Laws of Thought - Amateur Logician

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Laws of Thought

Law of Identity: What is, is.
— That is, a thing has the attributes it has.

Law of Contradiction: Nothing can both be and not be.
— That is, contradictory judgements cannot both be true.

Law of Excluded Middle: Everything must either be or not be.
— That is, one of two contradictory judgements must be true.

These three laws are the backbone of traditional logic.

No sense of a judgement or proposition would be possible, if the Law of Identity weren’t true. Validity and invalidity would become meaningless, if the Law of Contradiction weren’t true.

The Law of Identity is about “sameness.” It’s a reflexive law, as a thing is always equivalent to itself. Something has the attributes it has. Socrates is the man he is (or was). Four is equal to four. X = X.

Every judgement or proposition exemplifies the Law of Identity. For example, “Socrates is a man” presupposes this law. If that judgement didn’t presuppose it, then it could be the case that “Socrates is not a man.” Socrates is what he is; he is not what he is not.

We would be unable to identify anything without this principle. It makes it possible to form ideas about things. It declares that a thing is what it is. It allows us to perceive an individual as a unique thing. Stable judgements can be formed because of it.

Misunderstandings of the Law of Identity. . .

This law doesn’t only exemplify the judgement that “a man is a man.” That would be a so-called tautology (where subject and predicate are the same).

The law is more expansive than that. To judge “Socrates is a man” affirms some kind of identity. The predicate “man” is a real attribute of Socrates. This is not a tautology. Hence, the Law of Identity is not just about tautological judgements.

The Law of Contradiction is about violations of this “sameness.” Nothing can both be and not be in the same exact manner. It’s impossible that Socrates is both a man and a non-man. Those judgements contradict each other, and so cannot be simultaneously attributed to the subject.

Predicates P and non-P are contradictory vis-à-vis each other. Thus they cannot be judged to be attributes of the same subject in exactly the same way at the same time.

A contradiction takes place between the propositions “S is P” and “S is not P” when P, the predicate, is the exact same in both propositions.

Misunderstandings of the Law of Contradiction. . .

It wouldn’t be a contradiction to claim a painting is blue and non-blue when applying the attribute of blue and non-blue in different ways. The painting might be partly blue and partly non-blue, for example.

And it wouldn’t be a contradiction to claim “Socrates is young” and “Socrates is not young” when applying the predicate “young” to Socrates during different periods of his life.

The Law of Excluded Middle, in a way, puts to use the Law of Contradiction. It’s also known as the Law of Bivalence. Ether something is or it is not. It cannot be both — but it will always be one of those contradictory options.

Either “S is P” or “S is not P” is true. Between the two judgements, one will be true and the other will be false. It’s a choice between having and not having, i.e., between being and non-being. So, the issue is over having existence or not having existence.

Misunderstandings of the Law of Excluded Middle. . .

The law only applies to two contradictory judgements. The attributes absolutely deny or exclude each other: P versus not P.

There’s no middle ground between contradictory judgements.

The law doesn’t apply to contrary judgements. Two contraries are opposite extremes that can have a middle ground. For example, “microscopic” is the contrary of “gigantic.” Something not being “gigantic” obviously doesn’t entail that it must be “microscopic.”

***

Contradiction Explosion & Non-Classical Logic
“Proving” the moon is made of cheese!

Once we pretend a contradiction is true, we can thereby derive anything whatsoever from that contradiction. Moreover, one contradiction allows us to derive another contradiction ad infinitum.

Modern propositional logic gives us an easy way to rigorously prove this…
Line 1: P and not P (premise)
Line 2: P (simplification, Line 1)
Line 3: P or Q (addition, Line 2)
Line 4: not P and P (commutative property, Line 1)
Line 5: not P (simplification, Line 4)
Line 6: Q (disjunctive syllogism, Line 3 and Line 5)

Line 1 violates the Law of Contradiction.
Ignoring that violation, we were able to derive anything whatsoever (Q).
We’ll let Q represent “the moon is made of cheese.”

Line 7: P or not Q (addition, Line 2)
Line 8: not Q (disjunctive syllogism, Line 7 and Line 5)
Line 9: Q and not Q (conjunction, Line 6 and Line 8)

Line 9 got us another explicit violation of the Law of Contradiction.
And we can keep deriving them.

Imagine somebody denying the Law of Contradiction…
Self-Undermining Claims & Non-Classical Logic!

Denying the Law of Contradiction.
We’ll call this denial the proposition P.
P = “the Law of Contradiction (LOC) is false.”

This raises the question: how do we know that P is true rather than false?

Since if LOC is categorically false, then it could be that P is true and non-P is true. That is, it could be true that LOC is false and it could be true that it is not the case that LOC is false.

Thus, in order to assert that P is true presupposes that LOC is true.
To declare that LOC is never the case is self-undermining.

Non-Classical Logic.

Believe it or not, some logicians have found a way around this. Rather than arguing the LOC is never the case, they will say that there are exceptions to LOC. This is not immediately self-undermining (unlike before).

There are “non-classical” logics that admit contradictions…

Dialetheism is the view that there can be statements that are both true and false. There can be true contradictions. Paraconsistent logics have developed around this view.

There seems to be two problems with this. First, logicians have to construct their system in such a way to avoid the problem of “contradiction explosion.” Thus the admitted contradictions are, it seems, artificially isolated. They haven’t done away with the problem. Second, what is a genuine example of something that is true but violates the Law of Contradiction? Most philosophers, Aristotelian and non-Aristotelian, would claim there are none. Any supposed example, on deeper analysis, doesn’t violate the law and hence has failed to understand or apply the law properly. There is no clear “real world” example we can point to.

To be fair, maybe it is best to be open to the possibility that there can be true contradictions. (For example, is the Liar’s Paradox an example? While there are no “real world” examples, are there self-referential propositions that are true contradictions?) Our default position, however, should be skeptical.

Rejecting the Law of Excluded Middle…?

A more promising “non-classical” logic is one that just drops the Law of Excluded Middle. Could we say there’s a third value between true and false? For example, a value of “unknown.” For example, a value of “somewhat true and somewhat false.”

From a traditional logic point of view, I believe, this fails to understand what the (traditional) Law of Excluded Middle is doing: when the question is of being, there is nothing in-between absolute existence and absolute non-existence. It doesn’t deny that we often don’t know the truth or falsity of some proposition.

But, contrary to what someone might suspect, I think an Aristotelian inspired view leaves room for non-classical logics. It’s just that metaphysical assumptions or interpretations of those logics might be slightly to radically different. There is nothing absurd per se, for example, with a logic that has more than two truth values. We can develop such systems without necessarily doing away with the (traditional) Law of Excluded Middle. In fact, the idea of a “spectrum” of being does exist, especially in Thomism. There is a spectrum of truth and things can more-or-less participate in it.

Indeed, Fulton J. Sheen writes in God and Intelligence in Modern Philosophy: “Philosophical errors are reducible in principle to exaggeration or defect. This is a corollary of the principle that being and truth are convertible. Nothing which is, is essentially false” (pp. 166-167).

Perhaps it should be noted, moreover, that none of this is to deny practical benefits that might arise out of a “non-classical” logic. However, the mere construction of them is not, by itself, evidence that these traditional laws are false.