What is the truth value of a unevaluated truth functional?

Question

I was trying to understand the truth functional definition of an implication "if A then B"/"if A, B". I was reading Stanford encyclopedia of philosophy article on indicative conditional and in the 3rd paragraph it seems they try to definition what an implication is. They discuss it in that paragraph but then they have the following statement (which I don't understand):

"If it's a square, it has four sides", said of an unseen geometric figure, is true, whether the figure is a square, a rectangle or a triangle.

what I don't understand is why the previous statement would be true (or actually take any boolean/logical value at all) if we don't actually have any figure to consider. For me it seems we are trying to define the boolean function f(A,B) = " if A, B" but in the context of the quote they give they essentially make A(x) a function of the shape and then ask me to consider what the truth value of g(x) = f(A(x),B(x)) would be. It seems really silly because we have not actually given the input x to A (or B in fact because "it" refers to the geometrical shape in question). Thus, for me regardless of what definition of implications I am trying to derive, I would leave that specific example as neither True or False, because we have not given the arguments to the function to be evaluated at all!

Am I right? Or am I missing something very simple? What is the purpose of that sentence in that article. It seems to be more confusing (and distracting) than helpful in convey whatever point they are trying to convey. How does it help them justify the definition of implication? Whats the use of that example in that paragraph?

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After reading a response I got on reddit to this it made me consider a part of a sentence in that paragraph that might be relevant to understand the whole better:

its is uncontroversial that "if A, B" is sometimes true when A and B are respectively (true,true), or (false,True), or (false,false).

It seems to me that that it is very controversial depending on what you want "if A then B" to mean (i.e. we are coming up with its truth funtional definition). In the section of the article we are deciding what an implication should mean. A quick view in the dictionary says:

1. the conclusion that can be drawn from something, although it is not explicitly stated.
2. a likely consequence of something.

So it would make sense to bring up examples that are relevant to that definition (otherwise why use "if A, B" to define something if its not going to mean sort of what it usually means). So it seems to me that its suppose to bring an uncontroversial example of why implications can be True when (true,true), or (false,True), or (false,false). If that is the case, can someone explain me how this example exemplifies this? At this point of the text we have not defined what an implication means and the example should be self evident, so I'd love to know why its self evident.

Answer 1

You seem to be struggling with truth-functional if. this is a pretty common problem and there's quite a few questions on the SE about it. It's also call "material implication" if that helps with your search (See

What's an intuition for material implication?

Material conditional: Why does the absence of the predicate validate the conditional?

Why does the material conditional have the truth table it does?

).

First, one key is to not get confused as to what we are doing with truth-functional if. Do not start with your intuitions about the word "if".

Instead, start with the following definition. The truth-functional definition of "if" is this:

A   B   A --> B
T   T      T
T   F      F
F   T      T
F   F      T

anything that uses the truth-functional definition of "if" follows this pattern.

In other words, whenever people use "if" in this way, they mean the above chart.

There are lots of uses of the word "if" in English (and cognates in other languages) that are not truth-functional. It'd be hard to catalogue all of them, but there's often two majors differences between language-if and truth-functional-if:

1. Some language-if rejects saying A -> B is true when A is false.
2. Some language-if rejects A -> B is true when B is true and A is false.

You seem to raise some issues with that.

The second issue seems to be a confusion about how first order logic works. There's no "evaluation" step in truth-functional logic. The entire proof is considered to happen instanter such that every step occurs in logical but not temporal sequence. Consequently, there's no waiting to evaluate even if there's pesky filling in. This is a feature (or maybe bug?) of basic truth-functional logic.

There's contemporary work being done to work with multi-valued logics or logics that can accept fractional values etc. but all of that comes after understanding the basics.