1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Logical entailment

  1. May 1, 2004 #1
    Please help me understand this:

    F [tex]\models \omega \:\text{(where}\: \omega\: \text{is any wff!)}[/tex]

    (That comes from Nilsson's "Artificial Intelligence, A New Synthesis", pg 225)

    How does that make any sense? There is no interpretation for which F is true.
  2. jcsd
  3. May 1, 2004 #2


    User Avatar
    Staff Emeritus
    Gold Member
    Dearly Missed

    False implies anything is a standard law of logic.
  4. May 1, 2004 #3
    Yes, clearly, if it said
    [tex]F \implies \omega[/tex]
    that would always be true.

    But apparently there is a distinction between implication and entailment, and I'm trying to understand what that distinction is.

    This is how he defines entailment:
  5. May 1, 2004 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Consider this:

    There are no interpretations in which F is true.

    Thus, it is trivial that ω is true for all interpretations in which F is true.
  6. May 1, 2004 #5
    Thanks Hurkyl. It's taking me a long time to respond because I'm trying to figure out what possible use there is to a statement like that [tex]\text{F}\:\models \omega [/tex]

    Can you explain the distinction between
    [tex] \text{P} \wedge \text{Q}\: \models \text{P}[/tex]
    [tex] \text{P} \wedge \text{Q}\: \implies \text{P}[/tex]

    Edit: added a related question:
    [tex] \text{P} \wedge \text{Q}\: \models \text{P}[/tex]
    true only because
    [tex] \text{P} \wedge \text{Q}\: \implies \text{P}[/tex]
    is a tautology?
    Last edited: May 1, 2004
  7. May 1, 2004 #6


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You would like

    [tex]P \wedge Q \models P[/tex]

    to be true right? What if P and Q are both false statements? ...

    I'm a little fuzzy in the formal logic department, but if I recall correctly, [itex]\Rightarrow[/itex] and [itex]\models[/itex] work out to be roughly equivalent.
  8. May 1, 2004 #7
    I don't think "what if P and Q are both false statements" is relevant. As I read that definition, (P and Q) logically entails P because P is true whenever (P and Q) is true.

    Unfortunately, "roughly equivalent" doesn't cut it on a final.

    Thanks anyway. I'll post back if I find out anything to clarify the difference.
  9. May 2, 2004 #8


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    P and Q can be any statements. It would be awkward (and somewhat redundant) to state "Whenever P and Q is satisfiable, [itex]P \wedge Q \models P[/itex]," would it not?

    I don't have my reference at the moment, so I may be wrong, but I seem to recall there being a theorem that says [itex]A \wedge B \wedge \ldots \Rightarrow P[/itex] if and only if [itex]A, B, \ldots \models P[/itex]. I don't remember it precisely, which is why I said "roughly" as a qualification. :smile:
  10. May 2, 2004 #9
    This is probably the theorem you were thinking of:

    [tex]{\phi_1, ... \phi_n} \models \phi \:\textrm{iff} \:\models (\phi_1, ... \phi_n) \Rightarrow \phi[/tex]

    [tex] \models \omega [/tex]
    by itself means [tex]\omega [/tex] is a tautology)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Logical entailment
  1. Non-Logical Logic (Replies: 26)

  2. The logic (Replies: 1)

  3. What is logic? (Replies: 2)