Universal Quantifier & Material Conditionals in First Order Logic: Explained

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Discussion Overview

The discussion revolves around the distribution of the universal quantifier over material conditionals in standard first order logic. Participants explore whether the implications (v)(Mv → Pv) and (v)Mv → (v)Pv are equivalent, and whether one can infer the other under certain conditions. The conversation includes technical reasoning and examples to illustrate the points raised.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether (v)(Mv → Pv) implies (v)Mv → (v)Pv, and vice versa.
  • One participant clarifies that the notation used in the original question may have been incorrect, suggesting that the second statement should involve a different variable: {v}Mv → (u)Pu.
  • Another participant provides an example involving students and grades to illustrate the difference between the two statements.
  • One participant expresses confidence that (v)Mv → (v)Pv can be inferred from (v)(Mv → Pv), but is uncertain about the reverse implication.
  • A counterexample is presented to challenge the validity of inferring (v)(Mv → Pv) from (v)Mv → (v)Pv, using a two-element set to demonstrate the logic.
  • Another participant agrees with the counterexample and seeks confirmation on the validity of the first implication.
  • One participant discusses applying 'modus ponens' to derive Pv for elements in the subset where Mv is true, suggesting that if (v)Mv is true, then the subset encompasses the entire set.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the implications of the statements discussed. There are competing views on the validity of the inferences, and uncertainty remains regarding the implications in both directions.

Contextual Notes

Participants note potential confusion in notation and the need for clarity in the variables used. The discussion highlights the importance of understanding the relationships between universal and existential quantifiers in logical expressions.

Shana
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Can anyone tell me if the universal quantifier distributes over material conditionals in standard first order logic? Does (v)(Mv → Pv) imply {v)Mv -> (v)Pv? Does the implication work the other way too? Thanks.
 
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If I understand your notation correctly your first sentence says "for all v, if Mv is true then Pv is true" while your second is "if there exist v such that Mv is true then Pv is true for all v". No, they are not at all the same. In particular the "v" in Pv is not the same as the "v" in Mv and it really should be {v}Mv->(u)Pu. That is "if Mv is true for some v, then Pu is true for all u.

For example, if M is "gets an A on every tests" and P is "gets an A for the course", your first statement is "If any student in the class gets an A on every test, then he/she will get an A for the course" while the second statement says "If any student in the class gets an A on every test, then every student in the class will get an A for the course". Not at all the same!

I'm not sure what you meant by "does the implication work the other way". Certainly M-> P is NOT the same as P->M. "If it rains today, I will take my umbrella" does not imply "if I take my umbrella it will rain today".
 
The first v in the second sentence is supposed to be universally quantified like in the first sentence. It looks like I made a typo and put in some kind of bracket instead of a left parenthesis. You read the first one right, but the second is supposed to be "if for all v, Mv is true, then for all v, Pv is true". All of the quantifiers are universal, none are existential.

Let me try to explain better. If (v)(Mv → Pv) is true, can you infer (v)Mv → (v)Pv? If (v)Mv → (v)Pv is true, can you infer (v)(Mv → Pv)?
 
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By the way I think its valid to infer (v)Mv → (v)Pv if (v)(Mv → Pv) is true. I think that it is not valid to infer (v)(Mv → Pv) from (v)Mv → (v)Pv. I would still like somebody to confirm for sure. I am more confident in the one I think is valid than the other. Both of the inferences could be valid.
 
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A counterexample for the second could be the two-element set {a, b}, where Ma = true, Pa = false (so that Ma -> Pa is false, thus (v)(Mv -> Pv) is false), and Mb = false (so that (v)Mv is false, and therefore (v)Mv -> anything is true).
 
Dodo said:
A counterexample for the second could be the two-element set {a, b}, where Ma = true, Pa = false (so that Ma -> Pa is false, thus (v)(Mv -> Pv) is false), and Mb = false (so that (v)Mv is false, and therefore (v)Mv -> anything is true).

Yup, your counterexample works. Thanks. Do you agree that the first is valid?
 
For any set, let S be the subset of v's for which Mv is true. Now, since (v)(Mv -> Pv), for each element in S you can apply 'modus ponens' and get Pv true for all v in S. In particular, if (v)Mv is true then S is the whole set.

P.S.: That is to say, "yes". :)
 
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