1.2 A Formal Language for Sentential Logic

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formal language	Comment. To represent similarities among arguments a natural language, logicians introduce formal languages. The first formal language we will introduce is the language of sentential logic (also known as propositional logic). In chapter 3 we introduce a more sophisticated language: that of predicate logic.
vocabulary	Definition. The VOCABULARY OF SENTENTIAL LOGIC consists of · SENTENCE LETTERS, · CONNECTIVES, and · PARENTHESES.
sentence letter	Definition. A SENTENCE LETTER is any symbol from the following list: A, ..., Z, A0, ..., Z0, A1, ...,Z1, ....
sentence variable	Comment. By the use of subscripts we make available an infinite number of sentence letters. These sentence letters are also sometimes called SENTENCE VARIABLES, because we use them to stand for sentences of natural languages.
connectives	Definition. The SENTENTIAL CONNECTIVES (often just called CONNECTIVES) are the members of the following list: ~, &, v, ->, <->.
	Comment. The sentential connectives correspond to various words in natural languages that serve to connect declarative sentences.
tilde ~	The TILDE corresponds to the English `It is not case that'. (In this case the use of the term `connective' is odd, since only one declarative sentence is negated at a time.)
ampersand &	The AMPERSAND corresponds to the English `Both ... and ...'.
wedge v	The WEDGE corresponds to the English `Either ... or ...' in its inclusive sense.
arrow ->	The ARROW corresponds to the English `If ... then ...'.
double-arrow <->	The DOUBLE-ARROW corresponds to the English `if and only if'.
	Comment. Natural languages typically provide more than one way to express a given connection between sentences. For instance, the sentence `John is dancing but Mary is sitting down' expresses the same logical relationship as `John is dancing and Mary is sitting down'. The issue of translation from English to the formal language is taken up in section 1.3.
) and (	Definition. The right and left parentheses are used as punctuation marks for the language.
expression	Definition. An EXPRESSION of sentential logic is any sequence of sentence letters, sentential connectives, or left and right parentheses.
	Examples.
	(P -> Q) is an expression of sentential logic.
	)PQ->~ is also an expression of sentential logic.
	(3 -> 4) is not an expression of sentential logic.
metavariable	Definition. Greek letters such as PHI and PSI are used as METAVARIABLES. They are not themselves parts of the language of sentential logic, but they stand for expressions of the language.
	Comment. (PHI -> PSI) is not an expression of sentential logic, but it may be used to represent an expression of sentential logic.
well-formed formula	Definition. A WELL-FORMED FORMULA (WFF) of sentential logic is any expression that accords with the following seven rules:
(1)	A sentence letter standing alone is a wff.
atomic sentence	[Definition. The sentence letters are the ATOMIC SENTENCES of the language of sentential logic.]
(2)	If PHI is a wff, then the expression ~PHI is also a wff.
negation	[Definition. A wff of this form is known as a NEGATION, and ~PHI is known as the NEGATION of PHI.]
(3)	If PHI and PSI are both wffs, then the expression (PHI & PSI) is a wff.
conjunction	[Definition. A wff of this form is known as a CONJUNCTION. PHI and PSI are known as the left and right CONJUNCTS, respectively.]
(4)	If PHI and PSI are both wffs, then the expression (PHI v PSI) is a wff.
disjunction	[Definition. A wff of this form is known as a DISJUNCTION. PHI and PSI are the left and right DISJUNCTS, respectively.]
(5)	If PHI and PSI are both wffs, then the expression (PHI -> PSI) is a wff.
conditional, antecedent, consequent	[Definition. A wff of this form is known as a CONDITIONAL. The wff PHI is known as the ANTECEDENT of the conditional. The wff PSI is known as the CONSEQUENT of the conditional.]
(6)	If PHI and PSI are both wffs, then the expression (PHI <-> PSI) is a wff.
biconditional	[Definition. A wff of this form is known as a BICONDITIONAL. It is also sometimes known as an EQUIVALENCE.]
(7)	Nothing else is a wff.
binary and unary connectives	Definition. &, v, ->, and <-> are BINARY CONNECTIVES, since they connect two wffs together. ~ is a UNARY CONNECTIVE, since it attaches to a single wff.
sentence	Definition. A SENTENCE of the formal language is a wff that is not part of a larger wff.
denial	Definition. The DENIAL of a wff PHI that is not a negation is ~PHI. A negation, ~PHI, has two DENIALS: PHI and ~~PHI.
	Example. ~(P -> Q) has one negation: ~~(P -> Q) It has two denials: (P -> Q) and ~~(P -> Q). (P -> Q) has just one denial: its negation, ~(P -> Q).
	Comment. The reason for introducing the ideas of a sentence and a denial will become apparent when the rules of proof are introduced in section 1.4.
parenthesis- dropping convention	Comment. For ease of reading, it is often convenient to drop parentheses from wffs, so long as no ambiguity results. If a sentence is surrounded by parentheses, then these may be dropped.
	Example. P -> Q will be read as shorthand for (P -> Q).
further conventions	Comment. Where parentheses are embedded within sentences we must be careful. For example, the expression P & Q -> R is potentially ambiguous between (P & Q) -> R and P & (Q -> R). To resolve such ambiguities, we adopt the following convention: ~ binds more strongly than all the other connectives; & and v bind component expressions more strongly than ->, which in turn binds its components more strongly than <->.
	Examples. ~P & Q -> R is read as ((~P & Q) -> R). P -> Q <-> R is read as ((P -> Q) <-> R). P v Q & R is not allowed, as it is ambiguous between P v (Q & R) and ((P v Q) & R). P -> Q -> R is not allowed, as it is ambiguous between (P -> (Q -> R)) and ((P -> Q) -> R).
Exercise 1.1	Which of the following expressions are wffs? If an expression is a wff, say whether it is an atomic sentence, a conditional, a conjunction, a disjunction, a negation, or a biconditional. For the binary connectives, identify the component wffs (antecedents, consequents, conjuncts, disjuncts, etc.).
i*	A
ii*	(A
iii*	(A)
iv*	(A -> B)
v*	(A -> (
vi*	(A -> (B -> C))
vii*	((P & Q) -> R)
viii*	((A & B) v (C -> (D <-> G)))
ix*	~(A -> B)
x*	~(P -> Q) v ~(Q & R)
xi*	~(A)
xii*	(~A) -> B
xiii*	(~(P & P) & (P <-> (Q v ~Q)))
xiv*	(~((B v P) & C) <-> ((D v ~G) -> H))
xv*	(~(Q v ~(B)) v (E <-> (D v X)))
Exercise 1.2	(a)* Rewrite all the sentences in i-xv above, using the parenthesis-dropping convention. Omit any parentheses you can without introducing ambiguity. (b) State whether each of the following is ambiguous or unambiguous, given the parenthesis-dropping conventions. In the unambiguous cases, write out the sentences and reinstate all omitted parentheses. In the ambiguous cases, write out some of the possible ways to reinsert parentheses to form wffs.
i*	P <-> ~Q v R
ii*	P v Q -> R & S
iii*	P v Q -> R <-> S
iv*	P v Q & R->~S
v*	P->R & S->T
vi*	P->Q->R->S
vii*	P & Q <-> ~R v S
viii*	~P & Q v R->S <-> T
ix*	P->Q & ~R <-> ~S v T->U
x*	P->Q & ~R->~S v T <-> U

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