# Tautology, Entailment and Consistency

Philosophy I · Logic · https://tryals.app/learn/philosophy-i/tautology-entailment-and-consistency

## What the Table Was For

Truth tables are a procedure; the **semantic relations** they compute are what logic actually cares about.

**Logical truth** is the base case. A **tautology** is true under every assignment, a **contradiction** under none, and a **contingency** under some but not all. Only contingent sentences carry information about the world, a tautology rules nothing out, which is why "it will either rain or not" is useless as a forecast.

**Entailment** is the central relation. A set of premises **entails** a conclusion when every assignment making all premises true also makes the conclusion true, written $\Gamma \models \varphi$. This is validity stated semantically, and it connects to the conditional through a result worth knowing: $\Gamma \models \varphi$ exactly when the conditional from the conjoined premises to $\varphi$ is a tautology. Argument-checking and sentence-checking are the same task in different clothes.

**Consistency** is the third relation. A set of sentences is consistent when *some* assignment makes them all true simultaneously. Inconsistency is a serious defect, because an inconsistent set entails **everything**: if the premises cannot all be true, no assignment makes them true with a false conclusion, so vacuously every conclusion follows. This is the principle of explosion, and it is why a contradiction in a theory is fatal rather than merely awkward.

Two relations that are easy to confuse:

| Relation | Holds when |
|---|---|
| $\varphi$ entails $\psi$ | Every model of $\varphi$ is a model of $\psi$ |
| $\varphi$ and $\psi$ are equivalent | They have exactly the same models |

Equivalence is entailment in both directions. Entailment alone is one-way: $P \wedge Q$ entails $P$, but $P$ does not entail $P \wedge Q$.

The final piece is that **an argument is valid if and only if its premises are inconsistent with the negation of its conclusion**. This equivalence is what licenses proof by contradiction: to establish a conclusion, assume its denial and derive an inconsistency.

> **Common pitfall:** treating an inconsistent set of premises as simply useless. It is worse than useless, it entails every sentence whatsoever, including the negation of anything you wanted to prove. A theory that has become inconsistent has not merely lost some conclusions; it has lost the ability to rule anything out.

## Practice questions

6 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Why does an inconsistent set of premises entail every sentence?

A. Every assignment makes the premises false, and false antecedents make conditionals true
B. A contradictory premise set semantically forces every conclusion to be a tautology
C. Standard deductive rules of inference break down completely when premises conflict
D. No assignment makes the premises true, so none makes them true with a false conclusion

**Answer:** D. No assignment makes the premises true, so none makes them true with a false conclusion

**Why:** Entailment is defined by the absence of a counterexample. If nothing satisfies the premises, no counterexample can exist, so every conclusion follows *vacuously*. This is the principle of explosion, and it is why contradiction is fatal to a theory.

Page: https://tryals.app/practice/philosophy-i/tautology-entailment-and-consistency/why-does-an-inconsistent-set-of-premises-entail-every-sentence

### 2. Which statements about entailment are correct?

A. P and Q entails P
B. Logical equivalence is entailment in both directions
C. Entailment can hold in one direction only
D. P entails P and Q

**Answer:** A. P and Q entails P; B. Logical equivalence is entailment in both directions; C. Entailment can hold in one direction only

**Why:** A conjunction entails each conjunct, but not conversely, knowing $P$ tells you nothing about $Q$. Equivalence is exactly mutual entailment, which is why it is the stronger relation.

Page: https://tryals.app/practice/philosophy-i/tautology-entailment-and-consistency/which-statements-about-entailment-are-correct

### 3. A tautology carries no information about how the world is.

**Answer:** True

**Why:** True, a tautology is true in every case, so it excludes nothing. "It will rain or it will not" is useless as a forecast precisely because no weather could falsify it. Only contingent sentences carry information.

Page: https://tryals.app/practice/philosophy-i/tautology-entailment-and-consistency/a-tautology-carries-no-information-about-how-the-world-is

### 4. An argument with premises Γ and conclusion φ is shown to be valid. What follows if we now combine Γ with the negation of φ into a single set?

A. The resulting set becomes contingent rather than valid
B. The set entails φ but fails to entail any other claims
C. The resulting set cannot be satisfied by any assignment
D. The premises alone must have been internally inconsistent

**Answer:** C. The resulting set cannot be satisfied by any assignment

**Why:** Testing validity by reductio works because establishing validity guarantees no assignment can simultaneously satisfy the premises and the denial of the conclusion. Inconsistency does not mean the original premises were flawed, nor does it selectively entail only original claims.

Page: https://tryals.app/practice/philosophy-i/tautology-entailment-and-consistency/an-argument-with-premises-and-conclusion-is-shown-to-be-valid

### 5. Sort each sentence by its semantic status.

**Answer:**

- Tautology: if P then P, P or not-P
- Contradiction: P and not-P
- Contingency: P and Q, if P then Q

**Why:** The classification is exhaustive: a sentence is true always, never, or sometimes. Note that the conditional is contingent, not a tautology, only a conditional whose consequent is guaranteed by its antecedent is a logical truth.

Page: https://tryals.app/practice/philosophy-i/tautology-entailment-and-consistency/sort-each-sentence-by-its-semantic-status

### 6. An argument has premises that are inconsistent. How many valid conclusions can be drawn from them?

**Answer:** -1

**Why:** Every sentence follows, so the count is unbounded, recorded here as $-1$ for "all of them". That total collapse of discrimination is the principle of explosion, and it is why inconsistency is a catastrophe rather than a gap.

Page: https://tryals.app/practice/philosophy-i/tautology-entailment-and-consistency/an-argument-has-premises-that-are-inconsistent-how-many-valid