Assuming Bob is a fairly rational person. If this is not the case, then is there a way to modify it? Also, is this the argument that Frege is making in "On Sense and Reference" that "the morning star" can't mean the same thing as "the evening star"?

  • 1
    It's not quite an answer, but when do two phrases ever mean exactly the same thing?
    – Cort Ammon
    Dec 23, 2014 at 4:52
  • under the Deflationary Theory of Truth. But if you like, we can get rid of "mean" and say "is equivalent to" in the sense that P if and only if Q. Dec 23, 2014 at 5:15
  • 2
    Frege's argument relies on the fact that "the morning star" and "the evening star" have the same reference (Bedeutung) but if we are not aware of this "contingent" fact, we cannot "equate" the tqo descriptions, and thus they have not the same sense. Meaning (for Frege) is "made of" both. Dec 23, 2014 at 7:16
  • Also, one thing to be careful about is distinguishing between actual beliefs as neurological concepts and the statements that express those beliefs. For Bob, P may mean the same as Q, but for Jack, who interprets language slightly differently due to linguistic ambiguity and uncertainty, P could mean something different from Q. Therefore, in no circumstance could you ever have two distinct statements that universally mean the same thing.
    – abhishek
    Dec 24, 2014 at 4:58
  • yes of course if he's rational, though if you want a strict proof you would have to operationalize 'rational'
    – user6917
    Jan 23, 2015 at 3:30

8 Answers 8


Your specific wording suggests the statement must be true, but a reworded version of the question may provide sufficient linguistic ambiguity to permit it to be false.

In our exchange in the comments, you mention that this question stems from the Deflationary Theory of Truth. If I may show my ignorance and quote wikipedia:

In philosophy and logic, a deflationary theory of truth is one of a family of theories that all have in common the claim that assertions that predicate truth of a statement do not attribute a property called truth to such a statement.

I will repeat your question here, just to put all of the necessary phrases in one place:

Assuming P means the same as Q and Bob believes P and is aware that P means the same as Q, can we conclude he believes Q?

Intuitively, it makes sense that we cannot conclude he believes in Q. Language has been evolving for a long time, and it would be risky to simply assume there is waste to be shaved off from it. While the "day-star" vs "evening-star" argument provides one argument, a common phrasing in religion provides a counterpoint:

Person 1: "Do you believe in God?" (P)

Person 2: "Yes"

Person 1: "Do you truly believe in God?" (Q -- alternatively worded "Is it true that you believe in God?")

Person 2 answers

That dialogue would be nonsensical or irrational under the argument that P and Q say the same thing.

The reason this is rational is because our knowledge is imperfect. We, on a regular basis, make decisions on things we think are true. However, we may not be comfortable with the transitive closure of those things we think are true; this is demonstrated in the classical style of debate which seeks to cause cognitive dissonance in the student in order for them to identify a "better" truth.

The logic of this position can be exemplified with Bayesian Inference. In Baysean Inference, each statement is in the form of a probabilistic statement. The truth value of P and Q may not be known a-priori, but "priors" can be used to estimate the distribution P and Q are drawn from.

There reaches a point where it is no longer reasonable to keep track of the statistics because the wording is no longer sufficiently terse to be optimal. At this point language will begin throwing around the word "true" in a sense which is less than the sense of "truth" used in Deflationary Theory of Truth.

Keeping track of these lesser flavors of "truth" is the key to my counter argument: your wording changes in the middle of the question to favor Deflationary thinking

Assuming P means the same as Q and Bob believes P and is aware that P means the same as Q, can we conclude he believes Q?

For P and Q, the phrasing is "believes." However, he "is aware" that P means the same as Q. I believe this wording choice is what forces Deflationary thinking, because it implies he is certain P and Q mean the same thing, but he merely believes P. In order to support Baysean style thinking, it must be changed:

Assuming P means the same as Q and Bob believes P and believes that P means the same as Q, can we conclude he believes Q?

Now we can see that the path from P to Q can be phrased as a Baysean inference. In this case, the connection between P and Q is imperfect, based on his statistical beliefs regarding that relationship. Those imperfect statistical beliefs can be sufficient to allow "Bob believes P" and "Bob believes Q" to have different truth values. Baysean inference would say they have different likelihoods of being true. A more frequentist approach would suggest that there is a cutoff for believability that P achieves but Q does not.

