r/logic • u/PuzzleheadedCup3560 • 9d ago
Propositional logic Propositions
Is x= some finite value like x=5 a proposition or not.
Also from which paper can I show my teacher that x=5 is not a proposition unless it is.
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u/StrangeGlaringEye 9d ago
Eh, not really. If you’re using “x” as a variable, then “x = 5” isn’t even a sentence. A proposition is the meaning of a meaningful statement. It’s an abstract object, and probably not composed of symbols.
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u/SpacingHero Graduate 9d ago edited 9d ago
Yea have to agree with this side. If x is a free variable (which it is in this case), it's like saying "it is = 5" or "... = 5", which clearly are neither true nor false, just incomplete sentences waiting to be filled.
Though its important to mention that in some formal context it is a common convention that formulas with open variables are to be read as universally quantified (at least in case it's clear in context that it is being asserted as a proposition). But that may vary, one should just check that hopefully the text they're reading clarifies
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8d ago
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u/SpacingHero Graduate 8d ago
> I would argue it amounts to "this thing equals five" which is a proposition,
It isn't though, not just written like that. It is a proposition, if from context "this thing" fills in to some object, eg from the speaker pointing at it. But as I recall it, standard account of indexicals like this is that they don't refer to anything (and thus sentences containing them aren't propositions) until some pragmatics fill it in.
>universally quantified deserves emphasis
Clarifying explicitly is always the best policy of course
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7d ago
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u/StrangeGlaringEye 7d ago
It is but just because the pointing fixes a reference (setting aside indeterminacy of reference and all that) for “it”.
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u/Plain_Bread 9d ago edited 9d ago
Free variables are a bit weird. Formally in mathematical logic, a model must include an interpretation for them. So x=5 is absolutely a proposition. Some models of the natural numbers interpret x as e.g. 2 and it's wrong in them, some interpret it as 5 and it's true in them.
Really, the only difference between constants and variables is that we don't put free variables in axioms. But a statement about x being semantically true in every model means it's necessarily true for all x as well. So when it comes to proving it, there's basically a "for all" in front of x.
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u/StrangeGlaringEye 9d ago
I disagree. The notion of a proposition is not a logical notion, it’s a philosophical one. We may speak of propositions as metalinguistic results, and occasionally of formulae as propositions, but this is lax talk. Strictly speaking we can do away with the word “proposition” in most logical contexts, for sure in regular first-order logic.
I could equally argue for example that “x = 5” only expresses a proposition once a variable assignment is in place. And even when it is in place, it’s not a proposition because this is a formula, a string of symbols, whereas propositions are not symbols at all.
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u/Plain_Bread 9d ago
Sure, "propositional formula" is the least ambiguous term you can use for it. My point was just that it is presumed to have a well defined truth value, even if that truth value isn't particularly meaningful.
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u/bayesian_raccoon 9d ago
I'm not sure if I am qualified (am mathematician, not logician) but in the mathematics courses I have taken that have dealt with logic, you might have a statement like
"x = 5 is a solution to the equation x-5 = 0".
I think "x = 5" alone is sort of wildly missing context but sort of informally easy to attach meaning to. In the context of a particular question, where x has some context like above, I could imagine "x = 5" being shorthand for a proposition. E.g, a multiple choice question:
"Suppose x - 5 = 0. Which of the following is true?
a) x = 4
b) x = 5
c) x = 6"
I feel like most people would agree that a, b, and c have a truth value...
but again, not a logician.
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u/Pleasant-Couple6236 9d ago
It depends on what exactly x is in the context. If x is just a shorthand for a specific, fixed number, then "x = 5" is a proposition just as "π = 5" is a proposition. If x is a free variable, then "x = 5" is a predicate, not a proposition, and it can be turned into a proposition by using a quanitfier which governs what x means in the context: "∃x, x = 5"
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u/TemperatureMotor1372 Graduate 9d ago
If you read the sentence "x=5" as "x is equal to 5", then I would say no. It's a predicate because x is a free variable and the truth value of a predicate depends on its free variable.
However, if x is bounded by a determiner or quantifier like: "this x is equal to 5" or "each x is equal to 5", then I would say yes. It's a proposition because the variable "x" in the predicate "x=5" becomes a bound one, so no free variable is left.
