19.2. Logical Functions and Laws of Thinking

The logical laws of thought are fundamental axiomatic rules upon which rational discourse itself is based. The rules have a long tradition in the history of philosophy.

The logical laws express the connections between different propositions. These types of connections are called logical functions. Commonly used logical connectives include:

1. Negation (not) (¬ or ~)

2. Conjunction (and) (&, or • )

3. Disjunction (or) ( ∨ )

4. Material implication (if...then) ( )

5. Biconditional (if and only if) ()

For example, the meaning of the statements “It is raining (P)” and “I am indoors (Q)” is transformed when the two are combined with logical connectives:

It is raining and I am indoors (P & Q)

If it is raining, then I am indoors (P  Q)

It is raining if I am indoors (Q  P)

It is raining if and only if I am indoors (P  Q)

It is not raining (¬P)

The law of identity states that an object is the same as itself: A ≡ A. Any reflexive relation upholds the law of identity. When discussing equality, the fact that "A is A" is a tautology. Aristotle identifies the principle in Book VII of the Metaphysics: “Now "why a thing is itself" is a meaningless inquiry (for — to give meaning to the question 'why '— the fact or the existence of the thing must already be evident —e.g., that the moon is eclipsed — but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical, unless one were to answer, 'because each thing is inseparable from itself, and its being one just meant this.' This, however, is common to all things and is a short and easy way with the question.)”

The principle of contradiction (“principium contradictionis” in Latin) is the second of the so-called three classic laws of thought. The oldest statement of the law is that contradictory statements cannot both at the same time be true, e.g. the two propositions A is B and A is not B are mutually exclusive. A may be B at one time, and not at another; A may be partly B and partly not B at the same time; but it is impossible to predicate of the same thing, at the same time, and in the same sense, the absence and the presence of the same quality. This is the statement of the law given by Aristotle. It takes no account of the truth of either proposition; if one is true, the other is not; one of the two must be false.

In the symbolism of propositional logic, the principle is expressed as: ~ (P & ~P).

According to Allan Bloom, "the earliest-known explicit statement of the principle of contradiction — the premise of philosophy and the foundation of rational discourse" — is given in Plato's The Republic where the character Socrates states, "It's plain that the same thing won't be willing at the same time to do or suffer opposites with respect to the same part and in relation to the same thing" (436B).

The traditional source of the Principle of Contradiction is Aristotle’s Metaphysics where he gives three different versions.

1) ontological: “It is impossible that the same thing belong and not belong to the same thing at the same time and in the same respect.”

2) Psychological: “No one can believe that the same thing can (at the same time) be and not be.”

3) Logical: “The most certain of all basic principles is that contradictory propositions are not true simultaneously.”

The law of excluded middle, also known as the principle of excluded middle or excluded middle is the principle that for any proposition, either that proposition is true, or its negation is. The principle can be expressed in either a logical or a semantic form. The semantic form uses the non-logical word "true", as above. The logical form uses only logical expressions "either", "or" and can be expressed by the formula "P ∨ ¬P": "either P or not P", where "P" is schematic for any proposition such as 'snow is white', 'Socrates is running' and so on.

The earliest known formulation of the principle is in the book On Interpretation by Aristotle, where he says that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false. He also states it as a principle in the Metaphysics book 3, saying that it is necessary in every case to affirm or deny, and that it is impossible that there should be anything between the two parts of a contradiction.

For example, if P is the proposition: “Socrates is mortal”, then the law of excluded middle holds that the logical disjunction: “Either Socrates is mortal, or it is not the case that Socrates is mortal” is true by virtue of its form alone. That is, the "middle" position, that Socrates is neither mortal nor not-mortal, is excluded by logic, and therefore either the first possibility (Socrates is mortal) or its negation (it is not the case that Socrates is mortal) must be true.

Basic and Derived Argument Forms
Name	Sequent	Description
Modus Ponens (The mode of putting)	The dog barks if it does not know the visitor well. The dog barks. Therefore dog doesn`t know the visitor well.	If p then q; p; therefore q
Modus Tollens	If the dog didn`t know the visitor well, then the dog would bark The dog didn’t bark Therefore the dog knew the visitor well	If p then q; not q; therefore not p
Hypothetical Syllogism		If p then q; if q then r; therefore, if p then r
Disjunctive Syllogism		Either p or q, or both; not p; therefore, q
Constructive Dilemma		If p then q; and if r then s; but p or r; therefore q or s
Destructive Dilemma		If p then q; and if r then s; but not q or not s; therefore not p or not r
Bidirectional Dilemma		If p then q; and if r then s; but p or not s; therefore q or not r
Simplification		p and q are true; therefore p is true
Conjunction		p and q are true separately; therefore they are true conjointly
Addition		p is true; therefore the disjunction (p or q) is true
Composition		If p then q; and if p then r; therefore if p is true then q and r are true
De Morgan's Theorem (1)		The negation of (p and q) is equiv. to (not p or not q)
De Morgan's Theorem (2)		The negation of (p or q) is equiv. to (not p and not q)
Commutation (1)		(p or q) is equiv. to (q or p)
Commutation (2)		(p and q) is equiv. to (q and p)
Commutation (3)		(p is equiv. to q) is equiv. to (q is equiv. to p)
Association (1)		p or (q or r) is equiv. to (p or q) or r
Association (2)		p and (q and r) is equiv. to (p and q) and r
Distribution (1)		p and (q or r) is equiv. to (p and q) or (p and r)
Distribution (2)		p or (q and r) is equiv. to (p or q) and (p or r)
Double Negation		p is equivalent to the negation of not p
Transposition		If p then q is equiv. to if not q then not p
Material Implication		If p then q is equiv. to not p or q
Material Equivalence (1)		(p is equiv. to q) means (if p is true then q is true) and (if q is true then p is true)
Material Equivalence (2)		(p is equiv. to q) means either (p and q are true) or (both p and q are false)
Material Equivalence (3)		(p is equiv. to q) means, both (p or not q is true) and (not p or q is true)
Exportation		from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true)
Importation		from (if q is true then r is true, if p is true) we can prove (if p and q are true then r is true)
Tautology (1)		p is true is equiv. to p is true or p is true
Tautology (2)		p is true is equiv. to p is true and p is true
Tertium non datur (Law of Excluded Middle)		p or not p is true
Law of Non-Contradiction		p and not p is false, is a true statement