Lecture 5 logical semantics

Lecture 5 logical semantics Lecture Five
Logical Semantics
Topics to Be Covered in This Session Introduction
Propositional logic
Composite logic
Logical connectives
Predicate logic
Argument structures
First order logic
Quantification logic
Universal quantifier
Existential quantifier
Logic and Semantics: introduction
‘Logic is concerned with Truth and Infe rence, that is, with determining the con ditions under which a proposition is tru e and the conditions under which one pro position may be inferred or deduced fro m other propositions’ (McCawley, 1981) Logic defined: by two key notions: entailment
calculus
Two more specific precise definitions: “A logic is a calculus for the entailme nt for the derivation of entailment
s.” (Encyclopedia of language and lingu istics)
“Entailment is a relation between asser tive sentences or, rather, between propo sitions expressed by assertive sentence s.” (Encyclopedia of language and lingu istics)
“A set of premises logically entail
s a conclusion if and only if every inte rpretation that satisfies the premises a lso satisfies the conclusion.”
Logic and Semantics: introduction Entailment for sentences:
“A sentence or a set of sentences P ent ails a sentence Q (P╞ Q) just in cas
e whenever P is true, Q must o
f necessit y also be true, on account of the meanin gs of P and Q – that is, for analytica l reasons.
An immediate complication:
occasional sentences and eternal sentenc es
Two examples:
Jack arrived late.
All humans are mortal.
In language use, most sentences are occa sional.
occasional sentence: not clear unless…eternal sentence: clear without
Logic and Semantics: introduction Semantics is concerned with the relation ship between meaning and linguistic stru cture.
Language:
syntactic form + logical representation logical representation = semantic conten t
Semantic triangle:
Semantic content = meaning relations bet ween language and real world
Semantic content:
Truth and truth value
Truth value conditions
Semantics and Logic: Semantic Triangle Note: Linguistic phenomenon that Logic i s interested in: Two examples:
Jack has been killed
Jack is dead
Sentence A entails Sentence B Compare:
Jack was shot in the heart
Jack died.
There is no entailment: not brought abou t by the semantic description but by phy sical causation
Semantics and Logic: Introduction Neighboring disciplines of semantics: Philosophers
Logicians.
Philosophers: 因为主客关系是哲学重要命题Logicians:
Relations: Logic-based semantics: Logica l Semantics: Also known as Formal Semant ics, Truth-conditional Semantics Logicians’ interest and concern: senten ce meaning:
relations between sentences / propositio ns
relations inside sentences / proposition s
Semantics and Logic: Introduction Earliest discussion of logic: Plato’s A cademy: Aristotle discovered:
Negation is used in logic to form a comp ound sentence the truth value of which i s the opposite of that of the simple sen tence it operates on. Thus, if It’s sno wing is true, then, It’s not snowing mu st be false.
p ～p
t f
f t
Negation word “not” functions differen tly according to whether the subject ter m is or is not quantified:
Greeks do not live in Britain =
all Greeks do not live Britain
No Greeks live Britain
Greeks live in Britain (is not true) Some Greeks do not live in Britain = Both “some Greeks do not” and “some d o” are both true
Semantics and Logic: Introduction ABPC
Aristotelian-Boethian Predicate Calculu s:
A: All F is G E: Al l F is not–G ≡ No F is G
I: Some F is G O: Som
e F is not–G ≡ Not all F is G
4 sentence types:
Types A and I (from Latin: affirmo = I a ffirm)
Types E and O (from Latin: nego = I den y)
Explain:
Type A entail Type I: Formally: A╞ I Type E entail Type O: Formally: E╞ O Where is logical semantics in meaning sy stems?
Communicative Meaning
Verbal communication n on-verbal communication
Inference
indefinite assertion explicaur
e implicature propositiona l attitude
meaning with truth valu
e meaning without trut h value
Semantics and Logic: Logical Forms Logical Forms: we use specialized symbol s whose status is like that of mathemati cs
((p ∧ q) & ~ p) → q is no more mysteri ous than:
(4 + 2) ÷ 2 = 3
Read:
if you add two apples to four and remov e half you will have three. Propositional Logic and Predicate Logic Propositional Logic (命题逻辑): alternat ive terms: Propositional Calculus or Sen tential Calculus:
how the truth of a composite propositio n is determined by the truth value of it s constituent propositions
concerned with the analysis of logical r elations in complex structures & betwee n sentences
Predicate Logic (谓词逻辑)
concerned with the internal structure o
f sentences
argument structure and quantifiers Proposition
What is proposition?
Lyons (1977) : “A proposition is what i s expressed by a declarative sentence wh en that sentence is uttered to make a st atement.”
Hurford and Heasley(1983): “A propositi on is that part of the meaning of a sent ence that describes a state of affairs.”From these definitions: proposition is d ifferent from sentence, proposition is a bstract.它是句子中关于客观世界的、关于事实以及事件状态的那部分内容，同时，命题只有内容，没有语音、语法外形。

