Introduction to Symbolic Logic

• In every language, an idea is communicated either verbally or in writing when words are combined together to create a sentence. This means that a sentence cannot exist without words. Also, a single word can form a complete sentence, and this sentence is called the sentence word.
• A complete sentence that expresses an idea that is acknowledged as either true or false is called a statement. Consider the following sentence as an example: the person reading this text is a living and literate person. This sentence can be acknowledged as a true statement because neither a dead person nor an illiterate person can directly read that sentence.
• In Antepraedicamenta, a text on logic attributed by Aristotle, the sentence word is described as a simple form of speech that lacks structure or composition and therefore cannot be acknowledged as either true or false. This means that the sentence word cannot be a statement.

Premise and Statement

• A statement that can be verified as either true or false is called a premise. The aforementioned sample sentence can be verified as a true statement, and consequently, can be described as a premise. This confirms that the premise has an inherent quality of truth or falsity. In other words, the premise can have only one of two mutually-exclusive qualities: truth (a true premise) or false (a false premise). For this reason, the premise is described as a declarative statement as it eliminates ambiguity about its expressed idea by affirming whether it is true or false.
• An idea derived from the premise and expressed as a statement is called the conclusion. This means that a conclusion can only exist if a premise exists.
• Usually, at least 2 premises are used to validate or invalidate a conclusion, which is usually derived from both premises.

Argument

• An argument is a set of premise(s) and conclusion in which the premise(s) provide reason(s) for accepting the validity of the conclusion.
• An argument is thus made up of two elements: an independent premise (or premises), and the derived conclusion. Therefore, an argument has a discernible structure that can best be described as a series of statements of which the first statement is a premise and the last statement is a conclusion. This allows for the argument to be presented as a single compound sentence that has at least 2 independent clauses, with the initial clause being a premise and the final clause being the conclusion. This unique structure of the argument explains why a collection of statements in a paragraph or compound sentence do not necessarily create an argument.
• An argument can have the conclusion as the first statement and the premise as the latter statement in a compound sentence.
• To argue is to present an argument. This involves providing reasons why the conclusion of the argument is valid.
• The process of creating an argument is known as argument formulation.

Conclusion Derivation by Deduction and Induction

• A conclusion can be derived from the premises in one of two ways: deduction or induction.
• If the premises are valid, then the conclusion derived from them must be valid. This is the basis of deduction which produces a deductively valid argument.
• The process of proving that the deductive argument is wrong is called invalidation.
• To invalidate a deductive argument, one needs to prove that one of the premises is false under certain circumstances, which can include hypothetical circumstances. Once one of the premises has been proven as false, then the conclusion can be false, hence resulting in an invalid argument.
• A deductively invalid argument is one where the premises are true but the conclusion is false. Normally, this can be identified by comparing the premises to the conclusion when assessing the deductive validity of an argument. Any conflict in the information contained in the premises and conclusion can prove the invalidity of the conclusion.
• The information contained in the premises can be interpreted, and this interpretation is then used to derive a conclusion. This argument is not deductively valid, but it can be a valid argument.
• Using interpretation of the premises to derive a conclusion is the basis of induction. This means that true premises do not guarantee the validity of the conclusion, rather they increase the probability of the conclusion being true. For this reason, the validity of the conclusion of an inductive argument is determined by its inductive strength which is equivalent to the probability of its validity.
• An inductive argument can be inductively strong or inductively weak based on its inductive strength.
• Inference is the process of making an induction. It allows for new knowledge to be derived from existing knowledge. This new knowledge comes at the cost of inductive strength (i.e probability of error in the conclusion). On the other hand, deduction serves to validate existing knowledge. Therefore, the inferred knowledge can be confirmed by deductive arguments.
• Inductive arguments form the basis of inductive reasoning. Outside logic, inductive reasoning is based on learning from experience e.g spices improve the flavor of food. This can then be projected to literature where the phrase “spiced up story” means that it differs from the original story because of the addition of new interesting details.
• An inductive argument is also called a probability argument.
• Rhetoric can be either argumentative or explanatory (expository).

