geometry, is not RELEVANT to draw either G or not-G as conclusion. Whereas Σ is a relevant set to draw that type of conclusion. So agnosticism can be defined as a certain position when somebody can draw neither G nor not-G from her/his Σ, which, I emphasize, is RELEVANT to producing such conclusion.
B:
I cannot disagree with you. But can you make this idea of  "relevance" more precise?
A:     
Not at the moment, though we can take it as something intuitively clear.
B:     
Yes, I must admit that on that particular respect my intuition converge with yours. But,.... yes..... sometimes I do have some undecided things. I assume that agnosticism is a position saying "this is not decidable from my Σ".
A:     
Yes, decidability is always enframed within a frame of relevance — so to speak.
B:     
Fine! For example — what I was saying — I am not decided on the abortion issue. But that is a moral issue — though some factual issues are relevant there — for example where to draw the line between killing and not killing. But is God, I mean G or not-G, a moral issue too?




Introducing Logic to Beginners



What is Logic? Well, I think you may already – though not very clearly – know it. Logic is something about arguments, or you can say it is a discipline about arguments. But we have to be more precise. It is not exactly a study about arguments tout court. We rather look for something subtler pertaining to arguments: why or how do we say an argument is correct or – more technically valid, and why or how do we say an argument is faulty or fallacious? Logic is something about that very subtlety pertaining to arguments.

An argument is faulty or wrong if you can refute it. How can you refute an argument? Here is an example. Imagine two persons: Beth who is a female and a feminist, and Ali who is a male and not very sympathetic to feminism. They have a dispute.

"It is very natural", says Ali,"that man dominates over woman"."For", he continues, "man is stronger than woman".

Beth couldn’t agree. "So, elephants should dominate over men; surely elephants are stronger than man", Beth retorts, sounding a little ironical.

Ali sees Beth’s point, he is – let us imagine – a rational man.

Now what is that Ali can see, and Beth can successfully show Ali? Beth, actually, brings up a parallel or similar argument. Ali’s argument – let us put it a little niftily – is:

Ali’s particular argument

        Men are stronger than Women

         Men dominate over Women

Parallally, Beth’s argument – which we shall call B-argument – is:

Beth’s particular argument

Elephants are stronger than Men

             Elephants dominate over Men

Here is the point or the subtlety that both Ali and Beth seem to agree upon: if we accept Ali’s particular argument then we should accept Beth’s particular argument, and vice versa. And furthermore, if any of these particular arguments is not acceptable then that indicates that there is something grossly mistaken: the very argument is invalid. Note a distinction is emerging here, on one hand there are "particular arguments" (those of Beth and Ali respectively) and on the other there is "the very argument" – which seems to be a little abstract and general. The point is this: if both the particular arguments are similar then they both share a general form, we may call the logical form of both the particular arguments. An argument, or more correctly, a particular argument has a general form; and the argument is correct or valid only if all the particular arguments sharing the general form are acceptable. In other words, a particular argument is valid if all the similar arguments – which are similar to that particular argument – are acceptable ( we assume that a particular argument is similar to itself). An allegedly correct argument can be shown to be incorrect by just citing an example which is similar to the allegedly correct argument but not acceptable. We may call the example counter example of the original argument. In other words an argument is invalid if it has counter example, if it is valid then there cannot be any counter example. Citing a counter argument is a good way of refuting an argument.

We may wonder now: what is the general form of both the particular arguments of Ali and Beth? Well! we may present it as follows.


The general form of Ali’s particular argument and Beth’s particular argument

X are stronger than Y

X dominate over Y

X and Y are a kind of variables,which – here – range over various classes or categories: like those of men, women, elephants and so on. They are – I would like to call them – soft part of the general form. The rest are – let us call them – hard part of the general form. So "are stronger than" as well as "dominate over" are hardparts; they are a kind of constants – or, perhaps in a broader sense – logical constants, which remain invariable across all the particular arguments.                                                                [ A note. We wrote "X are stronger than Y".But instead we could equally write "Xs are stronger than Ys " or "X is stronger than Y". The former sounds grammatically better – as because it duely shows the (concorded) number-inflexions of English nouns and verbs. But, this grammatical aspect is not relevant to logic, and we will hardly pay any attention to that.]

It turns out now – let us note – that the correctness or validity of a particular argument involves a general form of the argument and as well as other similar particular arguments, and the general form in turn has soft part as well as hard part. This way of seeing validity is the model-theoretic approach to logic; all the particular arguments sharing a common general form are just different models (by dint of having different values for the variables) of the general form. With this approach we need to have strong imagination – for we have to find out counter examples using our imagination and there seems to be little rule-following guidance for that. There is also another approach, for determining validity, which relies – at least apparently – more on rule-following guidance. This latter approach may be called proof-theoretic method. (Of course, we should remember, that the distinction between model-theoretic and proof-theoretic is more a modern phenomena).

