Part I – Chapter 1
About the Laws of Thought
The three “Laws of Thought” may be briefly explicated as follows:
- Thesis: there are certain appearances; appearances appear.
- Antithesis: there are incompatibilities between certain of these appearances; in such cases, one or both of them must be false.
- Synthesis: some remaining appearances must be true; find out which!
We can in this perspective consider dialectic as a fundamental form of thought, through which knowledge is made to progress on and on. It is not a mere detail, an occasional thought-process, but a driving force, an engine, of thought.
The laws are not mere information, but calls to cognitive action. They enjoin proactive and curative cognitive measures, to ensure (as much as possible at any given time) continued verification, consistency and completeness.
(i) The law of identity tells us to seek out the facts and sort them out as well as we can. The purpose of this law is to instill in people a healthy respect for facts, in the course of observation and judgment. It is essentially a call to honesty, and submission to the verdict of truth. People often think, or act as if they think, that ignoring or denying unpleasant facts or arguments will make them ‘go away’ – the law of identity says ‘no, they will not disappear, you must take them into consideration’.
Some people think that it is impossible for us to ignore that “A is A”. Far from it! All of us often do so – as when we refuse to look at or admit the evidence or a logical demonstration; when we avoid reality or evade it having glimpsed it; when we lie to ourselves or to others; and so forth. If the law of identity were always obeyed by us, there would be no need to formulate it. Logic states the obvious, because it is often shunned.
(ii) When the law of non-contradiction says to us “you cannot at once both affirm and deny a proposition”, it is also telling us that if we ever in the course of discourse encounter a situation where a proposition seems both true (for some reason) and false (for other reasons), we have to go back upstream in our discourse and find out where we went wrong in the course of it, and we have to effect an appropriate correction such as to eliminate the difficulty.
We are not just saying: “ah, there is a contradiction”, and leaving it at that, nonplussed. No, we are impelled to seek a solution to the problem, i.e. to resolve the contradiction. We are inferring that there must be something wrong in our earlier thinking that led us to this conundrum, some error of observation or reasoning that requires treatment. So long as this situation is tolerated, and we cannot pinpoint the source of error, the credibility of all related knowledge is proportionately diminished. Consistency must be restored as soon as possible, or we risk putting all subsequent knowledge in doubt.
(iii) Similarly, the law of the excluded middle does not just inform us that “no proposition can be claimed neither true nor false”. This law insists that if we find ourselves in such a situation, and it is indeed the case that both a proposition and its exact negation both seem false, we cannot let the matter rest or hope to find some compromise position – we have to eventually, as soon as possible, find good reason to opt for one side or the other. There is no logically acceptable middle ground, no avenue of escape.
These action implications inherent in the laws of thought may also be characterized as dialectical thinking. In this perspective, the “thesis” is our knowledge (or opinion) as it happens to be at a given time; the “antithesis” is the discovery of a logical flaw in that thesis, which causes us to have doubts about it and seek its review; and finally, the “synthesis” is the corrections we make in our premises, so as to resolve the difficulty encountered and obtain a less problematic new state of knowledge.
Axioms are not arbitrary, a-priori starting points of true human knowledge. They may be deductive or inductive, but in either case are to some extent empirical (in the large sense of ‘phenomenological’, i.e. without depending on any materialist or mentalist assumption concerning what is experienced).
Deductive axioms are established using certain positive or negative logical arguments, which we naturally find convincing. But even a deductive axiom relies on certain experiences, those that gave rise to the concepts and logical techniques involved in the proposition and its acknowledgment as an axiom.
The positive argument for an axiom is essentially dilemmatic: “whether this or that, so and so is true”. An example is the axiom that diversity exists. The mere seeming of diversity is itself a case of diversity, sufficient to establish the fact of diversity. It is no use arguing (like Parmenides or the Buddha) that this apparent diversity is an “illusion”, and that “all is really one” – because the coexistence of illusion and reality is itself an event of diversity. Thus, diversity truly exists, and cannot just be ignored. We might still try to uphold the thesis that reality is ultimately unitary, but only if we convincingly account for the fact of diversity.