Soundness and Argument Form

• The validity of an argument can be determined from its contents or its form. This means that the validity of any conclusion is determined by the contents of the premises or the form of the argument.
• The form is the logical structure of an argument that relates (or associates) the premises to the conclusion. It serves to create the semantics of the argument. Semantics refers to the study of relationships between ideas, statements, objects, and acts so as to find out how they are connected together to produce meaning or truth.
• It is the form of the deductive argument that ensures that its conclusion is validly true i.e a valid form preempts a false conclusion. This reveals that a valid argument form creates a valid argument if all its premises are true. Therefore, the form of a deductive argument can be used to determine its validity.
• A symbol represents an idea using a written form.
• The standard (or conventional) symbol used to represent an idea, concept, or object is called notation.
• The premise of an argument can be symbolized as P (uppercase P) and the conclusion be symbolized using the notation C (uppercase C). If there is more than one premise or conclusion in the argument, then each premise and conclusion can be labeled using a subscripted number e.g P_1, P₂, C₁, and C₂.
• A simple form of a deductive argument with two premises and a single conclusion can be represented as follows:
  1. Either P1 or P2 e.g Either Trump won or Biden won the 2020 elections.
  2. It is not the case that P1 (happened) e.g it is not the case that Trump won the 2020 elections.
  3. Therefore, P2 is the conclusion e.g Biden won the 2020 elections.
• In logic, the above argument uses premises that are mutually exclusive, which means that each of the premises is true, but the two premises cannot be true at the same time. Mutually exclusive premises are combined together using the connector OR.
• Another simple form of a deductive argument with two premises and a single conclusion can be represented as follows:
  1. P1 and P2 e.g Joe Biden is a Presidential Candidate and Joe Biden is a lifetime member of the Democratic Party.
  2. In the case that P1 is not true e.g it is not the case that Joe Biden is a Presidential Candidate.
  3. P2 is the conclusion that is valid e.g Joe Biden is a lifetime member of the Democratic Party.
• In logic, the above argument uses premises that are mutually independent, which means that each of the premises is true, and the two premises can be true at the same time. Mutually independent premises are combined together using the connector AND.
• An argument can be either valid or invalid, not true or false. Therefore, the argument has an inherent quality of being valid or invalid depending on the argument form and the truth/false quality of its premise(s).
• A sound argument is an argument that is valid and whose premises are true. Therefore, a sound argument has two inherent qualities: true premises and a valid conclusion.
• Any argument whose premises are not all true, or whose conclusion is not valid, is classified as an unsound argument. This means that a valid argument can have a false premise, and thus be an unsound argument.

Consistency

• In an argument made up of two or more premises, all the premises can be true, along with a valid conclusion. This argument is described as being consistent and sound because all its statements (i.e premises and conclusion) can be true. On the other hand, an argument made up of multiple premises can have some false premises and a valid conclusion, and this makes the argument inconsistent.
• If an argument has more than one premise, then the total number of premises in the argument is collectively designated as the set of premises.
• Consistency is the quality of an argument that determines if its premises are either all true or some of the premises are false. Unlike soundness, consistency does not consider the validity of the conclusion.
• Consistency can be related to validity and soundness as follows: an argument with two true premises and a valid conclusion is consistent, valid, and sound; while an argument with two premises, of which one is false, and a valid conclusion is an inconsistent, unsound, and possibly valid argument. This means that an argument can be consistent but invalid and unsound, or be inconsistent and unsound yet valid. However, an unsound argument that has at least one false premise cannot be consistent.
• The concept of consistency introduces the concept of possibility e.g can an inconsistent argument be possibly valid yet unsound?
• One of the models used to define the concept of possibility, and the related concept of necessity, is known as modal logic.
• Consistency allows for a new definition of what a valid argument is.
• A valid argument is an argument whose premises are true, and any attempt to describe its conclusion as false introduces inconsistency. This means that invalidating the conclusion renders one of the premises false.
• If one describes an argument as valid, two questions must be asked:
  1. What makes that person state that the argument is valid? Is it because the premises are true and the conclusion is valid? Answering these questions provides the context of discovery as it allows for investigating the truthfulness of the premises (as well as the validity of the conclusion of an inductive argument).
  2. What reasons does that person provide for stating that the argument is valid? Is it because the statements in the argument are valid, sound, and consistent? Answering these questions provide the context for justification, and this is the domain of logic.
• The logical form of an argument can be characterized in three ways, and they are:
  1. Validity.
  2. Soundness.
  3. Consistency.