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We say that the sentenceb is logically implied by the sentencea, if the conditional sentence having the sentencea as antecedent and the sentenceb as consequent may be obtained by substitution from a logical tautology.
We say that the sentenceb is an enthymematic consequence of the sentencea relatively to sentencec, ifb is a logical consequence of the conjunctiona andc but not ofa alone.
The grammarians distinguish three kinds of modi: modus realis, irrealis and potentialis. In some languages e.g. in Latin different moods and tenses are to be used in the antecedent and the consequent in the three modi. The difference in meaning between the modus realis and potentialis as formulated in the textbooks of grammar is not clear enough. It seems that this difference may be formulated as follows: a conditional sentence is in modo reali if it expresses that the speaker does not know the antecedent to be false but does not express the speaker's ignorance as to the truth of the same. A conditional sentence is in modo potentiali if it expresses that the speaker does not know the antecedent to be true and that he does not know it to be false. A conditional sentence is in modo irreali if it expresses that the speaker knows the antecedent to be false. The modus realis is hence used appropriately if we want to express that we do not know the antecedent to be false but we do not want to express whether or not we know the antecedent to be true. Hence it may be used appropriately both in case we know and in case we do not know the antecedent to be true, if only we do not know the antecedent to be false. The modus potentialis may be used appropriately if we neither know that the antecedent is true nor that it is false. For our analysis this difference between modus realis and modus potentialis is not essential. The difference, however, between the modus realis and potentialis on the one hand and the modus irrealis on the other is essential. This difference consists in that both the modus realis and the modus potentialis express the speaker's ignorance as to the falsehood of the antecedent while the modus irrealis expresses that the speaker knows the antecedent to be false. Hence if we refer in our analysis to the modus realis, both modus realis and modus potentialis are meant. The fact that a conditional sentence in modo reali (and potentiali) expresses the speaker's ignorance as to the falsehood of the antecedent while the modus irrealis expresses that he knows the antecedent to be false, makes it possible to use the former and not the latter in case a sentential function being neither true nor false is in the antecedent. I may say e.g. “If we boil water, we make it sterile”, but I cannot say “Had I boiled water, I should have made it sterile”, if in both cases any portion of water is meant and in the second case we want to express that we know the antecedent to be false. I may say however: “Had this portion of water been boiled, it would have been made sterile”, expressing thereby that I know the antecedent to be false with respect to a concrete portion of water. It should further be noted that referring in our analysis to conditional sentence in modo reali and potentiali we exclude conditional sentences in which the antecedent and consequent are sentential functions and not sentences. In order to apply the results of our analysis to conditional sentences with sentential functions in the place of the antecedent or the consequent it would be necessary to introduce some inessential changes. Besides all the types of the conditional sentences the so called inferential expression is to be distinguished. It may take one of the following forms: “p, thenq” “sincep, thenq” “q, becausep”. The first part of this expression we call premiss and conclusion the second. What states and what expresses the inferential expression? It states the state of things referred to in the antecedent (premiss) and that stated in the consequent (conclusion). It expresses that the speaker knows both the premiss and the conclusion to be true. Moreover it expresses the act of inference, namely that the speaker's assertation of the conclusion is motivated by his assertion of the premiss. Unlike the inferential expression a conditional sentence does not assert either its antecedent or its consequent; it also does not express that an act of inference has been made; it only expresses the readiness to make such an inference, i. e.it expresses a potential inference.
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Ajdukiewicz, K. Conditional sentence and material implication. Stud Logica 4, 135–153 (1956). https://doi.org/10.1007/BF02548913
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DOI: https://doi.org/10.1007/BF02548913