（蒋严、潘海华，1998）
Simple and Compound Propositions
A simple (or atomic) proposition is a pr oposition that contains no other proposi tion as a part. e.g.
it rains.
I’ll stay at home.
it is Tuesday.
The sky is blue.
A compound proposition is a propositio
n that is built up from two or more simp le propositions. e.g.
if it rains then I’ll stay at home.
it is Tuesday and the sky is blue.
are built up from the two simple proposi tions given above using “if...then” a nd “and”.
Logical connectives
Logical connective: a word or phrase usu ally belonging to the traditional gramma tical category of conjunction (and, o
r, therefore, since, but, before, as, ev en, though and if…then).
However, there are only five logical con nectives:
~ = negation (‘not’)否定关系
& = conjunction (‘and’)合取关系∨ = disjunction (inclusive ‘or’)析取关系
→ = implication (‘if…then’)蕴涵关系≡ = equivalence (‘if and only if… the n’)等同关系
More can be added:
Entailment
presupposition
Logical Connectives and Conjunctions (合取关系):&,∧
Conjunctions: Definitions:
logical conjunction or and is a two-plac e logical connective that has the valu
e true i
f both of its operands are tru e, otherwise a value of false.
Logical conjunction is an operation on t wo logical values, typically the value
s of two propositions, that produces a v alue of true if and only if both of it
s operands are true.
Conjunction: an illustration
an example of conjunction:
Everyone should vote. (Proposition A) Democracy is the best system of governme nt. (Proposition B)
Therefore, everyone should vote and demo cracy is the best system of governmen
t. (Conjunction of 2 Propositions) Conjunction elimination: inference fro
m any conjunction of either element of t hat conjunction.
A and B. (Conjunction of 2 Propositions) Therefore, A. (Elimination of B)
...or alternately,
A and B. (Conjunction of 2 Propositions) Therefore, B. (Elimination of A)
Logical Conjunction
In logical operator notation:
A ∧ B
∴ A
or alternately,
A ∧ B
∴ B
Conjunction in Natural Language
The logical conjunction and in logic i
s related to, but not the same as, the g rammatical conjunction and in natural la nguages.
Natural language: English “and” has pr operties not captured by logical conjunc tion. For example,
“and” sometimes implies order: “They g ot married and had a child”
in common discourse means that the marri age came before the child.
The word “and” can also imply a partit ion of a thing into parts, as
“The American flag is red, white, and b lue.”
Here it is not meant that the flag is a t once red, white, and blue, but rathe
r that it has a part of each color. Conjunction in Logic
Truth Table for Conjunction:
How to read the symbols?
First line: if both simple sentence
s p & q are true, the compound sentenc
e is true;
Next three lines: if either p or q or bo th are false, the compound sentence is f alse.
Test: 白娘子和许仙 / 小青
白娘子是人(F)& 许仙是人(T) = F
白娘子是蛇 (T) & 小青是蛇 (T) = T
f
f f
f
f t
f
t f
t
t t
p & q
p q
The Venn diagram of “A and B”(the red area is true)
Disjunction (析取关系): or, ∨
The disjunction is used in logic to crea te a compound sentence which is false on ly if both the simple sentences in it ar e false. It will therefore be enough tha t one sentence is true for the whole dis junction to be true.
p q p ∨q T T T
T F T
F T T
F F F
许仙是蛇 (F) or 白娘子是蛇。