This argument depended on rewording your question. If, in fact, "P means the same as Q" and "Bob is aware that P means the same as Q" are logical truths, then logically Bob must believe Q iff he believes P. There are predicates for which "Bob is aware" is possible. However, many rational philosophers argue that the questions of value are all such that we cannot "be aware" of their sameness. Thus, for this class of "interesting" questions your statement is true merely by trivialization: "If Bob is not aware of anything regarding this interesting class of questions, then the statement is true regardless of Bob's believe in P and Q because of the rules of logic." If the question is reworded to work around this triviality by changing "awareness" to "belief," then Baysean Inference shows a rational reason Bob may "believe P" and not "believe Q."

As an addendum, with respect to your comment:

You asked about a slightly different question, which pre-supposes the Deflationary Theory of Truth to hold. This is a difficult question to answer, difficult enough to be worth editing it into the answer instead of a comment, so that I may correct it if I get the wording slightly wrong. It is difficult because I have brought up an example with Person 1 and Person 2 regarding belief in God which I believe would be considered a rational line of questioning by a reasonable portion of the populace, and I'd like to avoid needing to define "fairly rational," which is used in your question. I find defining most of humanity as irrational is... uncomfortable.

The question is regarding language semantics and syntax, so my instinct is to turn to Model Theory and Proof Theory. Model theory is a strange little discipline designed to associate semantic truth to statements. Its dual is Proof Theory, which seeks to define syntactic manipulations which do not modify the truth of a statement. To pre-suppose Deflationary Theory of Truth is to admit a particular proof calculi, while the question is clearly directed at the semantic truth of the statement; this is much better approached in Model Theory. To be honest in my answer, I must phrase it in a format which may appear frustratingly circular.

If "Cort" believes that there exists a proof calculi which admits a series of operations on a model which can prove the truth value of your question to be "true," such as the Deflationary Theory of Truth, and "Cort" is aware that "there exists a proof calculi and a model" is the same as "for all proof calculi and all models", then you may conclude that 'Cort' believes you may reach the aforementioned conclusion rationally.

However, I would challenge that my awareness of the inherent sameness of "there exists" and "for all" is very limited and should not be relied upon in any rational discussion.

  • As a caveat: not everyone considers Bayesian Inference to be rational at all. Some consider it "questionable" while others call it "heretical witchcraft" :)
    – Cort Ammon
    Dec 23, 2014 at 15:53
  • OK, to address your Person 1 and Person 2 scenario: let's say truly believing in God means the same thing as believing in God, and further that Person 1 and Person 2 believe that truly believing in God means the same thing as believing in God - then I'd be inclined to say the dialogue wouldn't even occur. Yes, I see I have worded it to favor the deflationary thoeory of truth. If I modify my question and suppose that the deflationary theory of truth holds, then is my statement about belief preservation correct? Dec 24, 2014 at 0:00
  • @MathTeacher: I bring up that scenario because it is an example which occurs in real-life conversation between persons who we like to believe are "rational." This scenario simply makes me uncomfortable in accepting your initial proposition wholesale. For the full answer to your second question, where you assume the Deflationary Theory of Truth holds, I will have to direct you to an edit I made to address it. It was really too large of a response to fit in a comment. What you seek is the link between semantics and syntax, which is something we have sought a long time as humans.
    – Cort Ammon
    Dec 24, 2014 at 4:02

That's an interesting question, and I believe the answer lies on what you define as "has the same meaning as". Let's accept Russell's way, and define the meaning of some statement, take "Socrates is human", as his proposition. Then "Socrates is human" has the same meaning (i.e, is the same proposition) as "Sócrates é um homem", although they're different statements.

Now let's define "A believes p", for a proposition p, as the propositional function enter image description here. For the individual Bob, we have "Bob believes p". This is NOT a truth-function of p, since the truth-value of f does not depend on the value of p. I.e, p maybe true and yet Bob may not believe it. So, enter image description here, whatever value this matrix may have. The " ^ " symbol marks the terms in which variation is desired.

