The Foundations of Logical Reasoning: Structures, Fallacies, and Sound Reasoning

Plato and Aristotle in Raphael's fresco 'The School of Athens' (1509-1511)

Logic is one of, if not the single most, high-order principles when forming arguments. Sound logical structures are invaluable, whether in essay writing, argumentation, all the way through to self-assessment. Logic is the only way by which one can safely dissect and dissipate another’s argument or even come to its defence. Logic is necessary for mathematics, the sciences, philosophy and legal argumentation. I suppose each reader has of themselves a particularly kind opinion regarding their own capacity to form logical arguments; this may be so, but a focus on the purest logical structures will always improve the persuasiveness of any argument being put forward.

Logic at its simplest level is the study of reasoning, or, more specifically, correct reasoning. Logic is sub-divided into formal and informal logic. Formal logic can be defined as the study of deducing logical truths from premises; informal logic is the study of, amongst other things, informal fallacies within reasoning and critical thinking. Both are important to consider when constructing an argument, but it is formal logic which is indispensable.

Formal logic has its basis in premises and conclusions. A premise is, as it were, the building block of an argument. Premises are typically statements, such as: “all swans are white” or “cognac is smoother than grain whiskey.” A conclusion is that which can be validly deduced or induced from its respective premise, or premises. Both premises and conclusions must be what are termed ‘truth bearers’, that is, they are either true or false. Putting these simple elements together, we can form the basic most argument structure, which is known as modus ponens (MP), where we say: (P1) A, (P2) if A, then B, (C1) therefore B. This argument structure is always valid, that is, the premises lead to the conclusion without a gap in reasoning. That is not to say the argument is sound, where the argument is both valid (as above), but the premises are also true, and therefore the conclusion must be true, as the validity guarantees the conclusion follows from the true premises. A valid argument can look as follows: (P1) All swans are white, (P2) I own a swan, (C1) therefore my swan is white. It is perfectly valid; if it were true that all swans were white, that my swan is white would follow. However, (P1) is not true, and thus the argument is not sound. A sound argument can look as follows: (P1) all authentic champagne must come from the Champagne region of France, (P2) I have authentic champagne, (C1) therefore, my champagne comes from the Champagne region of France. Given both (P1) and (P2) are true, and they follow the MP logical structure, the conclusion (C1) is logically guaranteed to be true.

Like the MP structure, is the modus tollens (MT). It can be thought of loosely as an inverse MP, but not quite. The structure flows as follows: (P1) if A then B, (P2) not B, (C1) therefore not A. As with modus ponens, this structure is also subject to the classification of valid and sound status. A sound MT argument could run as follows: (P1) if all horses have four legs, then my horse will have four legs, (P2) my horse does not have four legs, (C1) therefore not all horses have four legs. This relies on a general truth, that where a statement is true, then its contrapositive is also true. However, where a statement is not true, this can be shown in reverse. Accordingly, the “target” of a MT argument is the if statement within the premises, as (P1) was above. As the reader may have gathered, this structure is particularly useful in undermining the premises within an argument and is something to consider when testing both the validity and soundness of one’s own arguments. To do so, one needs to first define the relevant premise which is suspected of being false, then apply the above structure. This is easier said than done but is the same operation we are doing when we ask “well, why exactly do you believe that?”, or more crudely “How exactly have you come to that?”

