3:16 PM
@Zanna Some of the popular Firefox extensions that selectively turn things off often have the effect of improving performance. I'm not sure it would help for the particular page, though. I use Privacy Badger and uBlock Origin. I also use Disable Javascript, which gives me a button I can click that immediately does what it says on the tin (per-site).
3:28 PM
@EliahKagan You might be thinking: what about sentences that are true by virtue of their logical structure but that match no valid truth-functional schema? For example, $\exists x\, Fx$ and $\exists y\, Fy$ express the same idea, just with a difference choice of variable. So each may be inferred from the other. So this is a valid argument: $$\exists x\, Fx$$ $$\therefore \exists y\, Fy$$
Thus that argument's corresponding conditional is a valid sentence: $$(\exists x\, Fx) \rightarrow (\exists y\, Fy)$$ But truth-functional logic is not, by itself, powerful enough to demonstrate its validity. That is, it's valid, but not truth-functionally valid.
You will notice that the most specific truth-functional schema that matches it is $p \rightarrow q$, which is not valid. So you cannot demonstrate the validity of $(\exists x\, Fx) \rightarrow (\exists y\, Fy)$ with a truth table, without first performing a variable substitution (which in this simple kind of case is sometimes called relettering) on one side or the other of the "$\rightarrow$".
A slightly less trivial and thus perhaps more compelling example is: $$\forall x\, Fx$$ $$\exists x\, Gx$$ $$\therefore \exists x\, (Fx \wedge Gx)$$ That is, if everything is $F$ and something is $G$, then something is both an $F$ and a $G$: $$[(\forall x\, Fx) \wedge (\exists x\, Gx)] \rightarrow \exists x\, (Fx \wedge Gx)$$ This is valid, in a way that clearly relies on the meaning of the quantifiers.
Truth-functional logic is sometimes used on its own, but it is more often used as part of a more powerful logic. Specifically, truth-functional logic is a fragment of quantification theory, which is a fragment of quantification theory with identity (where "$=$" and the logical truths about it, such as $\forall x\, (x = x)$, are available and are regarded to be built into the logic). Quantification theory with identity is usually what we mean by "first-order logic."
This can be subdivided further--I might have said that truth-functional logic is a fragment of monadic quantification theory (where predicates must be unary) which is a fragment of polyadic quantification theory (where predicates may be of any arity, at least any nonzero arity). Furthermore, first-order logic is the first-order fragment of second-order logic, and so on.
This is all perhaps less deep than it may sound. For example, first-order logic without function symbols is a fragment of first-order logic with function symbols, though it's rare to speak of it that way. The point is that different logics have different power to recognize validity...
Other kinds of schemata besides truth-functional schemata will come up later. You may have heard (and I may have mentioned, whilst showing set-builder notation) that ZFC is always axiomatized with at least one axiom schema, such as an axiom schema of replacement.
...On the other hand, there is something quite deep about it, because of the interaction between what is valid in a particular logical system and broader notions of deductive validity. One might try to tidy this up by claiming that validity is relative to one's choice of logic, but that doesn't seem right.
For example, suppose I claim that modus ponens is true only relative to various choices of logics. It still has to be meaningful to say of a logic that modus ponens holds in it. I might say, "If you are using (standard) truth-functional logic, and you have premises $p$ and $p \rightarrow q$, then you may infer $q$." Something like modus ponens is implicit in this statement of modus ponens.
More precisely, it is implicit in my belief that the person to whom I say it agrees with me about what it means. Likewise, if I seek to prove something, and in each deduction I cite what premises or intermediate conclusions I used and what rules or principle of inference I am applying, you can always respond by saying, "But why?"
People often claim that, in mathematics, all we know is that if various premises hold, then various conclusions hold. The way we know it, though, is through presystematic or metasystematic beliefs about what can be inferred, and if challenged on it, the best we can do is make an argument about what can be inferred from what. Those arguments themselves have premises and conclusions, though they may sometimes be inductive rather than deductive arguments.
