The Laws of Classical Logic

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Classical logic rests upon a foundation of axioms. The axioms of classical logic, are a set of a priori abstractions that humans create, in order to make categorical syllogisms; their existence is contingent upon sentient brains. Some may argue, like myself, that these laws have correlates to basic laws of metaphysics1 and that this accounts for the 'utility' of logic, but it does not follow that logical rules are rules for the universe - they are rules for arguments.

Contents

The Axioms of Classical Logic

  • The Law of Identity: For things, this law asserts that "A is A" or "anything is itself." For propositions: "If a proposition is true, then it is true."
  • The Law of Excluded Middle: For things, "anything is either A or not A." For propositions: "A proposition, such as P, is either true or false." We also refer to such statements as "tautologies"
  • The Law of noncontradiction: For things: "Nothing can be both A and not-A." For propositions: "A proposition, P, can not be both true and false."

All of our syllogisms rely on these laws - that any thing is equal to itself, that tautologies must be true, and that contradictions must be false. Classical logic holds that everything has a definite, non-contradictory nature. A metaphysical law of identity would hold that to be perceived or even exist at all it must have a definite, non-contradictory nature, but for our purposes, it is enough to say that If A, then A.


Self Evident Nature of Axioms

Seeing as axioms are the foundation to logic, it is not possible to use classical logic to justify them. However, Aristotle found this is unnecessary: the axioms of classical logic are held to be self evident. We hold that they are are self evident because all syllogisms rely on them, and because they can be defended through retortion.

A defense through retortion occurs whenever an argument must rely opon the very principle it seeks to overturn. Any attempt to form a syllogism to refute the axioms of classical logic will have to rely on them ,leading to a self refutation (we call this type of self refutation the Stolen concept fallacy.

Necessity and Contingency

An understanding of axioms can be further advanced if the concepts of necessity and contingency are introduced. A proposition is said to be a necessary proposition if it's negation necessarily entails a contradiction. Axioms are considered to be necessary propositions.

A proposition is said to be a contingent proposition if it can be either true or false. As we will learn later, Categorical Propositions are one type of statement that can be either true or false.

Controversy

While some logicians refer to these axioms as the "Three laws of thought", implying that all cognition relies on them, it is important to realize that this presumption is false: not all logical systems rely on these axioms, and some logical systems do not rely on axioms at all. Other forms of logic, such as Propostional Logic instead rely on rules or definitions defined within the system, such as Well Formed Formular.

What these axioms are NOT

It is also important to avoid conflating or confusing the so called "laws of thought" with set of nomological (Physical) laws for the universe. Logic is not cosmology. It is not descriptive of how the universe works'. It is prescriptive: it sets forth a method of examining arguments.

The universe is not 'logical', it merely is.

It is also important not to confuse classical logic with psychology. The so called laws of thought are not rules for human behavior, they don't even cover all human thought: in our dreams, we are able to imagine contradictions, like being both the victim and the attacker, or being both young and old at the same time - human thought contains rational, irrational and non rational thought - both logic and emotions, impulses and instincts.

Finally, there is no reason to hold that these axioms are "immaterial", or transcendental. Such claims are usually based on arguments from ignorance. An incomplete physical account for abstractions is not a positive argument for an immaterial account for abstractions. Second, 'immateriality' is a negative concept, and a negative definition devoid of a universe of discourse, is meaningless. Unless someone can show how something immaterial can exist, how something immaterial can interact with physical brains, and how something immaterial can act - at all without violating basic physics (the principle of conservation of energy) then the claim that these logical laws that people create are transcendent or immaterial remains incoherent - because the term "immaterial" is meaningless.

For Advanced Students

The following section delves into the matter of axioms more deeply, and requires knowledge of symbolic logic. Those following the Course in Logic 101 will learn everything required to grasp this section in later lessons, and can skip this section for now, if desired.

As stated above, some logical systems to not require any axioms at all. For example, the set of axioms for the sentential, or propositional, logic is {} - the empty set!

So how does such a system "get off the ground"?

It creates a set of rules, defined within the system:

Let the set of English capital letters be well-formed formular (WFFs), which may be appended
by zero to an infinite amount of primes to indicate different WFFs
If A and B are WFFs, then (A v B) is a WFF
If A and B are WFFs, then (A & B) is a WFF
If A is a WFF, then (~A) is a WFF
If A and B are WFFs, then define (A -> B) to be ((~A) v B)
If A and B are WFFs, then define (A <-> B) to be ((A -> B) & (B ->A))
No other strings are WFFs

That defines our "grammar" for our fake little language that will turn into the propositional logic.

Now define 21 rules of inference to allow us to move between WFFs:

Propositional_Logic.png

Then we define a function that maps each propositional variable to two values: "True" or "false" (or 1 and 0, or "your mom" and "your dad" - it doesn't matter from a formal point of view).

Then come the soundness (if you can derive a string from a set of strings above, then it must be "true") and completeness (all "true" strings can be derived from the rules above) proofs, and we're done! The system is both sound and complete.

All this in much more detail can be found at:

http://www.algebra.com/algebra/about/history/Propositional-logic.wikipedia#Axioms

The point is that there can be no axioms in this logic, the most basic of all modern logics (there are other formulations that do have axioms, though): everything is definitional. And how does one argue with a definition? My point is: the answer to the question "why doesn't everyone accept the axioms of logic?" is that it can be the case that there's nothing to accept. Literally.

This system even allows us to create a proof for the law of non contradiction:

Proof (by reductio):

1) (A & ~A) [Proposition]
2) A [Conjunction elimination from 1]
3) ~A [Conjunction elimination from 1]
4) ~(A & ~A) [Reductio, 1 - 3]

QED

This is a proof of the law of noncontradiction (LNC) using the simplest logical system there is, which is called sentential, or propositional logic. The rules used in the proof were cited in square brackets. The LNC is line 4 of the proof. In short, the argument shows that the LNC is derivable from logic in the most rigorous way possible: a deductive proof.

This said, there are cases where certain logics don't "work" (for example, "all men are mortal, Socrates is a man, therefore Socrates is mortal" is an invalid argument under the logic given above. I'll leave it as an exercise for the interested to figure out why)2.

Those taking the Course in Logic 101 will want to proceed to the next section: The Difference Between Believing and Knowing


Footnotes

1. Metaphysics is a term also invented by Aristotle; and it has to do with theories concerning how existence itself 'works'.

2. Ok, here's the answer: http://en.wikipedia.org/wiki/First-order_logic

Propositional logic is not adequate for formalizing valid arguments that rely on the internal structure of the propositions involved. For example, a translation of the valid argument

   * All men are mortal * Socrates is a man * Therefore, Socrates is mortal 

into propositional logic yields

   * A * B * therefore C 

which is invalid, as there is no connection between the premises and the conclusion.


References

  • Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.
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