Phil. of Logics: Week 02

Philosophy of

Logics

Week 2: Classical Sentential

Logic

"And," "Or," and "Not"

Sentential logic is the logic of "and," "or," and "not"; in other words, it is the logic of sentential connectives.

In some libraries, searching for "Mary AND NOT lamb" in a public terminal will find documents containing "Mary" but not those containing "lamb."

The logic of "and," "or," and "not" gives a taste of deductive

logic, with which we can compare inductive logic.

Later, we will discuss a more rigorous definition of deductive logic.

Sentences & referring or characterizing

expressions.

A declarative sentence is a sentence that makes a definite claim. A straightforward way of making a claim is to (1) identify what you are talking about, and (2) make a claim about it.

Thus, in the sentence "Socrates is bald," the proper name "Socrates" identifies who we are talking about and the

predicate "is bald" makes out claim about him. There are

non-declarative sentences, like “Are you hungry?” and “Get out of here!” Since we do not deal with these in this course, we will always mean declarative sentences by “sentences.”

In general, expressions that identify what we are talking about are called referring expressions and the expressions used to make factual claims about the things we are talking about are called characterizing expressions.

Kinds of Referring Expressions

Proper names: "Socrates," "Pluto," and so on. Pronouns: "I," "you," "he," "it," and so on.

Kinds of Characterizing Expressions

Although sentences are often constructed out of one referring expression, sometimes they are constructed out of more than one referring expressions, plus an expression that

characterizes the relation between the things referred to. For instance, the sentence "Mercury is hotter than Pluto" contains two referring expressions --- "Mercury" and "Pluto" --- and one characterizing expression --- "is hotter than."

Property expressions or one-place predicates: "is bald," "is

tall," "is hot," and so on.

Relational expressions or many-place predicates: "is hotter

The Structure of Molecular Sentences

"Socrates is wise." From these sentences, we can construct a variety of molecular sentences:

Socrates is not bald.

Socrates is bald and Socrates is wise. Socrates is bald or Socrates is wise.

Socrates is not bald or Socrates is wise.

Socrates is bald and Socrates is wise or Socrates is not bald and Socrates is not wise.

The words "not," "and," and "or" are logical connectives and are used together with referring and characterizing expressions to make complex factual claims.

The Truth Cond. of an Atomic Sentence

correctly characterizes the thing or things its referring expression(s) refer(s) to.

For instance, "Socrates is bald" is true just when "is bald"

correctly characterizes the person "Socrates" refers to, that is to say, just when Socrates is bald.

% “IFF” is the abbreviation of “if and only if.” When we write “A IFF B,” it means that “If A is true then B is also true, and if B is true, then A is also true.”

The Truth Cond. of a Denial

We deny or negate an atomic sentence by placing the word "not" at the appropriate place within it. We may deny a

sentence by placing a negation sign "~" in front of a letter that abbreviates a sentence. For instance, the denial or negation of "Socrates is bald" is "Socrates is not bald," (or more precisely, “It is not the Assignment that Socrates is bald). Then, "~P" is the denial of "P," which abbreviates "Socrates is bald."

The denial of a sentence is true IFF the original sentence is false, and the denial of a sentence is false IFF the original

sentence is true. We can symbolize this info. in this truth-table:

P ~P

Assignment 1: T F

The Truth Cond. of a Conjunction

We form the conjunction of two sentences by putting the word "and" between them. Each of the original sentence is then

called a conjunct. A conjunction is true just when both of its

conjuncts are true. Using the symbol "&" to abbreviate the word "and" we can represent this in the following truth-table:

P Q P&Q

Assignment 1: T T T

Assignment 2: T F F

Assignment 3: F T F

The Truth Cond. of a Disjunction

We form the disjunction of two sentences by putting the word "or" between them. Each of the original sentence is then called a disjunct. A disjunction is true just when one or both of its

disjuncts are true. Using the symbol "v" to abbreviate the word "or" we can represent this in the following truth-table:

P Q PvQ

Assignment 1: T T T

Assignment 2: T F T

Assignment 3: F T T

The Truth Cond. of a Molecular

Sentence 1

We may use the following procedure to construct a truth table for a molecular sentence:

Step 1: List all combinations of truth and falsity for the atomic constituent sentences.

