Inference and Assumptions

Definition

You will frequently see the words inference and assumption. Let us take a moment to define the meaning of each term since inferences and assumptions are related concepts in reasoning, but they have distinct meanings.

Inferences

An inference is a conclusion drawn from evidence or reasoning. It is a logical step based on information that is available. Inferences rely on data, observations, or premises. They are typically based on explicit information.

Example: If you see someone carrying an umbrella, you might infer that it will likely rain.

Assumptions

An assumption is something that is accepted as true without direct evidence. It is often taken for granted in reasoning or arguments. Assumptions may not be explicitly stated and can be based on beliefs, experiences, or generalizations rather than concrete evidence. Assumptions can often have a significant effect on the validity of the argument.

Example: Assuming that everyone in the meeting has read the report is an assumption that may not be true.

Summary

Inferences are conclusions based on evidence, while assumptions are beliefs accepted without direct proof. Inferences can be tested and verified, whereas assumptions may need to be challenged or examined for validity. In one sense, an assumption occurs “before” the argument, that is, while the argument is being made. An inference is made “after” the argument is complete and follows from the argument.

Who

Mathematical inference and assumptions are foundational in many fields, guiding decision-making, problem-solving, and theoretical development. Understanding these concepts is essential for anyone engaged in analytical and logical reasoning.

Scientists
Engineers
Economists
Computer Scientists
Statisticians

Inference and assumptions are integral to navigating everyday life, helping us make sense of our experiences, solve problems, and interact effectively with others. Being aware of these processes can enhance critical thinking and decision-making skills.

Communication
Decision-Making
Learning and Education
Problem-Solving
Social Interactions

What

Mathematics is really about proving general statements via arguments, usually called proofs. As you know from arguing with friends, not all arguments are “good”. A “bad” argument is one in which the conclusion does not follow from the premises, i.e., the conclusion is not a consequence of the premises. Logic is the study of what makes an argument good or bad.

Mathematical Inference

Rules of inference are logical tools used to derive conclusions from premises. They form the foundation of logical reasoning, allowing us to build arguments, prove theorems, and solve problems in mathematics, computer science, and philosophy. Understanding these rules is crucial for constructing valid arguments and ensuring that conclusions follow logically from given information. [1] Mathematical inference is essential for problem-solving and advancing mathematical theories. It allows mathematicians to build upon established knowledge and explore new concepts logically and rigorously.

Logical Reasoning

Inference involves using logical principles to derive new statements or conclusions from existing ones. This can include deductive reasoning (from general to specific) and inductive reasoning (from specific cases to general rules).

Types of Inference

Deductive Inference involves deriving specific conclusions from general principles or axioms. For example, if all squares are rectangles (premise), and we have a square (premise), we can infer that it is a rectangle (conclusion).

Inductive Inference involves making generalizations based on observations or specific examples. For instance, observing that the sum of two even numbers is always even can lead to the inference that this is a general rule.

Abductive Inference is a form of reasoning that seeks the best explanation for a set of observations. It’s less common in formal mathematics but can be applied in mathematical modeling.

Proofs

Inference plays a crucial role in mathematical proofs, where a series of logical steps are used to demonstrate the truth of a statement based on assumptions or previously established results.

Mathematical Logic

Inference is a central concept in mathematical logic, where formal systems are used to define rules of inference that dictate how new statements can be derived from existing ones.

Examples

Modus Ponens: If P implies Q (if it rains, the ground gets wet), and P is true (it rains), we can infer Q (the ground gets wet).

Transitive Inference: If A = B and B = C, we can infer that A = C.

Mathematical Assumptions

Mathematical assumptions are foundational statements or conditions that are accepted as true to build further reasoning, proofs, or theorems. Understanding and clearly stating assumptions is crucial in mathematics because they define the limits and scope of the conclusions that can be drawn from a given mathematical framework or argument. This clarity helps ensure that reasoning is valid and robust.

Foundational Basis

Assumptions often serve as the starting points for mathematical reasoning. For instance, axioms in geometry or algebra are assumptions that define the framework of a mathematical system.

Types of Assumptions

Axioms: These are basic assumptions that are universally accepted without proof. For example, Euclid’s axioms in geometry.

Postulates: Similar to axioms, postulates are assumptions specific to a particular mathematical theory.

Hypotheses: These are assumptions made in the context of a specific problem or theorem that, if true, allow for the conclusion to be drawn.