(T) = T
We can see that a disjunction is false i f both sentences are false; otherwise i t is true.
Disjunction
Two kinds of disjunctive relations: exclusive disjunction
inclusive disjunction:
Exclusive “or”
We can argue:
John is either at home or in his office John is at home
Therefore John is not in his office. This is what we call exclusive “or”. Logical disjunction is also concerned wi th inclusive “or”
Disjunction
Inclusive disjunction / “or”:
Ordinary language sometimes permits incl usive “or”: example: we might say: anyone who is female or over 65 Comment: not apply only to women under 6 5 and men over 65, also include women ov er 65.
In logical formula, exclusive or has t o be stated as:
(p ∨ q) & ~ (p & q)
Explanation: it adds, that is to say, t o inclusive ‘p or q’ the condition ‘n ot both p and q’
Disjunction (析取关系): Ⅴ
truth table for disjunction = or
Except first line where both are tru
e: I am a father = in relation to my so n / I am a son = in relation to my fathe r
In ordinary language, or = either… o
r = only one of the sentences is true (s ee next page)
f
f f
t
f t
t
t f
t
t t
p ∨ q
p q
The Venn diagram of “p or q”
(red is true)
Entailment（衍推关系）
entailment or logical implication is a l ogical relation that holds between a se t T of propositions and a propositio
n B when every model (or valuation or in terpretation) of T is also a model o
f B.
A entails
B =
whenever A is true, B is true
the information B conveys is contained i n the information A conveys
a situation describable by B is also a s ituation describable by A
'A and not B' is contradictory (can no
t be true in any situation) Entailment
“Entailment” can be illustrated by th e following two sentences, with Sentenc e A entailing Sentence B:
A: Tim married a Chinese artist. / Ti
m 娶了一个中国女画家。

B: Tim married a Chinese. / Tim 娶了一个中国女人。

In terms of truth value, the following r elationships exist between these two sen tences:
1) When A is true, B is necessarily tru e;中国女画家必定是中国人
(2) When B is false, A is false too;不是中国人必定不是中国画家
(3) when A is false, B may be true or fa lse; 可以是中国空姐，也可以是美国画家，未必一定是中国女人
(4) When B is true, A may be true or fal se. 中国女人未必一定是画家
Entailment is basically a semantic relat ion or logical implication, but we hav
e to assume co-reference o
f “Tim” in s entence A and sentence B, before we hav e “A entail B”
Entailment
Which of the sentences are entailed by t he 'a' sentences below?
a. Micky lives with Bill and Jane who ar e married to each other.
b. Micky lives with his parents.
c. Micky lives with two people.
d. Micky lives with exactly two peopl
e. Study also:
a. Jack saw a dog come through the doo r.
b. Exactly one dog came through the doo r.
c. There is a dog that Jack saw.
d. Jack is a person.
√√√√
The Venn diagram of Entailment
(the red area is true)
Implication (蕴涵关系): →
Implication refers to the relation betwe en the two propositions connected by th
e connective →. Illustration:
p q p →q
T T T
T F F
F T T
F F T The truth table shows that unless the an tecedent is true and the consequent is f alse the composite proposition will be t rue.
Implication: Premises & Conclusion
A favorite example in traditional logi
c textbooks:
All men are mortal
Socrates is a man
Therefore Socrates is mortal
Can be explained on the basis of premis e and conclusion
Here:
the conclusion (the 3rd sentence) follow s from
premises (the 1st two sentences):
the inference is logically valid Implication
Implication: Any true statement will imp ly any other true statement: e.g.:
If the horse is a mammal, the shark i
s a fish
If smoking leads to lung cancer, fever r esults in pneumonia.
Comment: We can infer the truth of q fro m that of p. Anything that does not conf orm to the rule of implication is consid ered incorrect: e.g.:
*If the horse is a mammal, he kills he
r brother.
Real natural language: Implication: fals e propositions
Examples of false propositions, we can i nfer (if I am invisible is false):
If I am invisible, no one can see me. (t rue)
If I am invisible, everyone can see m e. (false)
Though it seems quite absurd, it has som e correlation with usage in language:
If he is president, I am a Dutchman / I’ll eat my hat
尔康和紫薇的誓言：《上邪》：上邪，我欲与君相知，长命无绝衰。

山无棱，江水为竭，冬雷震震，夏雨雪，天地合，乃敢与君绝。

Comment: one false proposition implies a nother false proposition.
Summary: implication in logical sense do es not correspond exactly to the use o
f anythin
g in natural language. It owe
s its validity just to the truth functio ns assigned to it.
Implication
If… then
difference between entailment and implic ation
Austin’s sense of entailment:
In entailment, if. “p entails q” is tr ue, and. “p” is true,
Austin’s implication:
“p implies q” can be true, “p” can b e true and yet “q” can be false.
My saying that “the cat is on the mat” i mplies that I believe it to be so, ye
t I may say that the cat is on the mat a nd may not believe that it is.
The Venn diagram of Implication
(the dark area is true)
Implication
Truth Table of Implication: related t
o “if…then”
In ordinary language, we normally relat e sentences with ‘if…then’ only if t here is causal relationship between the m.
But not permissible in propositional log ic: why?
It takes no account of the nature of th
e sentences themselves
t
f f t
f t f
t f t
t t p → q p q。