Step 2: For each of those combinations, assign a truth-value, in accordance with the truth-tables in the earlier slides, to

each sub-sentence whose constituents (the negated

sentence, conjuncts, or disjuncts) are all assigned definite truth-values.

Step 3: If the whole sentence is assigned a truth-value, stop there. Otherwise, return to Step 2.

The Truth Cond. of a Molecular

Sentence 2

Here is an example of how to use the procedure on the previous slide: S W ~W ~SvW Assignment 1: T T F T Assignment 2: T F F F Assignment 3: F T T T Assignment 4: F F T T

Defined Connectives: Material

Conditional

We shall use “A->B” as the abbreviation of “(~A)vB.” Hence, “s->w” has the following truth-condition:

S W ~S ~SvW S->W

Assignment 1: T T F T T

Assignment 2: T F F F F

Assignment 3: F T T T T

Assignment: Completeness and

Consistency 1

We introduce the following definitions:

(Complete Assignments) An assignment a (of truth-values to atomic sentences) is complete if and only if, for any sentence

S, either S is true-according-to a or ~S is true-according-to a.

(Consistent Assignments) An assignment a (of truth-values to atomic sentences) is consistent if and only if there is no

sentence S such that both S and ~S are true according to a. So far, we have assumed that all assignments of truth values are complete and consistent. We call such assignments

Assignment: Completeness and

Consistency 2

In our basic classical logic, (i) a sentence A is false according to an assignment a IFF ~A is true according to a; and (ii) ~A is true according to a IFF A is not true according to a. Then, we can rewrite the conditions of completeness and consistency: (Complete Assignments*) An assignment a is complete if and only if, for any sentence A, either A is true-according-to a or A is false-according-to a.

(Consistent Assignments*) An assignment a is consistent if and only if no sentence is true and false at the same time,

Assignment: Completeness and

Consistency 3

From now on, we shall use these abbreviations:

“a⊨₁A” abbreviates “A is true-according-to a” and

“a⊨₀A” abbreviates “A is false-according-to a”

where A is an (atomic or molecular) sentence and a is any assignment (of truth-values to sentences). Then, an

assignment a is classical IFF, for any sentence A,

a⊨₁A and a⊭₀A; or a⊭₁A and a⊨₀A.

This condition holds just when a is complete&consistent, i.e.,

Semantic Values

So far, we have used “T” and “F” to denote truth and falsity. However, it is customary to let numbers 1 and 0 play the roles of truth-values. We will call those values “semantic values” (of declarative sentences).

Then, we can regard a classical assignment as a function v from the set S of sentences into our set V of semantic values, i.e., {1,0}.

From now on, we will build our truth-tables from these semantic values rather than truth-values.

Classical Assignment

Then, a classical assignment v is such that (i) v assigns exactly one of 0 and 1 to each atomic sentence A, and

(ii) for each molecular sentence A, v assigns a semantic value to A in accordance with these tables:

In other words, molecular sentences have these truth-values: Conjunction: a⊨₁A&B IFF a⊨₁A and a⊨₁B.

Disjunction: a⊨₁AvB IFF a⊨₁A or a⊨₁B.

Negation: a⊨₁~A IFF a⊨₀A.

General falsity condition: a⊨₀A IFF a⊭₁A .

~ 0 1 1 0 & 1 0 1 1 0 0 0 0 v 1 0 1 1 1 0 1 0

Classical Equivalence

Because a truth table tells us what factual claim is made by a molecular sentence, it can tell us when two sentences make the same factual claim. Here is an example:

Hence, W and (S&W)v(~S&W) are logically equivalent. In general, two sentences are classically equivalent IFF they have the same semantic value according to all classical

assignments. S W (S&W)v(~S&W) Assignment 1: 1 1 1 Assignment 2: 1 0 0 Assignment 3: 0 1 1 Assignment 4: 0 0 0

Classical Incompatibility

Truth-tables may also be used to show that two molecular

sentences make conflicting factual claims. Here is an example:

There is no assignment in which s&w and ~s&~w are both true. Thus, they make conflicting factual claims. In general, two

sentences are classically incompatible IFF there is no

classical assignment of truth-values to their atomic constituent sentences according to which both are true.