Context-Dependent

Assumptions may vary based on the mathematical context. For example, in Euclidean geometry, the parallel postulate is an assumption that does not hold in non-Euclidean geometries.

Testing Validity

While assumptions in mathematics are accepted for the sake of argument, they can be tested and examined. If an assumption leads to contradictions, it may need to be revised or discarded.

Examples

In calculus, we might assume a function is continuous when applying the Intermediate Value Theorem.

In statistics, one might assume that data are normally distributed when performing certain analyses.

Why

Inference

The ultimate goal for anyone learning math is to use it to make better and more confident decisions. The goal is to be able to make inferences from any given data sets. You cannot teach or learn this in a single class. Making inferences requires a good deal of experience and a consistent reflection method. You should not only be concerned that you are getting the correct answer but that you used all the different means available to make sense of whatever data you analyzed. [2]

Assumptions

Mathematical assumptions play a vital role in simplifying complex problems and establishing frameworks for analysis. They facilitate communication, enable proofs, guide research, and help apply mathematics to real-world situations. By providing a foundation for further exploration and understanding, these assumptions allow mathematicians and practitioners to work efficiently and effectively.

See Theoretical Knowledge Vs Practical Application.

How

Many of the References and Additional Reading websites, and Videos will assist you in understanding and applying inferences and assumptions.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

[1] “Rules of Inference.” 2024. GeeksforGeeks. August 28. https://www.geeksforgeeks.org/rules-of-inference/.

[2] “Math Worksheets Land.” 2024. Math Inference Worksheets. Accessed December 31. https://www.mathworksheetsland.com/7/28infer.html.

Additional Reading

Inference

“2.4: Rules of Inference.” 2022. Mathematics LibreTexts. Libretexts. April 17. https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book:_Friendly_Introduction_to_Mathematical_Logic_(Leary_and_Kristiansen)/02:_Deductions/2.04:_Rules_of_Inference.

Having established our set Λ of logical axioms, we must now fix our rules of inference. There will be two types of rules, one dealing with propositional consequence and one dealing with quantifiers.

“2.6 Arguments and Rules of Inference.” 2021. Mathematics LibreTexts. Libretexts. February 6. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/2:_Logic/2.6_Arguments_and_Rules_of_Inference.

In this section we will look at how to test if an argument is valid. This is a test for the structure of the argument. A valid argument does not always mean you have a true conclusion; rather, the conclusion of a valid argument must be true if all the premises are true. We will also look at common valid arguments, known as Rules of Inference as well as common invalid arguments, known as Fallacies.

“Discrete Mathematics – Rules of Inference.” 2024. tutorialspoint. Accessed December 31. https://www.tutorialspoint.com/discrete_mathematics/rules_of_inference.htm.

Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements. An argument is a sequence of statements. The last statement is the conclusion. and all its preceding statements are called premises (or hypothesis). The symbol “∴” (therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises. Rules of Inference provide the templates or guidelines for constructing valid arguments from the statements that we already have.

“Inference.” 2024. Wikipedia. Wikimedia Foundation. December 16. https://en.wikipedia.org/wiki/Inference.

Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word infer means to “carry forward”. Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle (300s BCE). Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular evidence to a universal conclusion.

Various fields study how inference is done in practice. Human inference (i.e., how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, and cognitive psychology; artificial intelligence researchers develop automated inference systems to emulate human inference. Statistical inference uses mathematics to draw conclusions in the presence of uncertainty. This generalizes deterministic reasoning, with the absence of uncertainty as a special case. Statistical inference uses quantitative or qualitative (categorical) data which may be subject to random variations.

“Rules of Inference.” 2021. CalcWorkshop. https://calcworkshop.com/logic/rules-inference/.

Have you heard of the rules of inference? They’re essential in logical arguments and proofs. Let’s find out why! While the word “argument” may mean a disagreement between two or more people, in mathematical logic, an argument is a sequence or list of statements called premises or assumptions and returns a conclusion. An argument is only valid when the conclusion, which is the final statement of the opinion, follows the truth of the discussion’s preceding assertions. Consequently, our goal is to determine the conclusion’s truth values based on the rules of inference.

Assumptions

Sayama, Hiroki. “2.4: What Are Good Models?” 2024. Mathematics LibreTexts. Libretexts. April 30. https://math.libretexts.org/Bookshelves/Scientific_Computing_Simulations_and_Modeling/Introduction_to_the_Modeling_and_Analysis_of_Complex_Systems_(Sayama)/02%3A_Fundamentals_of_Modeling/2.04%3A_What_Are_Good_Models%3F.

