In a less formal context we think of an implication as something intended to be understood (as when Ted asks whether Jane is implying he is lazy by pointing out that he is out of work). In formal logic implication names the relationship between something proposed (the antecedent) and something else (the consequent) that is necessarily true--not just possibly true or even probably true--when the antecedent is true. If we use P to represent Ted being out of work and Q to represent his being lazy, then PQC would express the relationship expressed in the sentence "If Ted is out of work then he is lazy." C here is a tight signal, meaning that the string QPC expresses a different story ("If Ted is lazy then he is out of work") which could be false even while the original sentence is true.

For Russell and the pioneers in modern symbolic logic this was the crucial relationship upon which everything else depended. The truth-table pattern is especially important to remember:
P Q || PQC
T T..........T
T F..........F
F T..........T
F F..........T
What it tells us is that for a conditional statement to be wrong we must first have the antecedent true and then the consequent turns out to be false. This gives us one of the useful substitution rules for symbolic logic:
not (if P then Q) is equivalent to P and not-Q (PQCN = PQNA).

Whenever the antedecedent is false the conditional is true (more properly, it is not inconsistent) regardless of the consequent. This relationship is referred to as material implication, and it is somewhat deceptive in that it gives the impression that somehow what is false actually proves anything else at all to be true.

There is a crucial parallel between a non-false conditional expression (a hypothetical situation, then) and any valid argument pattern. We define a valid pattern as one in which it is inconsistent to have true premises and a false conclusion, just as we define a false conditional as one in which we cannot have a true antecedent and a false consequent. Another way of expressing this is to say that a valid argument can always be rewritten as a tautology.

Return to Signals and Variables or look to Conditionals for different ways in which conditionals can be expressed.

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