S W S&W ~S&~W

Assignment 1: 1 1 1 0

Assignment 2: 1 0 0 0

Assignment 3: 0 1 0 0

Classical Tautology

There are some peculiar molecular sentences that make no factual claim whatsoever. If we say "Either Socrates is bald or Socrates is not bald" we have really not said anything at all

about Socrates. Let us see how this situation is reflected in the truth-table for "Sv~S":

In general, when a molecular sentence is true, no matter what the truth-values of its atomic constituent sentences are, that sentence is called a classical tautology.

S ~S Sv~S

Assignment 1: 1 0 1

Classical Contradiction

At the opposite end of the scale from a tautology is the type of sentence that makes an impossible claim. For instance, the sentence "Socrates is bald and Socrates is not bald" must be false no matter what the state of Socrates's head. This is

reflected in this truth-table:

In general, a sentence S is a classical contradiction IFF there is no classical assignment which makes S true.

S ~S S&~S

Assignment 1: 1 0 0

Classical Contingency

A sentence is classically contingent IFF it is neither a classical tautology nor a classical self-contradiction.

A classically contingent sentence is true or false depending upon the states of the objects referred to by the referring

expressions contained in it. For example, "Socrates is not bald" is false when Socrates is bald and true when he is not.

Argument

An argument is simply a sequence of (declarative) sentences. When an argument is given, we treat one of the sentences as the conclusion and the others as the premises.

It is customary to write an argument “A₁,...,A_n; therefore, B” in the form of

A₁,...,A_n//B

(where A₁,...,A_n are premises and B is the conclusion). In addition, we will use

A₁,...,A_n⊢_bcB

as the abbreviation of “A₁,...,A_n//B is valid in our basic classical logic.”

Classical Validity 1

An argument is classically valid IFF the conclusion is true according to every classical assignment which makes the premises all true. For example,

P//P&Q

is invalid. For see below:

According to assignment 2, P is true but P&Q is false.

P Q P&Q

Assignment 1: 1 1 1

Assignment 2: 1 0 0

Assignment 3: 0 1 0

Classical Validity 2

P//PvQ

is classically valid because there is no classical assignment according to which the premise, P, is true but the conclusion,

PvQ, is not true.

According to assignments 1 and 2, which makes P true, PvQ is also true. For another exercise, use the truth-table method to check the validity of

~P; PvQ//Q. P Q PvQ Assignment 1: 1 1 1 Assignment 2: 1 0 1 Assignment 3: 0 1 1 Assignment 4: 0 0 0

Classically Valid Arguments

• Modus Tollens: A->B,~B⊢_bc~A

• Disjunctive Syllogism: A v B, ~A⊢_bcB • Contraposition: A->B⊢_bc~B->~A

• Explosion: A,~A⊢_bcB

• v-Introduction: A⊢_bcA v B • &-Introduction: A, B⊢_bcA&B • &-Elimination: A & B⊢_bcA

• De Morgan: ~(A v B)⊢_bc~A&~B • De Morgan: ~(A & B)⊢_bc~A v ~B

• Double Negation Elimination (DNE): ~~A⊢_bcA. • Double Negation Introduction (DNI): A⊢_bc~~A

Classical Tautologies

Summary

● molecular sentences are constructed from atomic

sentences and the logical connectives "~," "&," and "v."

● The truth tables for "~," "&," and "v" show how the truth or falsity of molecular sentences depends on the truth or

falsity of their atomic constituent properties.

● With the aid of the truth tables for "~," "&," and "v," a truth table may be constructed for any molecular sentences. ● Classical equivalence, classical tautology, classical

self-contradiction, classical contingency can be defined with the help of truth tables.

● Similar claims hold for atomic and molecular properties.

● Classical validity can be also defined with the help of truth tables.