Deductive axioms are also justified negatively through paradoxical logic, i.e. by showing that their contradictories are self-contradictory. For example, “There is no diversity” is a claim to diversity (since it involves many words, many letters, many sounds, etc.), and therefore self-contradictory; whence, it is self-evident that “There is some diversity”. This argument may also be construed (as above) as dilemmatic in form: “whether you deny or affirm diversity, you affirm it”.
Inductive axioms rely on some generalization, or (more broadly) adduction, from experience; but such inductive process in their case is not ever likely to be in need of revision. Many truths of utility to epistemology are inductive, and yet once realized remain immutable; they thus behave largely like deductive axioms, and may by analogy be classed as inductive axioms.
For example, the fact that most of our beliefs are contextual is a non-contextual truth, though based on common observation. The awareness that most of our knowledge is empirical, and subject to revision as new experiences are encountered, that it is in constant flux, altering and growing – this is a broad observation that once realized will not be affected by any further empirical data. This observation is not useless, note well: it logically affects pursuit of knowledge, teaching us to remain aware of the non-finality of most of our beliefs.
But note also, the said principle of contextuality is pretty vague; it cannot by itself put specific knowledge in doubt (i.e. without some other more specific reason for doubt). Another example of such general but unspecific truth is the principle (derived from the law of the excluded middle) that “there is always some explanation”. This optimistic principle serves to encourage research, but does not tell us what the solution of the problem is specifically.
A (single) paradoxical proposition has the form “if P, then notP” or “if notP, then P”, where P is any form of proposition. It is important to understand that such propositions are logically quite legitimate within discourse: a (single) paradox is not a contradiction. On the other hand, a double paradox, i.e. a claim that both “if P, then notP” and “if notP, then P” are true in a given case of P, is indeed a contradiction.
The law of non-contradiction states that the conjunction “P and notP” is logically impossible; i.e. contradictory propositions cannot both be true. Likewise, the law of the excluded middle states that “notP and not-notP” is logically unacceptable. The reason for these laws is that such situations of antinomy put us in a cognitive quandary – we are left with no way out of the logical difficulty, no solution to the inherent problem.
On the other hand, single paradox poses no such threat to rational thought. It leaves us with a logical way out – namely, denial of the antecedent (as self-contradictory) and affirmation of the consequent (as self-evident). The proposition “if P, then notP” logically implies “notP”, and the proposition “if notP, then P” logically implies “P”. Thus, barring double paradox, a proposition that implies its own negation is necessarily false, and a proposition that is implied by its own negation is necessarily true.
It follows, by the way, that the conjunction of these two hypothetical propositions, i.e. double paradox, is a breach of the law of non-contradiction, since it results in the compound conclusion that “P and notP are both true”. Double paradox also breaches the law of the excluded middle, since it equally implies “P and notP are both false”.
These various inferences may be proved and elucidated in a variety of ways:
· Since a hypothetical proposition like “if x, then y” means “x and not y is impossible” – it follows that “if P, then notP” means “P and not notP are impossible” (i.e. P is impossible), and “if notP, then P” means “notP and not P are impossible” (i.e. notP is impossible). Note this explanation well.
We know that the negation of P is the same as notP, and the negation of notP equals P, thanks to the laws of non-contradiction and of the excluded middle. Also, by the law of identity, repeating the name of an object does not double up the object: it remains one and the same; therefore, the conjunction “P and P” is equivalent to “P” and the conjunction “notP and notP” is equivalent to “notP”.
Notice that the meaning of “if P, then notP” is “(P and not notP) is impossible”. Thus, although this implies “notP is true”, it does not follow that “if notP is true, P implies notP”. Similarly, mutadis mutandis, for “if notP, then P”. We are here concerned with strict implication (logical necessity), not with so-called material implication.
The reason why this strict position is necessary is that in practice, truth and falsehood are contextual – most of what we believe true today might tomorrow turn out to be false, and vice-versa. On the other hand, logical necessity or impossibility refer to a much stronger relation, which in principle once established should not vary with changes in knowledge context: it applies to all conceivable contexts.
· Since a hypothetical proposition like “if x, then y” can be recast as “if x, then (x and y)” – it follows that “if P, then notP” equals “if P, then (P and notP)”, and “if notP, then P” equals “if notP, then (notP and P)”. In this perspective, a self-contradictory proposition implies a contradiction; since contradiction is logically impermissible, it follows that such a proposition must be false and its contradictory must be true. This can be expressed by way of apodosis, in which the laws of thought provide the categorical minor premise, making it possible for us to exceptionally draw a categorical conclusion from a hypothetical premise.