Taking the above a step further and closer to the argumentation structures necessary for more complex subject matter, we consider the use of sub-conclusions. Consider again the MP structure. We had P1, P2 and terminated at C1. However, with a sound argument, C1 can be used as premise for a second argument, whether that is of the form MP or MT. When we do so, C1 becomes known as a sub-conclusion. There are two ways in which this sub-conclusion can be used at the simplest level. Either, C1 directly feeds into the subsequent argument as one of its premises, or, where there have been two distinct but related arguments which have generated two separate conclusions, these two conclusions, if sound, can be used to form a subsequent MP or MT argument, with C1 from argument 1 (Ag1) and C1 from argument 2 (Ag2) to form the following: (P1) C1, (P2) C2, (C2) therefore Z, where Z is the final conclusion of this dual-layered logical structure, or the ultimate conclusion. These are clearer within examples. An example of where a sound conclusion feeds directly into a subsequent argument can run as follows: Ag1 (P1) all mammals are warm-blooded, (P2) all dogs are mammals, (C1) therefore, all dogs are warm-blooded. Here, C1 is sound, and can be fed directly into a second argument: Ag2 (P1) All dogs are warm-blooded, (P2) my pet, Leo, is a dog, (C2) therefore, Leo is warm-blooded. An example of two sound conclusions being used to form a separate argument can run as follows: Ag1 (P1) all fruits contain natural sugars, (P2) an apple is a fruit, (C1) therefore, an apple contains natural sugars. Ag2 (P1) anything that contains natural sugars can ferment under the right conditions, (P2) an apple contains natural sugars, (C2) therefore, an apple can ferment under the right conditions. Here, C1 from Ag1 and C2 from Ag2 can be combined into a subsequent argument: Ag3 (P1) An apple contains natural sugars (C1), (P2) An apple can ferment under the right conditions (C2), (C3) therefore, apples are suitable for making cider.

The above can seem abstract but writing which is attempting to lead the reader to a conclusion, which fails to follow clearly linked premises through to a conclusion, will often read disjointed. See the following, which has the loose, intuitive ‘supporting your argument’ force behind it, but fails to fully adhere to a formally logical structure. For example: “I believe that everyone should exercise regularly because it keeps you healthy. My friend runs every morning, and he never seems to get sick. Plus, people who exercise tend to be happier, and happiness is important. So really, exercising is the best way to live a good life.” At first glance, the argument feels persuasive; it gives reasons, examples, and a positive conclusion. Yet, if we try to trace the logic formally, the premises don’t strictly guarantee the conclusion. Not everyone who exercises is immune to illness, and being happy is not logically sufficient to prove that exercise alone ensures a good life. The reasoning is intuitive and supportive, but the formal links are missing which makes it susceptible to counterargument and premise undermining, as we just did above.

Outside of adherence to the formal structures above, or the plentiful other examples thereof, there are also fallacies which must be avoided within reasoning, so as to ensure sound, logical arguments. Fallacies can be divided into formal and informal fallacies. Formal fallacies are errors in the logical structure of an argument itself, where the conclusion does not follow from the premises even if the premises are true. Informal fallacies are errors in reasoning that arise from the content, context, or language of the argument, rather than its formal structure. Fallacies are abundant, and cannot possibly all be considered here, but a few of the most prolific are: ad hominem, strawman, slippery slope, begging the question, appeal to authority and appeal to emotion.

The ad hominem (Latin, meaning ‘to the person’), is an informal fallacy whereby the speaker aims their statements at the individual they are speaking to, instead of the logic of said person’s reasoning for whatever they have stated. More colloquially, this arises where someone turns to insulting someone (which can be indirect), as either an effort to dismiss that person’s position, or worse, to influence the perception of that person’s capacity to reason. For example, suppose someone had just made the same argument we did above about the fermentation of apples, to which the person they are speaking to said: “I don’t really think you know what you are on about, you are not a chemist are you? What would you know about fermentation and science?” At first glance, someone attending this now heated discussion may think that is fair enough; he is not a scientist. But on considering the matter, it is clear the comment did absolutely nothing to the argument. The speaker could have tried proving that apples do not have sugar, or that not everything (an apple being one of them) which has natural sugars will ferment. However neatly the language used to commit this fallacy may be constructed, it is always and everywhere a fallacy.

