proposition≈ sentence meaning Truth value: truth or false Predicate logic (Pp 180) p (simple proposition)
one-place connective: negation ～ or ﹁ two-place connective: conjunction &
disjunction ∨ implication
equivalence ≡ or
Connective conjunction: similar to the English “and” Connective disjunction: similar to the English “or”
Connective implication/conditional implication: corresponds to the English “if…then” Connective equivalence/biconditional: corresponds to “iff…then”
C.f. Antonyms & “not”
? With complementary antonyms, the denial of one is the assertion of the other. ? With gradable, that is not necessarily the case. E.g. John isn’t old.
John is old.
C.f. Conjunction & “and” ? Conjunction
E.g. He missed the train and arrived late. ? “And”
E.g. He arrived late and missed the train.
*He missed the train and arrived late.
C.f. Implication & “if…then” ? Implication
E.g. If he is an Englishman, he speaks English.
If snow is white, grass is green. E.g. If snow is black, grass is green. ? “If…then”
E.g.? If snow is white, grass is green.
*If snow is black, grass is green.
In sum, propositional logic, concerned with the semantic relation between propositions, treats a simple proposition as an unanalyzed whole. E.g. All men are rational. Socrates is a man.
Therefore, Socrates is rational.
PREDICATE LOGIC/PREDICATE CALCCULUS studies the internal structure of simple propositions.
Question: How to analyze Socrates is a man?
Argument (主目): a term which refers to some entity about which a statement is being made
Predicate (谓词): a term which ascribes some property, or relation, to the entity, or entities, referred to
Socrates is the argument, and man is the predicate.
Token: M(s)
Note: A simple proposition is seen as a function (函数) of its argument. The truth value of a proposition varies with the argument. M(s) =1, M(c) =0
E.g. John loves Mary. L (j, m)
John gave Mary a book. G (j, m, b)
kill: CAUSE (x, (BECOME (y, (～ALIVE (y))))) C (x, (B (y, (～A (y))))) All men are rational.
1. All is the universal quantifier and symbolized by an upturned A—?in logic.
2. The argument men does not refer to any particular entity, which is known as a variable and symbolized as x, y. Notation: ?x (M(x) R(x))
“For all x, it is the case that, if x is a man, then x is rational.”
Some men are clever.
Some is the existential quantifier and symbolized by a reversed E—? Notation: ?x (M(x) & C(x))
C.f. Universal quantifier & existential quantifier 1. Quantifiers
2. Implication connective E.g.
All men are rational.
There is no man who is not rational.
Notation: ?x (M(x) R(x)) ≡～?x (M(x) & ～R(x)) (1) ?x (P(x))≡～?x (～P(x))
～?x (P(x))≡?x (～P(x)) ?x (P(x)) ≡～?x (～P(x)) ～?x (P(x)) ≡?x (～P(x))
(2) ?x (M(x) R(x))
M(s) ∴R(s)
(3) ?x (M(x)) R(x))
R(s) ∴R(s)
(4) ?x (M(x) & C(x))
M(s) ∴C(s)