I know you can prove multiple ways, but I actually don't know what each means.

Is semantics where you provide a paragraph proof and use objects and deduction where you do subproofs/elimations/intros to get to the conclusion?

  • Hmm, well they're not conceptual opposites: the binaries would be something like semantic-syntactic and deductive-inductive
    – Joseph Weissman
    Aug 30, 2014 at 13:14

1 Answer 1


Semantics is to do with meaning, and syntax is to do with form. Its quite possible to set up a deductive system and then formally prove something that is wrong; this because the semantics of the situation hasn't been taken into account.

To give a crude example one could formally develop a theory of happiness by scoring peoples happiness out of ten; and then you can ask questions about the total amount of happiness, or its maximum and so on; but the semantics of this situation, when thought about, is can you measure a qualitive thing such as happiness quantitively? If this is even possible, how to take account of people lying, or not being aware of the true state of their happiness: it isn't for nothing that 'know thyself' was inscribed in the forecourt of the Temple of Apollo at Delphi. All these are common objection to theory of Utilitarianism as developed by Bentham.

  • Not really my question. but I figure my initial tought was right. I know they overlap in some respects. BUT does providing a semantic or a deduction appeal to sound or completeness'
    – John
    Aug 30, 2014 at 16:05
  • Soundness/completeness are properties (or not) of formal deductive systems; when logicians build or investigate such systems they try to show that these properties hold as they are so desirable; these formal systems are built in two parts - syntax and semantics; these are only very loosely related to what you are asking in your question. Aug 30, 2014 at 16:42
  • Because they are built in two parts there are two forms of deduction - syntactical and semantic; and they need to correspond; a logical system would be of little use if I could syntactically prove a theorem, but when I prove it semantically, I either can't do it, or its shown its false. The two directions of this correspondence are called completeness and soundness. Aug 30, 2014 at 16:46
  • Only one comment : without a "semantics" appropriate for the language of the deductive system, how to say that something we have formally proven "is wrong" ? Without semantics we can only speak of "syntactical" properties like consistency. Sep 1, 2014 at 15:44
  • @Allegranza: I'm not specifically talking about the syntax-semantics paradigm in Modern Logic; that doesn't seem to be the concern of the OP; judging by how he put his question. Semantics does have a range of meanings other than the purely mathematical-logic one. Sep 1, 2014 at 15:50

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