If P, then (P and notP)
but: not(P and notP)
therefore, not P
If notP, then (notP and P)
but: not(notP and P)
therefore, not notP
· We can also treat these inferences by way of dilemma, combining the given “if P, then notP” with “if notP, then notP” (the latter from the law of identity); or likewise, “if notP, then P” with “if P, then P”. This gives us, constructively:
If P then notP – and if notP then notP
but: either P or notP
If notP then P – and if P then P
but: either notP or P
Paradox sometimes has remote outcomes. For instance, suppose Q implies P, and P implies notP (which as we saw can be rewritten as P implies both P and notP). Combining these propositions in a syllogism we obtain the conclusion “if Q, then P and notP”. The latter is also a paradoxical proposition, whose conclusion is “notQ”, even though the contradiction in the consequent does not directly concern the antecedent. Similarly, non-exclusion of the middle may appear in the form “if Q, then neither P nor notP”. Such propositions are also encountered in practice.
It is interesting that these forms, “Q implies (P and notP), therefore Q is false” and “Q implies (not P and not notP), therefore Q is false”, are the arguments implicit in our application of the corresponding laws of thought. When we come across an antinomy in knowledge, we dialectically seek to rid ourselves of it by finding and repairing some earlier error(s) of observation or reasoning. Thus, paradoxical argument is not only a derivative of the laws of thought, but more broadly the very way in which we regularly apply them in practice.
That is, the dialectical process we use following discovery of a contradiction or an excluded middle (or for that matter a breach of the law of identity) means that we believe that:
Every apparent occurrence of antinomy is in reality an illusion.
It is an illusion due to paradox, i.e. it means that some of the premise(s) that led to this apparently contradictory or middle-excluding conclusion are in error and in need of correction. The antinomy is never categorical, but hypothetical; it is a sign of and dependent on some wrong previous supposition or assumption. The apparent antinomy serves knowledge by revealing some flaw in its totality, and encouraging us to review our past thinking.
Contradiction and paradox are closely related, but not the same thing. Paradox (i.e. single not double paradox) is not equivalent to antinomy. We may look upon them as cognitive difficulties of different degrees. In this perspective, whereas categorical antinomy would be a dead-end, blocking any further thought––paradox is a milder (more hypothetical) degree of contradiction, one open to resolution.
We see from all the preceding (and from other observations below) the crucial role that paradox plays in logic. The logic of paradoxical propositions does not merely concern some far out special cases like the liar paradox. It is an essential tool in the enterprise of knowledge, helping us to establish the fundaments of thought and generally keeping our thinking free of logical impurities.
Understanding of the paradoxical forms is not a discovery of modern logic, although relatively recent (dating perhaps from 14th Cent. CE Scholastic logic).
Many people misunderstand what we logicians mean by ‘contradiction’. The contradictory of a term ‘A’ is its negation, ‘not A’, which refers to anything and everything in the universe other than A, i.e. wherever precisely A is absent in the world. The relation of contradiction between A and not-A is mutual, reversible, perfectly symmetrical.
The presence of something (A) excludes its absence (i.e. not A) in that very same thing, and vice versa, if all coordinates of space and time are identical. However, this does not exclude the logical possibility that the same thing may be partly A and partly not A. Thus, the law of thought ‘either A or not A’ can also be stated more quantitatively as “either ‘all A’ or ‘all not A’ or ‘part A and part not A”.
Some people appeal to this possibility of three alternatives as an argument against the laws of thought! But that is a misunderstanding – or worse, deliberate sophistry.
If something, e.g. ‘B’, implies but is not implied by not-A, it (i.e. B) is as ‘incompatible’ with A as not-A is, but it is not contradictory to A: it is merely contrary to A. The contradictory not-A of A differs from A’s contraries in that the absence of not-A implies A, whereas in the case of mere contraries like B (or B1 or B2… etc.) this added logical relation of ‘exhaustiveness’ does not apply.
