# Identifying propositions?

I have been asked to identify whether the following sentence is a proposition or not: (in accordance with this definition)

"Tomorrow is Monday."

For any given day of the week, this sentence will either be true or false, but it most definitely can never be both. Hence, I concluded that it is a proposition but my answer is incorrect.

What is the flaw in my logic? What is the best way to get my head around this and solve these types of questions with ease? (Please keep in mind that I only just graduated high school.)

Thanks ever so much! :) Regards.

Edit 1: After giving the matter some more thought, I have identified the flaw in my logic, which is that the sentence may assume a positive(true)/negative(false) value depending upon the day on which it is told. Hence, it does not qualify as a proposition, correct?

Edit 2: Alas! I have come across yet another definition of the term "proposition", which doesn't match the way that I think about it. Even though my book states that the sentence " Tomorrow is Monday" is not a proposition, this definition seems to contradict this. Please help!

• :-) this is what happens when you try to make complex things simpler: you cause approximation and error. As you already noticed there are countless definitions: the most solid ones unfortunately require a theoretical background which you weren't provided with. For your exam I can only suggest that you take your book's definition and only when in doubt use en.wikipedia.org/wiki/Sentence_(logic) which will probably require a little reasearch on your side in exchange for a deeper view on the topic. Sep 5, 2016 at 14:22
• Yes, I realize this in vain :( I will. Thanks for your help :)
– user361896
Sep 5, 2016 at 14:29
• @Lorenzo: P.S: Congratulations on being able to comment now!
– user361896
Sep 5, 2016 at 14:35
• @Mauro ALLEGRANZA: I'm not sure if you will be able to look it up. It's one of them study packages that I have received from a coaching institute here, in India.
– user361896
Sep 5, 2016 at 14:36
• One fortunate thing: once you move into studying mathematics, many of these issues disappear, because of the way mathematical terminology is set up. For example, statements about mathematical objects usually don't refer to the past or future. Some of the generalities you encounter at the beginning of a logic course are important in principle but rarely come up in actual mathematical logic. Sep 5, 2016 at 17:53

The usual idea of mathematical statement is that of a statement that does not change its truth value with time; e.g.

$2 + 2 = 4$.

The statement:

"Tomorrow is Monday."

is not true today, because tomorrow will be Tuesday.

• Yes, OK, thanks! ^.^
– user361896
Sep 5, 2016 at 11:20

Actually most natural language "statements" are logically intractable since models are actually "cultures". Furthermore most logical or mathematical statements require a model or axiomatic interpretation to gain "meaning" and thus a truth value. Finally we need a language definition for any proposition (in logic or math or anywhere) to even be considered a tautology or a contradiction (which by definition is a statement whose truth is model-indipendent)

So, your issue doesn't depend on the fact that the proposition's truth value may change in different "models": that is part of the game (think of a function assuming different values given different domains: similarly a statement is a truth meta-function).

Your issue depends on the fact that you can have people in different time-zones such that the same interpretation of the same statament yields different truth values.

I doubt these sort of "tricks" are of any pedagogic use since the level of assumptions required to solve your exercise is even bigger than what we just did (can we attribute other's people statements a truth value without knowing the model they use? do we use classical physics as a model of the world where our proposition is interpreted? Is 00:00am today or tomorrow? etc... )

But anyway... sooner or later in your logic course - depending on its scope - you'll be presented with the concepts of language/denotation/interpretation/model.

Then you'll understand why some propositions are meaningful even if they include variables and why we want to distinguish them from most people's mumbling which has no truth value even when only using constants.

@Mauro Allegranza Can't comment yet so I'm commenting here: whatever a mathematical statement is, I'm sure it's quite problematic to invoke the concept of time for its definition and I doubt it is used in the "usual" definition.

• Wow, your answer is very detailed. In truth, I have only just graduated high school and as of now, am not taking advanced courses in logic. I am preparing for a competitive examination and so, encountered this doubt. I did manage to understand most of your premise but unfortunately, at my level, I have been told that sentences involving time variables such as "today" and "tomorrow" aren't propositions because they may assume different truth values under different circumstances. Same is the case for sentences containing words such as "here" and "there".
– user361896
Sep 5, 2016 at 13:47
• I have edited my question. Please check.
– user361896
Sep 5, 2016 at 13:54

Leaving the full sentence alone for a moment, the word "tomorrow" only refers to a particular day once supplied with a context (the day the word is spoken or considered).

So, if you assume that some context will be supplied at the time that "tomorrow is Monday" will be evaluated for truth, then in that context it will be either true or false. Then you'd say it's a proposition.

If you don't assume that some context will be available, then the phrase cannot even in principle be evaluated for truth. Then you'd refuse to accept it as a proposition.

As to how this would arise in practice, suppose you're given some mess of statements like "Tomorrow is Monday. Yesterday I saw a bus. All buses are red. I haven't seen anything red in 8 days", and asked to make some kind of sense out of them. If we assume that all the statements are evaluated on the same day then this seems contradictory, and on the face of it I appear to be lying to you (I mean, for starters, it's Monday today as I write this). But what if a week elapsed between the first statement and the last? Then I could have been telling the truth each time. I can say "Tomorrow is Monday. Tomorrow is Tuesday" and both statements be true, if the full stop between them happens exactly at midnight. The context completely changes how we evaluate them, and whether we can evaluate them at all.

