Tabla de contenido
To derive conclusions from known premises, we’ll use the rules of logical inference, fundamental tools that guarantee that if the premises are true, the conclusion will also be true.
In this guide, you’ll learn the main inference rules, from the classic Modus Ponens to constructive dilemmas, with step-by-step examples.
What is Logical Inference?
Logical inference is the process of obtaining valid conclusions from a set of premises, following rules that guarantee the validity of reasoning.
Formal Definition
A rule of inference is a valid argument form that allows deriving a conclusion from premises. If the premises are true, the conclusion will necessarily be true.
Notation
We use the symbol ⊢ (called “turnstile”) to indicate “derives” or “concludes”:
| Notation | Meaning |
|---|---|
| p, q ⊢ r | From p and q, r is derived |
| Γ ⊢ φ | From the set Γ, φ is derived |
Note about the comma: In inference notation, the comma means “AND” or “together with.” So “p, q ⊢ r” reads: “From p and q, r is derived.”
Difference: Equivalence vs Inference
It’s important not to confuse these concepts:
| Concept | Symbol | Meaning |
|---|---|---|
| Equivalence | ≡ | The expressions have the SAME truth value (interchangeable) |
| Inference | ⊢ | The conclusion is DERIVED from the premises (consequence relation) |
Example of the difference:
- Equivalence: p → q ≡ ¬p ∨ q (they’re the same, can be interchanged)
- Inference: (p → q) and p ⊢ q (from both premises together, q is concluded)
Difference: Conditional vs Implication
Just as we differentiate the biconditional (↔) from the equivalence symbol (≡), we must also distinguish between:
| Concept | Symbol | Type | Truth value |
|---|---|---|---|
| Conditional | → | Logical connective | Can be T or F (depends on p and q) |
| Implication | ⇒ | Tautology | Always T |
The Conditional (→)
It’s a logical connective that forms compound propositions. Its truth value depends on the values of p and q:
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Example: “If it rains, I get wet” can be true or false depending on the situation.
The Implication (⇒)
It’s a tautological relation between propositions. We say that p implies q (p ⇒ q) when the conditional p → q is a tautology (always true).
p ⇒ q means that [(p → q) is a tautology]
Example: Inference rules are implications:
- Modus Ponens: [(p → q) ∧ p] ⇒ q (always true)
- Hypothetical Syllogism: [(p → q) ∧ (q → r)] ⇒ (p → r) (always true)
Analogy with Equivalence
| Relation | Connective | Tautology |
|---|---|---|
| If…then | → (conditional) | ⇒ (implication) |
| If and only if | ↔ (biconditional) | ≡ (equivalence) |
In summary:
- We use → inside formulas as a connective
- We use ⇒ to indicate that one formula logically implies another (the relation is a tautology)
What about the ⊢ symbol?
The symbol ⊢ (turnstile) is similar to ⇒, but with a nuance:
| Symbol | Name | Focus |
|---|---|---|
| ⇒ | Implication | Semantic – “It’s a tautology” (truth by truth values) |
| ⊢ | Derivability | Syntactic – “It’s provable” (a formal proof exists) |
In classical propositional logic, both coincide: everything provable is true and vice versa. So, for practical purposes, you can consider them equivalent in this context.
The 4 Fundamental Rules
These are the most important and commonly used inference rules.
1. Modus Ponendo Ponens (M.P.P.)
Modus Ponens (from Latin “the mode that, by affirming, affirms”) is perhaps the most fundamental inference rule.
Symbolic Form
p → q (If p, then q)
p (p is true)
─────
∴ q (Therefore, q)
In words
If we have a conditional and affirm the antecedent, we can conclude the consequent.
Examples
Example 1:
- Premise 1: If it rains, I get wet
- Premise 2: It rains
- Conclusion: I get wet ✓
Example 2:
- Premise 1: If you study, you pass the exam
- Premise 2: You study
- Conclusion: You pass the exam ✓
Example 3 (symbolic):
- Premise 1: p → (q ∨ r)
- Premise 2: p
- Conclusion: q ∨ r ✓
Explanation
The conditional p → q is only false when p is T and q is F. If we know that p → q is true and p is true, then q MUST be true (otherwise, the conditional would be false).