I think a more plausible view, than the view that logical validity can only be judged relative to some specific choice of logic, is that words very often mean things. Finding out what they mean is an empirical activity, though not always a scientific activity. When people learn a language, they're learning facts about patterns of speech tend to be associated with particular states of mind.
In particular, when young children learn their first languages, they are discovering real features of the world. If one wishes to justify logic, one's recourse is linguistics, psychology, and sociology. In other words, xkcd 435 is totally wrong. This insight is no way due to me. This is essentially the view (or a part of the view) taken by Quine, though he didn't state it the form of a disagreement with a web-comic. I think Quine is right about this.
Thus I do not mean to suggest that we could not use words like "and" to mean "or", or phrases like "for all" to mean "exactly seven." The reality of our language and culture would make those choices impractical for numerous purposes, but we could do it.
But it is very easy to mistake that for a stronger "could," such as the "could" of "All of society could mean Y by X" in the absence of consideration about what it would actually take to bring this about, who has the (or more) power to make such things happen, and who does not.
Logical truths can be regarded as those that are true by virtue of the meaning of the logical constructions they contain--to state it slightly imprecisely, by virtue of the meaning of the logical words and phrases they contain.
So to bring this back to symbolic formal logic and formal mathematics, recall that some symbols are considered logical, like the truth-functional sentential connective and quantifiers (and parentheses), while others are considered non-logical, like $\in$, $+$, and $\mathrm{paintbrush}$.
I mentioned that in a schema, we have given meanings to the logical symbols but not the non-logical symbols. One might regard a choice logic as an assignment of meanings to logical symbols. In that case, "Validity is relative to one's choice of logic," becomes a far weaker but plausible claim. Of course there is the question: who decides which symbols are the logical symbols and which are not?
That is, couldn't I have a logic in which "Purple cars are rare in Manchester," it is a logical truth?
One response is to say that that one could, but that in doing so, one would be picking a meaning of those words that causes them no longer to describe any feature or empirical reality. While I suspect this objection is correct, it's a bit heavy for my tastes. It has what might be called a large attack surface. After all, couldn't it be that we only think it is logically possible for purple cars to be common in Manchester?
Although I think the answer to this question is "no," I don't think it's necessary to establish this to be convinced that it is unreasonable to consider "Purple cars are rare in Manchester" a logical truth.
I think, instead, that a conceptual framework in which "Purple cars are rare in Manchester" is fundamental is not a logic in the sense we typically mean by logic.
This is an extreme example, but what about $7 + 3 = 10$? Could it be reasonable to regard that a a logical truth? I think not, and I think most people today think not... but this possibility was taken seriously at around the turn of the 20th century, and is related to logicism, the idea that mathematical truths are true because they arise directly out of logic. Note, though, that one could be a logicist without actually holding that $7 + 3 = 10$ is a logical truth.
But the need, in proving major theorems of mathematics, for axioms that strongly appear not to be logical truths or the mere introduction of definitions causes logicism to decrease markedly in popuarity.
Note that, at least in ordinary language, there are disagreements about the meaning of logical words and phrases. I recall a long-running dispute with a friend of mine who is not an astronaut. (I am also not an astronaut, unfortunately, but that isn't really relevant to this story.) I insisted that every time he has visited the moon, he has died. After all, he has not been to the moon, so it is vacuously true that every time he has been there he has died.
But he disagrees, on the grounds that he has never been to the moon, so it is not so that every time he has been to the moon he has died. Although the disagreement is, at the surface level, about the meaning of words like "all" and "every," but it is also in effect about what is useful or elegant or illuminating for those words to mean.
The meaning I prefer is the one that corresponds to how $\exists$ and $\forall$ are used in symbolic logic. I feel that I am on very solid ground in saying that these usages are the practically wise ones for those symbols in logic, so that edge cases don't break the rules of passage for "$\neg$" across quantifiers, i.e., so edge cases don't prevent $(\neg \forall x\, Fx) \leftrightarrow (\exists x\, \neg Fx)$ and $(\neg \exists x\, Fx) \leftrightarrow (\forall x\, \neg Fx)$ from being valid.
