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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 1st 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI find this wording at [[predicate]] rather odd: >...a predicate is a statement... Philosophers would say 'is blue' is a predicate, while 'x is blue' is a statement. If a (unary) predicate over $X$ is to be thought of as a subject of $X$, so, if in a topos, a map $X \to \Omega$, isn't the associated statement $1 \to X \to \Omega$, for some (perhaps variable) arrow $1 \to X$. In philosophy, it's never the case that >The term proposition may be used synonymously with ‘predicate’. With 'statement' yes, but not with 'predicate'.

   I find this wording at predicate rather odd:

   …a predicate is a statement…

   Philosophers would say ’is blue’ is a predicate, while ’x is blue’ is a statement.

   If a (unary) predicate over $X$ is to be thought of as a subject of $X$, so, if in a topos, a map $X \to \Omega$, isn’t the associated statement $1 \to X \to \Omega$, for some (perhaps variable) arrow $1 \to X$.

   In philosophy, it’s never the case that

   The term proposition may be used synonymously with ‘predicate’.

   With ’statement’ yes, but not with ’predicate’.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 1st 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSuggestions? Would 'expression' be better?

   Suggestions? Would ’expression’ be better?

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeOct 1st 2011
   • (edited Oct 1st 2011)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex>Philosophers would say 'is blue' is a predicate, while 'x is blue' is a statement. I think that mathematicians find ‘is blue’ too nebulous; you need a placeholder. At the very least, the predicate should be ‘$-$ is blue.’ or the function $(x \mapsto {}$‘[$x$] is blue.’$)$. The cleanest approach (as stated in the article but perhaps not with adequate explanation) is to specify the context first. This makes it clear that we don't have two distinct predicates ‘$x$ is blue.’ and ‘$y$ is blue.’ … unless in a context where both of these variables appear, in which these *should* be distinguished! (I don't know how to do that with just ‘is blue’.)

   Philosophers would say ’is blue’ is a predicate, while ’x is blue’ is a statement.

   I think that mathematicians find ‘is blue’ too nebulous; you need a placeholder. At the very least, the predicate should be ‘$-$ is blue.’ or the function $(x \mapsto {}$‘[$x$] is blue.’$)$.

   The cleanest approach (as stated in the article but perhaps not with adequate explanation) is to specify the context first. This makes it clear that we don’t have two distinct predicates ‘$x$ is blue.’ and ‘$y$ is blue.’ … unless in a context where both of these variables appear, in which these should be distinguished! (I don’t know how to do that with just ‘is blue’.)

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 1st 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexA predicate is a term applicable to those entities which share some property. It refers to a concept. For Frege it is 'unsaturated', waiting for a name to turn it into a proposition. It's $F(_)$ rather than $F(x)$. Could we say then that a predicate names an arrow in a topos with codomain $\Omega$, or a subobject of an object, and a proposition/statement names an arrow $1 \to \Omega$, or a subobject of $1$?

   A predicate is a term applicable to those entities which share some property. It refers to a concept. For Frege it is ’unsaturated’, waiting for a name to turn it into a proposition. It’s $F(_)$ rather than $F(x)$.

   Could we say then that a predicate names an arrow in a topos with codomain $\Omega$, or a subobject of an object, and a proposition/statement names an arrow $1 \to \Omega$, or a subobject of $1$?

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeOct 1st 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex>For Frege [a predicate] is 'unsaturated', waiting for a name to turn it into a proposition. It's $F(_)$ rather than $F(x)$. It's interesting that you say this, because I just came online to remark upon it. In mathematical logic, there seems to be a trend away from an approach dealing with a function $P$ from terms to propositions, so that one writes $P(x)$ (for $x$ a variable) or more generally $P(t)$ (for $t$ a term), to an approach where $P$ is the entire statement that previously would have been written $P(x)$ and so one must write $P[t/x]$ (meaning the result of substituting the term $t$ for the variable $x$ in the proposition $P$) for what would previously have been written $P(t)$. Paul Taylor even goes so far as to write $[t/x]^* P$ to emphasise that this is a pullback. >Could we say then that a predicate names an arrow in a topos with codomain $\Omega$, or a subobject of an object, and a proposition/statement names an arrow $1 \to \Omega$, or a subobject of $1$? I'm not sure if you mean anything by saying ‘names’ here instead of ‘is’. Anyway, one certainly could use terminology this way, but why do we need two words for the latter? (I'm not sure that I knew that ‘statement’ was used in a technical sense by anybody.) Another terminology, which I associate (possibly unfairly) with dumbed down secondary-school courses, is ‘open sentence’ (with one or more free variables) and ‘closed sentence’ (without).