From what was said about propositions, we have evidence for assuming the relation "has the same meaning as" as being equivalent to "has the same truth-value for every occurrence in logical matrixes of whatever order". I.e, we can't say that "Socrates is human" is the same as "Sócrates é homem", but, since they're the same propositions, every function in which the statement "Socrates is human" occurs will have the same truth-value (if the function is truth-function) as "Sócrates é homem". So, the condition we are looking for is this enter image description here. I.e, this relation is read "p has the same meaning as q".

Since you assume that Bob believes p, then enter image description here will be a particular truth-function of p. Since p has the same meaning as q, then enter image description here. So, Bob will believe it.

The whole point is that, if p and q have the "same meaning", then there's no distinction at all between then (in semantical terms of course). So, to suppose that Bob wouldn't believe q would be the same as suppose I don't have my second leg, having it at the same time. Regarding Frege, I'm not really sure if that's what he meant. Frege is difficult, and not a very good place to start on subject. Even Russell, who is very didactic, has some cumbrous text: Stephen Neale, a specialist on Russell's Theory of Descriptions, said that the worst thing a professor can do is give Russell's text "On Denoting" to his linguistic students. I recommend the following texts:

From G.E.Moore:


http://www.hist-analytic.com/ramseymoore2.pdf (with Ramsey)

From Russell:



https://archive.org/details/PrincipiaMathematicaVolumeI (speacilly the preface to the second edition).

Have a nice day.


What does it mean to believe P?

P is a sequence of words. I read the words, and according to the words, my knowledge, my prejudices, and various thought processes, I decide to believe P or not.

If I am aware that P means the same as Q, I should logically believe Q as well if I believe P (or I should stop believing P if I don't want to believe Q). But that's what I should do if I was a perfectly rational person.

It is quite possible that because Q is a different sequence of words, I will decide not to believe Q when I believed P. I am not saying that this is very rational, but it is quite possible.

So for the question asked: We can not assume that Bob believes Q. We might assume that Bob should believe Q, but we cannot assume that he does.


The answer is a conditional YES. We can conclude that Bob believes Q, if the following conditions are met: 1) P and Q must be absolutely equivalent, 2) Bob must understand the "equivalence" of P and Q, and 3) Bob must be "perfectly" rational. If any of these conditions is not met, the answer is NO.

My example would be:

1) Bob believes that "predestined events do not happen." (P)
2) Bob is aware that "predestined events do not happen" (P) is equivalent to believing in "free will" (Q). Therefore, we can conclude that Bob believes in "free will" (Q).


You can simply answer with an counterexample, assuming 'fairly rational' and 'religious' are not mutually exclusive.

Many Christians firmly believe:

1) God can predict the future flawlessly and in complete detail.

They understand that this means that events are fixed and cannot be changed.

But they also believe:

2) People can take action right now that will change the future in drastic and important ways.

And most of them reject the extravagant notions of extra-temporal beings that theologians have come up with to make the two agree.

So, from direct observation, we know the answer is "no".

The question about the morning and evening star persists as a deeper aspect of reference than is involved in belief.

There are genuine questions in grammar, in logic, and in law about whether phrases that are known to have equal referents can be substituted for one another under various conditions.

Frege wants to clearly separate the kind of situations where equals can always be substituted for equals, from those where they sometimes cannot. We want mathematics and certain kinds of discussion of ontology to be limited only to the former cases.

  • I'm not quite seeing how what you say is a counter-example. Is your point that 1) and not-2) are equivalent, but many christians believe 1) and don't believe not-2) (instead believe 2))? If so, would you say that these Christians also BELIEVE that 1) and not-2) or equivalent? Dec 23, 2014 at 23:53
  • OK, so if we label 'events are fixed and cannot be changed' as 1a, people believe 1, realize 1 implies 1a, believe the implication is valid and that 1a is equivalent to not-2. So 1 and not-2 are not equivalent, but they imply things that are equivalent, and people do believe the implication. And yes, they believe and act on both propositions, valuing both detailed prophesy and free will.
    – user9166
    Dec 24, 2014 at 15:59
  • @jobermark: I disagree that your counterexample is valid. Just because God is CAPABLE of "predicting the future flawlessly and in complete detail," it does not mean that He DOES it! This is what makes 1), NOT equivalent to not-2)! It would be illogical for the SAME person to believe on both 1) and 2).
    – Guill
    Dec 29, 2014 at 22:31
  • @Guill Official Catholic dogma is both 1 and 2, so we are not talking about 'choose one' the same people believe both. This why theologians have had to say remarkably odd things about time. Cf. Augustine and 'A Course in Miracles' as two examples. And 2 would remove the CAPABILITY of 1, not just its actuality so this is no argument.
    – user9166
    Dec 29, 2014 at 23:52
  • i can't follow that, sorry. it's not even clear that the two ideas are contradictory, let alone hat christian bob knows they are
    – user6917
    Jan 23, 2015 at 3:34

