# A = A . . . is it important?

In the Nihilism thread, Aleta has asked whether A = A is of real-world importance.

Given the depth of the breakdown in reason that we are seeing, I think this is important to take up.

I took a moment to suggest an answer, which I think I should headline:

______________

KF, 111: >>Aleta,

100: >>can anyone give an example of a logical argument that uses A = A to help advance the argument?

Examples from math are easy. In solving 2x – 5 = 17, students write 5 = 5 in order to “add 5 to both sides of the equation”, invoking a principle [–> an axiom held to be self-evident, in fact] from Euclid that “if equals are added to equals, then the wholes are equal.”

But is there any example of a use of A = A in a logical chain of reasoning about the real world that adds anything to the argument? A = A is true, but outside of pure logic, is it ever useful?>>

103: >>we use such things as the transitive property, logical conditionals, the law of the excluded middle, etc., all the time. However, A = A, it seems to me, doesn’t advance any argument, so I’m thinking it’s not used in real-world chains of reasoning as opposed to its uses in math such as I mentioned.>>

First, in thinking, arguing and speaking we must recognise and respect distinct identities if we are to be sound. Echoing Aristotle, truth says of what is [A], that it is [A = A], and of what is not [~A] that it is not [~A != A].

That immediately brings to bear all three laws: identity is recognised, antithesis between a thing and not the thing is respected, dichotomy is understood.

And, the three are inseparable.

We recognise distinct identity A, and the world partition:

W, the world

A in W, a distinct thing

~A in W, what is not that thing

W = {A|~A}, partition that exhausts W

LOI: (A=>A) = 1, where in this case single implication entails equivalence.

LNC: (A AND ~A) = 0

LEM: x in W => x is in A or else in ~A due to the partition

In very practical terms, many issues hinge on recognising that a distinct identity pivots on core characteristics so that relabelling not-a-thing as though it were the thing fails. Likewise, relabelling a thing as though it were not the thing, fails too.

Lincoln’s folksy example of relabelling the tail of a sheep a “leg” and insisting that a sheep must now be reckoned as having five legs is aptly illustrative of the problem.

And, it speaks quite directly to the situation of policy making, liberty and justice and war. Notice his second inaugural:

AT this second appearing to take the oath of the Presidential office there is less occasion for an extended address than there was at the first. Then a statement somewhat in detail of a course to be pursued seemed fitting and proper. Now, at the expiration of four years, during which public declarations have been constantly called forth on every point and phase of the great contest which still absorbs the attention and engrosses the energies of the nation, little that is new could be presented. The progress of our arms, upon which all else chiefly depends, is as well known to the public as to myself, and it is, I trust, reasonably satisfactory and encouraging to all. With high hope for the future, no prediction in regard to it is ventured.

On the occasion corresponding to this four years ago all thoughts were anxiously directed to an impending civil war. All dreaded it, all sought to avert it. While the inaugural address was being delivered from this place, devoted altogether to saving the Union without war, insurgent agents were in the city seeking to destroy it without war—seeking to dissolve the Union and divide effects by negotiation. Both parties deprecated war, but one of them would make war rather than let the nation survive, and the other would accept war rather than let it perish, and the war came.

One-eighth of the whole population were colored slaves, not distributed generally over the Union, but localized in the southern part of it. These slaves constituted a peculiar and powerful interest. All knew that this interest was somehow the cause of the war. To strengthen, perpetuate, and extend this interest was the object for which the insurgents would rend the Union even by war, while the Government claimed no right to do more than to restrict the territorial enlargement of it. Neither party expected for the war the magnitude or the duration which it has already attained. Neither anticipated that the cause of the conflict might cease with or even before the conflict itself should cease. Each looked for an easier triumph, and a result less fundamental and astounding. Both read the same Bible and pray to the same God, and each invokes His aid against the other. It may seem strange that any men should dare to ask a just God’s assistance in wringing their bread from the sweat of other men’s faces, but let us judge not, that we be not judged. The prayers of both could not be answered. That of neither has been answered fully. The Almighty has His own purposes. “Woe unto the world because of offenses; for it must needs be that offenses come, but woe to that man by whom the offense cometh.” If we shall suppose that American slavery is one of those offenses which, in the providence of God, must needs come, but which, having continued through His appointed time, He now wills to remove, and that He gives to both North and South this terrible war as the woe due to those by whom the offense came, shall we discern therein any departure from those divine attributes which the believers in a living God always ascribe to Him? Fondly do we hope, fervently do we pray, that this mighty scourge of war may speedily pass away. Yet, if God wills that it continue until all the wealth piled by the bondsman’s two hundred and fifty years of unrequited toil shall be sunk, and until every drop of blood drawn with the lash shall be paid by another drawn with the sword, as was said three thousand years ago, so still it must be said “the judgments of the Lord are true and righteous altogether.”