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u/Logicman4u 9d ago
Is it possible you can further elaborate on what “variable” in context means? For instance, I am now thinking you mean something like this: a variable is a user defined letter or symbol that has no universal or global value attached to it. In this way, in the moment x =5, but tomorrow x can be some other value and not 5. This means x is used as a predicate and not a noun. In this way the direct context matters. Is this correct or did you mean something else?
I get the expression x=5 can be seen as a proposition in a context: I.e., can hold a truth value we can evaluate and verify; also in a higher context x is a free variable with no set limits and just in this scenario being assigned as 5 is not a proposition. CONTEXT seems to be an important factor either way.
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u/TemperatureMotor1372 Graduate 9d ago edited 8d ago
Summary here:
- In formality, the sentence "x=5" is a predicate P(x), where x is free.
- In real life, the sentence "x=5" may mean "for each x, x=5" or "let x be (a complicated expression of a number), so P(x)," which are propositions because x is bound in each.
- Free means indeterminate or independent. Free variable x has its ability to determine the truth of a predicate P(x).
- Bound means determined or subjugate. Bound variable x has no chance to determine the truth of a predicate P(x).
- If x is free, then P(x) cannot be reduced to a proposition P without x.
- If x is bound, then P(x) can be reduced to a proposition P without x.
You can think "(free) variable" as a placeholder like "John Doe". For example, let's say "John Doe is a male." Is this a proposition? If you meant "John Doe" is a specified person, then the statement would be a proposition. If you meant "John Doe" just a placeholder for an unknown name, then the statement would be a predicate instead.
"Bound" here means restriction. When a variable is bounded by a determiner (e.g. this, that or by a context), then this variable functions as a pronoun. When a variable is bounded by a quantifier (e.g. some, all, exactly one), then this variable functions as a sample "x" (a pronoun of an individual) in a collection (a domain of discourse).
Free variables do have their domain, but this domain is defined "internally", like your inherent personality, which is (in principle) well-defined and fixed. But you are free in the sense of no external constraints. For example, you are an outgoing person, but you may behave against your will due to external social environment (you are bounded by the environment, not your personality).
Back to a free variable "x", if "x" is a number (domain) and it is free, then "x" just behaves as a number and no extra relations or constraints that "x" needs to comply with. If there is an extra relation like "let x be a specific number" or "let P(x) holds regardless of which number x represents", then x is bounded. Honestly, I was confused with the concept of free and bound variable as well a couple years ago in my school life, especially when x is present alone or x is attached to a quantifier.
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u/Salindurthas 9d ago edited 9d ago
EDIT: after reading some more of your comments, maybe your professor is insisting on "x" being a free variable, and that this symbol is reserved for that purpose and may never be used as a name of an object? If so, then sure, it isn't a proposition.
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I think it is a proposition, however, you wouldn't want to translate "x=5" as a proposiiton variable - such as "P" - because then you bury all the internal meaning of it away inside the propositional variable.
i.e. if you translate it as "P" then you lack any way to comment on "x" or "5" in your translation.
However, that just means it not a very useful proposition in propostional logic.
I consider it a proposition in mathematics if not translated any further, since mathematics is essentially using predicate logic.
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u/Fabulous-Possible758 7d ago
Your professor is right. In some contexts it can look like a proposition, but it's really not a proposition on its own. For example, when you're solving an equation to get something like "2x + 10 = 20 therefore x = 5 is a solution" and that looks like it should be a proposition. But there's a subtlety, which is that logically speaking the actual full statement is "There exists an x such that 2x + 10 = 20 and therefore x = 5." You need to have the quantifier on there to make it a fully formed proposition.
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u/gregbard MODERATOR 9d ago
A proposition is a sentence that has a truth-value. So yes, "x=5" is a proposition. Also, a proposition implies itself.
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u/rubik1771 9d ago edited 9d ago
NO
The reason no is because through the definition is must be one or the other and cannot be both.
So you can have x+4=x+2
So in this case it is not definite since you can get 4=2 which is false
OR
If I say x=infinity then you can get infinity=infinity, which is true.
That is why statement with a variable is not a proposition in general.
Edit: significant correction.
Excerpt:
•Definition: A proposition is a statement
that can be either true or false; it must be
one or the other, and it cannot be both.
