## The Propositional Calculus

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If every proposition is true, then truth is trivial. It does nothing.
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Well, to me, modal logic is part and parcel of propositional logic

This is untrue. Propositional calculus does not even have first-order quantifiers (forall, exists) to have modal operators. And modal operators are behaviorally analogous to quantifiers (and are rightly a kind of quantifier that doubles as an operator). Zero order logics, like prop. calc, by definition do not quantify over anything.

You might be conflating model-theoretic valuations (of prop. calc) or interpretations (of FOL) with the nodes that exist in possible world semantics for modally extended versions of FOL (like FOL₌ + S5).

A wff is a validity just in case it is modeled by all valuations (or interpretations).
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:ok:
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proof of consistency

Consistency follows from soundness. Proving soundness is not deep. We ordinarily just do induction on the length of derivations.

inconsistent language - or theory, if you prefer

'inconsistent language' doesn't make any sense.

A language is not something that can be consistent or inconsistent.

It's important to understand why that is the case:

A language doesn't make assertions, perforce, it doesn't assert contradictions. Sentences made in a language do make assertions, and a theory is a set of sentences closed under derivability, so theories are consistent or inconsistent.

Informally, think of it this way, for example with a natural language:

English doesn't make assertions. English is used to make assertions.

[with an inconsistent theory] every theorem can be deduced; on in which everything is true.

That is wrong in two ways:

(1) It is not informative. It should be, "With an inconsistent theory, every sentence can be deduced". By definition, a theorem is a deducible sentence, so with every theory (consistent or inconsistent), of course every theorem can be deduced, because being deducible is what it means for a sentence to be a theorem.

(2) It is not the case that every sentence is true in an inconsistent theory. Sentences are not even true or false in theories. Rather, sentences are true or false (and never both) per any given model. An inconsistent theory asserts contradictions, but that doesn't make the contradictions true. Indeed contradictions are false in all models.

if a contradiction is true in our system, then anything is derivable.

Same thing. Sentences (including contradictions) are not true or false in theories. Rather they are true or false in any given model. Meanwhile, a model of a theory is a model in which every theorem of the theory is true. But a contradiction is not true in any model, so an inconsistent theory has no models. There are models for the language of an inconsistent theory, but those are not models of the theory.

What we do say is, "From a contradiction, we may derive any sentence." But there is no such thing as a "contradiction that is true in a theory". Again: sentences are not true or false in a theory (sentences are true or false in models), and (in ordinary propositional logic) there is no such thing as a "true contradiction".
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'trivial' in mathematics, in the context of this discussion, is not a formal notion. What mathematicians mean is that a trivial theorem is one that we can see the proof of it without much effort. If theorem has such an easy, simple, short proof that one would grasp it in an instant, then we say the theorem (or sometimes, the proof of the theorem) is trivial.

Also, for example, we might say an inconsistent theory is trivial, because every sentence is a theorem, so, given that a theory is inconsistent, there is no work involved in determining which sentences are theorems.

Another example, the empty set is a function, since the empty set vacuously satisfies the definition of 'is a function'. We could say it is the most "trivial" function. In a case such as the empty set being a function, we might also say "it's vacuously the case that the empty set is a function".

'Triviality' is not a deep notion in this context. It's just a way for mathematicians to point out that they recognize that certain claims are correct but quite obviously so, or that certain objects have certain properties but in a not very informative way.
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Doubtless all true, and should be accounted for. In my defence this thread is not intended to be so formal but to get on with outlining what is going on.

By all means, rain on the parade, but perhaps not so heavily as no one shows up.

Consistency follows from soundness. Proving soundness is not deep. We ordinarily just do induction on the length of derivations.
Some simplified detail might be fun.
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this thread is not intended to be so formal but to get on with outlining what is going on.

Confusing fundamental concepts is not mere informality. And any outline based on such fundamental confusions cannot be other than itself more confusion.

I'm not doing any raining. I'm giving you information and explanation.

Some simplified detail might be fun.

Are you suggesting that I provide more detail, or are you suggesting that you might be providing more detail?
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If every proposition is true, then truth is trivial. It does nothing.

Wanna run something by you if it's ok with you. True that if contradictions (p & ~p) are allowed, "every proposition is true" but every refers not to logically independent propositions like "some swans are not white" and "Socrates was bald" but to logically dependent propositions like "all swans are white" and "some swans are not white" (contradictions). So the argument from the principle of explosion ( ex contradictione sequitur quodlibet) is, in fact, the circular argument: contradictions are unacceptable because contradictions are unacceptable. :chin:
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if contradictions (p & ~p) are allowed, "every proposition is true"

No, that is not the case. I explained in detail why.

Now Banno's misconception has been inherited by you.