With malice toward none, with charity for all, with firmness in the right as God gives us to see the right, let us strive on to finish the work we are in, to bind up the nation’s wounds, to care for him who shall have borne the battle and for his widow and his orphan, to do all which may achieve and cherish a just and lasting peace among ourselves and with all nations.

A big part of the run-up to the US civil war of the 1860’s pivoted on refusing to recognise that a black man is just as much a man as any other and so if by endowment of the Creator there is an unalienable right to life, liberty and pursuit untrammelled of happiness/ purpose one is here for, then surely a black descriptive modifier does not alter the fact that a man is a man. And if more was needed, many a white man fathered children upon a black slave woman; often under conditions where because of her enslaved status she had no recourse of refusal and no respect of being wife or even concubine.

{I add, a portrait of the Evans family likely just before the American Civil War, and — please read the painful story here — this was an actual marriage between a white man and a free black woman:}

The leg of a sheep is a leg, the tail of a sheep hath not the characteristics of and cannot be a leg.

So, if a black enslaved woman can have fertile children when her white massa’s son or massa himself takes a fancy to her, even under circumstances tantamount to rape, that already shows that we are “of one blood” . . . one common kind of life as “the life of the flesh is in the blood.” (That ties to Ac 17, in the same scriptures read by all.)

Refusal to recognise and respect distinct identity (here, held in common by whites and blacks) was the pivot of what led to war. In a context where a key argument for union in the Federalist Papers was that union was the best means of securing the peace, the common good and a prosperous future. As the history of regional divisions, alienation and wars in Europe so clearly showed.

In our day, it is highly arguable that refusal to recognise that the unborn child is as we all once were and at minimum should have the benefit of doubts is an engine driving the worst holocaust in human history, which in turn enmeshes the public, parliaments, courts, police and more across our whole civilisation in the terrible blinding and benumbing power of bloodguilt.

As one direct consequence, in the leading country of our civilisation, leadership from the party of mass abortion is implicated in many other things and the other party on average is half-hearted.

Some kinds go not out save by scripture so let me put it this way: they say peace, peace, when there is no peace.

In such a context where there is a habit of refusal to acknowledge that the right hand side restates what is on the left hand side . . . what equals means . . . we now face a policy of enabling the leading terrorism sponsoring nation in the world in its determination to acquire nuclear weapons as the effective means of intended genocide.

Linked to that there is a refusal to acknowledge that per foundational history, texts and policy sustained for 1400 years, there has been a major militarily backed or implemented IslamIST expansionism. Indeed, I find it interesting in a saddening, sobering way, how — while the media are ever so eager to pounce on all sorts of convenient stories — there has been persistent refusal to have a sustained focus on things like the Muslim Brotherhood’s 100 year global conquest plan (which I first found online on Sept 11 2001, along with recipes for building nukes when I sought to understand what was going on), or the papers documenting the settlement-jihad strategy (mass immigration of the violently hostile is invasion), and more.

{I add, here is that eye-opening image first seen by me on Sept 11, 2001, note the insert bottom-right:}

So, I find that being willing to recognise distinct identity and acknowledge that a certain implement with a long handle and a wide thin blade at one end shaped for digging into and lifting the soil is a spade is pivotal.

In short, are we willing to acknowledge that a spade is indeed a spade?

Scripture, again, from 2700 years ago:

Isa 5:18 Woe to those who draw iniquity with cords of falsehood,
who draw sin as with cart ropes,
19
who say: “Let him be quick,
let him speed his work
that we may see it;
let the counsel of the Holy One of Israel draw near,
and let it come, that we may know it!”

20
Woe to those who call evil good
and good evil,
who put darkness for light
and light for darkness,
who put bitter for sweet
and sweet for bitter!

21
Woe to those who are wise in their own eyes,
and shrewd in their own sight!

22
Woe to those who are heroes at drinking wine,
and valiant men in mixing strong drink,
23
who acquit the guilty for a bribe,
and deprive the innocent of his right!

24
Therefore, as the tongue of fire devours the stubble,
and as dry grass sinks down in the flame,
so their root will be as rottenness,
and their blossom go up like dust;
for they have rejected the law of the Lord of hosts,
and have despised the word of the Holy One of Israel.