2. Modus Tollendo Tollens (M.T.T.)
Modus Tollens (from Latin “the mode that, by denying, denies”) works with the negation of the consequent.
Symbolic Form
p → q (If p, then q)
¬q (q is false)
─────
∴ ¬p (Therefore, p is false)
In words
If we have a conditional and negate the consequent, we can conclude the negation of the antecedent.
Examples
Example 1:
- Premise 1: If it rains, the street is wet
- Premise 2: The street is NOT wet
- Conclusion: It’s NOT raining ✓
Example 2:
- Premise 1: If the engine works, the car starts
- Premise 2: The car does NOT start
- Conclusion: The engine does NOT work ✓
Example 3 (symbolic):
- Premise 1: (p ∧ q) → r
- Premise 2: ¬r
- Conclusion: ¬(p ∧ q) ✓
Relation to Contraposition
Modus Tollens is an application of the contraposition law:
- p → q ≡ ¬q → ¬p
If ¬q is true and ¬q → ¬p is equivalent to p → q, then ¬p must be true.
3. Hypothetical Syllogism (H.S.)
The Hypothetical Syllogism allows chaining conditionals like links in a chain.
Symbolic Form
p → q (If p, then q)
q → r (If q, then r)
─────
∴ p → r (Therefore, if p, then r)
In words
If p implies q, and q implies r, then p implies r (transitivity).
Examples
Example 1:
- Premise 1: If I don’t study, I don’t pass
- Premise 2: If I don’t pass, I repeat the course
- Conclusion: If I don’t study, I repeat the course ✓
Example 2:
- Premise 1: If it rains, the streets get wet
- Premise 2: If the streets get wet, there’s accident risk
- Conclusion: If it rains, there’s accident risk ✓
Example 3 (long chain):
- p → q
- q → r
- r → s
- Conclusion: p → s ✓
Analogy
It’s like a domino chain: if the first pushes the second, and the second pushes the third, then the first pushes the third.
4. Disjunctive Syllogism (D.S.) / Modus Tollendo Ponens (M.T.P.)
The Disjunctive Syllogism works with disjunction and the negation of one option.
Symbolic Form
p ∨ q (p or q)
¬p (p is false)
─────
∴ q (Therefore, q)
In words
If we have a disjunction and negate one alternative, we can conclude the other.
Examples
Example 1:
- Premise 1: I go to the movies or I stay home
- Premise 2: I do NOT go to the movies
- Conclusion: I stay home ✓
Example 2:
- Premise 1: The suspect is John or Peter
- Premise 2: It’s NOT John
- Conclusion: It’s Peter ✓
Example 3 (symbolic):
- Premise 1: (p ∧ q) ∨ r
- Premise 2: ¬(p ∧ q)
- Conclusion: r ✓
Variant
It also works by negating the other alternative:
p ∨ q
¬q
─────
∴ p
Construction Rules
These rules allow building more complex propositions from simpler ones.
5. Simplification (Simp.)
From a conjunction, we can extract any of its components.
Symbolic Form
p ∧ q
─────
∴ p (or also ∴ q)
Example
- Premise: John is tall AND thin
- Conclusion: John is tall ✓
- Conclusion: John is thin ✓
6. Addition (Add.)
From any true proposition, we can form a disjunction.
Symbolic Form
p
─────
∴ p ∨ q (for any q)
Example
- Premise: Today is Monday
- Conclusion: Today is Monday OR tomorrow it rains ✓
Note: This may seem strange, but it’s logically valid. If we know p is true, then “p or anything” is also true.
7. Conjunction (Conj.)
Two true propositions can be joined in a conjunction.