When contradictories are placed in a disjunction, ‘either A or not-A’, the disjunction involved signifies both mutual exclusion (‘or’, meaning ‘not together’) and exhaustiveness (‘either’, meaning ‘and there is no other alternative’). It intends: if ‘A’, then not ‘not-A’; and if not ‘A’, then ‘not-A’.
On the other hand, any number of contraries can be placed in a disjunction: ‘A or B or B1 or B2… etc.’, so that the presence of any disjunct implies the absence of all the others; but such disjunction is not exhaustive, unless we specify that the list of contraries in it is complete. If that list is indeed complete, then the negation of all but one of the disjuncts implies the affirmation of the remaining one. Thus, ‘not-A’ can be equated to the exhaustive disjunction of all things in the world ‘contrary to A’.
Something different from A, e.g. ‘C’, is not necessarily contradictory or even contrary to A. The mere fact of difference does not imply incompatibility. Different things (like A and C) may be compatible, i.e. capable of coexistence in the same thing, at the same time and place. ‘Difference’ simply signifies that we are able to distinguish between the things concerned: i.e. they are not one and the same when they appear before our consciousness. ‘Similar’ things may be the same in appearance, but not one (e.g. two instances of the same kind); or they may be one (i.e. parts of a single whole), yet not the same.
Thus, for example, the logical relation between the colors black and white depends on how precisely we focus on them. They are different, since distinguishable. Since they may coexist on different parts of the same surface, they are broadly compatible. However, as such or per se, they are contrary; that is to say: if I perceive a surface or part of surface as totally white, and you perceive the very same place and time as totally black, our claims are incompatible. This irreconcilability is not a contradiction, however, because it is possible for a surface to be neither black nor white.
The expression ‘contradiction in terms’ refers to a compound term composed of incompatible elements, such as ‘A and not A’ or ‘A and B (where B is contrary to A)’. Such a mixed-up term may be said to be paradoxical, as well as internally inconsistent, since it implies that contradiction is possible, so that the laws of thought are denied by it, and then (by generalization, if you like) ‘anything goes’ including denial of the ‘A and not A’ conjunction.
For example, the term “illusory reality” is a contradiction in terms. On the other hand, note, terms like ‘an inhuman human’ or ‘an anti-Semitic Jew’ are not strictly speaking contradictions in terms; they refer to natural possibilities of conjunction, only the terminology used makes them superficially seem contradictory (i.e. there are people who behave inhumanly, or Jews that hate their own people).
The proposition ‘A is not A’ (or ‘some thing that is A is also not A’), being self-contradictory, implies ‘A is A’, its contradictory form. This statement should be explicitly acknowledged, though obvious, because it correlates two important concepts, viz. ‘internal inconsistency’ and ‘the logic of paradoxes’.
The statement ‘A is not A’ is logically impossible, because it both affirms and denies the same thing. Therefore, the opposite statement is true. That statement, i.e. ‘A is A’, is logically necessary, because even its contradictory ‘A is not A’ implies it.
Whoever claims ‘A is not A’ is admitting ‘A is A’ – ipse dixit, he himself said it! Whereas, whoever claims ‘A is A’ is consistent with himself.
Self-contradiction consists of three items:
- The proposition in question, call it P.
- The admission that it is an assertoric statement, i.e. one that affirms or denies something.
- The admission that all assertoric statements involve claims to consciousness, to knowledge, to truth, etc.
Thus, given P (e.g. “reality is unknowable”), admit that P implies “this is an assertion” – but all assertions imply some knowledge of reality – therefore, P implies non-P. There is a process from P to its negation, which Logic demands we acknowledge. That demand cannot be refused without committing the very same self-contradiction. This is not a circular or ad infinitum proof, but an appeal to honesty, without which no dialogue is possible.
That all assertoric propositions assert is an aspect of the Law of Identity. The Law of Non-contradiction may be discerned in the argument: All assertions assert something; P is an assertion; therefore, P asserts; whence, if P denies asserting, P implies non-P. The Law of the Excluded Middle is also implicit here, in the awareness that we have no choice but to firmly disown P.
Contradictions appear in discourse in many guises. They are not always overt, but may be hidden in the fact of making a statement or in the standards of judgment used.
A claim may be paradoxical because it inherently entails its own contradiction, although it does not on the surface seem to be self-inconsistent. Such implication is not always formal but requires awareness of the meaning of the terms used. This form of indirect self-contradiction has been called “the Stolen Concept fallacy”.
