11

Consider the following truth table, which serves to define the logical connective ⇔,

P | Q || P⇔Q
T | T || T
T | F || F
F | T || F
F | F || T

According to the above truth table, the logical connective ⇔ is defined as the binary operation which takes as its arguments two statements P and Q, and produces another statement which has truth value "true" when both the truth values of P and Q are the same, and "false" otherwise.

Having defined in the above way, one can prove (using truth tables) that the statement P ⇔ Q is equivalent to (that is, it has the same truth table as) the statement (P ⇒ Q) ∧ (Q ⇒ P). In showing these statements P ⇔ Q and (P ⇒ Q) ∧ (Q ⇒ P) are "equivalent" using their truth tables, we are invoking a "truth functional" notion of equivalence.)

However, does there exist a more rudimentary notion of equivalence, separate from the truth table definition of equivalence? I can try to suggest why this might be desirable. For example, it is commonplace to consider the statement 3>2 "equivalent" to the statement 3-2>0, not only because both statements are true (and hence have the same truth values, allowing us to say (3>2)⇔(3-2>0) is "true" using the above definition of ), but also in that they are saying the same thing in terms of their content or meaning! Therefore, we might want to pick a new symbol such as and say (in a stronger sense) that (3>2)≡(3-2>0).

On the other hand, consider the statements 3>2 and 4+6=10. Since both statements are true, we could write (using the definition of above) that (3>2)⇔(4+6=10), which seems counter-intuitive. Or, for another example, suppose the statements "the sky is blue" and "the grass is green" are both always true. Then since both statements have the same truth value, we could again write (using the definition of above) that "the sky is blue" ⇔ "the grass is green" is true.

Here's my point: Although in the examples above we've exhibited truth-functional equivalence, we probably wouldn't think of these statements as "equivalent" in terms of their meaning, since they are saying completely different things. E.g., the content of 3>2 and 4+6=10 are unrelated to each other.

Am I missing something here? Is there a more rudimentary notion of equivalence, separate from the truth table definition? Should we use the symbols and differently, perhaps reserving for statements equivalent in their content or meaning (like 3>2 and 3-2>0) and not merely for statements such as 3>2 and 4+6=10)?

Thanks for your thoughts!

  • 1
    Good question! Which allows for good answers! – DBK Apr 25 '16 at 21:39
  • @DBK Thank you. There are very good answers here! Regretfully, I am slow to parse them. Before I select an answer, I would like to be sure I understand the difference between and (as the difference seems not to be what I thought it was)! – Mathemanic Apr 26 '16 at 0:03
9

There is a difference between semantic consequence expressed by truth tables, and syntactic consequence in a deductive system, some authors use ⊨ for the former and ⊢ for the latter, and the corresponding difference in equivalence. The latter can be used to capture what you are describing somewhat. In Kant's theory of conceptual containment equivalence "not only because both statements are true, but also in that they are saying the same thing in terms of their content" is called analytic equivalence, and it can be formalized as follows.

Note that in your examples truth of the statements depends on using the laws of arithmetic. One can select a subset of them relative to which 3>2 and 3-2 > 0 are still provably equivalent, but 4+6=10 is unprovable. More generally, you can designate some axioms you accept as "analytic" (say laws of pure logic only), and others as "synthetic" (say arithmetical laws). Equivalence is declared analytic if only analytic axioms are used to derive it. However, with purely logical analyticity 3 > 2 and 3 - 2 > 0 are not analytically equivalent. And they shouldn't be if you think about it. The latter involves 0 and subtraction, whereas the former does not, so it does add additional arithmetical content and properties. But you can get what you want by moving some of arithmetic into the analytic column. For example, Frege thought that all of arithmetic is analytic, and Kant thought that none of it is, see Was Locke right that analytic knowledge is vacuous?.

The root of the problem is the definition of material conditional (and equivalence) in terms of truth tables. "The material conditional does not always function in accordance with everyday if-then reasoning... One problem is that the material conditional allows implications to be true even when the antecedent is irrelevant to the consequent." Conditional that takes "relevance" into account is called indicative conditional, and a theory of that can not be as simple or formal as that of material conditional exactly because it has to take "meanings" into account, and those are a handful.

5

I get, what you are saying, but implication in classical logic has nothing to do with the "meaning" of propositions. In particular, 3>2 and 4+6=10 are in fact equivalent statements.

The reason for having two symbols to represent logical equivalence goes roughly like this: inside a given theory of mathematics A we construct a logical system B. It (B) simply consists of (a set of) strings of symbols called "propositions", like Q ⇒ P or P∧P along with semantics or a syntactical calculus for deriving what one calls "valid propositions" (in your case it's 'truth tables'). The symbol is then just a character, then may or may not occur in these strings and inside of B it means, that two propositions are equivalent. On the other hand, given two propositions P and Q (those are things in A, you cannot talk about individual propositions of B in B) we write P ≡ Q to say, that P⇔Q happens to be a valid proposition, so is simply a relation on the set of propositions, that tells you inside of A, that two propositions of B are equivalent. If you are not studying logic mathematically however you may simply "forget" that A and the symbol ever "existed", since you don't need it (most of the time).