Symbolic Form
p
q
─────
∴ p ∧ q
Example
- Premise 1: Mary is a doctor
- Premise 2: Mary is a mother
- Conclusion: Mary is a doctor AND a mother ✓
The Dilemmas
Dilemmas are argument forms involving multiple conditionals and a disjunction.
8. Simple Constructive Dilemma (S.C.D.)
Both conditionals have the same consequent.
Symbolic Form
(p → q) If p, then q
(r → q) If r, then q (same consequent)
p ∨ r p or r
─────────
∴ q Therefore, q
In words
If two different paths lead to the same destination, and we take one of them, we arrive at that destination.
Example
- If I study (p), I pass the exam (q)
- If I’m lucky (r), I pass the exam (q)
- I study OR I’m lucky
- Conclusion: I pass the exam ✓
9. Complex Constructive Dilemma (C.C.D.)
The conditionals have different consequents.
Symbolic Form
(p → q) If p, then q
(r → s) If r, then s (different consequent)
p ∨ r p or r
─────────
∴ q ∨ s Therefore, q or s
In words
If we have two conditionals with different consequents, and at least one antecedent is true, then at least one consequent is true.
Example
- If I win the lottery, I buy a house
- If I inherit money, I buy a car
- I’ll win the lottery OR inherit money
- Conclusion: I’ll buy a house OR a car ✓
Variant: Proof by Cases (Excluded Middle)
Special case where the antecedents are complementary (p and ¬p).
Symbolic Form
(p → q) If p, then q
(¬p → q) If NOT p, then q
─────────
∴ q Therefore, q
Foundation
It’s based on the Law of Excluded Middle: p ∨ ¬p (p is true OR p is false, there’s no third option). Since in both cases we reach q, then q is inevitable.
Example
- If it rains (p), I stay home (q)
- If it does NOT rain (¬p), I also stay home (I’m sick) (q)
- Conclusion: I stay home ✓ (no matter what)
10. Simple Destructive Dilemma (S.D.D.)
Both conditionals have the same antecedent, but different consequents.
Symbolic Form
(p → q) If p, then q
(p → r) If p, then r (same antecedent)
¬q ∨ ¬r NOT q or NOT r
───────────
∴ ¬p Therefore, NOT p
In words
If the same antecedent leads to two different consequents, and at least one of those consequents is false, then the antecedent is false.
Example
- If I study (p), I get good grades (q)
- If I study (p), I learn a lot (r)
- I do NOT get good grades OR I do NOT learn much
- Conclusion: I did NOT study ✓
11. Complex Destructive Dilemma (C.D.D.)
The conditionals have different antecedents and different consequents.
Symbolic Form
(p → q) If p, then q
(r → s) If r, then s (different antecedent)
¬q ∨ ¬s NOT q or NOT s
───────────
∴ ¬p ∨ ¬r Therefore, NOT p or NOT r
In words
If we have two conditionals with different antecedents, and at least one consequent is false, then at least one antecedent is false.
Example
- If John went to the movies, he saw the film
- If Peter went to the theater, he saw the play
- John did NOT see the film OR Peter did NOT see the play
- Conclusion: John did NOT go to the movies OR Peter did NOT go to the theater ✓
Variant: Reductio ad Absurdum (Reduction to Absurdity)
Special case where the same antecedent leads to contradictory consequents.
Symbolic Form
(p → q) If p, then q
(p → ¬q) If p, then NOT q
───────────
∴ ¬p Therefore, NOT p
Foundation
If assuming p leads us to both q and ¬q (contradiction), then p must be false. It’s the basis of proof by contradiction.
Example
- If this theory is correct (p), it predicts result X (q)
- If this theory is correct (p), it predicts result NOT-X (¬q)
- Conclusion: The theory is NOT correct ✓
Other Useful Rules
12. Absorption (Abs.)
Symbolic Form
p → q
─────
∴ p → (p ∧ q)
Example
- If it rains, I get wet
- Conclusion: If it rains, (it rains AND I get wet) ✓
13. Resolution (Res.)
Widely used in automated theorem proving systems.