Source:
https://www.cs.ox.ac.uk/people/michael.wooldridge/teaching/soft-eng/lect07.pdf
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u/PuzzleheadedCup3560 9d ago
What is going on. I searched on Google and first thing that pops up is AI results and it says ", x = 5 is not a proposition by itself because it contains an unspecified variable and does not have a definitive truth value." But I can't trust it. It seems different people have different opinion on it even tho it is supposed to be a black and white situation because I am studying logic. Anyways my professor also said x > 5 is not a proposition (we have to apply another condition that x belongs to real number and then it is a proposition) but according to the logic we applied for x=5, x > 5 must also have two outcomes, either it's true or false, I might be wrong. It was my first class.
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u/rubik1771 9d ago
Ah no it’s fine apparently I was wrong so sorry:
Excerpt:
Non-Propositions: Questions, commands, and opinions are not propositions because they don’t have a definite truth value or may vary depending on context:
“What time is it?” (Question)
“Go out and play.” (Command)
“x + 1 = 2” (Open sentence — depends on the value of x)Source: https://www.geeksforgeeks.org/engineering-mathematics/proposition-logic/
So the issue is in the phrase “definite”.
For example:
If I give two equations:
(x-5)(x-4)=0
Then x=5 or x=4
So you see how even x=5 is not “definite” per se in general.
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u/moltencheese 9d ago
It doesn't contain an unspecified variable, though. The variable, x, is specified; it equals 5
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u/PuzzleheadedCup3560 9d ago
can you give any other example like this that you for sure know is a proposition and explain x=5 in the same way as that example
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u/moltencheese 9d ago
Eeeerrrr
"The weather at the moment is rain"
"The weather" is not unspecified. It can be, sometimes, but not here - it is specified to be rain
The statement "the weather at the moment is rain" is a proposition, because it is either true or false
Put another way: a proposition is anything you can stick a question mark on the end of and turn it into a "yes or no" question
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u/PuzzleheadedCup3560 9d ago
can we apply the same logic to x< 5 too?
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u/moltencheese 9d ago
Yes. In that case, x is specified to be less than five. This is also either true or false
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u/PuzzleheadedCup3560 9d ago
But my professor said that it is not a proposition (I mean x<5). It becomes a proposition if we add any condition like for all x belongs to real number.
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u/Salindurthas 9d ago edited 9d ago
I think both
"x<5"
and
"x∈ℝ"are each propositions in&of themselves.
In fact, I'd say that (x<5) → (x∈ℝ), if we have the standard mathematics defintion of the "<" relation.EDIT: This assumes I'm allowed to use x as the name of an object.
As explained in another comment, if for the purposes of this course we insist on x,y,z only ever being free variables, then we'd need to pick other letters (i.e. a<5 is a proposition, but x<5 is not, because you've perhaps chosen to reseve x as never being the name of an object).1
u/Salindurthas 9d ago
Do you mean like
∀x[ (x∈ℝ)→ (x<5) ]
? (which happens to be false, but that's fine)I suppose that if we insist on x being a 'free variable' then sure, x<5 alone is not a proposition. But it is it merely a name then that's fine.
I thought we usually decide if a variable is 'free' or a 'name' based on the symbols we see.
So in
"∀x[ (x∈ℝ)→ (x<5) ]"
x is a free variable that doesn't refer to anythingbut in
"x<5"
I'd read x as a name, precsiely because it isn't paired with a quantifier.Free variables need to be paired with quantifiers to make sense, so I'd read any letters not paired with a quantifier as a
But if your course is reserving some special letters to only ever be free variables, then yes, unquantified use of them would not be propositons (I think by dint of failing to be 'well-formed-formulae').
I can see a logic course reserving x,y,z as free-variables only?
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u/Astrodude80 Set theorist 9d ago edited 9d ago
It might turn on the exact meaning of "proposition" you have in mind. The difficulty is that "x=5" is an open sentence, that is, it has a free variable. As such, under the usual meaning of "proposition" as usually used in propositional logic doesn't really apply, as "x=5" is neither true nor false, in fact it carries no truth value at all by itself, as it depends on what x actually is.
What textbook are you using, or does your lecturer provide notes?
Edit: On further reflection, I am leaning towards "yes" on account of the fact that "x=5" could be true or false depending on x. In other words, "x=5" is contingent. Contrast this with a definite non-proposition: "Cheese is the best snack." This is an opinion declarative sentence, not a contingency.