The language of propositions

The previous lesson found that validity depends on form, and then had to describe forms in English, which is the language whose vagueness caused the problem in the first place.

"He will resign or the board will force him out, and the shares will fall" has two readings, and they are not equivalent: one says a disjunction and a fall both hold, the other says either he resigns or the board acts and the shares fall together. Ordinary punctuation does not separate them. Before any form can be tested mechanically, the forms have to be written in a notation where that ambiguity cannot arise. This lesson builds that notation for the part of logic that treats whole statements as unanalysed blocks, and fixes the symbols used for the rest of the subject.

Atoms

Start by deciding what will not be analysed. A simple statement, or atom, is one containing no other statement as a part: "the shares fell", "Titan has a nitrogen atmosphere", "the alloy contains nickel". Each is written as a capital letter, and which letter is a free choice, though a mnemonic one saves work later: $S$ for the shares, $N$ for the nickel.

A compound statement is built from simpler ones with a connective: "the shares fell and the bond yields rose" contains two statements. Propositional logic studies exactly the structure that compounding creates, and it deliberately sees nothing inside an atom. "Socrates is a man" is a single letter to it, with no visible parts, which is the limitation that eventually forces the eighth lesson.

Choosing the atoms is the first real decision in any translation. Take them too coarse and structure the argument depends on disappears; take them too fine and you invent structure the English does not have. The rule of thumb: an atom should contain no word from the connective list below.

The five connectives

Five symbols do all the work, and each has a fixed meaning that never varies with context.

Negation, $\neg P$ , is read "not $P$ ", and it is true exactly when $P$ is false. English hides negation in many places: "it is not the case that", the prefix in "unhappy", the verb in "she denied it", and the quiet negation in "she failed to arrive".

Conjunction, $P \land Q$ , is " $P$ and $Q$ ", true when both parts are true. It is not only carried by "and". "But", "however", "although", "yet", "while" and a plain semicolon are all conjunctions as far as truth is concerned. "She is qualified but inexperienced" is true in exactly the circumstances that "she is qualified and she is inexperienced" is true. The contrast that "but" conveys is real and it makes no difference to truth, so the notation drops it. That is the first of several deliberate losses.

Disjunction, $P \lor Q$ , is " $P$ or $Q$ ", and it is inclusive: true when either part is true and also when both are. English sometimes means the exclusive version, "one or the other but not both", as in "the set menu comes with soup or salad". Logic fixes the inclusive reading as the meaning of $\lor$ and writes the exclusive one explicitly as $(P \lor Q) \land \neg (P \land Q)$ when it is wanted. Latin had two separate words, vel for the inclusive and aut for the exclusive, and the symbol $\lor$ is the first letter of vel.

The conditional, $P \to Q$ , is "if $P$ then $Q$ ". $P$ is the antecedent and $Q$ the consequent. Its truth conditions are strange enough to deserve their own lesson, which is the fourth.

The biconditional, $P \leftrightarrow Q$ , is " $P$ if and only if $Q$ ", true when the two sides have the same truth value. It is the standard form of a definition and of a mathematical criterion: a number is even if and only if it is divisible by two.

Truth-functionality, and what will not fit

Behind all five sits one restriction that decides what this language can and cannot say. A connective is truth-functional when the truth value of the compound depends on nothing but the truth values of its parts. Give me the value of $P$ and the value of $Q$ and I can compute $P \land Q$ without knowing what either says.

Plenty of English connectives fail this. "She resigned because the audit failed" cannot be evaluated from the two truth values alone: both parts can be true with the causal claim false, if she resigned for unrelated reasons. Causation is not a truth function, and neither is "before", since "he left before she arrived" and "she arrived before he left" can differ in truth while their parts do not change. Nor is "it is likely that", "the manager believes that", "it is obligatory that" or "it would have been the case that". Each has a whole branch of logic devoted to it, and none is here.

This exclusion is what makes the machinery of the next three lessons possible. Truth-functional connectives can be tabulated exhaustively, and tabulation is the whole method. The price is that any argument turning on causation, time, belief or obligation has that content flattened out of it in translation, and a valid formal argument may correspond to a bad English one for exactly that reason. Keeping track of what was thrown away is part of using the tool honestly.

Scope, brackets and the main connective

The ambiguity that opened this lesson is a question of scope: which connective governs which parts. Brackets settle it. $(P \lor Q) \land R$ says the disjunction and $R$ both hold; $P \lor (Q \land R)$ says either $P$ holds or the conjunction does. Those are different claims, and later lessons will show a case where one is true and the other false.

Every well-formed formula has one main connective, the one applied last when the formula was built and the one applied first when it is taken apart. Finding it is the first move in any analysis, and the procedure is mechanical: strip any outermost bracket pair, then the connective not enclosed in any remaining brackets is the main one. In $\neg (P \land Q)$ the main connective is the negation, so the formula denies a conjunction. In $\neg P \land Q$ it is the conjunction, so the formula asserts one thing and denies another. Those two are not equivalent, and mixing them up is the commonest translation error there is.

To reduce bracket clutter there is a precedence order: $\neg$ binds most tightly, then $\land$ , then $\lor$ , then $\to$ , and $\leftrightarrow$ binds least. So $\neg P \land Q \to R$ means $((\neg P) \land Q) \to R$ . Precedence is a convenience and never a defence: when a formula is hard to read, put the brackets in.