So, whether we allow "Tomorrow is Monday" as a proposition depends on the context in which we're going to evaluate it (how are we going to account for changing time?), not on the simple definition of "a proposition". If we can make it into something that's either true or false then we can use it as a proposition. If we're stuck saying, "well, it depends when you say it" then perhaps we can't (or perhaps we can, but it remains a variable in whatever logical operations we do to it).

Many real-world statements involving "is" are liable to change, even if they don't explicitly mention time, but usually we reason about them anyway, having implicitly made some assumption that we're reasoning about a particular fixed moment in time where our assumptions are true. So we allow things like "the sky is blue", "I have three donuts" or "I think therefore I am", ignoring that none of these will be true forever. At any given time either it's true or it's not, and usually that's good enough.

The good news is that once this introductory stuff is out of the way, it won't actually matter whether "Tomorrow is Monday" is a proposition or not, because that's not the kind of statement you'll be reasoning about. Mathematics is not the study of what day it is ;-) We're on much safer ground with "0 is an integer" or "$f$ is a continuous function".

Tbh I think the question is unfair: you shouldn't be asked to guess for yourself whether "Tomorrow is Monday" will or won't be an acceptable proposition, without knowing what you're going to do with it.

At this introductory level the important point should be that a proposition can be considered either true or false for the purposes of the logic you're about to do. Whether it's "actually" true or false in all contexts goes way beyond simple propositional logic. Saying that because it's time- or reference-dependent it's not a proposition is a sort of "gotcha!" that the textbook might regret if the same standard were applied to its other examples of "real-world" propositions!

Another important observation to learn from is that you've got two sources who've defined "proposition" to their own satisfaction in pretty much exactly the same words, but they've made different unspoken assumptions that lead to different decisions whether "Tomorrow is Monday" is a proposition or not. The lessons to learn are:

• definitions can be insufficient to cover all cases, and it's better to write definitions that don't allow any such differences of interpretation. When you encounter unspoken assumptions, speak them, exactly as you did in saying, "depending upon the day on which it is told".
• people can disagree about definitions, and that's fine as long as you don't mix up conclusions reached using one version of a definition with conclusions reached using a different version. Where possible stick with the definitions and examples in your books, but bear in mind any variant definitions you encounter in other sources. Never assume that just because the same word is used for something in different sources, it must refer to the same definition!
• Thanks! My textbook also lists another example: "x+y≥0". The answer that they have provided is "It is not a preposition, since this can be either true or false, depending upon the values of x and y." What do you make of this?
– user361896
Sep 6, 2016 at 0:08
• @KaumudiHarikumar: given that example it's clear that they aren't allowing variables in propositions, so work with that definition. In another context you could quite happily use propositional logic to analyse the syllogism: from $x > 0$ and $x > 0 \rightarrow y \leq 0$ deduce $y \leq 0$. But your book doesn't want to do that. Sep 6, 2016 at 7:45
• Okay. Thanks :)
– user361896
Sep 6, 2016 at 9:52

I don't know anything about the definition you refer to. Usually in philosophy a proposition is the factual content of a declarative sentence.

Your sentence is "Tomorrow is Monday" and it has a factual content. Its factual content is the respective proposition.

The same proposition can be expressed using a different sentence, "Tomorrow is the day after Sunday". Hence, the same proposition might be expressed by different sentences.

• Well, the definition in my book is "A proposition is a sentence that is either true or false but cannot be both" and in a subsequent example problem, it has mentioned that "Tomorrow is Monday" is not a proposition, since it can assume different truth values on different days.
– user361896
Sep 5, 2016 at 14:32
• Obviously your textbook conflates sentences which are defined syntactically in a language with propositions which pertain to semantics and have to do with meaning. However, for the sake of the discussion let's accept what your textbook says. In this case, any sentence - proposition except from logical truths and contradictions can be either true or false but not both. It depends on the intepretation of the language. Consider, for instance, the sentence "x is Monday". Its truth-value is eiter T or F but not both. It depends on how you understand x (interpretation). Sep 5, 2016 at 15:15
• ... The only sentences that are necessarily F are of the type "Sunday is Monday" and those which are necessarily T (independently of the interpretation ) is "Monday is Monday". Sep 5, 2016 at 15:17
• Exactly! Can you list any example of a sentence that is both, true AND false at the same time? What does it mean to "not be both"?
– user361896
Sep 6, 2016 at 0:09
• I am not sure that there are such sentencs. In the definition, the author just uses exclusive disjunction (either, or, not both) and that's all. What, of course, one might have in a language is uninterpreted sentences or in the context of a three-valued logic sentences taking a third truth-value. Sentences stating facts about incompatible (non-commuting) observables are examples of the latter case. For instance for any interpretation which makes true the sentence "the momentum of the electron is 3 units", the sentence "the position of the electron is 4 units" is neither tru nor false. Sep 6, 2016 at 5:50

The definition of proposition is that proposition is a declarative statement which can be determined either true or false, but not both.

Your statement is equivalent to this: $$1 \equiv x+1 \pmod 7,\quad x \in \mathbb{Z}$$ Its truth value depends on the value of $x$. It is a predicate, not a proposition.

Today is Sunday is not a proposition as it can have both the possibilities True and False. On Sunday the statement will be true and on Other days, it will be false. So you can not say if statement is true or false. Therefore It is not proposition.

For better understanding, Compare it with 2+2= 4 which is always true and 2+2=5 which is always false