Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is sound if it is valid and all its premises are true. One approach defines deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning.

Deductive logic studies under what conditions an argument is valid. According to the semantic approach, an argument is valid if there is no possible interpretation of the argument whereby its premises are true and its conclusion is false. The syntactic approach, by contrast, focuses on rules of inference, that is, schemas of drawing a conclusion from a set of premises based only on their logical form. There are various rules of inference, such as modus ponens and modus tollens. Invalid deductive arguments, which do not follow a rule of inference, are called formal fallacies. Rules of inference are definitory rules and contrast with strategic rules, which specify what inferences one needs to draw in order to arrive at an intended conclusion.

Deductive reasoning contrasts with non-deductive or ampliative reasoning. For ampliative arguments, such as inductive or abductive arguments, the premises offer weaker support to their conclusion: they indicate that it is most likely, but they do not guarantee its truth. They make up for this drawback with their ability to provide genuinely new information (that is, information not already found in the premises), unlike deductive arguments.

Cognitive psychology investigates the mental processes responsible for deductive reasoning. One of its topics concerns the factors determining whether people draw valid or invalid deductive inferences. One such factor is the form of the argument: for example, people draw valid inferences more successfully for arguments of the form modus ponens than of the form modus tollens. Another factor is the content of the arguments: people are more likely to believe that an argument is valid if the claim made in its conclusion is plausible. A general finding is that people tend to perform better for realistic and concrete cases than for abstract cases. Psychological theories of deductive reasoning aim to explain these findings by providing an account of the underlying psychological processes. Mental logic theories hold that deductive reasoning is a language-like process that happens through the manipulation of representations using rules of inference. Mental model theories, on the other hand, claim that deductive reasoning involves models of possible states of the world without the medium of language or rules of inference. According to dual-process theories of reasoning, there are two qualitatively different cognitive systems responsible for reasoning.

The problem of deduction is relevant to various fields and issues. Epistemology tries to understand how justification is transferred from the belief in the premises to the belief in the conclusion in the process of deductive reasoning. Probability logic studies how the probability of the premises of an inference affects the probability of its conclusion. The controversial thesis of deductivism denies that there are other correct forms of inference besides deduction. Natural deduction is a type of proof system based on simple and self-evident rules of inference. In philosophy, the geometrical method is a way of philosophizing that starts from a small set of self-evident axioms and tries to build a comprehensive logical system using deductive reasoning.

Definition

Deductive reasoning is the psychological process of drawing deductive inferences. An inference is a set of premises together with a conclusion. This psychological process starts from the premises and reasons to a conclusion based on and supported by these premises. If the reasoning was done correctly, it results in a valid deduction: the truth of the premises ensures the truth of the conclusion. For example, in the syllogistic argument "all frogs are amphibians; no cats are amphibians; therefore, no cats are frogs" the conclusion is true because its two premises are true. But even arguments with wrong premises can be deductively valid if they obey this principle, as in "all frogs are mammals; no cats are mammals; therefore, no cats are frogs". If the premises of a valid argument are true, then it is called a sound argument. It is necessary in the sense that the premises of valid deductive arguments necessitate the conclusion: it is impossible for the premises to be true and the conclusion to be false, independent of any other circumstances.

Deductive reasoning is studied in logic, psychology, and the cognitive sciences. There are two important conceptions of what this exactly means. They are referred to as the syntactic and the semantic approach.

The semantic approach suggests an alternative definition of deductive validity. It is based on the idea that the sentences constituting the premises and conclusions have to be interpreted in order to determine whether the argument is valid. This happens usually based only on the logical form of the premises. A rule of inference is valid if, when applied to true premises, the conclusion cannot be false. A particular argument is valid if it follows a valid rule of inference. Deductive arguments that do not follow a valid rule of inference are called formal fallacies: the truth of their premises does not ensure the truth of their conclusion.

Prominent rules of inference

Modus ponens

Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive rule of inference. It applies to arguments that have as first premise a conditional statement (<math>P \rightarrow Q</math>) and as second premise the antecedent (<math>P</math>) of the conditional statement. It obtains the consequent (<math>Q</math>) of the conditional statement as its conclusion. The argument form is listed below:

  1. <math>P \rightarrow Q</math>&nbsp; (First premise is a conditional statement)
  2. <math>P</math>&nbsp; (Second premise is the antecedent)
  3. <math>Q</math>&nbsp; (Conclusion deduced is the consequent)

In this form of deductive reasoning, the consequent (<math>Q</math>) obtains as the conclusion from the premises of a conditional statement (<math>P \rightarrow Q</math>) and its antecedent (<math>P</math>). However, the antecedent (<math>P</math>) cannot be similarly obtained as the conclusion from the premises of the conditional statement (<math>P \rightarrow Q</math>) and the consequent (<math>Q</math>). Such an argument commits the logical fallacy of affirming the consequent.