25
Therefore the anger of the Lord was kindled against his people,
and he stretched out his hand against them and struck them,
and the mountains quaked;
and their corpses were as refuse
in the midst of the streets.
For all this his anger has not turned away,
and his hand is stretched out still.

26
He will raise a signal for nations far away,
and whistle for them from the ends of the earth;
and behold, quickly, speedily they come!
27
None is weary, none stumbles,
none slumbers or sleeps,
not a waistband is loose,
not a sandal strap broken;
28
their arrows are sharp,
all their bows bent,
their horses’ hoofs seem like flint,
and their wheels like the whirlwind.
29
Their roaring is like a lion,
like young lions they roar;
they growl and seize their prey;
they carry it off, and none can rescue.

30
They will growl over it on that day,
like the growling of the sea.
And if one looks to the land,
behold, darkness and distress;
and the light is darkened by its clouds.

The price for apostasy and twisting truth into falsity, calling darkness light and light darkness is destructive defeat by way of fatally weakening oneself through marches of folly that “draw iniquity with cords of falsehood.”

A is A, A = A.

Calling A by its right name and acknowledging its right definition is a first step to wisdom.

A sound, thorough and accurate vocabulary or dictionary or encyclopedia — Wiki[d . . . ]pedia this speaks to you — is a revolutionary act of courage in an evil day.

A is A.

The tail of a sheep is not, cannot be a leg.

A leg is a leg.

It all starts with truthfully acknowledging (or with ill-advised refusal to acknowledge . . .) distinct identity.

A = A.>>

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A = A is indeed important and highly relevant in a world where it can be an act of sometimes almost suicidal courage to acknowledge, state and act on the simple truth. END

## 23 Replies to “A = A . . . is it important?”

1. 1
kairosfocus says:

Why is A = A important?

2. 2

People are hardwired for making choices. That’s the factual basis by which they are individuals and have rights. And as by logic and physics, what the agency of any decision is, is a matter of opinion. That’s the basis for freedom of opinion and religion.

3. 3
redwave says:

Are we faced with eristical arguments over A=A? Could we say that many opposing truth statements are “wranglers”?

Fuzzy logic does not supplant the laws of thought, that A=A, A does not equal ~A, either A or ~A.

Fuzzy logic proposes that A(x) = X for x = some number between 0 and 1, e.g., X = 0.1, 0.2, 0.3, … x, x is a member of X. This does not imply that A(x) does not equal A(x). Fuzzy logic extends the excluded middle by variable degrees.

Fuzzy logic treats situations when true or false questions can not be answered through identity, non-contradiction, and identical excluded middles. Fuzzy logic could not stand without the foundation from which it extends. In other words, A(x) = A(y) iff x=y. e.g., if x = 0.1, y=0.1, but if x=0.1, y=0.2, then A(x) does not equal A(y), but A(x) and A(y) are fuzzy in terms of A. This does not imply that A(0.1) = A(0.2) in fuzzy propositional logic.

Fuzzy logic does not deny that a red ball on the table is red, when red can not be confused with blue or green, but treats the occurrences when the detectors determine variable frequencies emitted from the surface of the ball over a range including red.

http://plato.stanford.edu/entries/logic-fuzzy/

As to infallibility … late 15th century, from French infaillible or late Latin infallibilis, from in- ‘not’ + Latin fallere ‘deceive’ (OED).

A=A does not deceive. Yet to those practicing eristical arguments, A is not A for the sake of arguing against A. Not even sophistic!

4. 4
Aleta says:

As I said in the Nihilism thread, kf has mis-interpreted my remarks, I think, as somehow challenging the validity of A = A. Here is what I wrote there:

A = A is fundamental, and it assumes/establishes distinct identity as a fundamental logical property. I’m not questioning that in the least bit.

I also understand its use in mathematics. What I can’t think of is a place in a chain or real-world reasoning where add a step that says A = A. Its so basic that it is not a step in a chain of reasoning as much as an underlying agreement that the things we are talking about retain their identity throughout the argument.

I am saying the same thing kf is saying when he says “First, in thinking, arguing and speaking we must recognise and respect distinct identities if we are to be sound.”

We are not in disagreement.

5. 5
daveS says:

Aleta,

This might be a stretch, but of course in indirect proofs, you try to derive a statement of the form A =/= A, which is false exactly because A = A is true. I’m not sure if that meets your requirements.

6. 6
bFast says:

Actually, I am finding this exercise in elementary logic to be painful, a distraction from reality, from our real purpose.

