Formal Logic

Some brief thoughts on why we need to know more than the basics

… what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence. – L. Wittgenstein, Tractatus Logico-Philosophicus

When we talk about logic in everyday discourse we often mean a mixture of formal and informal reasoning; that is, we mean both the analysis of forms of arguments that are considered to be valid (and sound), and those aspects of delivery and presentation that are persuasive and engaging. We might, for example, say something like ‘the logic for Brexit is flawed’ or ‘I’m not sure I follow the logic of the conclusion in your report’ without really distinguishing the validity of the arguments from the skilful rhetoric employed to express those ideas and arguments. However, providing others with valid reasoning and analysis is not the same thing as being a persuasive speaker or writer, as we know well from the world of politics, where our leaders get away with pronouncements that have no logical structure at all but appear to garner support from many followers nonetheless. So why should we be concerned about logic and sound reasoning when we are surrounded by examples of bad, but apparently persuasive, rhetoric?

However, I want to praise rhetoric not condemn it. Being able to craft and deliver language in ways that carry others with you is a skill. To be able to move large numbers is rarer. In fact, I do not believe we see good speakers and writers as often as we think. There is a considerable difference between presenting a lie or a distortion of the truth for short-term advantage (which is the trick of many politicians and press propagandists) and constructing an argument that works in a way that shifts our thinking, in and for itself. That is, there is a distinction between good rhetoric as the art of convincing through a well-crafted use of language and mere propaganda, where the latter is the use of misleading or out-and-out false information to manipulate opinion.

Once we accept that there are better and worse ways of presenting an argument we start to consider the first meaning of ‘logic’ we encountered; that is, a more formal study of how good arguments work and what we count as valid reasoning, aside from the mere presentational skill of the speaker. Moreover, it is not difficult to find reasons why some knowledge of this is useful in all walks of life. Examples of the fallacy of ‘affirming the consequent’ can be found in just about any newspaper you care to pick up, sometimes in quite subtle forms. [An unsubtle example: 1. If someone is the Prime Minister, then he or she visits the Queen once a week. 2. Prince William visits the Queen once a week. 3. Therefore, Prince William is the Prime Minister. Even if 1. and 2. are true, 3. does not follow.] There are plenty of other fallacies that we can learn to spot to ensure that we are not following flawed thinking in our dealings with others and to construct sound arguments for ourselves. Combined with learning argumentation theory more generally, especially in particular disciplines, such as within the law, we can significantly improve how we present arguments and information, and develop many essential skills for analysis in studying logic at this level.

Normally, this is where discussion of logic in professional life stops. However, formal logic, as most students of philosophy or computer science will know, goes much further. Formal logic abstracts away from natural language to explore the underlying structure of reasoning apart from the specific content, using the generation of artificial, logically simple languages that use a limited set of connections and symbols to express their grammar and rules. The question that is often asked is: of what benefit is this for professional life? Why would I need to know about this level of logic? This is what I want to answer here.

We say we live in a messy analogue world, where little is certain, and nothing is unchanging. Indeed we do, but we very rarely acknowledge that fact in the way we actually live our lives. (The closest I ever come to accepting this myself is when I run or meditate.) We impose theories, structures, models and meanings on the infinite complexity of reality. Some of this might be capturing and ‘carving reality at  its joints’ through metaphysics, mathematics or physics, but most of the systems we use for thinking are abstractions away from reality in an attempt to control or predict, in a limited way, some part of our world in (semi-)isolation from the rest, ignoring details and factors deemed irrelevant. This might be a manual of how a car engine works, a system to monitor profits from seasonal products in a supermarket, a prediction of the next general election, a theory of bee behaviour in hot climates, a set of machine instructions for an email router, the modelling of a corporation’s share value for the next two years, the rules of chess, an analysis of communication skills in the under-fives, an unconscious hypothesis of emotion formation in others, or the rendering of images of tower blocks in a flight simulator. So rather than in that messy analogue world, we live in a vast array of models and ideas that each programs the meanings we encounter and largely shapes our expectations and evaluations. Some of these models are well-formed and formal, most are ad hoc and informally structured, and the vast majority we are not even aware of. However, they are not only the basis of our tools for working and thinking but the basis of our thinking processes themselves. (I know I am asking the reader to accept a significant amount without demonstration or argument, but the theories of Daniel Dennett go some way to filling in some of the gaps on this last point.)

It seems to me that when we come to examine them, each of these systems and models has its own limited vocabulary that can be combined and manipulated according to the rules of the model. A computer programme can be a significant and creative construction, but it has a specific syntax and set of symbols it uses. Similarly, a financial model can be quite restricted in the mathematical components at its core. In other fields, looser theories (such as the one I am constructing here) have a much greater dependency on natural language, and yet they are all attempts to describe only a limited part of reality using a limited set of the tools available to us as human beings. So, initially, we can arrange and rank all our (conscious) models from the most formal to the least. Moreover, if we want to know how they actually work, at the deepest level, we need to understand the scope of what can be said, and what can be proved and shown in each model and theory because this defines their usefulness and application. Formal logic is the science of these limits and scope, the science of meaning, truth and rules. Logicians study how the rules within formal models can be shown to be consistent and how they could be interpreted as true or false. Students are usually exposed to two related logical languages with limited syntax and semantics: propositional and predicate symbolic logic, each of which demonstrates the greatest degree of logical coherence. For these systems of logic, it can be shown that they possess:

  • Validity: a false conclusion can never be correctly inferred from true premises.
  • Consistency: no theorem (a true statement derived from the rules of the language itself) contradicts another.
  • Completeness: if a statement is true it can be proved from the rules of the language or system.
  • Soundness: if a statement can be proved then it is true (cf. completeness).

These rules represent the gold standard of logical reasoning. The handful of languages normally studied in formal logic provide the rock-bottom test cases for understanding the properties of languages: meaning, interpretation, truth and reasoning. They are the exemplars of how syntactical and semantic rules actually work. As we move into more complex systems, such as arithmetic, not all these properties hold or require higher-level systems for us to apply them. However, the point of showing that they do not apply is as important for understanding the nature of those systems as showing that they do. My claim is that truly understanding these properties of formal languages provides the basis for thinking about any model or theory in any other language, be they in logic, mathematics or natural language. Exploring the relationships between the formal and the more informal understanding of reasoning we typically use is, strictly speaking, within the domain of philosophical logic. However, developing an awareness that any system, model or theory we use in work is indeed a system for thinking, that it can be analysed, that it could be made more logical or consistent, and that there are deep and powerful tools for doing this analysis, takes our reasoning to the next level. If necessary, when combined with empirical investigation, an understanding of formal logic can provide a valuable check on approaches to problems that rely on unfounded assumptions and unworkable modelling.

In our professional lives, we are surrounded by and endlessly use and manipulate information and systems that are based on formal logic. The worlds we occupy are structured by theories, models and systems that are more or less logically coherent. If our aim is to make the world a better place, then starting with an understanding of logic beyond the simple analysis of arguments is a necessity.

To find out more about learning aspects of formal logic for professional and business purposes, please do get in touch; I am always happy to help.

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