Simplicity of a model is really the key essence of what modeling is all about. The main reason why we want to build a model is that we want to have a shorter, simpler description of reality. As the famous principle of Occam’s razor says, if you have two models with equal predictive power, you should choose the simpler one. This is not a theorem or any logically proven fact, but it is a commonly accepted practice in science. Parsimony is good because it is economical (e.g., we can store more models within the limited capacity of our brain if they are simpler) and also insightful (e.g., we may find useful patterns or applications in the models if they are simple). If you can eliminate any parameters, variables, or assumptions from your model without losing its characteristic behavior, you should.

Validity of a model is how closely the model’s prediction agrees with the observed reality. This is of utmost importance from a practical viewpoint. If your model’s prediction doesn’t reasonably match the observation, the model is not representing reality and is probably useless. It is also very important to check the validity of not only the predictions of the model but also the assumptions it uses, i.e., whether each of the assumptions used in your model makes sense at its face value, in view of the existing knowledge as well as our common sense. Sometimes this “face validity” is more important in complex systems modeling, because there are many situations where we simply can’t conduct a quantitative comparison between the model prediction and the observational data. Even if this is the case, you should at least check the face validity of your model assumptions based on your understanding about the system and/or the phenomena.

Note that there is often a trade-off between trying to achieve simplicity and validity of a model. If you increase the model complexity, you may be able to achieve a better fit to the observed data, but the model’s simplicity is lost and you also have the risk of overfitting— that is, the model prediction may become adjusted too closely to a specific observation at the cost of generalizability to other cases. You need to strike the right balance between those two criteria.

Finally, robustness of a model is how insensitive the model’s prediction is to minor variations of model assumptions and/or parameter settings. This is important because there are always errors when we create assumptions about, or measure parameter values from, the real world. If the prediction made by your model is sensitive to their minor variations, then the conclusion derived from it is probably not reliable. But if your model is robust, the conclusion will hold under minor variations of model assumptions and parameters, therefore it will more likely apply to reality, and we can put more trust in it.

Simpson, Roy. “10.2: Axioms, Theorems, and Proofs.” 2024. Mathematics LibreTexts. Libretexts. June 3. https://math.libretexts.org/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/10%3A_Appendix_-_The_Language_of_Mathematics/10.02%3A_Axioms_Theorems_and_Proofs.

There is a strange creature in mathematics, not typically mentioned in lower division texts, called an axiom (or, in some texts, a postulate).

“Axiom.” 2024. Wikipedia. Wikimedia Foundation. December 24. https://en.wikipedia.org/wiki/Axiom.

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning.

In mathematics, an axiom may be a “logical axiom” or a “non-logical axiom”. Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example a + 0 = a in integer arithmetic.

Non-logical axioms may also be called “postulates”, “assumptions” or “proper axioms”. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.

Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be “true” is a subject of debate in the philosophy of mathematics.

“Mathematical Proof.” 2024. Wikipedia. Wikimedia Foundation. December 22. https://en.wikipedia.org/wiki/Mathematical_proof.

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish “reasonable expectation”. Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

“What Are the Assumptions in a Mathematical Model?” 2024. TutorChase. Accessed December 31. https://www.tutorchase.com/answers/a-level/maths/what-are-the-assumptions-in-a-mathematical-model.

Assumptions in a mathematical model are the conditions that must be met for the model to be valid.

Mathematical models are used to represent real-world situations and make predictions. However, these models are based on certain assumptions that may not always hold true. For example, a model that predicts the growth of a population assumes that the birth rate and death rate remain constant over time. If these rates change, the model may no longer be accurate.

Assumptions can be explicit or implicit. Explicit assumptions are those that are stated explicitly in the model, while implicit assumptions are those that are not stated but are still necessary for the model to be valid. For example, a model that predicts the trajectory of a projectile assumes that there is no air resistance. This assumption is not stated explicitly, but it is necessary for the model to be valid.

It is important to be aware of the assumptions in a mathematical model, as they can affect the accuracy of the model’s predictions. If the assumptions are not met, the model may need to be revised or a different model may need to be used. To understand how assumptions impact real-world applications, consider exploring real-world applications of mathematical models.

For those interested in developing their own models, it is crucial to grasp the importance of correctly establishing assumptions, which you can learn more about in creating models.

Additionally, evaluating the effectiveness and limitations of a model requires understanding its underlying assumptions. More on this can be found in analysing models.