# Why is notation in logic so different from algebra?

Introductory courses in philosophical logic that I've seen introduce specialised notation like:

For an outsider, this is highly confusing. I'm sure that everyone studying philosophy has been to high school and has seen algebraic problems like this:

or, using just symbols and no English words:

By analogy, the above syllogism could be equally well written as:

This notation is marginally longer, but it seems to me at least equally expressive and in line with the notation everyone is perfectly familiar with. So how did the above specialised notation emerge? To my understanding, some (many?) of the pioneers of formal logic have been mathematicians and certainly well-versed in ordinary algebra. Why would they need to invent a new notation, so different from the already established one?

Update:

An advantage of explicitly stating the truth value of a sentence is that it allows us to explicitly encode valid, but unsound arguments. Even more, we can use a placeholder for the truth value, like `x`, and investigate under which circumstances the argument is sound, or whether it is a tautology etc. Of course, this is trivial for a two-line argument with only one variable, but may become interesting as the number of statements and variables multiply.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Sep 6 at 15:37
• Very good. There is a lot of misuse of jargon within the field that seems to have no purpose but obfuscation. Commented Sep 6 at 18:48

The simple answer is that logic has different ideas and needs than mathematics, although, many modern thinkers work with notations of mathematical logic which use both. Symbolic mathematical notation goes back quite a long way, but modern symbolic logic arguably starts with Gotlob Frege and his Begriffsschrift in 1879. Here the original notation was rather distinct from modern notation. Russell and Whitehead's Principia Mathematica also introduced some new conventions in 1910. New symbols are created to differentiate new concepts from old ones. Thus, logicians originally came up with symbols that didn't look like math symbols because they wanted to show that math ultimately bottomed out in logic (in a programme called logicism). Having the symbols be similar or nearly identical would have made that goal more difficult since symbols are meant to simplify and clarify.

You are conflating statements and objects.

In your algebra example, you have two statements (`2x + y = 5` and `x = 3`) and from these you derive a third statement (`y = -1`). Working informally, `x` is an object and `3` is an object. The sentence `x = 3` is a statement. In set theory, objects are sets, and statements are logical propositions (equality in set theory is a meta-level equality). In type theory, objects are elements of types, and statements are elements of a different type. In category theory, objects are objects, and statements are morphisms, which can be objects also but in a different category. So statements and objects are distinct, or at the very least live in different levels (called universes).

In your logic example, you have two statements (`A -> B` and `A`) and derive a third (`B`). These are all statements - they are all logical propositions. There is no point writing `A = T` for two reasons. Firstly, your logic may not contain equality, so no such symbol exists. Secondly, claiming `A` is already saying "A is true". Writing `= T` is redundant. The equivalent of

`A -> B = T, A = T therefore B = T`

would be

`2x + y == 5 = T, x == 3 = T, therefore y == -1 = T`

where I have used `==` to mean "object equality" or "equality of integers", and I have used `=` to mean "equality of propositions".

So you see, in your logic example, no `= T` is needed, because they are already propositions, and by default are asserting their own truth.

As for your comment that this is highly confusing, I have yet to meet somebody who claims so. For a start, we do not use the horizontal bar (inference rules) in algebra. I have never seen this notation outside a logical or computer science context. Inference rules are also supposed to represent the axioms of our logical system, i.e. they are single steps. Your algebraic example is certainly not a single step, whereas the logical example is (it is Modus Ponens). You are correct that adding `= T` is equally expressive. It is also a colossal waste of paper and time as it is redundant and, as you have already said, equally expressive.

New notation was invented to talk about new things. You are comparing apples to oranges.

• Yup; `A→B` is not an equation. `A→B` is closer to being a function… or, at least, one cell from a lookup table for that function. It tells you that if you put `A` in as a value, you will get `B` out as a result. So, basically, `A→B` means `f(A)=B`. Just like how a Trig table will tell you `Sin(π/6)=1/2`. You don't then rewrite that as `Sin(π/6)=1/2=T`… Don't conflate things just because they look vaguely similar (if you squint a bit) Commented Sep 4 at 23:04
• @Chronocidal I don't see how `A+B` is any less a function than `A→B`: "In mathematics, an operation is a function which takes zero or more input values (...) to a well-defined output value. (...) Operations can involve mathematical objects other than numbers. The logical values true and false can be combined using logic operations, such as and, or, and not." en.wikipedia.org/wiki/Operation_(mathematics) Commented Sep 5 at 9:21
• @IgorF. `A→B` is Function, Input and Output. `A+B` is only Function and Input; that is to say, `A+B` can be written as `f(A, B)`, which lacks an Output. Adding the `=C` on the end as an Output gives `A+B=C` or `f(A, B)=C` . Whereas `A→B` becomes `f(A)=B`, which means that it already contains all three Objects, and so is already complete. Unlike `A+B`. Commented Sep 5 at 10:42
• @Chronocidal `A -> B` is NOT a function. The arrow symbol `->` is a meta-level operator; it is part of the language of first-order logic. The addition symbol is not (at least, not in first-order logic - it is, for example, in the logic of groups). You cannot "put in A as a value" or "get B out as a result". Its only meaning is that given by the rules, i.e. how it can be formed and how it can be eliminated. Commented Sep 5 at 23:11
• @JKusin "Every set has a statement about it". Every set has infinitely many statements about it, that's trivial and neither here nor there. "There's a set where x=3" - what does this even mean? Commented Sep 5 at 23:12

The expression A → B means that A → B is true.

The expression 2x + Y = 5 also means that 2x + Y = 5 is true.

If you write A → B = T instead of A → B to mean that A → B is true, then you should write 2x + Y = 5 = T instead of 2x + Y = 5 to mean that 2x + Y = 5 is true.

Humans are smarter than they realise and natural language is more difficult to improve than may seem.

The equals relationship goes two ways.

1+1 = 2 means that you can exchange 2 for 1+1, and 1+1 for 2.

The implication relationship does not go two ways.

P |= Q means, if you have P then you have Q. But it doesn't mean that if you have Q then you have P (that'd be the biconditional P=|=Q). Intuitively, if I have a cat than I have a mammal, but my owning a mammalian pet doesn't guarantee that it's a cat (maybe it's a dog or a badger or ...)

There are lots of other discontinuties between the stuff logic talks about, and the equals relation in arithmetic. I hope that this little example helps give a sense that these are really quite different conceptual worlds and that different notation is justified.