The work is licensed under
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2023-01-24
The work is licensed under
a Creative
Commons Attribution 4.0 International License.
“Logic” is technically defined by Oxford Languages as
reasoning conducted or assessed according to strict principles of validity
While this definition sounds fancy and scholarly, “logic” is much more common than this definition implies. Do you use the following English words, or their equivalent ones in a different language?
If you do, then you already have an intuitive grasp of logic.
A statement is a phrase or sentence that states a condition, something that is either true or false. Statements are important in everyday conversations. For example, “Today is going to be shiny!” is a statement. It states that (vaguely) the rest of today is without clouds and direct sunlight will reach the surface of the Earth. This statement can be true or false, it is implied that the person who is making the statement think the statement is true.
In logical thinking, it is crucial to differentiate a validated fact from claims. Logical deduction based on invalid assumptions may be correct from the logical perspective, but the conclusion is as invalid as the assumption.
This is a fancy term for words like “every”, “each”, “all”, “some”, “none”, etc.
Quantifiers are important in logic because they specify whether a statement applies to at least one, or every, instance of something.
Let us consider an example. The statement “system administrators are busy” is ambiguous because it is not quantified. One interpretation is “all system administrators are busy”, while the other interpretation is “*at least one” system administrator is busy.” In this case, most people understand the unspoken quantifier to be “all”.
This is a category of terms like “and”, “not”, “or”, “either”, etc.
These words are “connectives” because they connect smaller statements to composite statements.
For example, “today is going to be shiny” is a statement. “It will rain tomorrow” is another statement. Each of these two statements can be true or false. The connective “but” connects the two statements: “Today is going to be shiny, but it will rain tomorrow.”
In this example, the word “but” has the same meaning as “and”, meaning that the only way to make the composite statement to be true, today needs to be shiny, and tomorrow needs to be rainy.
Most implications are in the form of “if …, then …” For example, consider the implication of “if we get at least one inch of rainfall today, then there is no need to water the plants tomorrow.”
In this example, the component statement “we get at least one inch of rainfall today” is called the antecedent, whereas the component statement “there is no need to water the plants tomorrow” is called the consequent.
It is important to understand that an implication is a composite statement, in which the components are the antecedent and the consequent. This also means that an implication can be true or false. In this specific example, it is possible that the implication is false because some plants may need to be watered regardless of whether there is rain on the previous day. Or, if the plants are indoor, then the rain has no effect!
In fact, an implication can be expressed using logical operators. We will get this later.
“Necessary” is a fancier word for “needed.” For example, in order to use do laundry, a water supply is needed. This means that besides water, there may be other resources that are also needed. However, lacking water, one cannot do laundry.
“Sufficient”, on the other hand, is a fancier word for “enough.” Being caught driving under the influence (DUI) is reason enough to be arrested. This means that no other violation, such as speeding, is needed in addition to being caught DUI, for one to be arrested. Note that there is no “may”, “possibly”, or “probably” associated with “sufficiency.” In this example, being caught DUI by itself guarantees being arrested.
Both “necessary” and “sufficient” connect statements, similar to “implications.” In fact, “necessity”, “sufficiency”, and “implication” are all related. Assuming that \(A\) and \(B\) are statements, then:
Note that in most applications of an implication, there is an implied order. For example, in the statement “if \(A\), then \(B\),” it is usually implied that \(A\) has to occur before \(B\). This is not necessarily the case.
For example, consider the DUI example. “Timmy was not arrested last night” is a statement, and “Timmy was caught DUI last night” is another statement. In this case, “Timmy was not arrested last night” implies (is sufficient to deduce) “Timmy was not caught DUI last night.”
The main ingredients of logic are intuitive to most people. Daily conversations often apply logic.
If that is the case, why is it important to discuss logic?
STEM (science, technology, engineering, and math) is a general area of study where logic is important. The hierarchy can be over simplified as follows:
In logic, the words “and”, “or”, “not”, “implies”, etc. have specific meanings. For example, “or” refers to disjunction, but not “either or.” “Either or” is known as “exclusive or.”
More often than not, these words are replaced by symbols. The symbols used depend on the context. Let us just take conjunction as an example:
X && Y in C/C derived programming
languages.Regardless of the symbol, conjunction is defined by enumerating all four possible cases:
Mathematicians use logic as a tool to prove theorems. To prove a theorem is to start with statements that are assumed to be true, and logically conclude that a specific is also true as a result. Another way to look at theorem proving is that it is a (often long) chain of implications leading from axiomatic statements to the statement (theorem) that is being proven.
Scientists use logic (and math) to prove or disprove theories. The process of a proof is unlike that in math because in science, axiomatic statements must be evidence-based or supported by evidence.
Engineers use logic to design artifacts so that the artifacts meet a specific set of criteria. The criteria, also known as the specifications of an artifact or project, also use logic as a part of the language. When things do not work, engineers apply logical thinking to track down the problem and derive solutions.
Technologists, like engineers, also utilize logic as a language to describe the specifications of an artifact. Also like engineers, technologists use logic to diagnose problem and find solutions.
Logic is actually more commonly understood and utilized than what people are led to believe. However, in order to utilize logic both for communication and as a tool, formalization is necessary for clarity, precision, and rigor. We will explore formalized logic in another module.