The following is an example of an argument using modus ponens:

  1. If it is raining, then there are clouds in the sky.
  2. It is raining.
  3. Thus, there are clouds in the sky.

Modus tollens

Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (formula) and the negation of the consequent (<math>\lnot Q</math>) and as conclusion the negation of the antecedent (<math>\lnot P</math>). In contrast to modus ponens, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following:

  1. <math>P \rightarrow Q</math>. (First premise is a conditional statement)
  2. <math>\lnot Q</math>. (Second premise is the negation of the consequent)
  3. <math>\lnot P</math>. (Conclusion deduced is the negation of the antecedent)

The following is an example of an argument using modus tollens:

  1. If it is raining, then there are clouds in the sky.
  2. There are no clouds in the sky.
  3. Thus, it is not raining.

Hypothetical syllogism

A hypothetical syllogism is an inference that takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:

  1. <math>P \rightarrow Q</math>
  2. <math>Q \rightarrow R</math>
  3. Therefore, <math>P \rightarrow R</math>.

In there being a subformula in common between the two premises that does not occur in the consequence, this resembles syllogisms in term logic, although it differs in that this subformula is a proposition whereas in Aristotelian logic, this common element is a term and not a proposition.

The following is an example of an argument using a hypothetical syllogism:

  1. If there had been a thunderstorm, it would have rained.
  2. If it had rained, things would have gotten wet.
  3. Thus, if there had been a thunderstorm, things would have gotten wet.

Fallacies

Various formal fallacies have been described. They are invalid forms of deductive reasoning. An additional aspect of them is that they appear to be valid on some occasions or on the first impression. They may thereby seduce people into accepting and committing them. One type of formal fallacy is affirming the consequent, as in "if John is a bachelor, then he is male; John is male; therefore, John is a bachelor". This is similar to the valid rule of inference named modus ponens, but the second premise and the conclusion are switched around, which is why it is invalid. A similar formal fallacy is denying the antecedent, as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore, Othello is not male". This is similar to the valid rule of inference called modus tollens, the difference being that the second premise and the conclusion are switched around. Other formal fallacies include affirming a disjunct, denying a conjunct, and the fallacy of the undistributed middle. All of them have in common that the truth of their premises does not ensure the truth of their conclusion. But it may still happen by coincidence that both the premises and the conclusion of formal fallacies are true. This issue belongs to the field of strategic rules: the question of which inferences need to be drawn to support one's conclusion. The distinction between definitory and strategic rules is not exclusive to logic: it is also found in various games. The hallmark of valid deductive inferences is that it is impossible for their premises to be true and their conclusion to be false. In this way, the premises provide the strongest possible support to their conclusion. However, in a more strict usage, inductive reasoning is just one form of ampliative reasoning. For abductive inferences, the premises support the conclusion because the conclusion is the best explanation of why the premises are true.

The support ampliative arguments provide for their conclusion comes in degrees: some ampliative arguments are stronger than others. Ampliative reasoning is very common in everyday discourse and the sciences.