   For Frege [a predicate] is ’unsaturated’, waiting for a name to turn it into a proposition. It’s $F(_)$ rather than $F(x)$.

   It’s interesting that you say this, because I just came online to remark upon it. In mathematical logic, there seems to be a trend away from an approach dealing with a function $P$ from terms to propositions, so that one writes $P(x)$ (for $x$ a variable) or more generally $P(t)$ (for $t$ a term), to an approach where $P$ is the entire statement that previously would have been written $P(x)$ and so one must write $P[t/x]$ (meaning the result of substituting the term $t$ for the variable $x$ in the proposition $P$) for what would previously have been written $P(t)$. Paul Taylor even goes so far as to write $[t/x]^* P$ to emphasise that this is a pullback.

   Could we say then that a predicate names an arrow in a topos with codomain $\Omega$, or a subobject of an object, and a proposition/statement names an arrow $1 \to \Omega$, or a subobject of $1$?

   I’m not sure if you mean anything by saying ‘names’ here instead of ‘is’. Anyway, one certainly could use terminology this way, but why do we need two words for the latter? (I’m not sure that I knew that ‘statement’ was used in a technical sense by anybody.)

   Another terminology, which I associate (possibly unfairly) with dumbed down secondary-school courses, is ‘open sentence’ (with one or more free variables) and ‘closed sentence’ (without).

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 1st 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI actually like the terms 'open' and 'closed', because one can connect them with topological meanings. If one writes down string diagram (or even better, surface diagram) representations of terms in a predicate calculus, then free variables correspond to open input and output strings. The act of quantifying (say, existentially quantifying) can be likened to the act of "capping off" an open string to make it closed; you get a closed term by capping off or binding all the open strings. This idea really goes back to Peirce and his system Beta.

   I actually like the terms ’open’ and ’closed’, because one can connect them with topological meanings. If one writes down string diagram (or even better, surface diagram) representations of terms in a predicate calculus, then free variables correspond to open input and output strings. The act of quantifying (say, existentially quantifying) can be likened to the act of “capping off” an open string to make it closed; you get a closed term by capping off or binding all the open strings. This idea really goes back to Peirce and his system Beta.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 1st 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexRe 5, 'names' rather than 'is' , I was trying to indicate the philosophers' distinction between the words used to designate a property or concept and the property or concept itself. It's really quite a mess because there's the further issue of intensional or extensional identity, e.g., is the concept 'has a heart' the same as 'has a liver' if they have the same extension. As for statement/proposition, these have to made to fit with sentence and assertion. There's no unanimity. The Wikipedia [page](http://en.wikipedia.org/wiki/Proposition) shows some of the confusion. If it can all be boiled down to arrow talk, things could get a lot clearer.

   Re 5, ’names’ rather than ’is’ , I was trying to indicate the philosophers’ distinction between the words used to designate a property or concept and the property or concept itself. It’s really quite a mess because there’s the further issue of intensional or extensional identity, e.g., is the concept ’has a heart’ the same as ’has a liver’ if they have the same extension.

   As for statement/proposition, these have to made to fit with sentence and assertion. There’s no unanimity. The Wikipedia page shows some of the confusion.

   If it can all be boiled down to arrow talk, things could get a lot clearer.

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeOct 2nd 2011
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   Author: TobyBartels
   Format: MarkdownItexThink about the latter paragraph of comment 4. Here we see explicitly that a proposition is a special case of a predicate, since $1$ is an object. But given an object $A$, we may pass to the slice category over $A$, and now we see that a predicate is a proposition in a different context. I have written [[proposition]] almost entirely anew; take a look.

