Truth Functional Logic

© 2011 By Paul Herrick

A Survey of the Main Branches of Logic

A truth-functional argument is one like this:

  1. If it is daylight then the sun is out.
  2. It is daylight. 
  3. Therefore the sun is out. 

And like this:

  1. If we travel to New York, then we will visit Harlem.
  2. If we visit Harlem then we will drop in on cousin Clarence.
  3. So, if we travel to New York, then we will visit cousin Clarence.

And like this:

  1. Either we will have burritos or we will eat hamburgers.
  2. But we will not have burritos.
  3. Therefore we will eat hamburgers.

Logicians in the Stoic school of philosophy, during the 3rd century B.C., meeting on the painted porch in downtown Athens, were the first to discover and systematize the logical standards for this type of argument, and for a long time it was called "Stoic logic." The system was updated and expanded in the late 19th century by the German logician Gottlob Frege (1848-1925). It is now called “truth-functional logic” for reasons that will be explained shortly.

The logical details are a little harder to explain, compared to the details of categorical logic, even when simplified greatly. For one reason, they are more mathematical in nature. But let’s give it a try.

A declarative sentence is compound if it contains within itself one or more sentences and one or more sentence operators. (By sentence I will always mean “declarative sentence.”) A sentence is simple if it is not compound. A sentence operator is a word or phrase that joins one or more sentences into a compound sentence. A compound sentence is thus composed of one or more sentence operators joining one or more sentences into the compound. In the following compound sentence, and is the operator joining the two simpler sentences into the compound:

Plato founded the Academy, and Aristotle founded the Lyceum.

This type of compound sentence is called a “conjunction.” The simple sentence to the left of the operator, “Plato founded the Academy,” is called the “left conjunct,” and the simple sentence to the right of the operator, “Aristotle founded the Lyceum,” is called the “right conjunct.” The operator (and) is called the “conjunction operator” because it is used to form a conjunction.

A sentence will be said to have the “truth-value” of “true” if it expresses a true proposition, and it will be said to have the “truth-value” of false if it expresses a false proposition. These definitions will allow us to express some complicated points more economically.

Now, a compound sentence is truth-functional if the truth-value of the compound as a whole is a strictly defined function of the truth-value or truth-values of the sentence or sentences inside it. A sentence operator is truth-functional if the compound it forms is a truth-functional compound. Finally, an argument is truth-functional if (a) it is composed of truth-functional sentences and (b) its validity is a strict function of nothing but the abstract arrangement of the truth-functional operators in its sentences.

Now we can say: a truth-functional conjunction is “any conjunction formed with an operator asserting that both conjuncts are true and nothing more.” The compound sentence, “Plato founded the Academy, and Aristotle founded the Lyceum,” is a truth-functional conjunction because it asserts that both of its conjuncts are true and its truth-value as a whole is a function of the truth-values of its parts according to the following exact rule covering all possible cases:

  • If the left conjunct is true and the right conjunct is true, then the conjunction as a whole is true.
  • If the left conjunct is true and the right conjunct is false, then the conjunction as a whole is false.
  • If the left conjunct is false and the right conjunct is true, then the conjunction as a whole is false.
  • If the left conjunct is false and the right conjunct is false, then the conjunction as a whole is false.

This rule, or function, applies, of course, to any conjunction formed with and when the and asserts that both conjuncts are true and nothing more. Logicians express this general rule more compactly on a device called a “truth-table.”

Where P, Q, and R are variables ranging over all declarative sentences and the symbol & represents a conjunction operator joining P and Q:

P Q   P & Q
T T     T  
T F     F  
F T     F  
F F     F  
Truth-Table for Conjunction

In this table, P represents the left conjunct, Q represents the right conjunct, and the formula P & Q represents the conjunction as a whole.

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Or

The word or can also be used as a sentence operator joining two sentences into a compound, and when it is, the compound sentence formed is called a “disjunction,” as in “We’ll eat hot dogs, or we’ll eat hamburgers.” In this compound, “We’ll eat hot dogs” is called the “left disjunct,” and “We’ll eat hamburgers” is the “right disjunct.”

A truth-functional disjunction is any disjunction formed with an operator asserting that one or the other of two disjuncts is true and nothing more. There are two types of disjunction. An inclusive disjunction asserts that one or the other or both of the disjuncts is true, while an exclusive disjunction asserts only that one or the other but not both of the disjuncts is true. In what follows, let us assume we are speaking of an inclusive disjunction.

The compound sentence, “We’ll eat hot dogs, or we’ll eat hamburgers” is a truth-functional (inclusive) disjunction because it asserts that one or the other of its disjuncts is true or both, and its truth-value as a whole is a function of the truth-values of its parts according to the following exact rule covering all possible cases:

  • If the left disjunct is true and the right disjunct is true, then the disjunction as a whole is true.
  • If the left disjunct is true and the right disjunct is false, then the disjunction as a whole is true.
  • If the left disjunct is false and the right disjunct is true, then the disjunction as a whole is true.
  • If the left disjunct is false and the right disjunct is false, then the disjunction as a whole is false.