If you want your implication to respect the meaning of the propositions, you have to take a look at "relevance logic" (http://plato.stanford.edu/entries/logic-relevance/). I can't give you any further information regarding this, unfortunately.

3

I think what you're picking up on is the Fregean distinction between sense and reference. The referent of a term t is the object that t picks out 'in the world', whereas the sense of t is, roughly, something like the idea associated with t. (VERY roughly. Frege calls the sense of the term the 'mode of presentation' of the term; one way to think of sense is as the way in which a term refers. Note also that by calling the sense of a term an 'idea' I don't mean that sense is something mental).

Frege identifies the referent of a (well-formed) sentence with the truth value of that sentence. This is consistent with your recognition that '3>2' and '4+6=10' are truth-functionally equivalent, and yet 'don't mean the same thing', i.e. are synonymous (I take it this is what you're trying to capture when you talk about content). Frege will explain this by appealing to a difference in sense between the two sentences (better: the propositions expressed by those sentences). Similarly, the sentences 'all bachelors are unmarried men' and 'all vixens are female foxes' are both (necessarily) true, hence truth-functionally equivalent, but we wouldn't want to say that they're synonymous. And again Frege will explain this by appealing to the distinction between sense and reference.

For more you can check out http://plato.stanford.edu/entries/frege/#FreLan for a discussion of Frege's philosophy of language. Frege himself is an incredibly clear writer, and you could probably go straight to his "On Sense and Reference" (just google it - you can find it all over the place online), where he lays out this distinction in some detail.

3

Actually the two symbols in question, and , have very different behaviors in propositional logic. For 99% of situations, you can interchange them and get away with it. However, in the last 1% the difference is essential to the use of propositional logic.

The difference is that the meaning of is formally defined in the definition of propositional logic, while is not. is related to the implication operator by the definition iff A ⊢ B AND B ⊢ A then A ≡ B. Implication, , is different from the conditional operator because the meaning of is formally defined and is informally defined.

Why this wierdness? It comes up when one tries to define the rules of the language of propositional logic. I could not use the statement (A⇔B) ⇔ ((A⇒B) ^ (B⇒A)) to define the operator because I'd end up using it in its own definition! This circular loop is impossible to get away from, no matter how many clever operators you formally define. The solution is that is used informally to define , and then is used to define :(A⇔B) ≡ ((A⇒B) ^ (B⇒A))

Once we define in this way, we can kinda-sorta get away with treating and as the same thing. However, in this initial bootstrapping phase, the difference is essential to the construction of propositional logic and cannot be understated.

  • Cort, thanks for your answer! Now I'm wondering not only the difference between and , but now the difference between and as well! In my math classes I've only seen used before, not ever . A few follow-up questions : The symbol is defined by its truth table, correct? Does have a different truth table than ? I noticed you said is not formally defined, but can't everything in logic be formalized? You're piqued my interest in the construction of propositional logic; is there a text you might recommend that explains this construction in more detail? Many thanks! – Mathemanic Apr 25 '16 at 22:28
  • @EthanAlvaree The technical answer is that is not defined by a truth table and operates on statements, not values, but once you have fully defined all operators in propositional logic, you find that you can construct a truth table by feeding it statements that are proven true and statements that are proven false, and you will find that truth table looks identical to the truth table that defines . Note that I intentionally approached the "truth table" of a little backwards. As it turns out, not everything in logic can be formalized. The behavior of the "implies" operator, , – Cort Ammon Apr 26 '16 at 0:48
  • is the one little piece that cannot be formalized into its own language. It's just so mindnumbingly simple that most people are willing to handwave it away as a self-evident truth. If you try to get rid of it, you find yourself quickly running in circles or extending off to infinite numbers of steps, or more commonly, you end up formalizing it in a different language (like English), for which there is no formalization. – Cort Ammon Apr 26 '16 at 0:55
  • The wikipedia page on Propositonal Logic actually touches on these issues, but does not dig into their deeper meaning, because for 99% of propositional logic use, its not necessary. – Cort Ammon Apr 26 '16 at 0:57
  • For an amusing ride: consider trying to solve this issue by formally defining in a "higher order" propositional logic, a logic which we assume is already defined. We can then define that language's and so forth. In theory, if you continue this to infinity, you should eventually get to the "true" behavior of . However, there's a catch. To describe that trip to infinity, you need to be able to count. You need to be able to make proofs regarding basic arithmetic (such as proving 2+3=5). Alfred Tarksi proved that such a language cannot define its own semantics. You can – Cort Ammon Apr 26 '16 at 0:59
0

For purposes of logic, which may consider mathematical, political, or philosophical arguments, or coded computer data, it is more general to consider only the truth value of statements entirely apart from their meaning. In similar fashion, we usually find it more useful to consider 2+3=5 regardless of whether the numbers are of pigs, square millimicrons, raindrops, or occurrences of the letter r in a line of Shakespeare's collected works, rather than having to define separate arithmetics for a braces of pigeons and pairs of shoes.

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