We will now move to Aristotle’s way of dealing with some special kind of arguments, the arguments which involves classes or categories (Aristotle must have preferred the latter term) as soft parts, and certain class relations as hard parts. Take two classes X and Y. They may either overlap or do not overlap with each other, or in other words one of them may include or exclude the other. Furthermore this relation of overlapping/not-overlapping, or the relation of inclusion/exclusion can occur in two ways: completely or partiallly. So, eventually, we have criss-crossing of two binary modes: between inclusion/exclusion on one hand and completely/partly on the other. The result is a classification of four – as 2x2 = 4 – types of hardparts. This can be shown in the following table



overlap

or

inclusion

do not overlap

or

exclusion


complete

All Xs are Ys


A(X,Y)

No Xs are Ys


E(X,Y)


partial

Some Xs are Ys


I(X,Y)

Some Xs are not Ys


O(X,Y)

Note – as shown on the table – the four types of hardparts or hardpart formats are symbolized as A(X,Y), I(X,Y), E(X,Y) and O(X,Y) respectively; we may also write simply A, E, I, and O – if we can ignore the softparts. The sentences with these general forms are known as categorical propositions. (Warning. Here the adjective "categorical" is quite different from the "categorical" of Kant’s "categorical imperative". "Categorical" in the latter sense means "unconditional", which is the standard meaning; but in our present sense "categorical" means "pertaining to categories or classes". ) Note that the hardparts of categorical propositions are different from the hardparts "X are stronger than Y" and "X dominate over Y"; sentences having the latter general forms are not categorical propositions. There are special kind of particular arguments which have categorical propositions as premises – in fact there should be two such premises – and categorical propositions as conclusion. Such particular arguments are known as syllogisms. Here are two examples:

Syllogism 1

Mammals are animals

Men are mammals

Men are animals

Syllogism 2

Some mammals are animals

Some men are mammals

Some men are animals

Now comes the question of validity. Are these valid syllogisms? Answer: Syllogism 1 is valid, but not Syllogism 2. But how and why? We will check it by model-theoretic means: that is by dint of finding counter examples. First, we abstract out the general forms of the respective syllogism – as shown below.

the general form of Syllogism 1

All X are Y

All Z are X

All Z are Y

the general form of Syllogism 2

Some X are Y

Some Z are X

Some Z are Y

Take the the general form of Syllogism 1. Is it possible to have three classes X, Y and Z so that X is included in Y and X again includes Z but – having that scenario – Z is not included? We have to use our imaginative faculty now; the following diagram might be helpful.

 

So, is it possible to have three classes X, Y and Z so that X is included in Y and X again includes Z, but – having that scenario – Z is not included in Y? No it is impossible. We cannot have three classes X, Y and Z so that X is included in Y and X again includes Z, but Z is not included in Y. We cannot have any counter examples like that? Syllogism 1 is, therefore, a valid argument.

Syllogism 2

Some mammals are animals

Some men are mammals

Some men are animals

the general form of Syllogism 2

Some X are Y

Some Z are X

Some Z are Y

We will now move to Syllogism 2. Is it possible to have classes X Y and Z so that some Xs are Ys and as well as some Zs are Ys but no Zs are Ys? Again resorting to diagrams is a good aid. The following diagram shows that such situation, which must be a counter-example of Syllogism 2, is possible.

Some fashion-models (=: X) are, for example, females (=: Y). And some males (=: Z) are fashion-models (=: X). But that doesn’t mean that some males(=: Z) are females (=: Y). In fact no males are females. Thus we have a counter-example,and Syllogism 2 turns out to be invalid. But note that all the premises and the conclusion of Syllogism 2 are true, despite that it is invalid. Lesson: having all the premises and conclusion as true does not render an argument valid.

The point is: there is a kind of necessity connection between a conclusion and the corresponding premises. But this – the necessity connection – doesn't require that anything whether it is one of the  premises or the conclusion to be true. Interestingly, we do often talk about the necessity connection when we see an argument is invalid. Ali and Beth, for example, could dispute like this.

Ali         It is very natural that man dominates over woman. For man is stronger than woman.

Beth :     Not necessarily!

This time – unlike what she did in our previous example – Beth doesn’t ironically remark, "So, elephants should dominate over men; surely elephants are stronger than man". Instead she is taciturn, but she seems to be certain that Ali’s argument is not valid; the argument is not too tight to stop counter examples.


In fact it is very much possible that an argument can be valid having its conclusion and as well as its premises false. Consider this.

Syllogism 3

Muslims are Hindus

Christians are Muslims

             Christians are Hindus

Nothing is true here, but still it is valid, the necessity connection is there. The whole point of what we called model-theoretic approach is that if an argument is valid then there cannot be a model where the conclusion can be false given all the premises are true, or in other words if only ( it is a big "if", a very critical point ) the premises can be true in any model then the conclusion must be true in that model. So, if by some means or (re-) interpretation or modelling or whatever (but we won’t enter in that technicality here) it becomes true that Muslims are Hindus and Christians are Muslims then there (which is a model, or an interpretation, or a possible-world) Christians must be Hindus.



Ċ
Shaheen Islam,
Jan 1, 2010, 8:15 PM
Ċ
Shaheen Islam,
Jan 1, 2010, 8:19 PM
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