So the argument from the principle of explosion ( ex contradictione sequitur quodlibet) is, in fact, the circular argument:

What argument are you referring to?

The proof that from a contradiction all statements are provable is not circular.

Speaking of ... the above is petitio principii.
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I don't know what that means.

So, if you posted it to make us even, you succeeded.
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For any wff, or for that matter any sentence in a natural language, ρ and σ

(ρ & ~ρ ) ⊃σ

...but every refers not to logically independent propositions like "some swans are not white" and "Socrates was bald" but to logically dependent propositions like "all swans are white" and "some swans are not white"

:brow: Not sure what "logically dependent" is doing here.

If (Socrates is bald and socrates is not bald) then swans are green.
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It's alright. My brain can't grok the point of ex contradictione sequitur quodlibet.

Not sure what "logically dependent" is doing here.

Does the truth/falsity of "Socrates was bald" depend on the truth/falsity of "Some swans are not white"?,
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Has anyone mentioned that there's a name for this -- the principle of explosion -- and that it is a direct consequence of how the material conditional is defined? Every material conditional with a false antecedent is true, whatever the consequent. (I think the terminology I've always heard used for cases like this is that these conditionals are "vacuously true", which would also apply to the equivalent disjunction.)

Some nonstandard logics are motivated precisely by the wish to avoid the principle of explosion by defining implication otherwise.
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Indeed, it is apparent in the final two rows of the truth table:

$\begin{array} {|r|r|}\hline ϕ & ψ & ϕ ⊃ ψ \\ \hline T & T & T \\ \hline T & F & F \\ \hline F & T & T \\ \hline F & F & T \\ \hline \end{array}$

(ϕ &~ϕ) is only ever false, so we are looking at the bottom two lines. In both, ϕ is true.

Hence regardless of the meaning of ϕ, if (ϕ &~ϕ) then ϕ is true.
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the final row

The third and fourth rows.

I suppose it's additionally a consequence of bivalence, since every consequent must land you on row 3 or row 4 and nowhere else.
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I explained in detail why.

But perhaps not all that clearly. If one sets (ϕ &~ϕ) as true, then since (ϕ &~ϕ)⊃ρ where ρ is any wff, every wff would be true.

yep. Fixed.

...bivalence...

Rejecting bivalence presumably implies rejects (ϕ &~ϕ)⊃ρ.

Are you suggesting that I provide more detail...
Well, you are better informed than I. If you did it might be interesting.
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Rejecting bivalence

Not something we need to address.

I suppose I should have put my point this way: short-circuiting is, here anyway, an unofficial procedure. If the truth table is our definition of implication, then there is no option not to consider the truth-value of the consequent, even though it's unnecessary. (We can short-circuit.) So it's almost worth pointing out that every consequent gets you to row 3 or to row 4 because no third is given.
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But perhaps not all that clearly.

No, with utmost clarity. You can go back to the posts.

If one sets (ϕ &~ϕ) as true

One does not do that.

Again, statements are true or false (and not both) per a given model.

If a theory has a statement of the form P & ~P, then the theory has no model.

If one says "'P & ~P' is true", then one has simply stated a falsehood. It doesn't follow from that falsehood that everything is true.

You are conflating the syntactics with the semantics.

Yes, syntactically:

P & ~P |- Q

But semantically:

The above syntactical principle doesn't provide that Q is true in any particular model. All it does (via the soundness theorem) is provide that Q is true in any model in which P & ~P is true. But there are no models in which P & ~P is true.

then since (ϕ &~ϕ)⊃ρ where ρ is any wff,

Please go back to my post in which I explained with exactitude why that is not the case.

Get a good book on mathematical logic to learn the notions of provability, truth in a model, entailment, etc.
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You can go back to the posts.

I did.

If a theory has a statement of the form P & ~P, then the theory has no model.

That's, indeed, rather the point. We don't do that because it undermines the enterprise at hand.

Get a good book on mathematical logic to learn the notions of provability, truth in a model, entailment, etc.

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Seems is correct that we cannot have a less formal discussion of propositional calculus. It's either too rich for some or too poor for others. I think that a shame.

I've no intention of writing another logic text that will satisfy @TonesInDeepFreeze. End of tread, I suppose.
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I wouldn't call the points @TonesInDeepFreeze has made "quibbles" but I would call them "helpful". On the other hand, I wonder how accessible any of this is to someone who has no background at all in logic.

Ah, I see you've reached the same conclusion.
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Yeah,"quibbles" was a bit too pejorative.
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Anyone wishing to pursue the topic further might look at Carnap.
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I liked the word "informal" in your previous post, it's just that propositional calculus is a formal system. It's a branch of mathematics.

