Implications and necessary conditions

Find the case the rule forbids

Suppose a workshop rule reads “If a strip is copper, it goes in the blue tray.” Take a single strip and ask what finding would prove the rule false. A copper strip in the red tray would do it, provided the trays are mutually exclusive. A steel strip in the blue tray would tell you nothing about a violation. The sentence tells you the destination of every copper strip. It leaves the destination of steel open. Imagining a physical object often makes an abstract implication (implicazione) easier to read.

Call being copper the starting condition and being in the blue tray the consequence. Being copper is sufficient for the blue tray under this rule. Being in the blue tray is necessary for being copper. Read “necessary” as a requirement that every instance must satisfy. The requirement can be shared by other objects. If all copper strips have a stamp as well, steel strips can still have the same stamp. A list of necessary features is useful for ruling candidates out even when it cannot identify them uniquely.

Work forwards or use the contrapositive

From a known copper strip, follow the rule forwards to its tray. From a strip known to be outside the blue tray, reason backwards to the absence of copper. For if the strip were copper, the rule would have put it in blue. This backwards inference is the contrapositive. It reverses the order and negates both conditions. Reversing the order alone gives the converse. A steel strip in blue shows precisely why the converse can fail.

Try a four-row table if the language gets tangled. Record whether an object satisfies the starting condition and whether it satisfies the consequence. Both true is allowed. Both false is allowed. A false start with a true consequence is allowed too. Only a true start with a false consequence breaks the rule. Cross out that row. When given additional evidence, remove rows that disagree with the evidence, then inspect the survivors. A conclusion must hold in every surviving row to count as a deduction.

Translate small connecting words

In “the cabinet opens only if the latch is raised”, the words after “only if” give the requirement. Starting from an open cabinet lets you infer a raised latch. A raised latch by itself settles nothing about the cabinet opening. In “the cabinet opens if the green button is pressed”, pressing the button suffices. With both sentences together, a press guarantees an opening, which guarantees a raised latch. State both intermediate links before trying to skip one. If a question says “if and only if”, it explicitly grants the reverse direction as well.

Treat “unless” by writing what happens when the exception is absent. “The bell rings unless the switch is muted” commits you to ringing whenever the switch is unmuted. On that statement alone, the behaviour when muted is unspecified. Read any extra sentence before deciding the exceptional case. Similarly, ordinary logical “or” allows both conditions unless the statement excludes that possibility. For “and” you need both conditions together. If a release needs both a signature and a seal, finding only a signature leaves the seal to establish.

Check a deduction with one object

To test a proposed conclusion, deliberately try to defeat it. Keep every given rule true and arrange for the conclusion to be false. Write down one object with its features to show that the arrangement really is possible. If the rule concerns every item, check that your chosen object obeys every rule that applies to it. A story that breaks a premise cannot disprove a deduction. This method is especially helpful for a conclusion containing “must”. One permissible exception is enough to defeat “must”, even if the conclusion seems reasonable in everyday life.