Example. Find the main connective of $\neg (P \to Q) \lor (R \land \neg S)$ , and say what kind of statement it is.

There is no outermost bracket pair enclosing the whole formula, so scan for a connective outside all brackets. The first $\neg$ applies to $(P \to Q)$ only, and $\land$ and $\neg S$ sit inside the second bracket pair. The $\lor$ is the only connective outside every bracket, so it is the main connective and the formula is a disjunction. Its left disjunct denies a conditional and its right asserts a conjunction.

Now you. Find the main connective of $(P \lor Q) \to \neg (R \leftrightarrow S)$ .

Answer

The arrow. The $\lor$ is inside the first bracket pair and the $\leftrightarrow$ inside the second, so only $\to$ sits outside both, and the formula is a conditional whose consequent is a negated biconditional.

Translating

Translation is where care is repaid, and a few English constructions cause nearly all the trouble.

"Only if" is not "if". "You may vote only if you are registered" does not say registration gets you a vote; it says a vote requires registration. So " $P$ only if $Q$ " is $P \to Q$ , with the part after "only if" as the consequent, exactly opposite to the placement in " $P$ if $Q$ ", which is $Q \to P$ . Combining the two gives the reason "if and only if" is a biconditional.

"Unless" means "if not". "The flight leaves unless the fog thickens" is $\neg F \to L$ , and since that is equivalent to $L \lor F$ , translating "unless" as $\lor$ is also correct. Readers argue about whether "unless" is exclusive; treat it as inclusive unless the sentence says otherwise, and note the choice.

"Neither $P$ nor $Q$ " is $\neg P \land \neg Q$ , both denied. "Not both $P$ and $Q$ " is $\neg (P \land Q)$ , which is weaker: it permits either one alone. The difference between those two formulas is a law with a name, and it arrives in the fifth lesson.

"Provided that", "given that" and "assuming that" all introduce an antecedent, so they behave like "if". A sufficient condition is an antecedent and a necessary condition is a consequent, which the fourth lesson makes precise.

Example. Translate: "The contract is valid unless it was signed under duress, but it is void if either party lacked capacity."

Atoms: $V$ for the contract being valid, $D$ for it being signed under duress, $C$ for either party lacking capacity. Take the clauses in turn. "Valid unless $D$ " is "if not $D$ then valid", so $\neg D \to V$ . "But" is a conjunction. "It is void if $C$ " puts the antecedent second, so it is $C \to \neg V$ , using $\neg V$ for void, since void and valid are contradictories here. The whole translation is

(\neg D \to V) \land (C \to \neg V)

Note what the translation had to decide: that "void" is the negation of "valid" rather than a third state, and that "either party lacked capacity" is one atom rather than a disjunction over two parties. Both choices are defensible and both are choices, which is why translation is judgement rather than calculation.

Now you. Translate: "The alarm sounds only if the door is open or the glass is broken, and it does not sound now."

Answer

Atoms: $A$ for the alarm sounding, $D$ for the door being open, $G$ for the glass being broken. "Only if" puts its clause in the consequent, giving $A \to (D \lor G)$ , and the second clause is $\neg A$ . The whole is

(A \to (D \lor G)) \land \neg A

A common error is to write $(D \lor G) \to A$ , which claims an open door sets the alarm off, something the sentence never says.

Two more habits worth forming

Negations of compounds repay slowing down. "It is not true that the fee is refundable and the deadline is fixed" is $\neg (R \land F)$ , not $\neg R \land \neg F$ . The English is denying a package, and denying a package leaves both individual claims open.

Scope over "and" inside a negation catches almost everyone, and so does the English habit of dropping repeated subjects. "The valve is neither open nor stuck" expands to two atoms with two negations, and "the valve is not open or stuck" is genuinely ambiguous in English between $\neg (O \lor S)$ and $\neg O \lor S$ . When the source is ambiguous, the honest thing is to translate both and note that the argument may depend on which was meant.

Example. Translate "not both of the tanks are full" and "neither tank is full", and say how they differ.

With $A$ and $B$ for the two tanks being full, the first is $\neg (A \land B)$ and the second is $\neg A \land \neg B$ . They differ in the case where exactly one tank is full: the first sentence is then true and the second false. So the second implies the first and not the other way round, which is the general relation between denying a conjunction and conjoining two denials.

Now you. Translate "the report is neither timely nor accurate" and "the report is not both timely and accurate", and give a situation that separates them.

Answer

With $T$ and $A$ : the first is $\neg T \land \neg A$ , the second is $\neg (T \land A)$ . A report that is timely but inaccurate separates them, making the second true and the first false.

What the language now has, and what it lacks

There is now a syntax: atoms, five connectives, brackets, a precedence order, and a translation practice with its known traps. Every formula built by those rules is well formed, and every well-formed formula has exactly one main connective and one unambiguous reading.

What is missing is any way to evaluate one. Nothing so far says whether $(A \to (D \lor G)) \land \neg A$ can be true, or whether one formula follows from another. The connectives were chosen to be truth-functional precisely so that this can be answered by computation rather than by intuition, and the computation is a table with $2^{n}$ rows for $n$ atoms. That is the next lesson, and it makes validity decidable for everything this language can say.