The truth-table for the truth-functional disjunction follows, with the symbol v standing for a truth-functional disjunction operator:

P Q   P v Q
T T     T  
T F     T  
F T     T  
F F     F  
Truth-Table for Inclusive Disjunction

If This, Then That

The words If…then can also be used as a sentence operator to join two sentences into a compound, and when they are, the compound sentence is called a “conditional sentence,” as in, “If we go swimming, then we’ll get some exercise.” In this type of compound, “we go swimming” is called the “antecedent,” and “we’ll get some exercise” is called the “consequent.” The name (conditional) derives from the fact that the antecedent states a condition leading to, or supporting in some way, that which is expressed in the consequent.

A truth-functional conditional is one formed in such a way that the truth-value of the whole is a precise function of the truth-values of the parts. The following function is assigned to the conditional in this branch of logic:

  • If the antecedent is true and the consequent is true, then the conditional as a whole is true.
  • If the antecedent is true and the consequent is false, then the conditional as a whole is false.
  • If the antecedent is false and the consequent is true, then the conditional as a whole is true.
  • If the antecedent is false and the consequent is false, then the conditional as a whole is true.

The corresponding truth-table follows, with the symbol > representing the “if, then” operator:

P Q   P > Q
T T     T  
T F     F  
F T     T  
F F     T  
Truth-Table for the Truth-Functional Conditional

In this table, P represents the antecedent, Q represents the consequent, and the formula P > Q represents the conditional as a whole.

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Not

One more truth-function is worth noting. The phrase, “It is not the case that” can be used as a sentence operator, and when it is applied to a sentence, the resulting compound is called the “negation” of the sentence to which the operator was applied. Thus:

Original sentence: Negation of the sentence:
It is raining. It is not the case that it is raining.
Jan is President. It is not the case that Jan is President.

A negation is a truth-functional compound when the truth-value of the compound is a function of the truth-value of the sentence inside it by the following exact rule:

If the original sentence is true, then the negation is false.
If the original sentence is false, then the negation is true.

The truth-table is obvious. Where P represents the original sentence and the symbol ~ (called “tilde”) represents the negation operator:

P   ~ P
T   F  
F   T  

Now, using these items as building blocks, several common forms of valid truth-functional reasoning can be specified with precision. Each of these logical forms was first expressed in symbols and integrated into a deductive system of logic by the ancient Stoic logicians in the 3rd century B.C. Using the truth-tables, each can be shown to be a valid pattern of reasoning, with the precision of mathematics. Except for the last form, the names were assigned by logicians in the cathedral schools of Europe during the Middle Ages:

MODUS PONENS

  1. P > Q
  2. P
  3. Therefore Q

Example:

  1. If this is Seattle, Washington, then this is part of the United States.
  2. This is Seattle, Washington.
  3. Therefore, this is part of the United States.

MODUS TOLLENS

  1. 1. P > Q
  2. ~Q
  3. Therefore ~P

Example:

  1. If this plane is a jet, then it has at least one jet engine.
  2. It is not the case that this plane has at least one jet engine.
  3. Therefore, this plane is not a jet.

DISJUNCTIVE SYLLOGISM

  1. P v Q
  2. ~P
  3. Therefore Q

Example:

  1. We shall stay or we will go home.
  2. It is not the case that we will stay.
  3. Therefore, we will go home.

HYPOTHETICAL SYLLOGISM

  1. P > Q
  2. Q > R
  3. Therefore P > R

Example:

  1. If it rains, then the roof will get wet.
  2. If the roof gets wet, then the ceiling will leak.
  3. Therefore, if it rains, then the ceiling will leak.

THE “NOT-BOTH” FORM:

  1. Not both P and Q
  2. But P
  3. Therefore ~Q

Example:

  1. It is not the case that both Ann and Bob are home.
  2. But Ann is home.
  3. Therefore, it is not the case that Bob is home.

This is just a sampling, of course. The complete system is so precise, and it forms a lovely whole.

Incidentally, the rules of truth-functional logic are among the basic rules that open and close the “logic gates,” or circuits, inside the modern digital computer. When the pioneers of computer science were first figuring out how to link electric circuits together so that they would obey precise algorithms designed to compute exact solutions to problems, they discovered that the rules of truth-functional logic are essential. This means that the circuits inside your laptop, cell phone, and the device that plays your digital music all obey the laws of truth-functional logic! The ancient Stoic logicians could not have known it when they first discovered the logic of truth-functions, but the logical theory they discovered is part of what drives every digital computer on the planet today.

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