The strawman fallacy arises where an individual levels a counterargument against a weakened caricature of an opponent’s argument, hence the name strawman. This is a fallacy given that, as the argument being challenged is not really what the opponent has said, the damage done to it by the strawman attack is nil. This is an easy fallacy to fall into, especially with complex subject matter, which can be difficult to understand, or which has contextual variables that someone has not considered. For example, geo-political issues are almost inherently context-based, and some individuals, such as policy makers or world leaders, have access to far more relevant information regarding the actual circumstances, so from the outside, it is hard to construct a true counterargument without risking straw manning their opponent. For example, if one politician says, “We need to increase funding for education to improve literacy rates,” and an opponent responds, “My rival just wants to throw money at schools without thinking about results,” the response misrepresents the original argument. The first politician never claimed that money alone would solve the problem; the counterargument attacks a weakened version, creating a strawman. The same fallacy can also arise innocently, negligently, or even recklessly within academic work, as individuals, faced with a particularly complex subject matter, fail to sufficiently understand a particular writer’s reasoning, condense it down in an inappropriate manner, then level some criticism at the confused strawman argument, which no one had made.

A slippery slope fallacy arises where someone claims that, if X, then Y, without a formal connection between the two. The argument runs more on intuition and can resemble a formally logical structure but lacks the validity which gives it strength. For example, someone could say: “If we allow drinking at 16, it won’t be long before they are all doing drugs!” This may or may not be true, but the statement itself cannot get you there, yet had it been structured formally, the conclusion could have been stronger and clearer. Supposing the data were true, then the argument could have been reworded as follows: (P1) 89% of those who drink alcohol under the age of 18 end up trying, or using, illegal drugs, (P2) where something has an 89% chance of occurrence, we can safely say that it is very likely, (C1) therefore, if we allow 16-year-olds to drink alcohol, it is very likely they will try or use illegal drugs. Slippery slopes arise frequently when an individual has in their mind an intuitive causal link between two events but is failing to explain clearly how one leads to the other.

Begging the question arises where a premise itself has not been established as true prior to starting the relevant argument. Suppose we looked again at the apple fermentation argument from above, but it flowed as follows: Ag1 (P1) anything that contains natural sugars can ferment under the right conditions, (P2) an apple contains natural sugars, (C1) therefore, an apple can ferment under the right conditions. C1 is then fed into a second argument, as follows: Ag2 (P1) An apple contains natural sugars (C1), (P2) an apple can ferment under the right conditions (C2), (C3) therefore, apples are suitable for making cider. However, as this nexus of reasoning relies on a premise that has not been independently established as true, P1 from Ag2 is unsupported and begs the question. This fallacy is also prevalent within academics, especially younger students, who either innocently presume that a given statement is taken for granted as true by all, or less innocently look to slip in a premise they cannot themselves support with an argument but intuitively believe is true.

The appeal to authority fallacy arises where someone uses an individual’s credentials, or some equivalent “authority” they have concerning a certain matter, as a premise in arguing that the person is correct. A loose inverse of this also arises at times when someone makes an ad hominem argument concerning someone’s credentials, as we saw in the scientist example above. An appeal to authority can be used either with respect to oneself or someone else. For example, someone could say: “I am a solicitor, so of course I would know. Under English Law the death penalty is still allowed when someone has committed multiple murders.” This is, of course, false. With respect to someone else, the argument could look like: “My Dad is a surgeon, and he says that people only have one big nerve that runs down their spine, which branches off to the rest of the body.” This too is false, but as with the death penalty argument, you can see how the authority and credentials of the individual are being used as an implicit premise within the argument. This fallacy, however, must be considered with a touch of nuance; certain individuals are better placed or informed regarding certain subject matter, which can lend them a certain level of deference when an individual who does not understand the subject matter is discussing something with them. Ultimately, everyone trusts others to an extent on this basis, no matter how critical we may believe our thought is. For example, we trust a neurosurgeon is correct when he says they need to make an incision in the frontal lobe to remove a tumour, and implicitly, this is at least partly on the strength of both his qualifications and track record of success.