For instance, the skeptical claim “I know nothing” may be rejected as self-contradictory, because as soon as someone makes it – someone who understands and intends the meaning of the terms “I”, “know” and “nothing” – that is by itself proof absolute that the person concerned “knows” something, whence the original claim (of total ignorance) is shown up to be unavoidably contradictory and thus necessarily false.
Thus, in cases of this sort, the tacit implication involved is that one of the terms used (knowing nothing) implicitly includes the act in question (knowing that I know nothing), as a case in point contradictory to the explicit claim. (Rephrasing the said statement as “I do not know anything” does not change its underlying assumptions, needless to say.)
There are countless examples of such inherent self-contradiction. Saying “I have nothing to say” is saying something. Claiming “We have no memory” is self-contradictory, because each term in it presupposes a word, concept and background experiences remembered by the speaker – and the hearer too. An amusing common example is “I do not speak a word of English”!
Another important form of covert self-inconsistency is the use of a double standard. This consists in applying less stringent standards of judgment to one’s own discourse than to the discourse of one’s intellectual opponents. A lot of philosophical, and particularly political and religious, discourse resorts to such inequitable methodology.
The contradiction involved in a double standard is apparent the moment we step back and view its user’s knowledge and methodology as a whole. In this wider perspective, the user of a double standard is clearly inconsistent with himself, even if his discourse viewed piecemeal may superficially seem self-consistent.
Whole philosophies may be based on such fallacious reasoning. For instance, Phenomenalism sets as a general standard a limitation of knowledge to sensory data without allowing extrapolations from them to assumed external material objects – yet it does not criticize its own adductions using the same rigid standard.
There are two ways this fallacy may be committed: one may use relaxed standards on one’s own discourse, while seemingly applying universal norms to one’s opponents’ discourse; or one may appear to apply universal norms to oneself, while concocting overly strict norms for them. One may exempt oneself from the usual logical rules, or one may make unusual logical demands on others.
In either case, the holder of a double standard is in conflict with logic’s requirement of uniformity. An assumption of reason is that all humans are epistemologically on the same plane. Equity is an aspect of ‘common sense’. Experience and logic have to be used to convince oneself and others, not sophistical manipulation or authority.
Standards of judgment have to be fair and universal; all discourse must be equally treated. If differences are advocated, they have to be convincingly justified. The principle of equality admittedly involves generalization; but the onus of proof is on any proposed particularization of it.
An example of a double standard is the appeal to cultural relativism. One may seek to rationalize ideas or thought processes that are contrary to ordinary reason, by claiming them to belong to a different cultural framework. Such tolerance seems on the surface friendly and open-minded, but it is proposed without full consideration of its negative human and epistemological implications.
The three Laws of Thought must not be construed as some prejudice of Aristotle’s, which some scientific discovery – like the particle-wave duality or the relativity of space-time measurements – could conceivably raise doubt about or displace. These laws of thought are intended as perfectly neutral; they make no direct, specific ontological or epistemological claim, but rationally sort out the very act and concept of such claims – whence their name.
These laws express the ways we assimilate complex experiences, and resolve difficulties in the course of thought (concepts, propositions and arguments). Only by such logic can we ‘make sense’ of the world around us and in us. By making these truths explicit, Aristotle made it possible for humans to henceforth consciously practice the logic they were already unconsciously tending to.
These laws exclude, ab initio, the notion that something could both have and lack some property, or neither have nor lack it – at the same place and time and in the same respects. The latter specification, which Aristotle clearly and repeatedly stressed, is often ignored by those who consider these laws expendable.
That, say, a stone is blue on one side and red on the other, is not a contradiction, since the different colors are in different parts of it. That over time the colors may change is not an antinomy either: the concept of time is intended to ensure that. That you and I view the same object from different angles, and see different aspects of it, is no surprise. That my view of the world and yours are not quite identical, is quite understandable in view of the different context of experience and thought we each have.
The laws of thought do not evade or deny the appearance of contradictions or unsolved problems; they just tell us that such appearances are illusions, not realities. They are designed precisely to help us take such apparent discrepancies into consideration and resolve them in some way. We continue to need the same laws of thought in the more complex cases uncovered by modern physics.
