Math and logic symbols

Propositional logic, or 'sentence logic' in clearer terms, is the most gentle of the forms of logic that you can learn. It also forms the basis for every other type of classical logic, whether that's predicate logic, set theory, lambda functions, or all of standard mathematics. In short, if you want to start anywhere, start with sentence logic.

There are several mathematical conventions I wish I'd known when I first began to seriously study sentence logic. I'll give the most important of those conventions here.

Mathematical conventions (or logical ones — they really stem from the same source) are things that we've all agreed to adhere to. However a convention doesn't necessarily have to be adhered to. It just makes mathematical and logical communication easier if you do.

For example, though classical logic has no causality (see below), there are in fact other forms of logic that disregard one or more of these conventions (e.g., fuzzy logic, causal logic, intuitionistic logic, modal logic, and so on).

However if you want to learn the kind of logic that gets computers to run, gets rockets to re-land, and matches flawlessly with physics, then classical logic is the way to go. And learning sentence logic first will give you a sense of how it all hangs together: predicate logic is really just sentence logic with a few extras.

Paul Teller's Book

This last fact has been made explicit in Paul Teller's amazing Modern Formal Logic Primer (see Volume 2, pp. 20--21, available here for quick reference).

This book, which the author has been kind enough to make available for free, is one of the best books on the topic I have ever read. If you prefer a hardcopy, you can very probably still get the out-of-print title at Amazon.com.

If you're just starting, read through the first three chapters of Volume 1. Do all the exercises. Then read through this webpage again, and finally read the third chapter again. My hope is that doing this will give you a solid basis for sentence logic. You can extend this basis well into predicate logic, set theory, and discrete (and even continuous) mathematics.

And always remember that nothing ever happens over night.

The Mathematical/Logical Conventions, for Any Classical System

Here they are, the more important mathematical conventions implicit in the study of logic:

  1. Truth Values. Every sentence letter is semantically either true or false, not both ('mutual exclusivity'), and not neither ('exhaustivity' — all parts of the system, without exclusion, must have a truth value. That is, if and only if ('iff') you include semantics, which are truth values. Logic consists of syntax — the sentence letters — and semantics — the truth values. It is possible to do just syntax without semantics. One of the (many) systems for this is Natural Deduction).

    • If the sentence letter is both true and false, you have a contradiction.

    • If the sentence letter is neither, you are no longer doing logic. To be within the system of logic, it must be either true (T, \(\top\)), false (F, \(\bot\)), or both (a contradiction).

  2. Law of the Excluded Middle. There are no shades or grades of truth and falsity in classical logic. Either the sentence letter is completely true, or the sentence letter is completely false.

    • Classical logic looks at all of the possibilities (i.e., T and F) for each and every sentence letter. It isn't about whether or not the moon is really truly made of cheese. Rather, classical logic considers both cases, one where it is absolutely true that the moon is made of cheese, and one where that proposition is completely false. The truth table reflects the fact that logic represents all possible cases.
  3. Form, Not Content. Logic is based on the form of an argument: what its truth-value is under all truth values possible for each sentence letter.
    • Logic does not deal with an argument's content: is the statement really true? Logic does not care about this. It only gives (absolute, categorical) truth-values based on every possible circumstance: if the statement is T, this will be the result (based on the other statements being T or F). If the statement is false, then that will be the result.
      • The actual truth of sentences in an argument is known as the argument's soundness. Though there is a word for an argument which is composed of empirically true sentences, soundness is beyond logic. Whenever you start thinking that a logical argument's sentences are really true or really false, you're no longer doing logic. To do logic, you must consider both the case where each sentence is true, and the case where each sentence is false. In other words, you must consider all truth values for all sentences.
  4. Universality Only. 2 is 2 is 2, regardless of your cultural background. Similarly, the language of logic is entirely composed of true universals. Natural languages like English or Mandarin have an infinity of nuances that get lost in translation. But logic is only about universals: when you make a statement in the formal system of logic, the background of the reader doesn't matter. It means 1 and only 1 thing.
  5. No Physical Space. Pure logic and mathematics are abstract; that is, they exist only in the mind, with no manifestation in the physical world.
    • Therefore in any mathematical/logical system, it does not matter how you arrange symbols on the page. If a formula is written left to right, top to bottom, big or small font, etc., this will not have an effect on how that formula is to be interpreted. Again, this is because logic (and mathematics) is only in thought, not in space or place.
  6. No Time. Nothing can be said to occur before or after anything else in logic or mathematics. For example, if I travel from London to Montreal, as far as logic is concerned I am not in London before I am in Montreal. This is again because logic/mathematics is based on thought (form), and not content (actual things occurring in time).
  7. No Causality. Nothing occurs be-cause of anything else as far as logic and mathematics are concerned. This is especially important with the implication connective '\(\longrightarrow\)'. For example, if I am bleeding because a dog bit me, and I want to convert this statement into logical symbols, the logical symbols are no longer bound by one event happening because of another.

There are other, less important conventions, but these are the ones which cause the most trouble for beginners. Especially because except for the Law of the Excluded Middle, these conventions aren't mentioned anywhere that I have seen.