@EliahKagan Similarly, there is disagreement about to what extent the conditional, "$\rightarrow$", in formal logic, captures conditional sentences in ordinary language. I think, at least in English and some other languages, that ordinary-language conditionals, when they are in the indicative mood, are usually captured extremely well by $\rightarrow$.
(Conditionals that don't make claims cannot be expressed in terms of "$\rightarrow$", of course, nor can conditionals in the subjunctive mood, like "If I were a bird, I would fly.")
I do think people who deny that indicative conditionals in English are usually well-captured by "$\rightarrow$" are mistaken. But I don't think I am on as solid ground in arguing this as I am in saying that "$\rightarrow$", in the meaning we give it, is both the most suitable truth-functional sentential connective for conditional reasoning, and that it is in practice extremely useful when doing logical manipulations and formal thinking.
(Note that I am not saying the symbol "$\rightarrow$" has these virtues; the practical considerations surrounding what symbols we should use overlap with but are often not the same as the considerations surrounding what meanings we should express with individual symbols. There are some good practical arguments for the older symbol "$\supset$" over "$\rightarrow$", though also some good arguments against it.)
5:19 PM
Modern formal logic is done symbolically. But formal logic is very old and has not always been associated with the use of formal languages. Aristotelian logic is syllogistic, which is to say that it is concerned with the deductive validity of syllogisms, which are arguments like "All ravens are swans. Socrates is a raven. Therefore Socrates is a swan," and "All creatures with hearts have livers. Sea sponges don't have livers. Therefore sea sponges don't have hearts."
These are just a couple syllogistic forms; there are others. Those syllogisms are both valid, by the way, though the first one is unsound. (Recall that, to be sound, an argument must be valid and all its premises must be true.) Aristotle thought that, to assert that something holds for all things with a particular property, one effectively is asserting that there is some such thing. That is, he thought "all implies some," as it is usually put.
5:53 PM
A couple things to note. I had meant to link to this part of the earlier conversation, which is very relevant both to the above example and to the general issue of what logical symbols mean.
Also, I want to reiterate my usual request for your thoughts, questions, and corrections, and also mention that philosophical objections are welcome. They have always been welcome--or I have always intended that, anyway--but it is particularly in connection with this material that I feel I should mention that explicitly. (I hope I would remember to say this even if I didn't know you had studied philosophy in school and that you read philosophy, have opinions of particular philosophers, etc.)
I regard all logic to be philosophy (though I don't construe logic so broadly as to regard most of set theory to be part of logic), though it may be that most or all formal logic is also mathematics. I don't want to give the impression that I think only now have things become philosophical.
The things I have said recently about philosophy of logic--inevitably affected by my own biases and interests--have been for the purpose of expounding and clarifying the logic itself. I have tried to limit them to claims I believe are well-grounded. But that does not mean I succeeded even at that; and even if so, I recognize they are not universally accepted.
6:44 PM
7:40 PM
> But it turned out to be needed, in order to prove major theorems in mathematics, to have some axioms that strongly appeared neither to be logical truths in any reasonable sense of that phrase, and also that did not appear to be tantamount to the formalization of a definition. This caused logicism in the philosophy of mathematics to decrease markedly in popularity (though I think it would be overly strong to claim that logicism has been refuted).
The axiom of infinity -- which, in set theory, is an axiom that effectively says there's an infinite set -- is such an axiom. Some people do argue that one should develop mathematics in a way that does not require an axiom of infinity, but that's not the view that caught on, nor is that how foundation of mathematics is usually done.
The axiom of choice is another example; one can manage to do quite a bit without it, but it facilitates a number of important general results that mathematicians like to have, such as the theorem that every vector space has a basis. (The "C" in ZFC stands for the axiom of choice. The system like ZFC but without the axiom of choice is called ZF.)
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