The Basics of the Laws of Logic
It is almost an inevitability that in any discussion with a person engaging in presuppositional apologetics that the phrase “the Laws of Logic” will be uttered as some prescriptive way to somehow “prove” or at least validate the existence of God. While they are also known as “laws of thought”, they are really merely descriptive principles of logic or axioms from which classical logic is predicated upon that extend from propositions to ontological states. As analytical propositions they are a priori knowledge which is known by a priori justification independent of experience.
Often these laws of logic can often extend to other forms of logic or semantic interpretations, however, many forms of logic deny or exclude them outright such as paraconsistent logic (a logical system which allows for contradictions without falling into the principle of explosion; ex contradictione quodlibet), dialetheism (a type of paraconsistent logical system which allows for true contradictions), fuzzy logic, intuitionistic logic, and other forms of logical that allow for truth value gluts (a sentence is both T and F) and truth value gaps (a sentence is neither T nor F).
It is generally held that there are 3 traditional, historical, or canonical laws of logic.
The law of Identity:
The is what some consider the most foundational of all the law of logic axioms. Socrates implied it in Plato’s Theaetetus by asking the question “”Then do you think that each differs to the other, and is identical to itself?”. Russell more explicitly described it as “Whatever is, is” a shortened version of Parmenides philosophy of “whatever is is, while Leibniz referred to it as “Everything is what it is”, and what is not cannot be”. Aristotle considered it to be the most fundamental law and obvious truth.
Mathematically the Law of Identity can be represented as:
Which is read as “For all x: x=x” where “=” represents equality and/or identity.
Unlike other laws of logic, the law of identity is related to terms and not propositions and isn’t used in propositional logic. It more informally can merely be stated as x=x, a=a, or A is A as all relate the same concept of something is itself. Identity is a type of binary relationship which is between the object of equality and itself. This is very closely related to a second order logical principle known to as what Leibniz referred to as identity of indiscernibility:
∀x∀y[∀F(Fx ↔ Fy) → x=y]
Read as for “for all of x and y, if x and y have the same properties then x is identical to y” where “Fx” represents the properties of x. (Capital letters tend to represent properties, while lower case represent subjects and referential expressions).
This can also be more explicitly defined by:
x=y =𝒹ₑ𝒻 (∀F)(Fx ↔ Fy)
Where x is the same as y by definition given they have exactly the same properties. Ex: .999… = 1 because “.999…” is just a different type of signifier (an infinite decimal expansion) representing “1” as both have exactly the same properties (they both exist at the same exact point on the real number line and are the same exact value).
The law of Non-Contradiction (LNC):
The LNC is that a proposition can not be both true and false at the same time. Propositionally LNC can be defined tautologically as:
LNC =𝒹ₑ𝒻 ¬(P Λ ¬P)
Meaning that given any proposition it can not be both true and false at the same time, or given any two propositions “A is B” and “A is not B” are mutually exclusive. I tend to use, merely by personal choice, capital “P” or say “A is B” to infer all or any proposition and “p” when referring to a specific proposition..but to the best of my knowledge there is no standard convention on this and ¬(P Λ ~P) and ¬(p Λ ~p) would represent the same thing.
This can also be expressed in terms of metatheory as:
(∀P) ~ (T(P) Λ T(~P))
This would be read as for all propositions it must be the case that the proposition is true or it’s negation is true (as in negation of p is equivalent to p is false).
The Law of Excluded Middle (LEM, tertium non datur):
By use of one of DeMorgan’s laws you can derive from the LNC the Law of Excluded Middle, that a proposition must be either true or false:
DeMorgan’s law: ¬(P Λ Q) ↔ (¬P V ¬Q)
Given ¬(P Λ ~P) you can derive LEM by:
¬(P Λ ¬P) = ¬P V ¬ ¬P
¬P V P (double negation rule)*
Propositionally the LEM can then be defined tautologically as:
LEM =𝒹ₑ𝒻 ¬P V P
Or explicitly by law as always true:
P V ¬P ≡ T
*Double negation rule also known as double negation elimination ¬¬P ⇒ P (⇒ means “can be replaced with”), ¬¬P ↔ P (Biconditional) or ¬¬P ⊢ P (Sequent notation). In intuitionistic logic double negation rule of A≡ ¬(¬A) does not hold s.t. ~ ¬¬P ⊬ P at least so far as that p can not be derived directly from double negation.
Just for fun I proved you can derive LEM from LNC using a natural logic checker (https://proofs.openlogicproject.org/) you can also go play around with: