Symbolic Logic

To fully understand deductive reasoning we need a general theory of deduction:

1. Explain the relations between premises and conclusions.

2. Provide techniques for discriminating between valid and invalid.

Early on, we examined classical logic (Aristotelian logic): Square + Syllogisms

We now look at modern logic: Symbolic Logic

These two systems have similar aims, but proceed in very different ways.

Modern logic uses an artificial symbolic language.

Why? Because natural languages—like English—often use vague words, double meaning, metaphors, etc.

A symbolic language overcomes these issues—symbols facilitate our thinking and keep us focused on the logical structure of arguments.

For example:The Indo-Arabic numerals we use today is an improved symbolic language.

Think about multiplying Roman numerals: CXIII multiplied by IV. 113 x 4 is much easier to multiply!

Classical logic has a serious limitation: To test arguments, you must translate them into syllogisms.

Let’s look into the logical symbols:

As usual, we shall use relatively simple arguments.

The boy has a red hat or he has a white hat.The boy does not have a white hat.Therefore the boy has a red hat.

…or

If the moon is made of cheese, then Mr. Robinson is an elf.Mr. Robinson is not an elf.Therefore, the moon is not made of cheese.

Arguments of this type may contain compound statements.

Statements are either simple or compound.

Simple Statement: A statement that contains no other statement as a component.

Example: “Charlie is tall” is a simple statement.

Compound Statement: A statement that contains one or more statements as components.

Example: “Charlie is tall and Charlie is intelligent” is a compound statement.

There are different compound statements, each of which requires its own logical symbol.

CONJUNCTION

We can form a compound statement by way of conjunction, by using the word “and”.

So, the compound statement, “Charlie is tall and Charlie is intelligent” is a conjunction whose first conjunct is “Charlie is tall” and whose second conjunct is “Charlie is intelligent”.

BUT BE CAREFUL!

One might suppose that “Lincoln and Grant were contemporaries” is a conjunction.

Nope! In this case the word “and” simply expresses a relationship between the nouns.

Also, if I say, “He took off his shoes and got into bed” I intend to express a temporal succession.

But if I say, “He went to bed and took off his shoes” I am expressing a conjunction.

The symbol we use to connect statements conjunctively is the symbol “+”.

Thus the previous conjunction can be written as “Charlie is tall + Charlie is intelligent”.

Later we will translate these sorts of statements into symbols so that we can express arguments and then test their validity by constructing a truth table.

Just as a preview: The compound statement above may be a premise to an argument:

Charlie is tall and Charlie is intelligent.Therefore, Charlie is intelligent.

The above argument can be stated symbolically. We can symbolize the statement “Charlie is tall” using the letter “T”

So we stipulate that T = “Charlie is tall.”

And we can use I = “Charlie is intelligent.”

Then we can state the argument as such:

T + I∴ I

Truth Value

We know that every statement is either true or false. Therefore, we say that every statement has a truth value.

So, the truth value of the compound statement is determined by the truth value of its components: if both its conjuncts are true, the conjunction is true, otherwise it is false.

For this reason, a conjunction is said to be a truth-functional compound statement.

And its conjuncts are truth-functional compounds.

And the symbol “+” is a truth-functional connective.

Given any two statements there are only 4 possible sets of truth values they can have.

Let 2 statements be P and Q.

When P is true and Q is true = “P + Q” is true.

When P is false and Q is true = “P + Q” is false.

When P is true and Q is false = “P + Q” is false.

When P is false and Q is false = “P + Q” is false.

…or

P Q P+Q ______________

T T TF T FT F FF F F

…in other words, a conjunction is true if and only if both its conjuncts are true.

The above table is what modern logicians call a truth table. A truth table shows all possible truth values of compound statements.

NOTE: The English words but, yet, also, still, although, however, moreover, nevertheless, and the comma and semicolon, can also

conjoin two statements into a compound statement, and therefore they can be represented by the “+” symbol.

NEGATION

The negation of a statement is typically indicated with the word “not”.

Negating a statement also can be done by prefixing a sentence by “it is false that…”

Modern logicians use the symbol “–”.

This is an easy one to represent:If a statement is true, then its negation must be false, and vice versa.

Example:

Let P stand for “popcorn is good” and –P (Not P) for the negation of that statement, i.e., popcorn is not good.

If it is true that popcorn is good, then it must be false that popcorn is not good, right? And the other way around.

So,

P –P _________

T FF T

DISJUNCTION

The disjunction of two statements is formed by adding the word “or” between them statements. The two statements are called disjuncts.

For the disjunction we use the symbol “v”.

A disjunction is false only if both its disjuncts are false.

P Q P v Q ______________

T T T F T T T F T F F F

CONDITIONAL STATEMENT

When two statements are combined by the words “if” and “then”, the resulting compound is a conditional statement.

The statement that follows the word “if” is called the antecedent.

Example:

“If I graduate, then I will get a job.” Here, the statement “I graduate is the antecedent.

And the sentence that follows the word “then” is called the consequent. Here the sentence “I will get a job” is the consequent.

A conditional statement asserts that in any case in which its antecedent is true, its consequent is also true.

It does not claim that its antecedent is true, but only that if it is true, then the consequent is also true.

NOTE: A conditional statement and its truth value must be determined by the implication.

The implication is the relation between the antecedent and the consequent.

Consider the following conditional statements, each of which asserts different implications.

1. If all humans are mortal and Socrates is a human, then Socrates is mortal: The consequent here follows by logical necessity.

2. If Marc is a bachelor, then Marc is unmarried: Here the consequent follows by virtue of meaning.

3. If you place a bar of soap into a tub filled with water, then the bar of soap will float on the water: Here the consequent follows inductively and the truth must be empirically determined.

4. If I lose the game, then I will eat my hat: Here the consequent is merely a decision stipulated by the speaker.

Any conditional statement is false if its antecedent is true and its consequent false. In any other case the statement is true.

The symbol use for the conditional is the horseshoe, “⊃”

P Q P ⊃ Q_________________T T TF T TT F FF F T

You noticed that in the event that P and Q are false, the conditional is true! And when P is false and Q true, the conditional is true!

Let me explain: Say that our conditional statement is the following:

“If you graduate, then I will buy you a car.” (If P then Q)

Now think about it: Say that you do not graduate. Then I will not buy you a car. Right? But it is still true that if you graduate, I’ll buy you a car. You failed to graduate, but I did not lie.

MATERIAL EQUIVALENCE

Two statements are said to be materially equivalent when they have the same truth value, that is, when they are both true or both false.

The truth-functional connective symbol is the three-bar sign, “≡” and is pronounced “if and only if” often abbreviated as “iff”.

P Q P ≡ Q_________________T T TF T FT F FF F T

Punctuation

Just like in normal language I need punctuation, I need it in logic too.

“The teacher said John is intelligent.”

…or

“The teacher, said John, is intelligent.”

…and

“2x3+5”

…or

“2x (3+5)”

So in symbolic logic we use parentheses, brackets, braces, etc.

Let’s see how:

“If the President vetoes the bill or if Congress balks and the people protest, then the chances of passing this year’s new tax amendment are not good.”

Step 1: Identify simple statements and assign letters:

V C

“If 1the President vetoes the bill or if 2Congress balks and 3the P G

people protest, then 4the chances of passing this year’s new tax amendment are not good.”

Step 2: Rewrite simple sentences:

1: The President vetoes the bill = V2: Congress balks = C3: The people protest = P4: The chances of passing this year’s new tax amendment are good = G

Notice I did not include “not” because it is a negation and I want only the simple sentence.

Step 3: Rewrite using the letters:

If V or if C and P, then not G

C and P should be grouped together, so (C + P)

If V or if (C and P), then not G

Now, the whole statement is a conditional:

If V or if (C and P) is the antecedent.

then not G is the consequent.

Therefore, they should be connected with the horseshoe ⊃

If V or if (C and P) ⊃ not G

To further indicate that the antecedent and consequent are separated we use brackets (because we already used parentheses).

[If V or if (C and P)] ⊃ not G

Now we need to get rid of words and convert them into operators:

[V v (C + P)] ⊃ –G