   Think about the latter paragraph of comment 4. Here we see explicitly that a proposition is a special case of a predicate, since $1$ is an object. But given an object $A$, we may pass to the slice category over $A$, and now we see that a predicate is a proposition in a different context.

   I have written proposition almost entirely anew; take a look.

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 3rd 2011
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   Author: David_Corfield
   Format: MarkdownItexI still think that anyone coming from philosophy will be very confused but what they will see as a conflation of 'proposition' and 'predicate'. I wonder what others think. Oddly, despite this, we don't even have the pages [[proposition]] and [[predicate]] talking to each other. 'categorial logic' could link to [[logic]].

   I still think that anyone coming from philosophy will be very confused but what they will see as a conflation of ’proposition’ and ’predicate’. I wonder what others think. Oddly, despite this, we don’t even have the pages proposition and predicate talking to each other.

   ’categorial logic’ could link to logic.

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeOct 3rd 2011
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    Author: Mike Shulman
    Format: MarkdownItexThe usage I am familiar with in mathematical logic is that a proposition is a statement containing no free variables, whereas a predicate is a statement possibly containing free variables. Would it be sufficient for the philosophers, David C, if we added a remark like > In philosophy, the variable in a predicate is usually elided, so that one writes simply "is blue" rather than "$x$ is blue". The mathematical version which contains a variable is more precise, as it specifies exactly where in the statement the entity in question is to be substituted. ?

    The usage I am familiar with in mathematical logic is that a proposition is a statement containing no free variables, whereas a predicate is a statement possibly containing free variables. Would it be sufficient for the philosophers, David C, if we added a remark like

    In philosophy, the variable in a predicate is usually elided, so that one writes simply “is blue” rather than “$x$ is blue”. The mathematical version which contains a variable is more precise, as it specifies exactly where in the statement the entity in question is to be substituted.

    ?

11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeOct 3rd 2011
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    Author: zskoda
    Format: MarkdownItexHaving a free variable looks like just a trick of axiomatic mathematics, not something what is universal beyond mathematics to other disciplines. In linguistics indeed one starts with say a semantic entity which is represented by say a verb phrase and then the knowledge of language puts it into a semantic relation with certain closed and certain open classes of other entities which it can apply to. Human is thursty. in a surrealistic stretch computer can also be thirsty. By no means emptiness (abstract notion) can be thirsty, though poetry can put it into action.

    Having a free variable looks like just a trick of axiomatic mathematics, not something what is universal beyond mathematics to other disciplines. In linguistics indeed one starts with say a semantic entity which is represented by say a verb phrase and then the knowledge of language puts it into a semantic relation with certain closed and certain open classes of other entities which it can apply to. Human is thursty. in a surrealistic stretch computer can also be thirsty. By no means emptiness (abstract notion) can be thirsty, though poetry can put it into action.

12. • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeOct 3rd 2011
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex>Oddly, despite this, we don't even have the pages [[proposition]] and [[predicate]] talking to each other. Yes we do; they are (and have always been) the same page! That you see the latter as different is just the dreaded cache bug.

    Oddly, despite this, we don’t even have the pages proposition and predicate talking to each other.

    Yes we do; they are (and have always been) the same page! That you see the latter as different is just the dreaded cache bug.

13. • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeOct 3rd 2011
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex>I still think that anyone coming from philosophy will be very confused but what they will see as a conflation of 'proposition' and 'predicate'. Will be they be confused by the last paragraph under ## Predicates ##, where predicates are viewed as __propositional functions__? I have just added a justification of that view to the paragraph on toposes in ## category-theoretic logic ##.

    I still think that anyone coming from philosophy will be very confused but what they will see as a conflation of ’proposition’ and ’predicate’.

    Will be they be confused by the last paragraph under ## Predicates ##, where predicates are viewed as propositional functions? I have just added a justification of that view to the paragraph on toposes in ## category-theoretic logic ##.

