1
A connective is a table
¬, ∧ and ∨ are each defined by what they do to truth values, row by row. English words such as “and” and “or” are used to read them, and where the words carry more than the table, the table wins.
Lesson · Arguments and Validity · Propositional logic
Truth Values and the Connectives
Not, and and or are defined by the truth value they give, whatever the English words suggest. The one that surprises people is or, which logic reads as true when both parts are true; the exclusive reading is a separate connective, and the tables differ in exactly one row.
¬p true when p is false
p ∧ q true when both are true
p ∨ q true when at least one is true
p ⊕ q true when exactly one is true
or in logic includes both
Three connectives and the exclusive or
Core ideas
What this lesson establishes
A connective builds a bigger sentence from smaller ones, and in this course the bigger sentence's truth value depends on nothing but the smaller ones' truth values.
2
Or includes both
p ∨ q is true when p is true, when q is true, and when both are. It is false only when both are false. The exclusive reading, true when exactly one is true, is written p ⊕ q.
3
Compound sentences nest
¬p ∧ ¬q negates each part and then joins them. A table for it has a column for each smaller piece, so that each column is read from the ones before it.
The material
The tables
Four tables for two sentences, side by side, and the English each one usually carries.
Definition · Negation
¬p is true when p is false and false when p is true. “It is not the case that” is the safe English for it.
Definition · Conjunction
p ∧ q is true exactly when both p and q are true. “And” is the usual word, and so, for truth value, are “but” and “although”: they add a contrast that the table does not record.
Definition · Disjunction
p ∨ q is false exactly when both are false. It is inclusive: true also when both are true.
p q | ¬p p ∧ q p ∨ q p ⊕ q
---------------------------------------
T T | F T T F
T F | F F T T
F T | T F T T
F F | T F F F
The two right-hand columns agree except in the first row. That row is where English leaves a gap and logic closes it. “You may have soup or salad” on a menu usually means one of them, so the exclusive reading; “you will be fined if you are late or absent” does not mean you escape the fine by being both, so the inclusive reading. Logic picks the inclusive one and writes the exclusive one separately.
A compound formula is built up, and its table is built the same way. For ¬p ∧ ¬q, first fill the column for ¬p, then the one for ¬q, then combine them. Only one row of the four makes it true: p false and q false. This is the sentence “neither p nor q”.
In the lab, choose a formula for Formula A and read the table. The lab spells the connectives with a tilde for not, an ampersand for and, a vertical bar for or and a caret for exclusive or; the rows and the verdicts are the same. Switch the menu to compare two formulas, and the lab shows the rows where their columns differ.
Example · Menu logic
A diner asks for soup and salad both and is told the menu says soup or salad. If the menu's “or” is the logician's, the diner has complied with it. If it is the exclusive one, the diner has not. The sentence does not say which, which is why contracts and laws that use “or” often add “or both” or “but not both”.
Interactive
Truth table
Every assignment, evaluated
Method
Building a table by hand
The same four steps for any compound sentence over two letters.
1
List the cases
Write the four rows for p and q: both true, p only, q only, neither. Always in the same order, so that columns can be compared.
2
Add a column for each smaller piece
Work from the inside out. For ¬p ∧ ¬q, the pieces are ¬p and ¬q, and only then the whole.
3
Fill each column from the ones it depends on
A cell is decided by the definition of its connective and by the cells to its left. Never by what the English sounds like.
4
Read the answer in the last column
The rows with T are the cases in which the whole sentence is true. Count them, and compare with another formula by comparing columns.
Reading the tables
Score 0 / 0Choose the answer you can justify.
Soup or salad
Let p be “you have soup” and q be “you have salad”. The menu says p or q.
soup salad | inclusive (∨) exclusive (⊕)
T T | T F
T F | T T
F T | T T
F F | F F
Four rows, and the columns agree in three. The first row is the only place the menu's meaning matters: both soup and salad. On the inclusive reading it is allowed, and on the exclusive reading it is not. Everything else about the menu is settled by either reading.
Common mistakes
1
Reading “or” as exclusiveIn logic p ∨ q is true when both parts are true: the first row of the table says T. An exclusive reading belongs to a different connective, p ⊕ q, and the two columns differ in that one row. Reading ∨ as “one or the other but not both” will make a valid argument look invalid later in this course.
2
Treating “but” as something other than a conjunction“She is clever but lazy” carries a contrast, and the table for p ∧ q does not record it. For truth value, the sentence is true exactly when she is clever and she is lazy. Whatever else it suggests is not part of what the connective computes.
3
Reading a compound sentence in the wrong order¬p ∧ q negates p alone; ¬(p ∧ q) negates the conjunction. The two have different tables, and the second is the subject of a lesson ahead. Parentheses decide the order, and a table is built one piece at a time to keep it visible.
Check your understanding
✓
Finish when you can build a table and say what each row means.
Given a formula in ¬, ∧ and ∨, build its table row by row, state how many rows make it true, and point to the row where the inclusive and exclusive readings of or part.
Note: Every connective in this lesson is truth-functional: the truth of the whole is fixed by the truth of the parts. Not every English connective is. Whether “p because q” is true depends on more than the truth of p and of q, so no table can be drawn for it, and this course treats “because” as a signpost in an argument rather than as a connective inside one.