# What is wrong with the following logical sentence?

In first order logic (with ∀ meaning "for all" and 37 being the value in Celsius)

``````∀x∀t ( Fever(x) <=> Temperature(x,t) and t > 37 )
``````

Edit, clarification of what I think this sentence wants to be:

For all persons x and all temperatures t, person x has a fever if and only if the temperature of the person is t and t is above 37.

When you rewrite the equivalence this becomes 2 sentences:

1) ∀x∀t ( Fever(x) => Temperature(x,t) and t > 37 )

Which says, I think, for all persons x and temperatures t, if x has a fever then x has a temperature t and t is above 37. This sentence is correct.

2) ∀x∀t ( Temperature(x,t) and t > 37 => Fever(x) )

Which says, I think, for all persons x and temperatures t, if all persons x have a temperature t and t is above 37, x has a fever.

I think the second sentence is wrong (can have a fever without everybody else having fever) but I'm not sure.

What the second sentence wants to express is more like this I think (with ∃ being "there exists"):

``````∃x∃t ( Temperature(x, t) and t > 37 and Fever(x) )
``````

The second sentence means this:

For every person x and temperature t, if x has a temperature t above 37, x has fever.

I would say is false, because one could be in an oven at any given temperature above 37, without having real fever, because fever is supposed to be caused by an illness.

The equivalence is false.

• Hmmm, there is a certain falseness to the equivalence just by the semantics, but I don't think that is the point (the sentence is proposed in a formal logic course and focusses less on semantics and more on the logic ). So, purely from a logical viewpoint, something must be wrong I think. Or isn't it, and is the only fault from the informal interpretation? – Sven Jan 2 '14 at 10:41
• @Sven Semantics give definitions of what we are comparing, and logic shows the falsability of the equivalence. I don't see how could you focus only in logic, please explain further. – Natxo Jan 2 '14 at 14:41
• I have to choose my words more carefully, especially in this context. However english isn't my first language so it is a bit difficult. What I meant to express was that it seems to me that the cause of the equivalence being "wrong" was because of a wrong use of logic (things like using implications while you want a conjunction for example) instead of the informal interpretation. – Sven Jan 2 '14 at 16:00
• @Sven, as i implicity pointed out and Chris explicity did, you got the second sentence wrong. The reformating done by Lie Ryan clarifies a bit more the logical sentence. ∀x∀t doesn't mean for all people "at the same time". You have to take separately each person and for that person, each temperature makes a different case to evaluate. I think this was your mistake. – Natxo Jan 3 '14 at 9:04
• Yes, my reading of the sentence is wrong, I understand. My point is that I've been told that the equivalence itself is wrong (so regardless of my eventual reading). What I tried to do was give a correct interpretation of the wrong sentence, so proving the equivalence is wrong. However the interpretation itself is wrong. But I assume your reason for the wrongness of the equivalence (fever is illness, temperature above 37 is not necesarily illness) is correct, even though I'd hoped the fault would lie in a wrong use of logic. – Sven Jan 3 '14 at 10:29

In structural terms, the second sentence isn't incorrect, you just read it incorrectly.

You read it as "For all persons x and temperatures t, if all persons x have a temperature t and t is above 37, x has a fever."

The second "all persons" is incorrectly inserted. The correct reading would be: "For all persons x and temperatures t, if x has a temperature t and t is above 37, x has a fever."

NOTE: The sentence that would actually correspond to what you read would be:

!x!t( !x(Temp(x,t) and t > 37) => Fever(x))

Which doesn't make a whole lot of sense in any context.

As the t isn't mentioned in the part of Fever(x), we can push it inside. Then we become:

∀x(Fever(x)<=>∀t(Temperature(x,t) /\ t>37))

This means that person x has fever iff all objects from the universe are the temperature of x and above 37.

A better sentence would be:

∀x(Fever(x)<=>∃t(Temperature(x,t) /\ t>37)).

Then you're saying that person x has fever iff he has a temperature above 37.

Usually, if you have a ∀, then you need an implication (=>). If you have ∃, then you shouldn't have an implication.