14. • CommentRowNumber14.
    • CommentAuthorTobyBartels
    • CommentTimeOct 3rd 2011
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex>'categorial logic' could link to logic. Right now, [[categorial logic]] and [[categorical logic]] redirect to [[internal logic]], but this doesn't seem right to me. (The link that I wrote was to [[category-theoretic logic]], which remains unfulfilled.)

    ’categorial logic’ could link to logic.

    Right now, categorial logic and categorical logic redirect to internal logic, but this doesn’t seem right to me. (The link that I wrote was to category-theoretic logic, which remains unfulfilled.)

15. • CommentRowNumber15.
    • CommentAuthorTobyBartels
    • CommentTimeOct 3rd 2011
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex>By no means emptiness (abstract notion) can be thirsty, though poetry can put it into action. Wondering whether emptiness is thirsty seems to me to be a type error. (The poetry apparently supplies a conversion between types.)

    By no means emptiness (abstract notion) can be thirsty, though poetry can put it into action.

    Wondering whether emptiness is thirsty seems to me to be a type error. (The poetry apparently supplies a conversion between types.)

16. • CommentRowNumber16.
    • CommentAuthorzskoda
    • CommentTimeOct 3rd 2011
    • (edited Oct 3rd 2011)
    • PermaLink
    Author: zskoda
    Format: MarkdownItexRight, you can try to make it type error in a synchronic system, but the language is changing under the influence of semantics, including typically changing the rules for such barriers for certain lexical or semantic groups or idioms.

    Right, you can try to make it type error in a synchronic system, but the language is changing under the influence of semantics, including typically changing the rules for such barriers for certain lexical or semantic groups or idioms.

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 4th 2011
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    Author: David_Corfield
    Format: MarkdownItexRe 12, ah, the dreaded cache bug. The idea is to have what's at 'Proposition' as the future page? Why not title it 'Predicate' and then describe whatever we eventually decide about its relation to 'proposition'? At [[predicate]], we have >The term proposition may be used synonymously with ‘predicate’, or it may be restricted to the case when there are no free variables, recognising that at least some people see a proposition as a special case of a predicate, so suggesting 'predicate' is the lead term.

    Re 12, ah, the dreaded cache bug. The idea is to have what’s at ’Proposition’ as the future page? Why not title it ’Predicate’ and then describe whatever we eventually decide about its relation to ’proposition’? At predicate, we have

    The term proposition may be used synonymously with ‘predicate’, or it may be restricted to the case when there are no free variables,

    recognising that at least some people see a proposition as a special case of a predicate, so suggesting ’predicate’ is the lead term.

18. • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 4th 2011
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexRe 10, adding something along those lines sounds good, when we get the page straight.

    Re 10, adding something along those lines sounds good, when we get the page straight.

19. • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 4th 2011
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexLots of philosophy students train on Hodges' [Logic](http://books.google.co.uk/books?id=xIcTc2d4y_AC). He writes >a predicate is defined to be a string of English words and individual variables, such that if the individual variables are replaced by appropriate designators, then the whole becomes a declarative sentence with these designators as constituents. example >$x$ loves a bit of night-life. Not quite sure why 'English' has to be there. But he then adds >The word **predicate** is often ued by grammarians and philosophers in ways which are at varaince with the definition we have given. For example, some people use the word to mean **property** or **quality**. He uses 'declarative sentence' rather than 'proposition'.

    Lots of philosophy students train on Hodges’ Logic. He writes

    a predicate is defined to be a string of English words and individual variables, such that if the individual variables are replaced by appropriate designators, then the whole becomes a declarative sentence with these designators as constituents.

    example

    $x$ loves a bit of night-life.

    Not quite sure why ’English’ has to be there.

    But he then adds

    The word predicate is often ued by grammarians and philosophers in ways which are at varaince with the definition we have given. For example, some people use the word to mean property or quality.

    He uses ’declarative sentence’ rather than ’proposition’.

