Translation Begins with Truth Conditions

The tempting method is to replace words mechanically: and becomes $\wedge$, or becomes $\vee$, and if becomes $\to$. This works only when the grammar already exposes the intended logical structure. Ordinary language often leaves scope, direction, and exclusivity implicit.

A sound translation instead preserves truth conditions. For a sentence $S$ and its translation $\tau (S)$, the aim is

$$S\text{ is true in a situation}⟺v(\tau (S))=1$$

under the corresponding valuation $v$. The words guide the translation, but the decisive question is which combinations of truth values the sentence permits and forbids.

Begin with whole claims

Choose an atom for each elementary claim whose internal structure is irrelevant to the argument. Record the meaning before writing any connectives:

$$L:=\text{I stay late at work},F:=\text{I finish my assignments}.$$

The same claim must receive the same atom throughout. Different claims should not be compressed into one atom merely because they occur in the same sentence.

Atomicity is relative to the problem. “The server is available” may be one atom if nothing depends on why it is available. If the argument distinguishes power, network, and process availability, those conditions need separate atoms.

Translate the constraint, not the cue word

The following readings are standard in classical propositional logic:

Natural-language form	Formula
$A$ and $B$	$A\wedge B$
$A$ but $B$	$A\wedge B$
$A$ or $B$	$A\vee B$
either $A$ or $B$, but not both	$(A\vee B)\wedge \neg (A\wedge B)$
not both $A$ and $B$	$\neg (A\wedge B)$
neither $A$ nor $B$	$\neg A\wedge \neg B$
if $A$, then $B$	$A\to B$
$A$ if $B$	$B\to A$
$A$ only if $B$	$A\to B$
$A$ if and only if $B$	$A\leftrightarrow B$
$A$ unless $B$	$\neg B\to A$, equivalently $A\vee B$

These are defaults, not a claim that natural language is perfectly rigid. When context supports several readings, state the chosen reading rather than hiding the ambiguity.

The words but, although, and however carry contrast or emphasis. Plain propositional logic does not represent that pragmatic difference, so “$A$, but $B$” has the same truth conditions as the conjunction $A\wedge B$.

Small words determine implication direction

The implication

$$A\to B$$

forbids exactly the case $A=1$ and $B=0$. It says that whenever $A$ holds, $B$ must hold. Consequently, these phrases have the same direction:

$$\begin{array}{rl}& \text{if }A\text{, then }B,\\ & A\text{ is sufficient for }B,\\ & B\text{ is necessary for }A,\\ & A\text{ only if }B.\end{array}$$

The word if introduces a sufficient condition; only if introduces a necessary condition. This is why

$$\text{“}A\text{ if }B\text{”}=B\to A,\text{“}A\text{ only if }B\text{”}=A\to B.$$

Reversing $A\to B$ to $B\to A$ is the converse error. Nothing about “If it rains, the ground is wet” says that rain is the only possible cause of wet ground.

Scope comes before simplification

Negation applies to a particular constituent. Parentheses should expose that constituent before any algebraic rewriting:

$$\begin{array}{rl}\text{It is not true that }A\text{ and }B& ⇝\neg (A\wedge B),\\ \text{not }A\text{, and }B& ⇝\neg A\wedge B,\\ \text{neither }A\text{ nor }B& ⇝\neg (A\vee B)\equiv \neg A\wedge \neg B.\end{array}$$

In particular,

$$\neg (A\wedge B)\not\equiv \neg A\wedge \neg B.$$

“Not both” permits either claim to hold alone; “neither” requires both to be false. Applying De Morgan’s rules before identifying the intended scope often turns one reading into the other.