UD is full of A=A and idiots that won’t comprehend this — on the premise that in doing so they make themselves out to be humble/wise. It is failing miserably for them, they are proving that they can’t logic their way out of a paper bag, therefore that anything they believe is truly unreasoned.

I got into a bit of a “real” conversation with Learned Hand on a previous thread, but before he had to admit that his position vis-a-vis abortion was not logically supportable, he returned to A=A babble.

I miss Uncommon Descent, the site where reasonable conversation about ID and evolution takes place. It has degraded into a flailing kindergarten class in logic.

7. 7
Mung says:

A=A in the real world. Just try to post on an internet blog without it.

8. 8
kairosfocus says:

BF, I hear your concern. I am afraid this is what is behind a lot of the lock-out mentality at later levels. So, it needs to be cleaned up for record to have room to understand what warrant is, what that means in the context of scientific investigations and education, and on origins/cosmology. Beyond, lies science in society. KF

9. 9
Eugen says:

Thanks Kairos

Articles and discussions about the rules of logic are new and interesting to me. I’m not too familiar with the topic. It’s hard to believe somebody would doubt them but it turns out it’s the atheists. If they doubt that shouldn’t they doubt themselves?

In the beginning atheists were amusing to me but now they seem creepy. Especially after discussions on recent disaster of Planned Parenthood abortion videos.

10. 10
dgosse says:

“Sometimes 2+2=5” Lawrence M. Krause (Stupid Math Tricks)

11. 11
kairosfocus says:

DG, any student who so abused rounding off and error bars would at once be rewarded with the red-X. KF

12. 12
kairosfocus says:

Eugen, that is precisely the point. Those who vaunt their brilliance (even aspiring to be termed “the brights”) are here found in the lists in stout opposition to the first principles of reason. And, they are so polarised that they are projecting confusion to those who are pointing out the error. Beyond this point, when such pop up to trot out tired objections to design inferences, my basic response will in part draw out that these are the ilk that will not accept self-evident principles of right reason when it suits them to object so how can we take seriously their objections to things of lesser possible warrant? For, they have ruled themselves as being beyond the pale of responsible, reasonable discussion, on science much less science in society and ethics. With the worst holocaust in human history being an ongoing reality propped up by selective hyperskeptical dehumanising of the as yet unborn. KF

13. 13
kairosfocus says:

F/N: I think I should comment on ways to express the truth A is A and some thoughts on where it points:

W = {A|~A}

Where x in W is such that x = A, by the partition x != ~A.

A = A, as opposed to ~A

More illuminatingly,

(A => A) = 1

That is, as A is on both sides of the implication, this is a case where implication does entail equivalence, i,e,

A = A is equivalent to (A => A) = 1

If A is so, then A is so is always true.

A if and only if A.

Sounds like “just an identity” or a tautology.

It is more than that, as the context is that A is such that it has distinct identity in the world, we see a world partition and such has consequences.

But what is distinct identity?

It is to be possible, requiring coherent core attributes, it is to be such that once actualised other things will be different — in nature, functions and operations, location, timing, specific material composition if that is relevant, etc. Two peas from a pod closely resemble but are quite different. Identical twin girls I once taught are often confused when one visits from the UK. But they are different, have different life stories and much more.

By contrast there is just one null set, though we may often refer to it in various contexts.

To see where this heads, first notice that if “two” initially thought to be separate “things” are instead identical — one and the same as the evening and morning star were found to be differing manifestations of Venus, they will be indiscernible. That is, you cannot tell them apart, though you may have labelled the same entity say A as x, or as y when you arrived at it from different ways. Which, is NOT the same as x implies y and y implies x, or that if x is present y must be also and the converse.

The twins I speak of are thus closely resembling rather than genuinely identical despite the common term. Last I looked at the two together, one was in the driver’s seat and the other in the passenger seat. Their children were in the passenger seats. and though I have known the girls much longer than those kids have, I assure you they instantly know mummy and auntie. Which BTW are a case of differing and overlapping labels that in context attach to diverse things in a perfectly coherent way. Language truly is a strange thing.

A more demanding issue is the principle of identity of indiscernibles.

If two things are not distinct they do not hold separate identities, in effect.

When we speak of them in different contexts we are in effect pointing to them rather than separately instantiating. But it is possible to put up certain phrasings and contexts that can land one in contradictions. The issue is that close resemblance or close similarity of component parts and their arrangement do not constitute identity.