Lastly, we consider the appeal to emotion fallacy. This fallacy arises when someone attempts to persuade an audience by manipulating their feelings rather than providing logically sound premises. The argument relies on eliciting fear, pity, anger, or any other strong emotion to make the conclusion seem correct, rather than demonstrating it through evidence or reasoning. For example, a speaker might say, “If we do not donate to this charity, countless innocent children will suffer and die,” implying that failure to act is morally reprehensible. While the situation may be tragic, the argument does not actually establish that donating is the most effective or logically justified course of action; it merely leverages emotional responses to push the audience toward a conclusion. Appeals to emotion are particularly common in political speeches, advertising, and fundraising campaigns, where the goal is to persuade quickly and viscerally. Like other fallacies, they can feel compelling and intuitive, but they bypass the formal logical connections necessary for a sound argument.

Fallacies, as noted, are plentiful and these are only a few. At their core, they are a break in the formal logical structure all sound arguments must follow. Adherence then, to clear, cogent and formally structured logical arguments, in the absence of any formal or informal fallacies, will make for the most convincing and sensible arguments.

To illustrate the practical utility of these logical tools, consider the following example. Suppose a debater at a genteel university society argues: “Napoleon’s retreat from Moscow in 1812 was entirely due to his own incompetence; if he had simply planned better, he would have won the campaign.” At first, this might sound persuasive, but by applying the tools of formal logic, one can dissect the argument. The key premise is: “Napoleon’s failure was solely due to poor planning.” A counterargument using modus tollens could proceed: (P1) If Napoleon’s failure was solely due to poor planning, then external factors, such as the Russian winter, scorched-earth tactics, and supply shortages, would be irrelevant. (P2) The Russian winter, scorched-earth strategy, and logistical difficulties clearly had a decisive impact. (C1) Therefore, Napoleon’s failure was not solely due to poor planning. By identifying the flawed premise and showing that the conclusion does not logically follow without considering other established facts, the speaker’s assertion is undermined.

A more nuanced example arises when an argument contains hidden assumptions within one or more premises, or when multiple conclusions feed into a final claim. Suppose a student claims: “Britain won the Battle of Trafalgar because Admiral Nelson’s strategy was brilliant, and therefore British naval supremacy was guaranteed for decades.” This is a dual-layered argument. The first layer is: (P1) Nelson’s strategy at Trafalgar was brilliant, (P2) brilliant strategy leads to victory, (C1) therefore, Britain won the battle. The second layer builds on the first: (P1) Britain won the battle (C1), (P2) winning this battle guarantees naval supremacy for decades, (C2) therefore, Britain enjoyed long-term naval dominance. The hidden assumption is buried in P2 of the second argument - “winning this battle guarantees naval supremacy” - which is not established and is historically contestable. By examining each premise critically, one can see that while the argument feels persuasive, the ultimate conclusion does not necessarily follow, illustrating how layered reasoning can conceal unverified assumptions and why logical scrutiny is essential for both writers and critics.

Hopefully by this point, the reader will understand the importance of adhering to a strictly logical structure within argumentation. It ensures clarity, cogency, sound reasoning and defence against any potential counterargument. It portrays the writer as an individual intimately familiar with the subject matter and, without it, as someone just dipping their toes into it. Logical arguments are more convincing, which is invaluable for individuals seeking to convey an idea to their reader. A barrister must make logical submissions within court; else they risk a counterargument from opposing counsel undermining their entire case. Medical professionals must make logically sound conclusions regarding diagnoses; else they risk either performing the wrong procedure or prescribing the wrong medication. Students ought to employ as strict a logic as they can within their written assessments, else they risk being marked down for lack of clarity or supported arguments. Academics must consider the internal logic of their own work, to ensure both that it is immune from undermining by a counterargument, as well as devoid of any fallacies which erode credibility. Politicians and media sources must ensure their publicised voices are heard making only logical statements, to avoid either misleading or mischaracterising individuals or events. Ultimately, logical reasoning is a discipline necessary for all who speak and write, as without it, the words can mislead, confuse, or fail to persuade, no matter how compelling they may seem on the surface.