The Logical Conditional: If-Then ('\(\longrightarrow\)')

The logical if-then is not the causal if-then of natural languages like Japanese or Swahili. We tend to think 'if event A happens, then event B will happen because of event A. If event A did not occur, event B would not have occurred either', and this is reflected in all the natural languages on this planet.

But this is not what is meant by the logical 'if-then'.

In logic, the following statements are logical truths: 'If my grandmother owns a set of wingèd snakes, then 2+2=4'. This is true under all circumstances if we go by standard conventions in the mathematics (e.g., '+' means addition, base 10 digits, etc.). Remember that for the final truth value of the if-then connective ('\(\longrightarrow\)'):

  1. All that matters are the truth values of its inputs, and not whatever those sentences mean in natural language.
  2. Even if we were to treat the meaning of natural languages, like Russian or English, as somehow relevant (several logics do, though not classical logic), remember that logic has no causality.

Given two inputs A and B (A can be any sentence whatever, as can B), the logical connective '\(\longrightarrow\)' (the logical if-then) is completely defined as follows:

\(A\) \(B\) \(A \longrightarrow B\)
\(\top\) \(\top\) \(\top\)
\(\top\) \(\bot\) \(\bot\)
\(\bot\) \(\top\) \(\top\)
\(\bot\) \(\bot\) \(\top\)

In the above table, \(\top\) = 'true' and \(\bot\) = 'false'. I wrote 'true' and 'false' in this way to reinforce that any symbols can stand for truth and falsity: {1, 0}, {⋆, ♪}, {♀, ♂}, etc. Furthermore, although these two symbols are in 1:1 correspondence with actual truth and actual falsity, in themselves, the symbols mean absolutely nothing.

{\(\top\), \(\bot\)} are conventional, as are {0,1}, {F, T}, and {t, f}. The only reason we think of them as being true and false is because of our natural language giving meaning to everything.

\(A\) is known as the Antecedent or Hypothesis. \(B\) is known as the Consequent or Conclusion. (I much prefer referring to them as 'Hypothesis \(\longrightarrow\) Conclusion', rather than 'Antecedent \(\longrightarrow\) Consequent', especially because 'consequent' in English very strongly implies causality, which classical logic does not have).

If you notice, any time the hypothesis \(A\) is False, the conclusion is True. And if the conclusion \(B\) is True, then no matter what the \(A\)ntecedent is, the entire if-then statement is true. This latter fact is the reason for `If my grandmother owns a set of wingèd snakes, then 2+2=4' being a logical truth. The equals sign of '2+2=4' is constantly true: it is logically true. Thus what we're really saying is: `If my grandmother owns a set of wingèd snakes, then \(\top\)'. More abstractly: \(A \longrightarrow \top\), and looking at the table, anything \(\longrightarrow \top\) is a logical truth.

One last thing: \(A \longrightarrow B\) is a statement, the main connective of which is implication ('\(\longrightarrow\)'). Whenever you see \(A \longrightarrow B\), if you have an additional \(A\) on its own line in natural deduction, that means you can immediately substitute the \(A\) with a \(B\). The other way around — having a \(B\) and substituting with \(A\) — is not allowed. Officially, it's a fallacy: 'affirming the consequent', which means treating the consequent, the conclusion of your argument, as though it were the hypothesis — the points that you must make to get to that conclusion.

The Logical Bi-Conditional: If and Only If ('Iff', '\(\longleftrightarrow\)')

The biconditional ('\(\longleftrightarrow\)') is much more what we mean by 'if-then' in natural language. That's just a coincidence: the truth values of the biconditional work out so that if A is true, B is also true; if A is false, B is also false. However, remember that classical logic (and by extension mathematics, set theory, lambda notation, etc.) has no causality at all. It just happens to be that the definition of the biconditional operator is on a 1:1 correspondence with causality. But the biconditional operator itself belongs to a system without causality.

Here is the table defining the biconditional:

\(A\) \(B\) \(A \longleftrightarrow B\)
\(\top\) \(\top\) \(\top\)
\(\top\) \(\bot\) \(\bot\)
\(\bot\) \(\top\) \(\bot\)
\(\bot\) \(\bot\) \(\top\)

A Final Word

I don't think there is any better book that has been published than Paul Teller's to ease you in to a study of logic. Universities I'm familiar with usually use either Copi (Copi, I., Cohen, C., and McMahon, K. (2010). Introduction to logic. Routledge, NY, 14th edition), or The Logic Book (Bergmann, M., Moor, J., and Nelson, J. (2013), McGraw-Hill Education, Columbus, OH, 6th edition).

In my opinion, they're missing out, if for only this reason: the language used in both the texts just mentioned is academic, formal, dry, at times even pompous. This is no way to speak to anyone learning logic for the first time. Teller's book, on the other hand, is like being with him one-on-one. It has of course its flaws (all things do), but within its pages, there is no sense that logic is impenetrable, or knowledge that you either get or you don't. He shows that logic is something innate in being human. And you may as well try Teller's book: it's free.