Consider a world in which there are only three air-filled red balls on a table, the balls being of same shape and composition. Presumably with a rectangular table location will distinguish. But what of balls symmetrically distributed on a circular table that does not have a texture that “fingerprints” location. Worse, just the three balls floating there.

It would then seemingly be impossible to distinguish the balls. They are in a space, W. An observer can see three separate balls. Close the eyes and spin, like in a shell game. Look again, which is which?

Even if you were able to look at molecular level and discern micro-state, due to the randomness, if after enough time you were to look back you could not tell the balls apart that way. They “forget” initial conditions.

For some purposes, the balls are perfectly substitut-able, patently.

But, recall, such balls are contingent.

Pop one with a pin, paint another half blue.

The three are now no longer closely resembling.

So, they were not identical. They had distinct identities, and fates.

The null set is not like that.

Related, two people can have and even believe the same proposition in mind, e.g. 2 + 3 = 5. We can signal it to one another. But we cannot prick it or paint it to make it different in the different minds; the claim being affirmed is not made up of differing molecules, it is fundamentally different from properties of matter. The glyphs and symbols we use are in effect pointers under a code to the inherently mental matter at stake, a proposition that is being affirmed.

This is a clue that matter-energy in space-time does not exhaust reality.

Information and propositions are clearly immaterial, they are mental, and the mV potentials in brains or voltage states in computer registers etc are not the same.

They are pointers to the realities under whatever system of signs or symbols is there.

(And yes, I am implying that nerve cell impulses and firing are signals and computational processes but we must look elsewhere in a different category of reality to see the abstract information or proposition involved.)

It also opens the door for us to ponder that while we may hold some propositions and thoughts in common, our minds are indeed discernible and distinctly identifiable, as those twin girls J and J are. We do not just differ in our bodies but in our inner conscious selves.

Distinct identity is significant.

KF

14. 14
daveS says:

KF,

I’m not clear on your notation here. What does (A => A) = 1 stand for?

And is A a formula/proposition or just an arbitrary element of some set?

I have kept in mind your world partition example where A is a red ball, so the world is the union of A and A-complement, but in that case I don’t know how to interpret A => A.

15. 15
Aleta says:

I think he means that “A implies A is true” is an alternative way of saying A = A.

16. 16
daveS says:

Aleta,

Thanks, that’s what I initially assumed, but if A is a red ball for example, A cannot be true or false, so the meaning of A => A is still unclear to me.

More explicitly, I understand “the red ball = the red ball” but I don’t know what to make of “the red ball implies the red ball”.

17. 17
Mung says:

Aleta: I think he means that “A implies A is true” is an alternative way of saying A = A.

Actually, what he said is the exact opposite of that.

Or perhaps he said both what you say he said and the exact opposite of what you say he said and he also claimed that they are both the same.

Do carry on.

18. 18
kairosfocus says:

DS, (A => A) = 1 is equivalent to A = A. If you wish expansion on an illustrative concrete case, If a red ball A (on a table) exists, then . . . KF

19. 19
daveS says:

KF,

Thanks, but … that doesn’t add much to my understanding.

Can you write out in English what (A => A) = 1 translates to where A is a red ball?

20. 20
kairosfocus says:

DS, Pardon I thought giving IF A . . . THEN . . . would be enough. The key point is that A is on both sides of the implication so this is a case where implication implies equivalence (A and A will be T-T and F-F together on the Truth Table) so the = 1 part is where there is simply “is a tautology.” The trick is, this connects to A denoting some distinct thing, i.e. it addresses reality. If A is so, A is so, or, A is A. The fundamental context is that A as a distinct thing in the world imposes a world partition: W = {A|~A}. Distinct identity is what directly leads to A being itself and not what is not itself, no x in W being both A and ~A, and such an x being A or else ~A. KF

21. 21
daveS says:

KF,

Thanks for coming back to this.

Are you really talking about conditionals of the form P(A) => P(A) then (as opposed to A => A), where P is a predicate?

As I stated above, I don’t know how to make sense of “the red ball => the red ball”, since “the red ball” has no truth value.

If you apply a predicate to A however, that problem is solved.

22. 22
kairosfocus says:

DS, A is broad enough to enfold a proposition, especially one closely anchored to a reality, say A = p(R), R = the red ball, and p(R) say there is a red ball on the table. KF

23. 23
daveS says:

KF,

I think we’re more or less on the same page then, as long as we’re clear that the A’s in “A = A” and “A => A” are different; one stands for a thing in the world, while the other stands for proposition(s) about that thing.