An important drawback of deductive reasoning is that it does not lead to genuinely new information. It has been suggested that this problem can be solved by distinguishing between surface and depth information. On this view, deductive reasoning is uninformative on the depth level, in contrast to ampliative reasoning. But it may still be valuable on the surface level by presenting the information in the premises in a new and sometimes surprising way. On this view, deductive inferences start from general premises and draw particular conclusions, while inductive inferences start from particular premises and draw general conclusions. This idea is often motivated by seeing deduction and induction as two inverse processes that complement each other: deduction is top-down while induction is bottom-up. But this is a misconception that does not reflect how valid deduction is defined in the field of logic: a deduction is valid if it is impossible for its premises to be true while its conclusion is false, independent of whether the premises or the conclusion are particular or general. Because of this, some deductive inferences have a general conclusion and some also have particular premises. In a meta-analysis of 65 studies, for example, 97% of the subjects evaluated modus ponens inferences correctly, while the success rate for modus tollens was only 72%. On the other hand, even some fallacies like affirming the consequent or denying the antecedent were regarded as valid arguments by the majority of the subjects. In an often-cited experiment by Peter Wason, four cards are presented to the participant. In one case, the visible sides show the symbols D, K, 5, and 7 on the different cards. The participant is told that every card has a letter on one side and a number on the other side, and that "[e]very card which has a D on one side has a 5 on the other side". Their task is to identify which cards need to be turned around in order to confirm or refute this conditional claim. The correct answer, only given by about 10%, is the cards D and 7. Many select card 5 instead, even though the conditional claim does not involve any requirements on what symbols can be found on the opposite side of card 5. as in "If the card does not have an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card has an A on the left". The increased tendency to misjudge the validity of this type of argument is not present for positive material conditionals, as in "If the card has an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card does not have an A on the left". This is done by applying syntactic rules of inference in a way very similar to how systems of natural deduction transform their premises to arrive at a conclusion. But there are also alternative accounts that posit various different special-purpose reasoning mechanisms for different contents and contexts. In this sense, it has been claimed that humans possess a special mechanism for permissions and obligations, specifically for detecting cheating in social exchanges. This can be used to explain why humans are often more successful in drawing valid inferences if the contents involve human behavior in relation to social norms. But the subject of deductive reasoning is also pertinent to the computer sciences, for example, in the creation of artificial intelligence. Deductive inferences are able to transfer the justification of the premises onto the conclusion. Some theorists hold that the thinker has to have explicit awareness of the truth-preserving nature of the inference for the justification to be transferred from the premises to the conclusion. One consequence of such a view is that, for young children, this deductive transference does not take place since they lack this specific awareness.

History

Aristotle, a Greek philosopher, started documenting deductive reasoning in the 4th century BC. René Descartes, in his book Discourse on Method, refined the idea for the Scientific Revolution. Developing four rules to follow for proving an idea deductively, Descartes laid the foundation for the deductive portion of the scientific method. Descartes' background in geometry and mathematics influenced his ideas on the truth and reasoning, causing him to develop a system of general reasoning now used for most mathematical reasoning. Similar to postulates, Descartes believed that ideas could be self-evident and that reasoning alone must prove that observations are reliable. These ideas also lay the foundations for the ideas of rationalism.

Deductivism

Deductivism is a philosophical position that gives primacy to deductive reasoning or arguments over their non-deductive counterparts. This way, the rationality or correctness of the different forms of inductive reasoning is denied. Some forms of deductivism express this in terms of degrees of reasonableness or probability. Inductive inferences are usually seen as providing a certain degree of support for their conclusion: they make it more likely that their conclusion is true. Deductivism states that such inferences are not rational: the premises either ensure their conclusion, as in deductive reasoning, or they do not provide any support at all. For example, a chicken comes to expect, based on all its past experiences, that the person entering its coop is going to feed it, until one day the person "at last wrings its neck instead". According to Karl Popper's falsificationism, deductive reasoning alone is sufficient. This is due to its truth-preserving nature: a theory can be falsified if one of its deductive consequences is false. So while inductive reasoning does not offer positive evidence for a theory, the theory still remains a viable competitor until falsified by empirical observation. In this sense, deduction alone is sufficient for discriminating between competing hypotheses about what is the case.

Natural deduction

The term "natural deduction" refers to a class of proof systems based on self-evident rules of inference. The first systems of natural deduction were developed by Gerhard Gentzen and Stanislaw Jaskowski in the 1930s. The core motivation was to give a simple presentation of deductive reasoning that closely mirrors how reasoning actually takes place. In this sense, natural deduction stands in contrast to other less intuitive proof systems, such as Hilbert-style deductive systems, which employ axiom schemes to express logical truths. Natural deduction, on the other hand, avoids axioms schemes by including many different rules of inference that can be used to formulate proofs. These rules of inference express how logical constants behave. They are often divided into introduction rules and elimination rules. Introduction rules specify under which conditions a logical constant may be introduced into a new sentence of the proof. It was initially formulated by Baruch Spinoza and came to prominence in various rationalist philosophical systems in the modern era. It gets its name from the forms of mathematical demonstration found in traditional geometry, which are usually based on axioms, definitions, and inferred theorems. An important motivation of the geometrical method is to repudiate philosophical skepticism by grounding one's philosophical system on absolutely certain axioms. Deductive reasoning is central to this endeavor because of its necessarily truth-preserving nature. This way, the certainty initially invested only in the axioms is transferred to all parts of the philosophical system. A different criticism targets not the premises but the reasoning itself, which may at times implicitly assume premises that are themselves not self-evident.