1
Or is inclusive
p ∨ q is true when both are true. English "or" is often exclusive ("soup or salad"), and reading ∨ that way is the single most common error in this course.
Lesson · Logic and Proof · Propositional logic
Logical Connectives
Four connectives, each defined by exactly what it does to truth values. The one that causes trouble is ∨, which is inclusive — and the one English does not have a clean word for is ⊕.
¬p true exactly when p is false
p ∧ q true exactly when both are true
p ∨ q true when at least one is true (INCLUSIVE)
p ⊕ q true when exactly one is true (exclusive)
Four connectives, four definitions
Core ideas
What this lesson establishes
A connective is defined by its truth table and by nothing else. Any intuition about the English word is a convenience, not the definition.
2
Connectives are functions on truth values
∧ takes two truth values and returns one. It does not know or care what the propositions are about, which is why p ∧ q and q ∧ p are the same.
3
Precedence exists to save parentheses
¬ binds tightest, then ∧, then ∨. So ¬p ∧ q means (¬p) ∧ q. When in doubt, write the parentheses.
The material
The four basic connectives
Each defined by a truth table, then read back into English.
Definition · Negation
¬p ("not p") is true when p is false and false when p is true. It is the only unary connective in this course.
Definition · Conjunction
p ∧ q ("p and q") is true exactly when both p and q are true — one row of the four.
Definition · Disjunction
p ∨ q ("p or q") is true exactly when at least one of p, q is true — three rows of the four. This is the inclusive or.
Definition · Exclusive or
p ⊕ q is true exactly when exactly one of p, q is true. It is false when both are true, which is where it parts company with ∨.
The two disjunctions differ in exactly one row of four, and that row is where natural language is ambiguous. "You may have coffee or tea" usually means ⊕; "an error occurs if the file is missing or unreadable" means ∨. Mathematics settles the ambiguity by decree: ∨ is always inclusive.
p q | ¬p p∧q p∨q p⊕q
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T T | F T T F ← the row that separates ∨ from ⊕
T F | F F T T
F T | T F T T
F F | T F F F
Two English constructions to watch. "Neither p nor q" is ¬p ∧ ¬q, which “Logical Equivalence” shows equals ¬(p ∨ q). "p unless q" is ¬q → p, which is “The Conditional and Its Relatives”'s business and is not p ⊕ q.
Theorem · Precedence
In order from tightest to loosest: ¬, then ∧, then ∨ and ⊕, then →, then ↔. So ¬p ∨ q ∧ r means (¬p) ∨ (q ∧ r). This is convention rather than mathematics, and parentheses always win.
Interactive
Truth table
Every assignment, evaluated
Method
Translating English into connectives
The verb tells you the shape; the small words tell you which.
1
Find the atomic propositions
Split the sentence into the smallest parts that are themselves true or false, and name them p, q, r.
2
Identify the connective words
"and", "but", "yet" and "however" are all ∧. "or" is ∨ unless the sentence forces exclusivity. "neither…nor" is a negated disjunction.
3
Decide whether the or is exclusive
Ask what happens when both hold. If the sentence is still true, it is ∨; if it becomes false, it is ⊕.
4
Parenthesise, then check a row
Write the formula with explicit parentheses and test it on the case you understand best. One wrong row means a wrong translation.
Connectives
Score 0 / 0Choose the answer you can justify.
Translating four sentences
Let p = "it is raining" and q = "the ground is wet".
"It is raining and the ground is wet." p ∧ q
"It is raining but the ground is not wet." p ∧ ¬q
"Neither is it raining nor is the ground wet." ¬p ∧ ¬q
"Either it is raining or the ground is wet,
but not both." p ⊕ q
"But" is ∧. It carries a contrast in English that logic simply discards, because the truth value of "p but q" is the truth value of "p and q" in every case. Losing that shade of meaning is a deliberate trade: what remains is checkable.
Common mistakes
1
Reading ∨ as exclusiveIt costs you one row in every truth table containing a disjunction, and it makes De Morgan's laws look wrong in “Logical Equivalence”.
2
Dropping parentheses around a negation¬p ∧ q and ¬(p ∧ q) disagree in two of four rows. When a negation governs more than a single letter, write the brackets.
3
Translating "but" as something other than ∧"But", "although" and "however" all signal contrast, which is a fact about emphasis. The truth conditions are exactly those of "and".
Check your understanding
✓
Finish when you can produce any of the four truth tables from memory and defend the ∨ row.
The test is not reciting the tables: it is being handed "either p or q" in a sentence and deciding correctly whether both holding keeps it true.
Note: ⊕ is the parity connective: p ⊕ q ⊕ r is true when an odd number of them are true. That is why it appears in checksums and error-detecting codes, and why Number Theory and Cryptography meets it again as addition modulo 2.