20. • CommentRowNumber20.
    • CommentAuthorHarry Gindi
    • CommentTimeOct 4th 2011
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    Author: Harry Gindi
    Format: MarkdownItexIt's an assembly of the second kind, of course! (cf. Bourbaki, Theorie des ensembles)

    It’s an assembly of the second kind, of course! (cf. Bourbaki, Theorie des ensembles)

21. • CommentRowNumber21.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 4th 2011
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexRe 14, can't we link to this [section](http://ncatlab.org/nlab/show/logic#categorytheoretic_logic_11), perhaps having expanded it?

    Re 14, can’t we link to this section, perhaps having expanded it?

22. • CommentRowNumber22.
    • CommentAuthorTobyBartels
    • CommentTimeOct 4th 2011
    • (edited Oct 4th 2011)
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItex>Why not title it 'Predicate' and then describe whatever we eventually decide about its relation to 'proposition'? Because the two things were *already* at one page. It used to be titled [[predicate]], but I decided that [[proposition]] would be a better name. Either way, one name has redirected to the other for months. >At [[predicate]], we have You mean at [version 5 of the page](http://ncatlab.org/nlab/revision/proposition/5). Once the cache is fixed, your link won't link to the old version anymore. >>The term proposition may be used synonymously with ‘predicate’, or it may be restricted to the case when there are no free variables, I changed that because I think that it's misleading. These people really just use ‘proposition’ only, keeping track of the context, because a predicate in one context $\Gamma$ is a proposition in an extension of $\Gamma$. From this perspective, the basic concept is that of proposition in context. A predicate is a special case of a proposition: a proposition in a context extended by free variables. A proposition is a special case of a predicate: a predicate in which the free variables do not appear. >He uses 'declarative sentence' rather than 'proposition'. Does he draw a distinction between these? >Re 14, can't we link to this [section](http://ncatlab.org/nlab/show/logic#categorytheoretic_logic_11), perhaps having expanded it? Maybe; it seems like it should have its own page, although not much else is on that page. In any case, I've put the redirects there for now.

    Why not title it ’Predicate’ and then describe whatever we eventually decide about its relation to ’proposition’?

    Because the two things were already at one page. It used to be titled predicate, but I decided that proposition would be a better name. Either way, one name has redirected to the other for months.

    At predicate, we have

    You mean at version 5 of the page. Once the cache is fixed, your link won’t link to the old version anymore.

    The term proposition may be used synonymously with ‘predicate’, or it may be restricted to the case when there are no free variables,

    I changed that because I think that it’s misleading. These people really just use ‘proposition’ only, keeping track of the context, because a predicate in one context $\Gamma$ is a proposition in an extension of $\Gamma$. From this perspective, the basic concept is that of proposition in context.

    A predicate is a special case of a proposition: a proposition in a context extended by free variables. A proposition is a special case of a predicate: a predicate in which the free variables do not appear.

    He uses ’declarative sentence’ rather than ’proposition’.

    Does he draw a distinction between these?

    Re 14, can’t we link to this section, perhaps having expanded it?

    Maybe; it seems like it should have its own page, although not much else is on that page. In any case, I’ve put the redirects there for now.

23. • CommentRowNumber23.
    • CommentAuthorTobyBartels
    • CommentTimeOct 4th 2011
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexPossibly we need an introduction for philosophers and linguists. But I'm sure that keeping explicit track of context is the key to using these concepts well.

    Possibly we need an introduction for philosophers and linguists. But I’m sure that keeping explicit track of context is the key to using these concepts well.

24. • CommentRowNumber24.
    • CommentAuthorTobyBartels
    • CommentTimeOct 4th 2011
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexI added a section about propositional and predicate logic.

    I added a section about propositional and predicate logic.

25. • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeOct 4th 2011
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex> I'm sure that keeping explicit track of context is the key to using these concepts well. Ditto.

    I’m sure that keeping explicit track of context is the key to using these concepts well.

    Ditto.

26. • CommentRowNumber26.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 4th 2011
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexOk, things are making better sense for me now. Re 22, Hodge doesn't use the word 'proposition' at all.

    Ok, things are making better sense for me now.

    Re 22, Hodge doesn’t use the word ’proposition’ at all.