A concrete real-world example:

Bob wants to buy a car. So he goes to the car dealer, picks a car and wants to pay. Then the car dealer makes a statement P which claims that it would be better financially for Bob to take out a loan for the car than to pay in cash. The statement P looks true, so Bob believes it. However, P only looks true. The car dealer, much more experienced in these things, has made statements that make the loan seem to be the better choice, but have hidden flaws.

Bob's friend Jim is better at maths. He takes the statement P apart and converts it to an equivalent statement Q. If is clear to Bob that statements P and Q mean the same things. However, the well-hidden flaws in statement P have chaned to very obvious flaws in statement Q. Therefore, Bob doesn't believe statement Q and pays his car in cash.

PS. Good comment. So we have a chronological sequence: Bob believes P. Bob believes P and Q are equivalent. Bob doesn't believe Q (in that order). What next? Bob now logically knows that P is wrong. This should make him stop believing it. Or can we simultaneously believe something and know that it is wrong?

Bob may still not be able to see the faults in P. Even though he logically knows that P is wrong, if the faults are hidden really well, he may still believe P (but not act on it because he is a reasonable man).

  • 1
    Once Jim explains Q (and it's equivalence to P), doesn't Bob stop believing that P is true? (this isn't specified in your answer.)
    – Dave
    Dec 26, 2014 at 4:51

No, we cannot conclude he believes Q. There are two possible reasons: one logical, one psychological.

  1. Logical reason:

Bob may not aware of modus ponens, and therefore he cannot infer from P and P==Q to Q.

  1. Psychological reason:

"Bob believes ϕ(x)" is an intensional function of ϕ(x), e.g., "Bob believes x is a man" does not imply "Bob believes x is a featherless biped." It is possible that Bob believes P, knows P==Q, and aware of modus ponens, but, for emotional reasons, he cannot bring himself to believe Q even after he utters "therefore Q." Bertrand Russell may call Bob a psychological monstrosity, but logic cannot stave off this form of illness. In other words, whether Bob believes Q or not depends only whether or not there is such a fact as Bob believes Q.

  • I feel like if we called Bob "fairly rational" then what you are describing wouldn't happen. Could we modify it further and say that there exists at least one person Bob who, provided belief in P and the equivalence of P and Q and the belief of that equivalence, must also believe Q? Dec 25, 2014 at 4:18
  • No, we cannot. An intensional function does not logically follow the premises. Whether it is true or not must be asserted in virtue of facts. In other words, whether Bob believes Q or not is not a logical consequence of those premises. Dec 25, 2014 at 4:28
  • It is possible that Bob believes Q, but possibility does not imply existence. Dec 25, 2014 at 4:35
  • @MathTeacher- Sometimes, even the most intelligent person hesitates to believe logical consequences. Bertrand Russell was incredulous when he first derived his paradox. It took him quite a while before he was convinced that the foundation then was flawed. Dec 25, 2014 at 17:39
  • Wouldn't he be failing to be a "rational person" for the time that he believed P but not Q?
    – Dave
    Dec 26, 2014 at 4:54

This is my answer, fwiw. It is not rational to deny the consequent, indeed it is rational to affirm it.

So Bob either has to be

  1. so stupid as to not understand the law of identity, intuitively, when he considers P and Q, which is possible people do make basic logical errors everyday.

  2. so mischievous he puts up with cognitive dissonance

  3. or just never considers P and Q.

3 seems unlikely if he knows they are identical. 1 seems unlikely to happen very often. 2 is possible though... it's difficult to put a figure on the chance that Bob does nevertheless not believe Q, but given that he is a "rational person" one can assume he does IMO.

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