The theory of relativity is precisely an attempt to rationalize the surprising empirical constancy in the velocity of light, whichever direction we measure it from. The theory is not a statement that there are no absolute truths, but a statement that such and such a way of looking at the surprising events discovered makes them rationally comprehensible. The theory affirms that this way is probably (i.e. inductively) the best explanation, and effectively denies those who contradict it (unless they come up with an inductively better explanation, more in line with the empirical findings). It does not deny the laws of thought, but is an application of them.
Similarly, the discovery that the same things may behave occasionally as particles and occasionally as waves does not constitute an argument against the laws of thought. Whether we interpret this duality epistemologically or ontologically, as due to different circumstances of observation or different material circumstances, it is affirmed to be a mysterious finding that must be faced. This realist attitude is precisely what the laws of thought demand. Any attempt to interpret the finding, one way or the other, is again an attempt to make the finding rationally comprehensible, so that we do not feel them logically impossible.
Under no circumstances may scientists or philosophers seriously claim the laws of thought to be abrogated. Such a claim is self-contradictory – because then its opposite is equally acceptable. It is therefore as if nothing has been said. It is the denial of reason, the institution of madness. The three laws of thought thus together constitute the most incontrovertible and universal frame of reference of rational thought.
Note also, the emphasis the laws of thought lay on existence. A common error of deniers of these laws is to regard ‘non-existence’ as just some other sort of existence, a parallel world or a location beyond space and time from which new existents come and to which finished existents go! These people are misled by linguistic habit into a reification of the word ‘non-existence’.
Whatever positively appears, exists to that extent. Existence becomes open to doubt to the extent that we add assumptions to appearance – i.e. we adductively guess what might lie beyond them. At this stage, the reality vs. illusion dichotomy arises. At this stage, too, the rational act of negation comes into play – when we say: this is apparent, but (since it gives rise to some antinomy) it is not real, it is illusory.
The ‘concept’ of non-existence thus has no direct empirical basis of its own. It is based on a rational act relative to experiences of existence. It is just a figment of the imagination, a mental dumping place for ideas that have failed the test of existential basis.
It is important to realize that the laws of thought are the motors of rational thought. They generate questions and the pursuit of answers; they feed curiosity and fuel research. If we are satisfied with the way things seem, however contradictory or incomplete they seem, thought is arrested. We lose perspective and become ignorant. We lose intelligence and become stupid. We lose touch with reality and become insane.
Consider the irrelevancy to science of a hypothetical denial of the laws of thought. For instance, according to Einstein’s theory of relativity, nothing can travel faster than light, yet it has been found that particles may affect each other instantaneously even though they are far apart. If in the face of such an apparent contradiction we just said: “oh, well, I guess the law of contradiction must be wrong!” and left it at that – would we be consoled? Clearly, not – this would not honestly solve the problem for us, but merely sweep it under the carpet. Our minds would not rest till some deeper, more convincing explanation was found.
Accepting contradiction is just simplistic and evasive. Similarly, with breaches of the law of the excluded middle: if you ask me a question, and inquire is X the answer or not X? and I reply, it is neither, but some third thing: will you be satisfied with such reply? Your knowledge of the issue at hand is not made complete by such reply; a gap remains, which can only be filled by either X or nonX. The law of the excluded middle is just a recognition of the inadequacy of such neither-nor replies.
Descartes’ “cogito, ergo sum” is composed of two self-evident propositions: “I think” (in the sense, I am conscious) and “I am” (I exist). For the contradictory of each of these propositions is self-contradictory, i.e. involves a stolen concept and gives rise to a paradox. Thus, “I am not conscious” could not be thought or said (or for that matter heard or understood) without being conscious. Similarly, “I am not” could not be expressed (or observed) without existing. Thus, Descartes was quite right in regarding these propositions as axioms; i.e. as first principles, which do not depend on prior principles.
Note moreover that these two clauses are axiomatically true independently of each other – So what about the ergo, which suggests that the sum follows from the cogito? Is the “therefore” perhaps meant to imply an order of knowledge, rather than an inference? One could formally deduce existence from consciousness, in the sense that a conscious being is a fortiori an existent being; but one would never in practice resort to such inference.
