# What is the truth value of a unevaluated truth functional?

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?

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.

• The conditional ("if this is a square, then it has 4 sides") is tautologically true -- the premise may not correspond to anything, but there is still a (positive) truth value of the consequence-relation itself Aug 28, 2017 at 1:15
• ^^ I can put this in an answer if helpful but wanted to sanity-check it's what you're actually curious about here Aug 28, 2017 at 1:17
• @JosephWeissman yes feel free to put it as an answer. I am also curious what the purpose of the sentence is as it just appears random to me (even if I understood it which I don't yet). Though, in my head it says "this", so its talking about a concrete shape, not just about the word "square", no? I think I see what you mean with what they tried to do but it seemed that their example failed cuz we need to know if "it" really is a square or not, no? Aug 28, 2017 at 1:24
• The conditional: "if this is a square, then it has four sides" it is not a (propositional) tautology. The issue is with the truth-functional definition of the conditional: the context is the assertion of the above sentence regarding "unseen geometric figure". Two cases: (i) if the figure is not a sqaure, then the antecedent is false, and thus the complete sentence is true. 1/2 Aug 30, 2017 at 11:30
• (ii) if the figure is a square, then by the same def of "square" (a specific kind of quadrilateral) we have that also the consequent is true, and again the complete sentence is true. 2/2 Aug 30, 2017 at 11:32

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".

``````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.

• is what you are saying that truth functionals are not exactly the same as boolean functions? Unevaluated boolean function are just that...unevaluated functions...I can't seem to be able to tell the difference. Aug 28, 2017 at 4:09
• As far as I'm aware, there's no real concept of "unevaluated" in truth-functional logic. I'm also not as confident as you are that there's always such a distinction in boolean functions. I'd assume there's such a distinction in some programming areas ... but I'd take "boolean" to refer more to a family of things than a single defined set. Aug 28, 2017 at 5:15
• In my model of truth functionals is that they are the same as boolean function. This model might be wrong because I am a computer scientist not a philosopher. However, for me its not so much the issue of boolean or not, but rather of functions itself. A truth functional f(A,B) seems to be a function that depends on two inputs A and B. It doesn't really matter if its a truth functionals or boolean but an evaluated function just means the function before its being used i.e. a symbolic description of f. Does that not exist in philosophy? Maybe functions are not used at all and thats my mistake. Aug 28, 2017 at 15:34
• In this case the function does not take or return a Boolean. The function "If(A, B)" takes two properties (functions of 'it' returning a Boolean -- Pythonishly something like "lambda it: it.sides() == 4", "lambda it: it in Squares" ) and returns a property (a function of 'it' returning a Boolean). Connectives like if, and, or, etc. when they act on statements with pronouns or variable references in them are acting on functions that return truth values, not on truth values themselves.
– user9166
Aug 28, 2017 at 17:20
• Otherwise what becomes of the unfilled 'slots' preventing evaluation? Most languages say 'Yup that function I cannot yet evaluate does exist -- TRUE!' which is not helpful. We want to avoid that, so 'free' variables have to be parameters to the property or sentence.
– user9166
Aug 28, 2017 at 17:21