This is not a beginner's game. You already know the five connectives. This game trains the three skills that separate students who pass logic from students who master it: scope, equivalence, and proof. Every rule below is explained in three layers — plain intuition, formal rule, worked example.
Skill 1 · Scope — Why Brackets Change Everything
Every formula has a main connective — the last connective you evaluate. Brackets tell you which connective that is. Get the scope wrong, and you get the meaning wrong.
Scope of Negation Critical
"It is not the case that P and Q" is not the same as "Not P, and not Q."
¬(P ∧ Q) ≠ ¬P ∧ ¬Q
Example: P = "I have a ticket", Q = "I have a seat."
¬(P ∧ Q) = "It is not the case that I have both a ticket and a seat" — I might have one but not the other.
¬P ∧ ¬Q = "I have no ticket and no seat."
These mean different things. Brackets matter.
Main Connective Foundation
The main connective is the one that determines the overall truth value of the formula. It sits outside all brackets.
In (P ∧ Q) → R, the main connective is →. In P ∧ (Q → R), the main connective is ∧.
Example: (P ∧ Q) → R means "If both P and Q, then R." P ∧ (Q → R) means "P is true, and if Q then R." Different claims.
Negation and Scope Advanced
A negation before a bracket negates the whole bracket. A negation before a single letter negates just that letter.
¬(P → Q) means "It is not the case that if P then Q." ¬P → Q means "If not P, then Q."
Example: P = "It rains", Q = "The match is cancelled."
¬(P → Q) = "It is not the case that if it rains, the match is cancelled."
¬P → Q = "If it does not rain, the match is cancelled."
Skill 2 · Equivalence — When Two Formulas Say the Same Thing
Two formulas are logically equivalent if they have the same truth value on every possible assignment. The three most important equivalences in first-year logic are De Morgan's laws and the conditional equivalence.
De Morgan's Law 1 Must Know
"Not (P and Q)" is the same as "Not P or not Q."
¬(P ∧ Q) ≡ ¬P ∨ ¬Q
Example: "It is not the case that you are both tall and fast" means "You are not tall, or you are not fast" (or both).
Proof intuition: If ¬(P ∧ Q) is true, then P ∧ Q is false. So at least one of P, Q is false. So ¬P ∨ ¬Q is true. The reverse works too.
De Morgan's Law 2 Must Know
"Not (P or Q)" is the same as "Not P and not Q."
¬(P ∨ Q) ≡ ¬P ∧ ¬Q
Example: "It is not the case that you are either tall or fast" means "You are neither tall nor fast."
Proof intuition: If ¬(P ∨ Q) is true, then P ∨ Q is false, so both P and Q are false. So ¬P ∧ ¬Q is true. Reverse works too.
Material Implication Advanced
"If P then Q" is the same as "Not P or Q."
P → Q ≡ ¬P ∨ Q
Example: "If it rains, the match is cancelled" means "Either it does not rain, or the match is cancelled."
Why this matters: This equivalence lets you translate every conditional into disjunction, which makes truth tables and proofs easier.
Contraposition Advanced
"If P then Q" is the same as "If not Q, then not P."
P → Q ≡ ¬Q → ¬P
Example: "If it rains, the match is cancelled" is the same as "If the match is not cancelled, it did not rain."
Note: The converse (Q → P) is NOT equivalent to P → Q. This is the most common error in first-year logic.
Skill 3 · Natural Deduction — The Rules of Proof
A proof is a sequence of lines, each justified by a rule. The four rules below are the ones you will use most in first-year logic.
Modus Ponens (MP) Rule
If you know "If P then Q" and you know P, you can conclude Q.
P → Q, P ⊢ Q
Example:
1. If it rains, the match is cancelled. (P → Q)
2. It rains. (P)
3. Therefore, the match is cancelled. (Q, by MP)
Modus Tollens (MT) Rule
If you know "If P then Q" and you know not Q, you can conclude not P.
P → Q, ¬Q ⊢ ¬P
Example:
1. If it rains, the match is cancelled. (P → Q)
2. The match is not cancelled. (¬Q)
3. Therefore, it did not rain. (¬P, by MT)
Disjunctive Syllogism (DS) Rule
If you know "P or Q" and you know not P, you can conclude Q.
P ∨ Q, ¬P ⊢ Q
Example:
1. The match is played at home or away. (P ∨ Q)
2. It is not played at home. (¬P)
3. Therefore, it is played away. (Q, by DS)
Hypothetical Syllogism (HS) Rule
If P leads to Q, and Q leads to R, then P leads to R.
P → Q, Q → R ⊢ P → R
Example:
1. If it rains, the match is cancelled. (P → Q)
2. If the match is cancelled, the refunds are issued. (Q → R)
3. Therefore, if it rains, refunds are issued. (P → R, by HS)
Question Types in This Game
Type A · Identify the main connective — given a formula, pick the main connective.
Type B · Evaluate under assignment — given truth values, compute the truth value.
Type C · Recognise equivalence — which formula is logically equivalent?
Type D · Justify a proof step — given a proof line, which rule was used?
Type E · Spot the error — a student's proof or translation is shown. Find the mistake.