Disclaimer

This article primarily explains propositional logic/statement logic/sentential logic/zeroth-order logic. However, some definitions are given using predicate logic/first-order logic (i.e. the statement “for any” is used). Given that the introductory course I am currently taking does not bother to distinguish them as different systems when it gives definitions, I won’t either for the time being.

We use symbols and definitions as the basis of logic. There are different standards, so I have chosen symbols and definitions based on my digital electronics coursework as well as the current philosophy course I am taking. This article tries to be self-contained and discipline-agnostic in that these tools can be used across philosophy, mathematics, and engineering.

There are colloquial definitions of logical terms and there are formal definitions. A common mistake is to rely on “intuition” of what we think terms mean, but in reality we often need to accept that we do not have enough experience to form any intuition or that we are using terms loosely and incorrectly.

Building Blocks

Sentence: A grammatical combination of words in a language.

Atomic sentence: A simple sentence.

Molecular sentence: A sentence made by combining together simple sentences with truth-functional connectives.

Truth-functional connective: Operators that combine or modify simple sentences, such as NOT, OR, AND, and implication. Also known as logical operators.

Statement: A declarative sentence. Either true or false, but not both.

Proposition: The meaning expressed by a statement. Either true or false, but not both. 1 2

Descriptors for Building Blocks

Tautology: For any proposition P, P is a tautology iff it is logically impossible for it to be false.

Contradiction: For any proposition P, P is a contradiction iff it is logically impossible for it to be true.

Descriptors for Relations Between Building Blocks

Sufficiency: Being F is sufficient for being G: For any x, if x is F then x is G.

Necessity: Being F is necessary for being G: For any x, if x is G then x is F.

Necessity and Sufficiency: Being F is necessary and sufficient for being G: For any x, (If x is F then x is G) and (If x is G then x is F). Abbreviated as: For any x, x is F iff x is G.

Entailment: For any proposition P and any proposition Q, P entails Q iff it is logically impossible for Q to be false while P is true.

Consistency: For any proposition P and any proposition Q, P is consistent with Q iff it is logically possible for P to be true while Q is true.

Inconsistency: For any proposition P and any proposition Q, P is inconsistent with Q iff it is logically impossible for P to be true while Q is true.

Arguments

Argument: A set of propositions consisting of a conclusion and at least one premise.

Premise: A proposition that expresses reasons meant to support the argument’s conclusion.

Conclusion: A proposition that expresses the position that the argument is meant to support.

Descriptors for Arguments

Validity: An argument is valid iff there is no logically possible situation in which its premises are true and its conclusion is false.

Soundness: An argument is sound iff it is valid and its premises are true.

Truth Tables for Truth-Functional Connectives

Given two propositions P and Q, we use the following symbols for truth-functional connectives:

  • ¬P: Not P (Negation)
  • P ∧ Q: P and Q (Conjunction)
  • P ∨ Q: P or Q (Disjunction)
  • P → Q: If P then Q (Conditional/Implication)
  • P ↔ Q: P if and only if Q (Biconditional/Equivalence)

We should note that AND and OR are commutative, associative, and distributive.

We should also note that P → Q means P is sufficient for Q, while Q is necessary for P.

Truth tables specify how the truth-value of a molecular sentence featuring connectives (right column) is determined by the truth- values of simple sentences on which the connective operates (left columns):

P Q ¬P P ∧ Q P ∨ Q P → Q P ↔ Q
T T F T T T T
T F F F T F F
F T T F T T F
F F T F F T T

Tautology

The proposition below is a tautology because it is logically impossible for it to be false.

For any proposition P, it is true or it is not true. This is also known as the Law of the Excluded Middle.

P ∨ ¬P

Contradiction

The proposition below is a contradiction because it is logically impossible for it to be true. Negating this proposition turns it into a tautology, the Law of Non-Contradiction.

For any proposition P, it is true and it is not true.

P ∧ ¬P

Rules of Inference

Rules of inference are rules according to which arguments of a certain form are always valid.

Given propositions P, Q, and R, and denoting premises with P[i]. and conclusions with C, we have the following valid arguments.

Modus Ponens:

If P, then Q. P, therefore Q.

P1. P → Q  
P2. P  
C. Q

Modus Tollens:

If P, then Q. Not Q, therefore not P.

P1. P → Q  
P2. ¬Q  
C. ¬P

Double Negation:

P, therefore not not P.

P1. P  
C. ¬¬P

De Morgan’s Laws:

Not P and Q combined is equivalent to not P or not Q. Equivalently, not P or Q combined is equivalent to not P and not Q.

P1. ¬(P ∧ Q)  
C. ¬P ∨ ¬Q

P1. ¬(P ∨ Q)  
C. ¬P ∧ ¬Q

Simplification:

P and Q, therefore P.

P1. P ∧ Q  
C. P

Adjunction:

P, Q, therefore P and Q.

P1. P  
P2. Q  
C. P ∧ Q

Addition:

P, therefore P or Q.

P1. P  
C. P ∨ Q

Modus Tollendo Ponens/Disjunctive Syllogism:

P or Q. Not P, therefore Q.

P1. P ∨ Q  
P2. ¬P  
C. Q

Hypothetical Syllogism:

If P, then Q. If Q, then R. Therefore, if P, then R.

P1. P → Q  
P2. Q → R  
C. P → R

Conclusion

With these fundamental logic tools, we are now prepared to make valid arguments across branches of philosophy. Stay tuned for the posts on epistemology!

  1. Propositions also consist of mental states, called propositional attitudes, that are directed at the proposition, but this is less important for our purposes. ↩

  2. If I say “The sky is blue” and my friend says “আকাশ নীল”, then we have said two different statements with the same underlying proposition. Or, if I say “My shirt is blue” and my friend says “Your shirt is blue”, we have also said two different statements with the same underlying proposition. However, for the purpose of defining an argument, the terms are often used interchangeably. In fact, throughout this text, they are used interchangeably. We are enforcing the same “language” through symbols and the subjects (I, You) are understood to mean the same thing when applicable, so they really do mean the same thing here. ↩