Either does not by itself exclude both

The disjunction $A\vee B$ is inclusive: it permits $A$ and $B$ to hold simultaneously. Exclusive disjunction requires an additional restriction:

$$\begin{array}{rl}A\vee B& \text{at least one},\\ (A\vee B)\wedge \neg (A\wedge B)& \text{exactly one}.\end{array}$$

The exclusive form is also equivalent to

$$\neg (A\leftrightarrow B).$$

Natural language sometimes uses “either” for emphasis and sometimes to suggest exclusivity. In a logic exercise, an explicit phrase such as “but not both” or “and I cannot do both” supplies the exclusive condition. If one sentence includes that phrase and another omits it, the contrast is usually deliberate. Do not add a constraint merely because it seems plausible in the real situation.

Therefore separates formulas from arguments

Words such as therefore, hence, and so do not denote logical connectives. They separate premises from a conclusion. If the premises translate to $P_1,\dots ,P_n$ and the conclusion to $Q$, the formal question is

$$P_1,\dots ,P_n\models Q.$$

This is an entailment claim, not another premise. To refute it, seek a countermodel:

$$v(P_1)=\cdots =v(P_n)=1,v(Q)=0.$$

For finitely many premises, the entailment holds exactly when

$$(P_1\wedge \cdots \wedge P_n)\to Q$$

is valid. Keeping the premises and conclusion separate nevertheless makes the structure of the argument easier to check.

Do not import facts that were not stated

A formalisation describes the given argument, not everything known about its subject. Common additions that change the problem include:

• interpreting “or” as exclusive without an explicit exclusion;
• replacing an implication by an equivalence because the consequent seems diagnostic of the antecedent;
• assuming that two activities cannot occur together because doing both sounds inconvenient;
• treating a causal statement as though it asserted both directions;
• silently assuming that at least one atom must be true.

These additions remove valuations. Since validity depends on which valuations remain possible, an apparently harmless assumption can turn an invalid argument into a valid one.

Check the translation before proving anything

The translation and the proof are separate tasks. A flawless proof of the wrong formula does not establish the original argument.

Truth-condition check

1. Write a glossary for all atoms.
2. Place parentheses according to grammatical scope.
3. For each connective, name the truth-value combinations it should forbid.
4. Read the formula back into plain language.
5. Test boundary cases, especially all-false, all-true, and the case excluded by an implication.
6. For an argument, try to make every premise true and the conclusion false.

A truth table is useful when uncertainty remains. It should confirm an intended reading, not compensate for never deciding what the sentence means.

Know when propositional logic is too weak

Bare propositional logic can combine complete claims, but it cannot expose every kind of structure. Quantification, identity, tense, modality, obligation, and causation require richer languages when those distinctions matter.

For example, “Every student submitted an assignment” may be treated as one atom, but then the formalisation cannot reason about a particular student. Preserving that internal relation requires first-order logic. Likewise, treating “possibly $A$” as an atom discards the modal relation between possibility and $A$.

The right language is the weakest one that still preserves the distinctions used by the argument.

Example

Staying late or walking the dog

Let

$$L:=\text{I stay late at work},F:=\text{I finish my assignments},W:=\text{I walk my dog}.$$

The argument is:

1. If I stay late, then I finish my assignments.
2. Either I stay late or I walk my dog, and I cannot do both.
3. Therefore, I either walk my dog or finish my assignments.

Its translation is

$$\begin{array}{rl}{P}_1& =L\to F,\\ P_2& =(L\vee W)\wedge \neg (L\wedge W),\\ Q& =W\vee F.\end{array}$$

The explicit phrase “I cannot do both” makes $P_2$ exclusive. The conclusion contains no such restriction, so $Q$ is an inclusive disjunction.

From $P_2$, either $W$ holds or $L$ holds. In the first case, $Q$ follows immediately. In the second case, $F$ follows from $P_1$, so $Q$ follows again. Therefore

$$P_1,P_2\models Q.$$

If the conclusion were instead translated as $\neg (W\leftrightarrow F)$, the argument would become invalid. The valuation

$$L=0,W=1,F=1$$

satisfies both premises but falsifies that exclusive conclusion. The countermodel does not expose a failure in the argument; it exposes a constraint that the mistranslation added.