Math Help - "Logically possible" vs "truth table-possible"

1. "Logically possible" vs "truth table-possible"

The sentence

$Larger(a,b) \& Larger(b,a)$

is "truth table-possible"; the row with both conjunctions stated as true will result in the sentence being stated as true in the table.

But this is not "logically possible", is it? I mean, it is not possible (under any circumstances in any world) that an object1 is larger than object2 which is larger than object1 itself.
Or is this just physical limitations which I should not care about in logic?

A logically possible sentence is always tt-possible, but a tt-possible sentence may or may not be logically possible.

Or is it the other way around?

2. "larger than" as an ordering of the real numbers (intergers, etc) would make a contradiction.
Trichotomy (mathematics) - Wikipedia, the free encyclopedia

If a < b, then b is not < a.

3. In talking about "logically possible," you impose restrictions on the meaning of "Larger." Presumably, "Larger" is a strict order. Every strict order is asymmetric, which means that Larger(a,b) & Larger(b,a) is always false.

This is a question of definitions or restrictions that one imposes on the concepts involved. It is possible to interpret "Larger" not as an order, but, say, as divisibility; then Larger(a,b) & Larger(b,a) is possible.

4. Originally Posted by emakarov
In talking about "logically possible," you impose restrictions on the meaning of "Larger." Presumably, "Larger" is a strict order. Every strict order is asymmetric, which means that Larger(a,b) & Larger(b,a) is always false.

This is a question of definitions or restrictions that one imposes on the concepts involved. It is possible to interpret "Larger" not as an order, but, say, as divisibility; then Larger(a,b) & Larger(b,a) is possible.
Would it be correct to say that since truth tables work at the propositional logic level only, and we're dealing with a predicate here that's at a lower level (the predicate logic level), that the truth table method simply won't be able to give us as much information as we might like?

5. the statement: P and Q, is "possibly true".

the statement: P and not P is never "possibly true".

the point being, is that you're missing some "implicit information", namely Larger(a,b) <--> ~(Larger(b,a)).

this has nothing to do with the statement Larger(a,b)&Larger(b,a), and everything to do with the predicate "Larger(-,-)".

6. Thanks for all replies!

Originally Posted by emakarov
In talking about "logically possible," you impose restrictions on the meaning of "Larger." Presumably, "Larger" is a strict order. Every strict order is asymmetric, which means that Larger(a,b) & Larger(b,a) is always false.

This is a question of definitions or restrictions that one imposes on the concepts involved. It is possible to interpret "Larger" not as an order, but, say, as divisibility; then Larger(a,b) & Larger(b,a) is possible.

In that case, the only tt-possible sentences which are not logically possible should involve identity..? I feel quite convinced that tt-possible does not imply logically possible but logically possible imlies tt-possible.

E.g., !(a=a) is tt-possible but it is not logically possible. Are there any other examples of a sentence being tt-possible but not logically possible?

I assume that a sentence cannot be logically possible and not tt-possible.
The way the truth table is constructed, it misses that the falsity of a=a is impossible. But it will never attribute truth to something which is not true; there is no atomic sentence which is never true under any circumstances. Right?

And the only atomic sentence which can never be false would be on the form a=a.

7. Originally Posted by Ackbeet
Would it be correct to say that since truth tables work at the propositional logic level only, and we're dealing with a predicate here that's at a lower level (the predicate logic level), that the truth table method simply won't be able to give us as much information as we might like?
This is a little vague, but, in a sense, yes. The reason Larger(a,b) /\ Larger(b,a) becomes always false is because in predicate logic one is able to formulate axioms (irreflexivity and transitivity) that use the inner structure of the propositions Larger(a,b) and Larger(b,a) to make their conjunction impossible. In contrast, in propositional logic, propositions lack any structure; they are atomic. So, in the language of propositional logic, one can form some finite number of axioms $\neg(\mathrm{Larger}(a_i,b_i)\land \mathrm{Larger}(b_i,a_i))$ for $i=1,\dots,n$, but one can't say using a finite number of axioms that all such conjunctions are false.

8. In that case, the only tt-possible sentences which are not logically possible should involve identity..? I feel quite convinced that tt-possible does not imply logically possible but logically possible imlies tt-possible.
The technical term for possible is satisfiable: "A formula is satisfiable with respect to a class of interpretations if it is possible to find an interpretation that makes the formula true." The difference between tt-possible and logically possible lies in the class of interpretations with respect to which one considers satisfiability.

There are two approaches to equality: first-order (i.e., predicate) logic with or without equality. In the first approach, equality is considered a logical symbol along with /\, \/, etc. and is interpreted as identity. In the second, it is a regular binary predicate symbol and can be interpreted in an arbitrary way, though additional axioms may force it to be an equivalence relation. From Wikipedia:
When this second convention is followed, the term normal model is used to refer to an interpretation where no distinct individuals a and b satisfy a = b. In first-order logic with equality, only normal models are considered, and so there is no term for a model other than a normal model.
So, I believe you call a formula tt-possible if it is satisfiable with respect to all, not necessarily normal, models, and you call a formula logically possible if it is satisfiable with respect to only normal models, i.e., models of first-order logic with equality. Now, suppose that we have two classes of models A and B such that A is a subset of B. If a formula is satisfiable w.r.t. A, then it is satisfiable w.r.t. B. Therefore, yes, if a formula is logically possible, then it is tt-possible, but not the other way around.

In the original example, by "logically possible" you presumably meant "satisfiable w.r.t. models where Larger is interpreted as a strict order." Again, since this is a subclass of all models, logically possible implies tt-possible.

I assume that a sentence cannot be logically possible and not tt-possible.
The way the truth table is constructed, it misses that the falsity of a=a is impossible. But it will never attribute truth to something which is not true; there is no atomic sentence which is never true under any circumstances. Right?
Yes, though I am not sure how this fact is used in showing that if a sentence is logically possible, then it is tt-possible.

9. First of all, thanks! I really appreciate the time and effort people put in!

Originally Posted by emakarov

There are two approaches to equality: first-order (i.e., predicate) logic with or without equality. In the first approach, equality is considered a logical symbol along with /\, \/, etc. and is interpreted as identity. In the second, it is a regular binary predicate symbol and can be interpreted in an arbitrary way, though additional axioms may force it to be an equivalence relation.
This is pretty much what I was "afraid of" from the start, the very reason I started the thread. But at least it feels good to know.
(Maybe I should have been able to find it out on wikipedia myself, but then again I would probaly have wanted some assurance about my interpretation of the article...).

I assume that a sentence cannot be logically possible and not tt-possible.
The way the truth table is constructed, it misses that the falsity of a=a is impossible. But it will never attribute truth to something which is not true; there is no atomic sentence which is never true under any circumstances. Right?
Yes, though I am not sure how this fact is used in showing that if a sentence is logically possible, then it is tt-possible.
It was supposed to "show" that something which is not logically possible can not be tt-possible.

Again, thanks.