If you want to raise the logical literacy of the forum, perhaps it would be better to aim at that dialect called "philosophical English," a dialect spoken by people familiar with formal systems. The traditional early chapters of a logic textbook try to show how the logical constants capture some of what we mean by familiar idioms. (The exception might be Kalish and Montague, because they're not kidding.) They give students exercises in "translating" English into the symbolism that's been defined.

But many philosophers today write in a style that's more like translated and then translated back. The style most of the SEP is written in, if it's not clear what I mean. So we're not quite talking about informal reasoning here, which is interesting in its own right but different. We're talking about a kind of semi-formal style, which aims at precision and explicitness.

In this case, the exercises would be a matter of making what you say more precise and more explicit, though still English. A guide to this style could embed a certain amount of the classical logic in everyday use in philosophy, but in English, not in mathematical notation.
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That was discussed, but I wish also to state for myself:

My comments here have not been mere quibbles. They are important considerations, especially for keeping clear the distinction between syntax and semantics.

The comments would be helpful for anyone who knows about the subject but can use some help with a refresher on these particular points. Or, if one doesn't know enough about the subject, then my comments may suggest learning more about the subject.
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The traditional early chapters of a logic textbook try to show how the logical constants capture some of what we mean by familiar idioms. (The exception might be Kalish and Montague, because they're not kidding.)

They don't give as many examples and exercises as most books, but they do give some.

What do you mean by "they're not kidding"?
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First, I can't speak to the later revision.

I've tried reading Montague, but it's pretty demanding, so he's still an aspiration for me. I do know a little about his views, in a general way, and his place in the history of logic and formal semantics. So I worked through a good chunk of the logic book, to see how he (and Kalish) handled classical logic, and there's a very different vibe to it from many logic textbooks.

Now I could be wrong, but this was my impression. Most logic textbooks go something like this: hey, you youngsters reason, everybody does, but I can show you a better way to do that, one that isn't hampered by the messiness and ambiguity of a language like English; we're going to show you a kind of language made just for reasoning; you'll recognize some of it from what you've been trying to do in English, and we'll show you how to take those groping attempts at reasoning in a medium not really suited for it, and instead do it in our nice clean system. (This is a vaguely Fregean conception, I guess.)

That is not what Montague seems to be up to at all. He and Kalish are obsessively precise about how the familiar English form relates to their notation -- something many logic textbooks swish by with some inadequate handwaving. (If memory serves, this is one of the things that Peter King will pillory a book for, playing it too loose with what exactly logical schemata are supposed to be, etc.)

And I think that's so because Kalish and Montague are not offering an alternative to reasoning in English -- a formal language you would translate some English into -- but an account of how the logical constants and quantifiers in English actually work. To put it plainly, I think this book presents something a lot more like a formal semantics of the logical constants and quantifiers. The result may look similar to what other textbooks are up to, because there is a formalism, but the relation of the formalism to the natural language English is quite different, and I think you can tell that it's different in the way the book is written.

So that's what I meant by "they weren't kidding." This book is not about a formal system someone invented that you might find a useful alternative to English; this book is a formal account of a subset of English, the words we use in connection with reasoning.

Does that make sense? Am I way off base?
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Of course, Monatague is famous for his work in formalizing natural language. It's an interesting question how much that involvement bears on the introductory textbook 'Logic: Techniques Of Formal Reasoning'. But I would be wary of thinking that the book suggests that natural languages lie down so easily that we can just read off its sentences always unambiguously into formal sentences. (I'm not saying that you're saying that is what the book suggests.)

I would guess that the authors would acknowledge that English (for example) has different senses of the connectives. For example, I would be surprised if the authors held that "if then" is always in English the material conditional.

I agree that the book is very careful indeed in how it states things and formulates things. I always recommend the book.
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I would be wary of thinking that the book suggests that natural languages lie down so easily that we can just read off its sentences always unambiguously into formal sentences.

Absolutely.

For example, I would be surprised if the authors held that "if then" is always in English the material conditional.

I don't happen to remember, but I would presume there's a way around needing to make such a claim. You can stipulate that your account applies to one way of using a word or a phrase, though there may be others, and still claim to have given an account of a subset of English usage, even if that subset doesn't take words as the joints it's carving at.

I agree that the book is very careful indeed in how it states things and formulates things. I always recommend the book.

I found it really fascinating.

I originally taught myself logic out of Quine's Methods of Logic, maybe the 3rd or 4th edition, and though he's meticulous, it's meant to be more accessible than the mathematical logic book. He has that breezy style and a lingering sort of logical positivist disdain for the unscientific, which English certainly is, so you're made to feel you're learning how to think more scientifically.

Kalish and Montague doesn't feel like that at all. It's a theory of what you were actually doing some of the time. There are very precise rules about what English words go where in the schemata, because it's intended to apply to, not replace English. So my memory of it is, anyway.
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