In practice, in my opinion, we are conscious of other things before we become conscious that we are conscious of them – so it would not be correct to place the “I think” before the “I am”. It could be argued that a baby may first experience inner states, but I would reply that such states are results of prior sensations. We may however support Descartes’ order, by considering it a logical one, in the sense that if the Subject did not have the power of consciousness, he or she would not be aware of existence. That is, it perhaps means: “I can think, therefore I can know that I am”.
But I think the correct interpretation is the following: when we are aware of something, any thing, this provides an occasion to become aware of oneself, i.e. that there is a Subject who is being conscious of that thing, whatever it is. Thus, the first clause of the sentence is not strictly: “I think”, but: “consciousness of things is taking place” (or “thought is occurring”). Whence the second clause is truly inductively inferred, i.e. we may well hypothesize that “there is something being conscious of things”, i.e. “thought has a Subject as well as an Object”, i.e. “there is an I” (or “I exist”).
It is the self that is inferred from the appearance of objects – reason argues: they must appear before someone. This is what distinguishes appearance from mere existence: it occurs through ‘cognition’ by ‘someone’. Thus, Descartes is justifying our habitual assumption of a cognizing Subject from the fact of cognition. It is not mere grammatical convention, he tells us, but “think” implies “I”.
Where does a material object begin or end, in view of the constant flow of particles and energy in and out of it, even (over a long enough time) in the case of apparent solids? We have to use the apparent limits of things as their space-time definition. Or more precisely, in acknowledgment of the above difficulties, their illusory limits. Thus, knowledge of matter is built on arbitrary, knowingly inaccurate, delimitations of “things”.
We can similarly argue concerning mental objects (i.e. images, sounds, etc.). At first thought, their limits seem obvious; but upon reflection, they become doubtful – imprecise and insecure. And this being the case, we cannot convincingly argue that the limits of material bodies are mental projections. If the limits of mental lines are unsure, then the limits of whatever they are intended to delimit are still unsure.
Ultimately, then, since we cannot even mentally delimit mental or material things, all delimitations are merely verbal artifices, i.e. claims we cannot substantiate. This remark concerns not only ‘borderline’ cases, but all material or mental objects.
These are very radical queries, productive of grave skepticism. They are principles of vagueness and doubt much more unsettling than the Uncertainty Principle, since they more basically question the validity of any geometry (and therefore, more broadly, of mathematics and physics).
When some Greek or Indian philosophers expressed skepticism at the possibility of human knowledge, this is perhaps what they were referring to. If one cannot delimit things, how can one produce precise concepts and propositions? And without precision, how can we judge them true or false?
Whereas denial of knowledge as such is self-contradictory, denial of accurate knowledge is not so. It is possible to observe the general vagueness of experience without denying the law of identity. If cloudiness is the identity of things, or we are simply incapable of sufficiently focusing our senses to get past such cloudiness, we simply remain stuck at that level of experience, like it or not.
The best counterargument I can muster is that phenomenological knowledge is still knowledge of sorts, and this can be used as a springboard to arrive at deeper knowledge, by means of adduction. That is, we can still formulate ontological hypotheses, capable of ongoing confirmation or rejection with reference to reason and experience, even if the epistemological status of the latter is at the outset merely phenomenological.
This does not directly overcome the difficulty of measurement, but it gives us some hope that we might succeed indirectly. I leave the issue open, and move on.
 “Check your premises”, Ayn Rand would say.
 Indeed, it could be argued that, since ‘deductive’ axioms all have some empirical basis (as already explicated), they are ultimately just a special case of ‘inductive’ axiom.
 For instance, Charles Pierce (USA, 1839-1914) noticed that some propositions imply all others. I do not know if he realized this is a property of self-contradictory or logically impossible propositions; and that self-evident or necessary propositions have the opposite property of being implied by all others. I suspect he was thinking in terms of material rather than strict implication.
 Our disagreement is not terminological, note. We have in the past agreed as to what experiences ‘black’ and ‘white’ correspond to; here, we suddenly diverge.
 By Ayn Rand and (I think) Nathaniel Branden.
 See Hamlyn, p. 137. The comments made here are not intended as an exhaustive analysis of the cogito statement, needless to say.
 I have already discussed this ontological issue in Phenomenology, chapter IV:5.