# Fallacy of Propositional Logic

Taxonomy: Logical Fallacy > Formal Fallacy > Fallacy of Propositional Logic

Alias: Fallacy of sentential logic*

### Exposition:

In logic, a proposition―or, "statement"―is a sentence that is either true or false. For instance, "it is raining" is a proposition. Some propositions contain other propositions as components, for example: "it is not raining" is a proposition as a whole which contains "it is raining." A proposition that contains one or more simpler propositions as components is called a "compound" proposition. So, "it is not raining" is a compound proposition. A proposition that is not compound, such as "it is raining", is called "simple".

Propositional logic is a system of formal logic that deals with the logical relations holding between propositions taken as a whole, and those compound propositions which are constructed from simpler ones with truth-functional connectives. For instance, consider the following proposition:

Today is Sunday and it's raining.

This is a compound proposition containing the simpler propositions:

 Today is Sunday. It's raining.

The word "and" which joins the two simpler sentences to make the compound one is a truth-functional connective, that is, the truth-value of the compound proposition is a function of the truth-values of its components. In other words, whether the whole sentence is true or false is determined by whether the simpler sentences that compose it are true or false. The truth-value of a conjunction―a compound proposition formed with "and"―is true if both of its components are true, and false otherwise. So, the compound sentence is true if "today is Sunday" and "it's raining" are both true, and false if one or both are false.

Propositional logic studies the logical relations which hold between propositions as a result of truth-functional combinations, for instance, the example conjunction logically implies that today is Sunday. In other words, if the whole sentence is true then it must also be true that today is Sunday. There are a number of other truth-functional connectives in English in addition to conjunction, and the ones most frequently studied by propositional logic are:

Truth-Functional Connectives
Name English Example
Disjunction or Today is Sunday or today is Saturday.
Negation not Today is not Sunday.
Conditional only if Today is Sunday only if yesterday was Saturday.
Biconditional if and only if Today is Sunday if and only if yesterday was Saturday.

Since a validating argument form is one in which it is impossible for the premisses to be true and the conclusion false, you can use the truth-functions to determine that forms in propositional logic are validating. For instance, the earlier example involving conjunction is an instance of the following argument form:

Simplification
Premiss Conclusion
p and q. p.

This form is validating because, no matter what propositions we put for p and q, if the premiss is true, then both p and q will be true, which means that the conclusion will also be true. Thus, to show that a propositional argument form is non-validating, all that you have to do is find an argument of that form which has true premisses and a false conclusion. Such an argument is called a "counter-example", and this method is used throughout the entries for the subfallacies, listed above, to show that the form of these fallacies is non-validating.

### Exposure:

A type of argument is a fallacy of propositional logic when two conditions are met:

1. Its propositional form is non-validating.
2. Its propositional form is similar enough to a validating form to be confused with it.

Another way to put this is that a propositional fallacy is a non-validating propositional form that appears to be validating. For this reason, each entry for a specific propositional fallacy―see the Subfallacies, above―includes a "Similar Validating Form", which is a validating propositional form similar enough to the fallacious form to be confused with it.

### Resources:

This discussion of propositional logic is by necessity brief, since I am only trying to give the minimal background required to understand the subfallacies above. For a lengthier explanation of propositional logic, see the following:

• Howard Pospesel, Introduction to Logic: Propositional Logic (Third Edition) (Prentice Hall, 1998). A good textbook on propositional logic for beginners.
• Peter Suber, "Propositional Logic Terms and Symbols" (1997). A concise class handout for a course in symbolic logic.