Symbolic Form
p ∨ q
¬p ∨ r
─────
∴ q ∨ r
Example
- John is a doctor OR an engineer
- John is NOT a doctor OR works at a hospital
- Conclusion: John is an engineer OR works at a hospital ✓
Inference Rules Summary Table
| # | Name | Form | Abbrev. |
|---|---|---|---|
| 1 | Modus Ponendo Ponens | p → q, p ⊢ q | M.P.P. |
| 2 | Modus Tollendo Tollens | p → q, ¬q ⊢ ¬p | M.T.T. |
| 3 | Hypothetical Syllogism | p → q, q → r ⊢ p → r | H.S. |
| 4 | Disjunctive Syllogism | p ∨ q, ¬p ⊢ q | D.S. |
| 5 | Simplification | p ∧ q ⊢ p | Simp. |
| 6 | Addition | p ⊢ p ∨ q | Add. |
| 7 | Conjunction | p, q ⊢ p ∧ q | Conj. |
| 8 | Simple Constructive Dilemma | (p→q), (r→q), p∨r ⊢ q | S.C.D. |
| 9 | Complex Constructive Dilemma | (p→q), (r→s), p∨r ⊢ q∨s | C.C.D. |
| 10 | Simple Destructive Dilemma | (p→q), (p→r), ¬q∨¬r ⊢ ¬p | S.D.D. |
| 11 | Complex Destructive Dilemma | (p→q), (r→s), ¬q∨¬s ⊢ ¬p∨¬r | C.D.D. |
| 12 | Absorption | p → q ⊢ p → (p ∧ q) | Abs. |
| 13 | Resolution | p∨q, ¬p∨r ⊢ q∨r | Res. |
Additional variants:
| Variant | Form | Based on |
|---|---|---|
| Proof by Cases | (p→q), (¬p→q) ⊢ q | Excluded Middle |
| Reductio ad Absurdum | (p→q), (p→¬q) ⊢ ¬p | Contradiction |
The Abbreviated Method (Reduction to Absurdity)
The abbreviated method allows verifying the validity of an argument without building the complete truth table.
What does it consist of?
It’s based on the principle of reduction to absurdity: if we assume the conclusion is false and reach a contradiction, then the argument is valid.
Method Steps
- Assume the conclusion is FALSE
- Assume all premises are TRUE
- Propagate the values to the subexpressions
- Look for contradictions:
- If there’s a contradiction → The argument is VALID
- If there’s no contradiction → The argument is INVALID
Step-by-Step Example
Verify: [(p → q) ∧ p] → q (Modus Ponens)
Step 1: We assume the main conditional is FALSE
- For → to be F, we need: antecedent T and consequent F
- So: (p → q) ∧ p = T, and q = F
Step 2: We propagate
- q = F ← already established
- (p → q) ∧ p = T ← then both components are T
- p = T ← from the conjunction
- p → q = T ← from the conjunction
Step 3: We verify consistency
- We have: p = T, q = F
- Then p → q should be F (T → F = F)
- But we said p → q = T
CONTRADICTION! ← The argument is VALID ✓
Advantages of the Method
| Advantage | Description |
|---|---|
| Speed | You don’t need to build the entire table |
| Efficiency | With many variables, tables are huge |
| Simplicity | You only look for a contradiction |
Application: Step-by-Step Proofs
Inference rules are applied in sequence to prove conclusions.
Example 1: Simple Proof
Premises:
- p → q
- q → r
- p
Prove: r
| Step | Proposition | Justification |
|---|---|---|
| 1 | p → q | Premise |
| 2 | q → r | Premise |
| 3 | p | Premise |
| 4 | q | M.P.P. (1, 3) |
| 5 | r | M.P.P. (2, 4) ✓ |
Example 2: With Hypothetical Syllogism
Premises:
- p → q
- q → r
- ¬r
Prove: ¬p
| Step | Proposition | Justification |
|---|---|---|
| 1 | p → q | Premise |
| 2 | q → r | Premise |
| 3 | ¬r | Premise |
| 4 | p → r | H.S. (1, 2) |
| 5 | ¬p | M.T.T. (4, 3) ✓ |
Example 3: With Disjunctive Syllogism
Premises:
- p ∨ q
- p → r
- ¬r
Prove: q
| Step | Proposition | Justification |
|---|---|---|
| 1 | p ∨ q | Premise |
| 2 | p → r | Premise |
| 3 | ¬r | Premise |
| 4 | ¬p | M.T.T. (2, 3) |
| 5 | q | D.S. (1, 4) ✓ |
Related Fallacies
It’s important to know the common errors that are NOT valid rules.
Fallacy of Affirming the Consequent
p → q
q ← INCORRECT!
─────
∴ p ← NOT valid
Incorrect example:
- If it rains, the street is wet
- The street is wet
- ❌ Conclusion: It’s raining (NO! The street could be wet for another reason)
Fallacy of Denying the Antecedent
p → q
¬p ← INCORRECT!
─────
∴ ¬q ← NOT valid
Incorrect example:
- If I study, I pass
- I do NOT study
- ❌ Conclusion: I do NOT pass (NO! I could pass another way)
Practice Exercises
Exercise 1: Identify the Rule
What inference rule applies in each case?
- From “p → q” and “¬q”, conclude “¬p”
- From “p ∨ q” and “¬p”, conclude “q”
- From “p → q” and “q → r”, conclude “p → r”
- From “p ∧ q”, conclude “p”
- From “p” and “q”, conclude “p ∧ q”
Exercise 2: Proofs
Prove the following conclusions using the inference rules:
- Premises: p → q, q → r, r → s, p. Conclusion: s
- Premises: p ∨ q, ¬p, q → r. Conclusion: r
- Premises: (p ∧ q) → r, p, q. Conclusion: r
Exercise 3: Abbreviated Method
Use the abbreviated method to verify if these arguments are valid:
- [(p → q) ∧ (q → r) ∧ p] → r
- [(p ∨ q) ∧ ¬p] → q
- [(p → q) ∧ q] → p (Is it valid?)
Answers
Answers to Exercise 1
- Modus Tollendo Tollens (M.T.T.)
- Disjunctive Syllogism (D.S.)
- Hypothetical Syllogism (H.S.)
- Simplification (Simp.)
- Conjunction (Conj.)
Answers to Exercise 2
1. Prove s:
| Step | Proposition | Justification |
|---|---|---|
| 1 | p → q | Premise |
| 2 | q → r | Premise |
| 3 | r → s | Premise |
| 4 | p | Premise |
| 5 | q | M.P.P. (1, 4) |
| 6 | r | M.P.P. (2, 5) |
| 7 | s | M.P.P. (3, 6) ✓ |
2. Prove r:
| Step | Proposition | Justification |
|---|---|---|
| 1 | p ∨ q | Premise |
| 2 | ¬p | Premise |
| 3 | q → r | Premise |
| 4 | q | D.S. (1, 2) |
| 5 | r | M.P.P. (3, 4) ✓ |
3. Prove r:
| Step | Proposition | Justification |
|---|---|---|
| 1 | (p ∧ q) → r | Premise |
| 2 | p | Premise |
| 3 | q | Premise |
| 4 | p ∧ q | Conj. (2, 3) |
| 5 | r | M.P.P. (1, 4) ✓ |
Answers to Exercise 3
- VALID – It’s Hypothetical Syllogism + Modus Ponens
- VALID – It’s Disjunctive Syllogism
- INVALID – It’s the fallacy of affirming the consequent
What’s Next?
In the next article, you’ll learn about mathematical proofs and the types of proofs.
Did you find this post useful? Leave me a comment with your questions or suggestions! And don’t forget to check out the next installment of this series on mathematical logic.